Dataset Viewer (First 5GB)
id
stringlengths 9
16
| raw
stringlengths 0
3.38M
| cleaned
stringlengths 0
3.38M
|
|---|---|---|
gr-qc/0606075
|
Tetrads in low-energy weak interactions
Alcides Garat11
1. Instituto de F´ısica, Facultad de Ciencias, Igu´a 4225, esq. Mataojo, Montevideo,
Uruguay.a)
(Dated: June 15th, 2006)
Tetrads are introduced in order to study the relationship between tetrad gauge states
of spacetime and particle interactions, specially in weak processes at low energy.
Through several examples like inverse Muon decay, elastic Neutrino-Electron scatter-
ing, it is explicitly shown how to assign to each vertex of the corresponding low-order
Feynman diagram in a weak interaction, a particular set of tetrad vectors. The re-
lationship between the tetrads associated to different vertices is exhibited explicitly
to be generated by a SU(2) local tetrad gauge transformation. We are establishing
a direct link between standard gauge and tetrad gauge states of spacetime using the
quantum field theories perturbative formulations.
a)[email protected]
1
arXiv:gr-qc/0606075v2 16 Oct 2025
I.
INTRODUCTION
We are trying to understand the underlying symmetries of different field architectures, by
showing explicitly the local geometrical structure of different kinds of groups of transforma-
tions of the Standard Model, specifically in their relationship to spacetime through specially
defined tetrads. In references1,2 we studied the local geometrical meaning of electromagnetic
local gauge transformations. In references3–5 we studied the local geometrical meaning of
SU(2) × U(1) and SU(3) × SU(2) × U(1) local groups of gauge transformations. Isomor-
phisms and homomorphisms were found that relate the standard model groups of local gauge
transformations with new groups of local geometrical transformations in four-dimensional
curved Lorentz spacetimes. These relationships can be explicitly displayed through the use
of appropriately defined tetrads. It is the purpose of this work, to make use of already de-
fined tetrads of different kinds1–5, in order to briefly show in an explicit way, the invariance
of the metric tensor associated to a low-energy weak interaction, under different kinds of
transformations. For instance, the invariance under electromagnetic local gauge transforma-
tions, the invariance under SU(2) local gauge transformations, the invariance under local
gauge transformations of the spinor fields6,7, etc. Since we are trying to “geometrize” the
local gauge theories, it is interesting in its own right, to understand as well, the geometries
that involve the standard fields associated with microparticle interactions. To that end, we
introduce what we call “tetrad Feynman calculus”. We are able to explicitly show how to
build a tetrad associated to a Feynman low-order diagram in low-energy weak interactions.
The massive weak interactions bosons must have an associated gravitational field as elec-
trons, muons and neutrinos also do have and even though these gravitational fields might
be weak, they possess the necessary geometrical structure that enables the local symmetries
of the standard model to be realized in an explicit fashion as it was analyzed in previous
manuscripts1–5. In high energy interactions where virtual phenomena becomes relevant, a
different approach is needed as we will discuss later on. We proceed to show how to assign
a tetrad to each vertex, for instance in inverse Muon decay, and elastic Neutrino-Electron
scattering. In the first two sections II and III we deal as an introduction with these tetrads
defined in a general curved four-dimensional Lorentzian spacetime. These two sections and
because of the construction nature of the new tetrads can also be identically developed in
flat Minkowski spacetimes. In section IV we will limit our analysis to flat spacetimes as an
2
example compatible with the foundations of quantum field theories. As a general thought
we strongly believe that the construction of tetrad fields, and metric tensors that explicitly
display the local symmetries of microparticle interactions, are hinting us over a possible re-
lationship or link, between General Relativity and Quantum Theories. In both subsections
IV A 1-IV A 2 of section IV we also demonstrate that it is possible to transform the tetrad
associated to a vertex in a particular diagram to the tetrad assigned to another vertex in
the same Feynman diagram through a local SU(2) tetrad gauge transformation at the same
spacetime point. It is clear that we envisage the spacetime of the interaction process as
a common spacetime for all participating objects in the region of interaction even though
in this first manuscript on the subject we are addressing the processes on a flat spacetime
background. Throughout the paper we use the conventions of references1,3,8. In particular
we use a metric with sign conventions -+++. The only difference in notation with8 will be
that we will call our geometrized electromagnetic potential Aα, where fµν = Aν;µ−Aµ;ν is the
geometrized electromagnetic field fµν = (G1/2/c2)Fµν. Analogously, f k
µν are the geometrized
Yang-Mills field components, f k
µν = (G1/2/c2) F k
µν.
II.
OVERVIEW OF NEW TETRADS AND SYMMETRIES FOR THE
ABELIAN CASE
The new tetrads have been designed on the basis of existence of antisymmetric second
rank tensors. In the particular case were Abelian non-null electromagnetic fields are present
in spacetime in addition to curvature or a gravitational field, or even when spacetime is flat
the new method involves a local duality rotation of these gauge fields. We then proceed to
introduce at every point in spacetime a duality rotation by an angle −α that transforms a
non-null electromagnetic field fµν into an extremal field ξµν,
ξµν = e−∗αfµν = cos(α) fµν −sin(α) ∗fµν,
(1)
where ∗fµν = 1
2 ϵµνστ f στ is the dual tensor of fµν. The local scalar α is the complexion
of the electromagnetic field and it is a local gauge invariant quantity. Extremal fields are
essentially electric fields and they satisfy,
ξµν ∗ξµν = 0 .
(2)
3
Equation (2) is imposed as a condition on (1) and then we find the expression for the
complexion that results in tan(2α) = −fµν ∗f µν/fλρ f λρ. We can also prove just using
identities valid in four-dimensional Lorentzian spacetimes that the condition (2) can be
rewritten as ξαµ ∗ξµν = 0.
In order to prove this equivalence between conditions we
will need an identity8 valid for two second rank antisymmetric fields in a four-dimensional
Lorentzian spacetime. This identity is given by,
Aµα Bνα −∗Bµα ∗Aνα = 1
2 δ ν
µ Aαβ Bαβ ,
(3)
When this identity (3) is considered for the case Aµα = ξµα and Bνα = ξνα we obtain,
ξµα ξνα −∗ξµα ∗ξνα = 1
2 δ ν
µ
Q ,
(4)
where Q = ξµν ξµν = −
q
TµνT µν according to equations (39) in reference8. Q is assumed
not to be zero, because we are dealing with non-null electromagnetic fields. It can be proved
that condition (2) plus the general identity (3), when applied to the case Aµα = ξµα and
Bνα = ∗ξνα provides the equivalent condition to (2),
ξαµ ∗ξµν = 0 ,
(5)
which is equation (64) in reference8. In geometrodynamics, the Einstein-Maxwell equa-
tions,
Rµν = fµλ f
λ
ν
+ ∗fµλ ∗f
λ
ν
(6)
f µν
;ν = 0
(7)
∗f µν
;ν = 0 ,
(8)
reveal the existence of two potential vector fields9 Aν and ∗Aν,
fµν = Aν;µ −Aµ;ν
(9)
∗fµν = ∗Aν;µ −∗Aµ;ν .
(10)
4
The symbol “;′′ stands for covariant derivative with respect to the metric tensor gµν
and the star in ∗Aν is just nomenclature, not the dual operator, with the meaning that
∗Aν;µ = (∗Aν);µ. The duality rotation given by equation (59) in8 fµν = ξµν cos α+∗ξµν sin α,
enables us to reexpress the stress-energy tensor in terms of the extremal field,
Tµν = ξµλ ξ λ
ν
+ ∗ξµλ ∗ξ λ
ν
.
(11)
It is now the right time to introduce the new tetrads that will diagonalize locally and
covariantly the stress-energy tensor (11).
U α = ξαλ ξρλ Xρ / (
q
−Q/2
q
Xµ ξµσ ξνσ Xν )
(12)
V α = ξαλ Xλ / (
q
Xµ ξµσ ξνσ Xν )
(13)
Zα = ∗ξαλ Yλ / (
q
Yµ ∗ξµσ ∗ξνσY ν )
(14)
W α = ∗ξαλ ∗ξρλ Y ρ / (
q
−Q/2
q
Yµ ∗ξµσ ∗ξνσY ν ) .
(15)
With all these elements put together, particularly equations (4-5) it becomes trivial to
prove that the tetrad (12-15) is orthonormal and diagonalizes1 the stress-energy tensor (11).
The vectors (12-13) with eigenvalue Q/2 and the vectors (14-15) with eigenvalue −Q/2. At
every point in spacetime the timelike and one spacelike vectors that for some geometries like
Reissner-Nordstr¨om are (12-13) generate a plane that we called blade one1,10. The other two
spacelike vectors (14-15) generate a local orthogonal plane that we called blade two. These
vectors are constructed with the local extremal field8 (1), its dual, the very metric tensor
and a pair of vector fields Xα and Y α that represent a generic gauge choice as long as the
tetrad vectors do not become trivial. We are aware then, that we still have to introduce the
vectors Xµ and Y µ. Let us introduce some names. The tetrad vectors have two essential
structure components. For instance in vector U α there are two main structures. First, the
skeleton, in this case ξαλ ξρλ, and second, the gauge vector Xρ. These do not include the
normalization factor 1/ (
q
−Q/2
q
Xµ ξµσ ξνσ Xν ). The gauge vectors must be anything
that do not make the tetrad vectors trivial and we mean by this that the tetrad (12-15)
diagonalizes the stress-energy tensor for any non-trivial gauge vectors Xµ and Y µ. It is then
possible to make different choices for Xµ and Y µ. The potential vector fields introduced in
equations (9-10) represent a possible choice in geometrodynamics for the vectors Xα = Aα
5
and Y α = ∗Aα. We do not mean that the two vector fields have independence from each
other, it is just a convenient choice. With this particular choice for the two gauge vector
fields we can define then,
U α = ξαλ ξρλ Aρ / (
q
−Q/2
q
Aµ ξµσ ξνσ Aν )
(16)
V α = ξαλ Aλ / (
q
Aµ ξµσ ξνσ Aν )
(17)
Zα = ∗ξαλ ∗Aλ / (
q
∗Aµ ∗ξµσ ∗ξνσ ∗Aν )
(18)
W α = ∗ξαλ ∗ξρλ ∗Aρ / (
q
−Q/2
q
∗Aµ ∗ξµσ ∗ξνσ ∗Aν ) ,
(19)
where the four vectors (16-19) satisfy the following algebraic properties,
−U α Uα = V α Vα = Zα Zα = W α Wα = 1 .
(20)
Using the equations (4-5) it is simple to prove that (16-19) are orthogonal vectors. We
think then about local electromagnetic gauge transformations. We notice that we can in-
terpret the independent local gauge transformations of the vector potentials introduced in
equations (9-10), that is, Aα →Aα + Λ,α and ∗Aα →∗Aα + ∗Λ,α as new choices for the two
gauge vector fields Xµ and Y µ. The first local gauge transformation leaves fµν invariant and
the second one leaves ∗fµν invariant, as long as the functions Λ and ∗Λ are scalars. Accord-
ing to Schouten, for non-null electromagnetic fields in Einstein-Maxwell spacetimes there
is a two-bladed or two-plane structure10 at every point in spacetime. These blades are the
planes determined by the pairs (U α, V α) and (Zα, W α). In manuscript1 it was demonstrated
that the transformation Aα →Aα + Λ,α generates a Lorentz transformation (except for one
discrete reflection) of the tetrad vectors (U α, V α) into ( ˜U α, ˜V α) in such a way that these
“rotated” vectors ( ˜U α, ˜V α) remain in the plane or blade one generated by (U α, V α). In the
same reference1 it was also proven that the transformation ∗Aα →∗Aα + ∗Λ,α generates a
“rotation” of the tetrad vectors (Zα, W α) into ( ˜Zα, ˜W α) such that these “rotated” vectors
( ˜Zα, ˜W α) remain in the plane or blade two generated by (Zα, W α). In manuscript1 it was
demonstrated that the group of local electromagnetic gauge transformations is isomorphic to
the group LB1 of boosts plus two discrete transformations on blade one, and independently
to LB2, the group of spatial rotations on blade two. Equations like
6
U α
(ϕ) = cosh(ϕ) U α + sinh(ϕ) V α
(21)
V α
(ϕ) = sinh(ϕ) U α + cosh(ϕ) V α ,
(22)
on the local plane one represent a local electromagnetic gauge transformation of the
vectors (U α, V α). The transformation of the two vectors (U α, V α) on blade one, given in
(16-17) by the “angle” ϕ in (21-22) is a proper transformation, that is, a boost. For discrete
improper transformations the result follows the same lines, see reference1. Analogously on
the local plane two,
Zα
(φ) = cos(φ) Zα −sin(φ) W α
(23)
W α
(φ) = sin(φ) Zα + cos(φ) W α .
(24)
Equations (23-24) represent a local electromagnetic gauge transformation of the vectors
(Zα, W α), the transformation of the two tetrad vectors (Zα, W α) on blade two, given in (18-
19), by the “angle” φ. It is straightforward to check that the equalities U[α
(ϕ) V β]
(ϕ) = U [α V β]
and Z[α
(φ) W β]
(φ) = Z[α W β] are true. These equalities mean that these antisymmetric tetrad
objects are gauge invariant.
In the Abelian case it was proved that the local group of
electromagnetic gauge transformations is isomorphic to both the local groups LB1 and LB2,
separately, independently. LB1 on the local plane one is a group composed by the tetrad
boosts SO(1, 1) and two different kinds of discrete transformations. One of the discrete
transformations is the full inversion or minus the identity two by two. The other discrete
transformation is not Lorentzian1 because it is a reflection or flip, a two by two matrix
with zeroes in the diagonal and ones off-diagonal. LB2 on plane two is the group of spatial
tetrad rotations on this plane, that is SO(2). It is worth reminding ourselves about a point
in mathematical language that could be loose or inaccurate but nonetheless immediate
to understand. With the purpose of exemplifying we can mention the isomorphisms in the
Abelian case1 between the local group of electromagnetic gauge transformations and the local
groups of tetrad transformations LB1 and LB2, separately, independently. The isomorphisms
strictly speaking are homomorphisms between the local algebra of scalars associated to the
local group of electromagnetic gauge transformations and the local groups LB1 and LB2,
independently. We know that between the local algebra of scalars and the local group of
7
electromagnetic gauge transformations there is a homomorphism, a homomorphism between
the real numbers R that is, the algebra of local scalars associated to the local group of
electromagnetic gauge transformations and U(1), that is, the local group of electromagnetic
gauge transformations. We give this relationship as implicitly understood even though we
talk about isomorphisms between the local group of electromagnetic gauge transformations
and the local groups of tetrad transformations LB1 and LB2, separately, independently.
We must also stress that the local transformations (21-22) are not imposed local boosts on
the vectors that define the local plane one. They are the result of local gauge transformations
of the vectors (U α, V α). For example, from reference1 a particular boost after the gauge
transformation would look like,
˜V α
(1)
q
−˜V β
(1) ˜V(1)β
=
(1 + C)
q
(1 + C)2 −D2
V α
(1)
q
−V β
(1) V(1)β
+
D
q
(1 + C)2 −D2
V α
(2)
q
V β
(2) V(2)β
(25)
˜V α
(2)
q˜V β
(2) ˜V(2)β
=
D
q
(1 + C)2 −D2
V α
(1)
q
−V β
(1) V(1)β
+
(1 + C)
q
(1 + C)2 −D2
V α
(2)
q
V β
(2) V(2)β
. (26)
In equations (25-26) the following notation has been used, C = (−Q/2)V(1)σΛσ/(V(2)βV β
(2)),
D = (−Q/2)V(2)σ Λσ/(V(1)β V β
(1) ) and [(1+C)2 −D2] > 0 must be satisfied. The notation Λα
has been used for Λ,α where Λ is the local scalar generating the local gauge transformation.
U α =
V α
(1)
q
−V β
(1) V(1)β
and V α =
V α
(2)
q
V β
(2) V(2)β
according to the notation used in paper1,
V α
(1) = ξαλ ξρλ Aρ
(27)
V α
(2) =
q
−Q/2 ξαλ Aλ
(28)
V α
(3) =
q
−Q/2 ∗ξαλ ∗Aλ
(29)
V α
(4) = ∗ξαλ ∗ξρλ ∗Aρ .
(30)
For the particular case when 1 + C > 0, the transformations (25-26) manifest that
an electromagnetic gauge transformation on the vector field Aα →Aα + Λα, that leaves
invariant the electromagnetic field fµν, generates a boost transformation on the normalized
tetrad vector fields
V α
(1)
q
−V β
(1) V(1)β
,
V α
(2)
q
V β
(2) V(2)β
. In this case cosh(ϕ) =
(1+C)
√
(1+C)2−D2, see also
equation (21). This was just one of the possible cases in LB1. Similar analysis for the vector
8
transformations (23-24) in the local plane two generated by (Zα, W α). See reference1 for
the detailed analysis of all possible cases.
Back to our main line of work we can write the electromagnetic field in terms of these
tetrad vectors,
fαβ = −2
q
−Q/2 cos α U[α Vβ] + 2
q
−Q/2 sin α Z[α Wβ] .
(31)
Equation (31) entails the maximum simplification in the expression of the electromagnetic
field. The true degrees of freedom are the local scalars
q
−Q/2 and α. We can also present
both local degrees of freedom as
q
−Q/2 cos α and
q
−Q/2 sin α. The object U[αVβ] remains
invariant1 under a “rotation” of the tetrad vectors U α and V α by a scalar angle ϕ like in
(21-22) on blade one. This is the way in which local gauge invariance is manifested explicitly
on this local plane. Analogous for discrete transformations on blade one. Similar analysis on
blade two. A spatial “rotation” generated by a gauge transformation of the tetrad vectors Zα
and W α through an “angle” φ as in (23-24), such that the object Z[α Wβ] remains invariant1.
All this formalism clearly provides a technique to maximally simplify the expression for the
electromagnetic field strength as in equation (31). It is block diagonalized automatically by
the tetrad (16-19). This is not the case for the non-Abelian SU(2) field strength. We do not
have an automatic block diagonalization. To this purpose a new algorithm was developed in
reference11. In the next section we study the construction of tetrads similar to the Abelian
case but in the non-Abelian environment.
III.
OVERVIEW OF NEW TETRADS AND SYMMETRIES FOR THE
NON-ABELIAN CASE
For the non-Abelian case we first would like to present a set of examples on how to
build extremal fields that are locally invariant under SU(2) gauge transformations. Let us
remember that in the Abelian case in the tetrads (12-15) the only dependence on gauge came
through the gauge vectors Xµ and Y µ. The tetrad skeletons as it was mentioned previously
are local gauge invariants. The advantage of this method is that when we introduce local
gauge transformations, the vectors that span the local plane one, do not leave this plane
after the transformation and analogous in the local plane two. This fact implies in turn that
the metric tensor, be non-flat or flat will not change and will remain invariant. It is then
9
important to show explicitly that we can construct extremal fields invariant under both the
Abelian and the non-Abelian gauge transformations. One example could be for instance
given by,
ζµν = cos β fµν −sin β ∗fµν ,
(32)
Following the Abelian pattern we can define the new complexion β, and to this end we
will impose the new SU(2) local invariant condition,
Tr[ζµν ∗ζµν] = ζk
µν ∗ζkµν = 0 ,
(33)
where the summation convention is applied on the internal index k. We are just using a
generalized duality transformation, and defining through it, this new local scalar complexion
β. Therefore, the complexion condition (33) is not an additional condition for the field
strength. We simply introduced a possible generalization of the definition for the Abelian
complexion, found through a new duality transformation as well. Then, we find the local
SU(2) invariant complexion β to be,
tan(2β) = −f k
µν ∗f kµν/f p
λρ f pλρ ,
(34)
where once again the summation convention was applied on both k and p. We can also
consider gauge covariant derivatives since they will become useful in the ensuing analysis.
For example, the gauge covariant derivatives of the three extremal field internal components,
ζkµν|ρ = ζkµν ; ρ + g ϵklp Alρ ζpµν .
(35)
where ϵklp is the completely skew-symmetric tensor in three dimensions with ϵ123 = 1,
with g the coupling constant. As in the previous section II the symbol “;” stands for the
usual covariant derivative associated with the metric tensor gµν.
Next we consider the
Einstein-Maxwell-Yang-Mills vacuum field equations,
Rµν = T (ym)
µν
+ T (em)
µν
(36)
10
f µν
;ν = 0
(37)
∗f µν
;ν = 0
(38)
f kµν
|ν = 0
(39)
∗f kµν
|ν = 0 .
(40)
The field equations (37-38) provide two electromagnetic potentials9, not independent
from each other, but due to the symmetry of the equations, available for our construction.
Aµ and ∗Aµ are the two electromagnetic potentials, see the comments made about the
Abelian potentials and the star nomenclature ∗Aµ in section II. Similar for the two Non-
Abelian equations (39-40). The Non-Abelian potential Akµ is available for our construction
as well12–19. With all these elements put together, we can proceed to define the auxiliary
antisymmetric field,
ωµν = Tr(∗ζ στ ζµν + ζ στ ∗ζµν) Tr(ζσρ
|ρ ∗ζτλ
|λ) .
(41)
This particular antisymmetric auxiliary field in our construction could also be alterna-
tively chosen to be,
ωµν = Tr(ζ στ ζµν) Tr(ζσρ
|ρ ∗ζτλ
|λ) .
(42)
We can choose this antisymmetric auxiliary field ωµν in many different ways, we just
show two examples. It is clear that (41) or (42) are invariant under SU(2) local gauge
transformations. Expressions (41) or (42) are nothing but explicit examples among many,
see for example reference3. Once our choice is made we perform a local duality rotation in
order to obtain the new extremal field. We remind ourselves through the algorithm created
in section II and reference1 that extremal fields are found through local duality rotations of
second rank antisymmetric tensors like in equation (1) because then we can use the equations
analogous to (4-5) to define an orthogonal tetrad. That is the core of this algorithm.
ϵµν = cos ϑ ωµν −sin ϑ ∗ωµν .
(43)
As always we choose this complexion ϑ to be defined by the condition,
11
ϵµν ∗ϵµν = 0 .
(44)
Thus we find the new local scalar complexion analogously to section II to be,
tan(2ϑ) = −ωµν ∗ωµν/ωλρ ωλρ .
(45)
We used our new algorithm to find a new kind of local SU(2) gauge invariant extremal
tensor ϵµν, that enables the construction of the new tetrad,
Sµ
(1) = ϵµλ ϵρλ Xρ
(46)
Sµ
(2) =
q
−Qym/2 ϵµλ Xλ
(47)
Sµ
(3) =
q
−Qym/2 ∗ϵµλ Yλ
(48)
Sµ
(4) = ∗ϵµλ ∗ϵρλ Y ρ ,
(49)
where Qym = ϵµν ϵµν and we assume this local scalar not to be zero. With the help of
identity (3), when applied to the case Aµα = ϵµα and Bνα = ∗ϵνα we obtain as in section II
the equivalent condition to (44),
ϵαν ∗ϵµν = 0 ,
(50)
It is a simple excercise using (3) for Aµα = ϵµα and Bνα = ϵνα, and (50), to prove that the
vectors (46-49) are orthogonal. As we did before in section II we will call for future reference
ϵµλϵρλ the skeleton of the tetrad vector Sµ
(1), and Xρ the gauge vector. In the case of Sµ
(3), the
skeleton is ∗ϵµλ, and Yλ the gauge vector. It is clear now that skeletons are gauge invariant
under SU(2) × U(1) as we announced at the start of this section. This property guarantees
that the vectors under local U(1) or SU(2) gauge transformations will not leave their original
planes or blades, keeping therefore the metric tensor explicitly invariant. Our final task in
this construction will be to define the gauge vectors Xσ and Y σ for the tetrad (46-49). A non-
trivial although useful choice that we can make is Xσ = Y σ = Tr[Σαβ E ρ
α E λ
β ∗ξ σ
ρ ∗ξλτ Aτ].
The nature of the object Σαβ is explained in section VI, Appendix II in reference3 and also
12
section VI. The object Σαβ is basically built with the Pauli matrices and the identity two
by two. The tetrad vectors E ρ
α inside the expression Tr[Σαβ E ρ
α E λ
β
∗ξ σ
ρ
∗ξλτ Aτ] can
be chosen to be the tetrad vectors that we already know from manuscript1 and section II
for electromagnetic fields in curved space-times. Following the same notation as in1 and
equations (16-19), we call E ρ
(o) = U ρ, E ρ
(1) = V ρ, E ρ
(2) = Zρ, E ρ
(3) = W ρ. The electromagnetic
extremal tensor ξρσ, and its dual ∗ξρσ are also already known from reference1 and section II.
We make use of the already defined tetrads built for Einstein-Maxwell spacetimes in order
to enable the use of the object Σαβ which is key in our construction. The key lies in the
translating property of this object between SU(2) local gauge transformations S and local
Lorentz transformations Λα
γ, see reference3 and notice from section VI that S−1 Σαβ S =
Λα
γ Λβ
δ Σγδ. We would like to study one more property of these chosen gauge vector fields
Xσ = Y σ = Tr[Σαβ E ρ
α E λ
β
∗ξ σ
ρ
∗ξλτ Aτ]. The structure E [ρ
α E λ]
β
∗ξρσ ∗ξλτ is invariant
under U(1) local gauge transformations. The electromagnetic extremal field property1,8,
ξµσ ∗ξµτ = 0 is useful in the contractions E ρ
α E λ
β
∗ξ σ
ρ
∗ξλτ. Because it is leaving in
the contraction of E ρ
α E λ
β with ∗ξρσ ∗ξλτ only the antisymmetric object E [ρ
2
E λ]
3 , which is
locally U(1) gauge invariant. Precisely because of property (5). Let us remember that the
object Σαβ is antisymmetric and contracted with the electromagnetic tetrads as Σαβ E ρ
α E λ
β
inside the local gauge vector, see section III.
In the first paper1 we proved that the group U(1) is isomorphic to the local group of
boosts plus discrete transformations on blade one that we called LB1. The same group U(1)
is isomorphic to SO(2), that we also called LB2 since it is related to local tetrad rotations
on blade two. This is a fundamental result in group theory alone, let alone in physics. We
proved in references3,4 that the local group of SU(2) gauge transformations is isomorphic
to the tensor product of three LB1 groups. Second, the local group of SU(2) gauge trans-
formations is isomorphic to the tensor product of three LB2 or SO(2) groups. All the local
gauge groups of the Standard Model have been proven to be isomorphic to local groups
of tetrad transformations in four-dimensional Lorentzian curved or flat spacetimes. The
no-go theorems of the sixties20–22 have been proven to be incorrect. Not because of their
internal logic but for the assumptions made at the outset of these theorems. We read in
reference22 “S (the scattering matrix) is said to be Lorentz-invariant if it possesses a sym-
metry group locally isomorphic to the Poincar`e group P.. . . A symmetry transformation is
said to be an internal symmetry transformation if it commutes with P. This implies that it
13
acts only on particle-type indices, and has no matrix elements between particles of different
four-momentum or different spin. A group composed of such transformations is called an
internal symmetry group”. The local electromagnetic gauge group of transformations U(1)
has been proven to be isomorphic to local groups of tetrad transformations LB1 and LB2
on both the orthogonal planes one and two. These local groups of transformations LB1
and LB2= SO(2) are composed of Lorentz transformations except in LB1 for an improper
discrete reflection, see reference1. Therefore the local Lorentz group of spacetime transfor-
mations cannot commute with LB1 or LB2 since Lorentz transformations on a local plane do
not necessarily commute with Lorentz transformations on another local plane at the same
point in spacetime. The local internal groups of transformations do not necessarily commute
with the local Lorentz transformations, because they are isomorphic to local groups of tetrad
transformations. Analogous results were proven for the non-Abelian cases SU(2)×U(1) and
SU(3) × SU(2) × U(1) Yang-Mills, see references3–5.
IV.
SPACETIME FEYNMAN CALCULUS
It is of fundamental importance to understand the geometry of spacetime when particle
interactions are taking place. Using the accumulated analysis for different kinds of gauge
theories carried out in1–5,11,23, we will show explicitly how to assign to different Feynman
diagrams in weakly interacting processes, different sets of tetrad vectors. The massive weak
interactions boson mediators have an associated gravitational field as well as electrons,
muons and neutrinos and even though these gravitational fields might be weak, they possess
the necessary geometrical structure that enables the local symmetries of the standard model
to be realized in an explicit fashion as it was analyzed in previous manuscripts1–5. We judge
relevant to understand that the transformations of colliding particles into the same or other
emerging particles can occur through the local transformation properties of gravitational
fields which will differ for different settings even though they exhibit analogous manifest
local invariance under the internal symmetries of the standard model. We remind ourselves
that in manuscripts1–5,11,23 all the local internal gauge symmetries of the standard model have
been proved isomorphic to local groups of tetrad transformations in four-dimensional curved
Lorentz spacetimes. However, in order not to get foundational contradictions with quantum
field theories at this stage of analysis we will assume that the kind of tetrads introduced in
14
sections II and III are defined in Minkowski spacetime. This choice of spacetime involves no
contradiction since the basic operation of local duality field transformation can be performed
in flat Minkowski spacetimes as well. We can define the tetrads in Minkowski spacetime and
prove that these tetrads define locally two orthogonal planes of stress-energy diagonalization.
These geometrical structures exist not only in curved spacetimes but also in flat spacetimes.
In essence local tetrad gauge states of spacetime would represent different microparticles
in their spacetime manifestation, that can transform through local gauge transformations
into other microparticles or other local tetrad gauge states of spacetime. The notation is a
replica of the notation in24, so we refer the reader to this reference. We also refer the reader
to24–28 for abundant literature citation, specially in the field of particle physics.
A.
Weak interactions
The existence of mediators as it was shown in1–4 is irreplaceable as far as we are concerned
with the construction of these kind of tetrads in weak interactions. In this case it is the
existence of local SU(2) “extremal” fields that allow us to build tetrads in weak processes.
There are interactions involving the massive mediators where any virtual effect is negligible.
For instance the W −as the mediator in inverse Muon decay. The Zo mediator in elastic
Neutrino-Electron scattering. This is important because the existence of virtual processes
would require a different approach.
We will analyze these processes through the use of
appropriately defined tetrads.
1.
Inverse Muon decay
Let us consider the process e−(1) + νµ(2) →νe(3) + µ−(4).
There are two vertices.
We invoke then the existence of the SU(2) tetrads introduced in3,4, specially the general
tetrad structure presented in the section “Gauge geometry” and also section III. We called
these general SU(2) tetrad vectors Sµ
(1) · · · Sµ
(4) and the structure of these latter tetrads was
introduced in equations (46-49) in the section III dedicated to the overview of these objects.
There was a remaining freedom in the choice of two vector fields, Xρ and Y ρ. It is exactly
through an appropriate choice for these two vector fields that we can identify a tetrad set for
each vertex at the same spacetime point. In addition to the previously introduced notation
15
and structures, let us call the non-null electromagnetic tetrads, following again the notation
in references1–4 and sections II-III, E ρ
α . There are local non-null electromagnetic tetrads in
both vertices at the same spacetime point since in one vertex we have an electron and in
the other vertex a muon. The indices α and β are reserved for locally inertial coordinate
systems. Then, we can proceed to define for the first vertex the two gauge vector fields,
Xρ = Y ρ = u(3) γα (1 −γ5) u(1) Eρ
α .
(51)
We are basically associating to the first vertex a current24 jα
−= u(3) γα (1 −γ5) u(1).
This current describes the process e−→νe + W −. For the second vertex we can choose for
instance,
Xρ = Y ρ = u(4) γα (1 −γ5) u(2) Eρ
α .
(52)
Again, we are assigning to the second vertex a current24 jα
−= u(4) γα (1 −γ5) u(2)
describing the process νµ + W −→µ−. It is evident from all the analysis in3,4 that the
geometrical transition from vertex one to vertex two and vice-versa, is an SU(2) generated
local gauge transformation. That is only allowed through the existence of massive mediators.
Following the ideas in3 we can start by choosing for instance,
Xρ = Y ρ = Tr[Σαβ E σ
α E λ
β
∗ξ ρ
σ
∗ξλτ Aτ]
(53)
The Σαβ objects are analyzed in appendix II in reference3 and sections III-VI in this
present paper, ξσρ are the electromagnetic “extremal” fields introduced in reference1 and
section II in this present paper, etc. Through a local SU(2) gauge transformation on blade
one, we can “rotate” the normalized version of vectors (46-47) on blade one, until Xρ in (53)
becomes Xρ in (51). We can also “rotate” the normalized version of vectors (48-49) on blade
two, until Y ρ in (53) becomes Y ρ in (51). Let us remember that the tetrad skeletons are
locally gauge invariant under U(1)×SU(2). This is just a sample of local gauge transforma-
tions of the normalized version of vectors (46-49). We proved in references1,3,4 that the maps
that both in the local plane one and two send the tetrad vectors that generate these planes
from an initial gauge vector into another gauge vector are injective and surjective maps. The
16
map that assigns local groups of gauge transformations into local groups of tetrad trans-
formations on either local orthogonal planes of stress-energy symmetry are isomorphisms.
Again we can start with (53) and appropriately “rotate” the tetrad vectors on blade one,
until they become the ones corresponding to Xρ given in (52). Similar for Y ρ in this second
case (52). It is evident then that (51) and (52) are connected through local SU(2) gauge
transformations on blades one and two, that in turn, leave invariant the metric tensor. That
is, these local gauge transformations exist because of transitivity. The local groups of gauge
transformations have been proven to be isomorphic to the local groups of tetrad transfor-
mations on the local orthogonal planes of symmetry. Given two sets of tetrads on the local
plane one, then there is a local gauge transformation that sends one set into the other and
vice-versa. Similar in the local othogonal plane two. These local orthogonal planes we re-
mind ourselves are the local planes of covariant diagonalization of the stress-energy tensor,
for the Abelian case and also for the non-Abelian case, see references1–5,11.
We can also notice that the vector fields (51-52) are not strictly vectors but pseudovectors
under local parity transformations, see reference24. But the metric tensor remains unaltered
under these local parity transformations. It is as if the geometry associated to the e−(1) and
νe(3) can be transformed through the existence of a massive mediator into the geometry
associated to the νµ(2) and µ−(4) without altering the spacetime. The vertices are local
tetrad gauge states of the same flat spacetime.
2.
Elastic Neutrino-Electron scattering
Now, we are considering neutral currents. In particular the interaction process νµ(1) +
e−(2) →νµ(3) + e−(4). As before we can assign to the first vertex the choice,
Xρ = Y ρ = u(3) γα (1 −γ5) u(1) Zρ
α .
(54)
The current jα
−= u(3) γα (1 −γ5) u(1), represents the process νµ(1) →νµ(3) + Zo. The
tetrad Zρ
α is built as follows. Following again the notation in24 we know we have available
a local vector field Zµ that results from the Weinberg rotation through the angle θw, in
addition to the standard electromagnetic local vector field Aµ. The rotation can be written,
17
Aµ = Bµ cos θw + W 3
µ sin θw
(55)
Zµ = −Bµ sin θw + W 3
µ cos θw .
(56)
The local tetrad field Zρ
α is present in both vertices at the same spacetime point, since the
massive neutral mediator is present in both local vertices and Zρ
α is a local tetrad associated
to this flat spacetime. The electro-weak mixing involves a weak isotriplet of intermediate
vector bosons W coupled to three weak isospin currents, and an isosinglet intermediate
vector boson Bµ coupled to the weak hypercharge current. If we follow all the steps in
reference1 and the method developed in section II in the present paper, we can build out of
the curl Zµ,ν −Zν,µ a new tetrad. This auxiliary local tetrad Zρ
α present in the definition of
gauge-vectors (54) would once more in its own construction involve the choice of two gauge-
vector fields, see reference1. We can choose for instance Zµ and Bµ as these two vector fields
needed in turn in the definition and construction of the local auxiliary tetrad Zρ
α. Then, the
tetrad that couples to the neutrino current is associated to the massive Zo.
The second vertex could be assigned a choice,
Xρ = Y ρ = u(4) γα (cV −cA γ5) u(2) Eρ
α ,
(57)
representing e−(2) + Zo →e−(4). For this particular interaction cV = −1
2 + 2 sin θw,
and cA = −1
2, where θw is again the Weinberg angle24. The massive mediator allows again
for a SU(2) local gauge transformation between the tetrad vectors chosen for vertex one
and the ones chosen for vertex two at the same spacetime point. The neutral current works
as a geometry mediator between the scattered particles keeping the spacetime invariant at
the same spacetime point. The vertices function as local tetrad gauge states of the same
spacetime.
V.
CONCLUSIONS
We have explored the possibility of assigning tetrads to Feynman diagrams. In interac-
tions where we can assume the existence of particles with associated local fields like Abelian
or non-Abelian gauge fields. At no step of analysis we have specified the tetrad themselves
18
making all these geometrical properties outstanding since they can be put forward with all
generality without the need to study case by case as long as the gauge fields are non-null
for example. We can think the massive weak interactions boson mediators to have associ-
ated gravitational fields as well as electrons, muons and neutrinos do and even though these
gravitational fields might be weak, they possess the necessary geometrical structure that
enables the local symmetries of the standard model to be realized in an explicit fashion as
it was studied thoroughly in previous manuscripts1–5. However we decided to consider a
background flat spacetime since gravitational fields would entail foundational contradictions
with standard quantum field theories. New concepts would have to be introduced and we
do not want to do this at this stage in analysis. We deem fundamental to understand that
the transformations of colliding particles into the same or other emerging particles through
elastic or inelastic processes can occur through the local transformation properties of space-
time which will differ for different settings through the new notion of tetrad gauge states
of spacetime. Having done this explicitly, a number of questions naturally arise. We want
these concluding remarks to be a summary of these open questions.
• The order of the formulations24−31.
We have worked out the low-order diagrams.
Then, what happens with higher order diagrams ?. The tetrads admit the choice of
two gauge vector fields Xρ and Y ρ, and the higher orders are additive exactly as in
the quantum theories in these vector fields available as a choice. But there is more to
understand. Do the higher order diagrams represent contributions coming from higher
order perturbative theories of a full relativistic formulation of these interactions23 in-
volving perturbations of the electromagnetic field, etc, for instance, or just perturba-
tive expansions in the gauge vector fields?. As an example of this kind of situation we
might want to qualitatively consider the quark decays b →s γ, see chapter XIV-7 in
reference32 for instance. There are several possibilities but there are certainly higher
order contributions to these kind of processes. Let us focus on the contribution that
involves a W −boson mediator in rare decays. There is the b →W −+ c vertex and
the subsequent c + W −→s vertex for example. Each one of these has an associated
current vector, let us call them for short jα
[bc] and jα
[cs]. Both vertices have particles with
electric charge so there is at each vertex an associated electromagnetic tetrad Eρ
[bc]α and
Eρ
[cs]α respectively. Then we can associate to each vertex in this higher order diagram
19
gauge vectors Xρ
[bc] = Y ρ
[bc] = jα
[bc] Eρ
[bc]α and Xρ
[cs] = Y ρ
[cs] = jα
[cs] Eρ
[cs]α. The jα
[bc] Eρ
[bc]α
contribution can then be added to the non-Abelian tetrad gauge vectors for vertex [bc]
and similar for jα
[cs] Eρ
[cs]α at the vertex [cs]. In this contribution there is also a photon
involved. This photon emission will change the stress-energy tensor and therefore will
be associated to the perturbed extremal field which is in turn a local duality rotation
of the perturbed electromagnetic field. These perturbation in the extremal field will
in turn perturb the tetrad skeletons. Therefore, the vertices in the diagram will be
associated to tetrad gauge states of the spacetime and the photon emission to tetrad
skeleton perturbations.
The b →s γ decays might involve contributions with top
quarks and the analysis will be similar, the b →s γ decays could include a loop of the
kind cc for example and we will proceed similarly as well. We will add to the gauge
vectors associated to the corresponding vertex, higher and higher contributions with a
corresponding expansion parameter. The currents at the corresponding vertex are the
key objects necessary to produce gauge vectors associated to vertices. There could be
that both, the perturbations in the skeletons and the perturbations in gauge vectors
proceed simultaneously as in the b →s γ case. On one hand the local orthogonal
planes of symmetry will tilt and on the other hand the tetrad vectors that span these
local planes will rotate inside them.
• A point outside the scope of this manuscript. The issue of “gauge gravity”. Since in
references1,3 it was explicitly proved that the Abelian and non-Abelian gauge theories
represent special symmetries of the gravitational field, we can ask about the meaning
of “gauge gravity”.
The electromagnetic field is associated to the LB1 and LB2
symmetries of the gravitational field, see reference1. The SU(2) group of local gauge
transformations is associated to the symmetries of the tensor product of three LB1
or three LB2 groups of transformations, see references3,4. Analogous for SU(3), see
reference5. Then, it is not obvious to understand what is the meaning of a statement
like, “casting the theory of gravity into a Yang-Mills formulation”. We have reason to
believe that we can truly cast the theory of gravity into a Yang-Mills formulation but
as said, it is not obvious and requires a whole new work.
• Another point outside the scope of this manuscript. The issue of quantum gravity. It
has been proved explicitly that metric tensors can be associated with microparticle in-
20
teractions. These constructions are possible by means of non-null Abelian tetrad fields,
and by means of SU(2) local non-Abelian tetrad fields. Perturbative formulations of
these tetrad field structures as in reference23 can take care of quantum fluctuations as
well. The quantum is connected through the existence of these tetrad fields to grav-
ity. A treatment for a curved spacetime where a gravitational field is present would
entail several new notions that we would like not to introduce at this stage of analysis.
Nonetheless we can advance that in curved spacetimes there will be local orthogonal
planes one and two and the isomorphisms between local gauge groups Abelian and
non-Abelian and local groups of tetrad transformations LB1 and LB2 would also ap-
ply in a similar fashion as to flat spacetimes1–5. The quantization will be reflected
through interactions that alter the local plane-symmetry structure by tilting these
local planes of symmetry. Continuously for continuous perturbations and discretely
in quantum settings situations. The main idea behind these quantum formulations
in curved spacetimes is that the local planes of stress-energy diagonalization are al-
ways and during quantum interactions local orthogonal planes of symmetry, because
quantum problems are basically confronted through perturbations analysis and these
perturbations lead to continuous or discrete evolution of the local planes of symmetry
either in flat or curved spacetimes. Continuous or discrete tilt symmetry evolution, see
reference23. We also established that during interactions of microparticles the tetrad
vectors that span the local orthogonal planes one and two might rotate inside them.
The tetrads of different nature that we were able to build in1–5 and the present work,
establish a link between the standard locally inertial flat field environment of the tra-
ditional standard quantum theories in weak interactions on one hand, and the curved
spacetime of gravity on the other hand. The point is the following, why are we using
in quantum gravity similar conceptual foundations to theories that are not formulated
in curved spacetimes but flat spacetimes ?.
• Another point outside the limited scope of this paper. The issue of the Higgs mecha-
nism. It is a device conceived in its relationship with the nature of mass, for instance
of the mass mediators. In the present tetrad environment we can ask if it is necessary,
or the mass comes into existence due to the presence of gravity ?. Is it possible that
the local Higgs field and its quantum fluctuations are related to the perturbations of
21
the local gravitational weak field scalar approximation on a flat Minkowski background
associated to the asymptotically curved spacetimes that in turn we can associate to
elementary microparticles ?.
• The issue of symmetry-breaking. It was proved in the general manuscripts1–4 that
the gravitational field when built with tetrads along the lines of expressions (46-49)
are manifestly invariant under local electromagnetic gauge transformations, and local
SU(2) gauge transformations as well. But when assigning a tetrad set to a vertex
in a low-energy weak process diagram in Minkowski spacetime, we make a particular
choice for the two gauge vectors Xρ and Y ρ. For instance, through associated currents
we choose a particular gauge, and a different one for each vertex, like in inverse Muon
decay or elastic Neutrino-Electron scattering. Then, we wonder if this gauge fixing
procedure could be the geometrical form of the standard symmetry-breaking process.
Hereby, we can see that it is the tetrad fields that bridge the two gauges associated
to the two vertices, through a local SU(2) gauge transformation, that in turn, leaves
invariant the metric tensor.
VI.
APPENDIX I
This appendix is introducing the object Σαβ that according to the matrix definitions
introduced in the references is Hermitic. The use of this object in the construction of tetrads
in section III enables the local SU(2) gauge transformations S, to get in turn transformed
into purely geometrical transformations. That is, local rotations of the U(1) electromagnetic
tetrads. The object σαβ is defined6,7 as σαβ = σα
+ σβ
−−σβ
+ σα
−. The object σα
± arises when
building the Weyl representation for left handed and right handed spinors. According to
reference7, it is defined as σα
± = (1, ±σi), where σi are the Pauli matrices for i = 1 · · · 3.
Under the (1
2, 0) and (0, 1
2) spinor representations of the Lorentz group this object transforms
as,
S−1
(1/2) σα
± S(1/2) = Λα
γ σγ
± .
(58)
Equation (58) means that under the spinor representation of the Lorentz group, σα
±
transform as vectors. In (58), the matrices S(1/2) are local objects, as well as7 Λα
γ. The
22
SU(2) elements can be considered to belong to the Weyl spinor representation of the Lorentz
group. Since the group SU(2) is homomorphic to SO(3), they just represent local space
rotations. It is also possible to define the object σ†αβ = σα
−σβ
+ −σβ
−σα
+ in a similar fashion.
Therefore, we can write,
ı
σαβ + σ†αβ
=
0
if α = 0 and β = i
4 ϵijk σk
if α = i and β = j ,
σαβ −σ†αβ =
−4 σi
if α = 0 and β = i
0
if α = i and β = j .
We then may call Σαβ
ROT = ı
σαβ + σ†αβ
, and Σαβ
BOOST = ı
σαβ −σ†αβ
and a possible
choice for the object Σαβ could be for instance Σαβ = Σαβ
ROT +Σαβ
BOOST. This is a good choice
when we consider proper Lorentz transformations of the tetrad vectors nested within the
structure of the gauge vectors Xµ and Y µ. For spatial rotations of the U(1) electromagnetic
tetrad vectors which in turn are nested within the structure of the two gauge vectors Xµ and
Y µ, as is the case under study in section III, we can simply consider Σαβ = Σαβ
ROT. These
possible choices also make sure the Hermiticity of gauge vectors. Since when defining the
gauge vectors Xµ and Y µ we are taking the trace, then Xµ and Y µ are real.
REFERENCES
1A. Garat, Tetrads in geometrodynamics, J. Math. Phys. 46, 102502 (2005). A. Garat,
Erratum: Tetrads in geometrodynamics, J. Math. Phys. 55, 019902 (2014).
2A. Garat, New tetrads in Riemannian geometry and new ensuing results in group theory,
gauge theory and fundamental physics in particle physics, general relativity and astro-
physics, Int. J. Mod. Phys. Conf. Ser., Vol. 45, (2017), 1760004.
3A. Garat, Tetrads in Yang-Mills geometrodynamics, Gravitation and Cosmology, (2014)
Vol. 20 No. 1, pp. 116-126. Pleiades Publishing Ltd. arXiv:gr-qc/0602049.
4A. Garat, The new electromagnetic tetrads, infinite tetrad nesting and the non-trivial emer-
gence of complex numbers in real theories of gravitation, Int. J. Geom. Methods Mod. Phys.,
Vol. 14, No. 9 (2017), 1750132.
5A. Garat, Tetrads in SU(3) × SU(2) × U(1) Yang-Mills geometrodynamics, Int. J. Geom.
Methods Mod. Phys., Vol. 15 no. 3 (2018), 1850045. arXiv:1207.0912.
23
6M. Kaku, Quantum Field Theory: A Modern Introduction (Oxford University Press, 1993).
7L. ´Alvarez-Gaum´e and M. A. V´azquez-Mozo, Introductory Lectures on Quantum Field
Theory (arXiv:hep-th/0510040).
8C. Misner and J. A. Wheeler, Classical physics as geometry, Annals of Physics 2, 525
(1957).
9N. Cabibbo and E. Ferrari, Nuovo Cim. 23, 1147 (1962).
10J. A. Schouten, Ricci Calculus: An Introduction to Tensor Calculus and Its Geometrical
Applications (Springer, Berlin, 1954).
11A. Garat, “Gauge invariant method for maximum simplification of the field strength in
non-Abelian Yang-Mills theories”, Int. J. Geom. Methods Mod. Phys., Vol. 12, No. 10
(2015), 1550104. arXiv:1306.2174.
12R. Gilmore, Lie Groups, Physics and Geometry (Cambridge University Press, 2008).
13R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (John Wiley &
Sons, 1974).
14J. Stillwell, Naive Lie Theory (Springer Science + Business Media, L.L.C., 2010).
15N. Carter, Visual Group Theory (The Mathematical Association of America, Inc, 2009).
16M.
Carmeli, Classical Fields: General Relativity and Gauge Theory (J. Wiley & Sons,
New York, 1982).
17C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).
18R. Utiyama, Phys. Rev., 101, 1597 (1956).
19T. W. B. Kibble, J. Math. Phys., 2, 212 (1961).
20S. Weinberg, Comments on relativistic supermultiplet theories, Phys. Rev. 139, B597
(1965).
21L. O’Raifeartagh, Lorentz invariance and internal symmetry, Phys. Rev. 139, B1052
(1965).
22S. Coleman and J. Mandula, All possible symmetries of the S matrix, Phys. Rev. 159, N5
1251 (1967).
23A. Garat, “Dynamical symmetry breaking in geometrodynamics”, TMF, 195:2, (2018),
313-328; Theoret. and Math. Phys., 195:2, (2018), 764-776. arXiv:1306.0602.
24D. Griffiths , Introduction to elementary particles (John Wiley & Sons, Inc. , 1987).
25T. P. Cheng and L. F. Li , Gauge Theory of Elementary Particle Physics (Oxford University
Press, 1989).
24
26W. Greiner and B. Mueller, Gauge Theory of Weak Interactions (Springer Verlag Gmbh,
1996).
27W. Greiner and B. Mueller, Quantum Mechanics, Symmetries (Springer Verlag, 1989).
28G. ,t Hooft, Renormalization of Gauge Theories (Lecture notes Erice, 1998).
29M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus
Books Publishing L.L.C., 1995).
30M. Srednicki, Quantum Field Theory (Cambridge University Press, New York, 2007).
31R.
Jackiw,
Fifty
Years
of
Yang-Mills
Theory
and
my
Contribution
to
it
(arXiv:physics/0403109, 2004).
32J. F. Donoghue, E. Golowich and B. R. Holstein, Dynamics of the Standard Model (Cam-
bridge monographs on particle physics, nuclear physics and cosmology) (Cambridge Uni-
versity Press, Cambridge, 2014).
25
|
Tetrads in low-energy weak interactions Alcides Garat11 1. Instituto de F ́ısica, Facultad de Ciencias, Igu ́a 4225, esq. Mataojo, Montevideo, Uruguay.a) (Dated: June 15th, 2006) Tetrads are introduced in order to study the relationship between tetrad gauge states of spacetime and particle interactions, specially in weak processes at low energy. Through several examples like inverse Muon decay, elastic Neutrino-Electron scattering, it is explicitly shown how to assign to each vertex of the corresponding low-order Feynman diagram in a weak interaction, a particular set of tetrad vectors. The relationship between the tetrads associated to different vertices is exhibited explicitly to be generated by a SU(2) local tetrad gauge transformation. We are establishing a direct link between standard gauge and tetrad gauge states of spacetime using the quantum field theories perturbative formulations. 1 arXiv:gr-qc/0606075v2 16 Oct 2025 I. INTRODUCTION We are trying to understand the underlying symmetries of different field architectures, by showing explicitly the local geometrical structure of different kinds of groups of transformations of the Standard Model, specifically in their relationship to spacetime through specially defined tetrads. In references1,2 we studied the local geometrical meaning of electromagnetic local gauge transformations. In references3-5 we studied the local geometrical meaning of SU(2) × U(1) and SU(3) × SU(2) × U(1) local groups of gauge transformations. Isomorphisms and homomorphisms were found that relate the standard model groups of local gauge transformations with new groups of local geometrical transformations in four-dimensional curved Lorentz spacetimes. These relationships can be explicitly displayed through the use of appropriately defined tetrads. It is the purpose of this work, to make use of already defined tetrads of different kinds1-5, in order to briefly show in an explicit way, the invariance of the metric tensor associated to a low-energy weak interaction, under different kinds of transformations. For instance, the invariance under electromagnetic local gauge transformations, the invariance under SU(2) local gauge transformations, the invariance under local gauge transformations of the spinor fields6,7, etc. Since we are trying to "geometrize" the local gauge theories, it is interesting in its own right, to understand as well, the geometries that involve the standard fields associated with microparticle interactions. To that end, we introduce what we call "tetrad Feynman calculus". We are able to explicitly show how to build a tetrad associated to a Feynman low-order diagram in low-energy weak interactions. The massive weak interactions bosons must have an associated gravitational field as electrons, muons and neutrinos also do have and even though these gravitational fields might be weak, they possess the necessary geometrical structure that enables the local symmetries of the standard model to be realized in an explicit fashion as it was analyzed in previous manuscripts1-5. In high energy interactions where virtual phenomena becomes relevant, a different approach is needed as we will discuss later on. We proceed to show how to assign a tetrad to each vertex, for instance in inverse Muon decay, and elastic Neutrino-Electron scattering. In the first two sections II and III we deal as an introduction with these tetrads defined in a general curved four-dimensional Lorentzian spacetime. These two sections and because of the construction nature of the new tetrads can also be identically developed in flat Minkowski spacetimes. In section IV we will limit our analysis to flat spacetimes as an 2 example compatible with the foundations of quantum field theories. As a general thought we strongly believe that the construction of tetrad fields, and metric tensors that explicitly display the local symmetries of microparticle interactions, are hinting us over a possible relationship or link, between General Relativity and Quantum Theories. In both subsections IV A 1-IV A 2 of section IV we also demonstrate that it is possible to transform the tetrad associated to a vertex in a particular diagram to the tetrad assigned to another vertex in the same Feynman diagram through a local SU(2) tetrad gauge transformation at the same spacetime point. It is clear that we envisage the spacetime of the interaction process as a common spacetime for all participating objects in the region of interaction even though in this first manuscript on the subject we are addressing the processes on a flat spacetime background. Throughout the paper we use the conventions of references1,3,8. In particular we use a metric with sign conventions -+++. The only difference in notation with8 will be that we will call our geometrized electromagnetic potential Aα, where fμν = Aν;μ-Aμ;ν is the geometrized electromagnetic field fμν = (G1/2/c2)Fμν. Analogously, f k μν are the geometrized Yang-Mills field components, f k μν = (G1/2/c2) F k μν. II. OVERVIEW OF NEW TETRADS AND SYMMETRIES FOR THE ABELIAN CASE The new tetrads have been designed on the basis of existence of antisymmetric second rank tensors. In the particular case were Abelian non-null electromagnetic fields are present in spacetime in addition to curvature or a gravitational field, or even when spacetime is flat the new method involves a local duality rotation of these gauge fields. We then proceed to introduce at every point in spacetime a duality rotation by an angle -α that transforms a non-null electromagnetic field fμν into an extremal field ξμν, ξμν = e-∗αfμν = cos(α) fμν -sin(α) ∗fμν, (1) where ∗fμν = 1 2 εμνστ f στ is the dual tensor of fμν. The local scalar α is the complexion of the electromagnetic field and it is a local gauge invariant quantity. Extremal fields are essentially electric fields and they satisfy, ξμν ∗ξμν = 0 . (2) 3 Equation (2) is imposed as a condition on (1) and then we find the expression for the complexion that results in tan(2α) = -fμν ∗f μν/fλρ f λρ. We can also prove just using identities valid in four-dimensional Lorentzian spacetimes that the condition (2) can be rewritten as ξαμ ∗ξμν = 0. In order to prove this equivalence between conditions we will need an identity8 valid for two second rank antisymmetric fields in a four-dimensional Lorentzian spacetime. This identity is given by, Aμα Bνα -∗Bμα ∗Aνα = 1 2 δ ν μ Aαβ Bαβ , (3) When this identity (3) is considered for the case Aμα = ξμα and Bνα = ξνα we obtain, ξμα ξνα -∗ξμα ∗ξνα = 1 2 δ ν μ Q , (4) where Q = ξμν ξμν = - q TμνT μν according to equations (39) in reference8. Q is assumed not to be zero, because we are dealing with non-null electromagnetic fields. It can be proved that condition (2) plus the general identity (3), when applied to the case Aμα = ξμα and Bνα = ∗ξνα provides the equivalent condition to (2), ξαμ ∗ξμν = 0 , (5) which is equation (64) in reference8. In geometrodynamics, the Einstein-Maxwell equations, Rμν = fμλ f λ ν + ∗fμλ ∗f λ ν (6) f μν ;ν = 0 (7) ∗f μν ;ν = 0 , (8) reveal the existence of two potential vector fields9 Aν and ∗Aν, fμν = Aν;μ -Aμ;ν (9) ∗fμν = ∗Aν;μ -∗Aμ;ν . (10) 4 The symbol ";′′ stands for covariant derivative with respect to the metric tensor gμν and the star in ∗Aν is just nomenclature, not the dual operator, with the meaning that ∗Aν;μ = (∗Aν);μ. The duality rotation given by equation (59) in8 fμν = ξμν cos α+∗ξμν sin α, enables us to reexpress the stress-energy tensor in terms of the extremal field, Tμν = ξμλ ξ λ ν + ∗ξμλ ∗ξ λ ν . (11) It is now the right time to introduce the new tetrads that will diagonalize locally and covariantly the stress-energy tensor (11). U α = ξαλ ξρλ Xρ / ( q -Q/2 q Xμ ξμσ ξνσ Xν ) (12) V α = ξαλ Xλ / ( q Xμ ξμσ ξνσ Xν ) (13) Zα = ∗ξαλ Yλ / ( q Yμ ∗ξμσ ∗ξνσY ν ) (14) W α = ∗ξαλ ∗ξρλ Y ρ / ( q -Q/2 q Yμ ∗ξμσ ∗ξνσY ν ) . (15) With all these elements put together, particularly equations (4-5) it becomes trivial to prove that the tetrad (12-15) is orthonormal and diagonalizes1 the stress-energy tensor (11). The vectors (12-13) with eigenvalue Q/2 and the vectors (14-15) with eigenvalue -Q/2. At every point in spacetime the timelike and one spacelike vectors that for some geometries like Reissner-Nordstr ̈om are (12-13) generate a plane that we called blade one1,10. The other two spacelike vectors (14-15) generate a local orthogonal plane that we called blade two. These vectors are constructed with the local extremal field8 (1), its dual, the very metric tensor and a pair of vector fields Xα and Y α that represent a generic gauge choice as long as the tetrad vectors do not become trivial. We are aware then, that we still have to introduce the vectors Xμ and Y μ. Let us introduce some names. The tetrad vectors have two essential structure components. For instance in vector U α there are two main structures. First, the skeleton, in this case ξαλ ξρλ, and second, the gauge vector Xρ. These do not include the normalization factor 1/ ( q -Q/2 q Xμ ξμσ ξνσ Xν ). The gauge vectors must be anything that do not make the tetrad vectors trivial and we mean by this that the tetrad (12-15) diagonalizes the stress-energy tensor for any non-trivial gauge vectors Xμ and Y μ. It is then possible to make different choices for Xμ and Y μ. The potential vector fields introduced in equations (9-10) represent a possible choice in geometrodynamics for the vectors Xα = Aα 5 and Y α = ∗Aα. We do not mean that the two vector fields have independence from each other, it is just a convenient choice. With this particular choice for the two gauge vector fields we can define then, U α = ξαλ ξρλ Aρ / ( q -Q/2 q Aμ ξμσ ξνσ Aν ) (16) V α = ξαλ Aλ / ( q Aμ ξμσ ξνσ Aν ) (17) Zα = ∗ξαλ ∗Aλ / ( q ∗Aμ ∗ξμσ ∗ξνσ ∗Aν ) (18) W α = ∗ξαλ ∗ξρλ ∗Aρ / ( q -Q/2 q ∗Aμ ∗ξμσ ∗ξνσ ∗Aν ) , (19) where the four vectors (16-19) satisfy the following algebraic properties, -U α Uα = V α Vα = Zα Zα = W α Wα = 1 . (20) Using the equations (4-5) it is simple to prove that (16-19) are orthogonal vectors. We think then about local electromagnetic gauge transformations. We notice that we can interpret the independent local gauge transformations of the vector potentials introduced in equations (9-10), that is, Aα →Aα + Λ,α and ∗Aα →∗Aα + ∗Λ,α as new choices for the two gauge vector fields Xμ and Y μ. The first local gauge transformation leaves fμν invariant and the second one leaves ∗fμν invariant, as long as the functions Λ and ∗Λ are scalars. According to Schouten, for non-null electromagnetic fields in Einstein-Maxwell spacetimes there is a two-bladed or two-plane structure10 at every point in spacetime. These blades are the planes determined by the pairs (U α, V α) and (Zα, W α). In manuscript1 it was demonstrated that the transformation Aα →Aα + Λ,α generates a Lorentz transformation (except for one discrete reflection) of the tetrad vectors (U α, V α) into ( ̃U α, ̃V α) in such a way that these "rotated" vectors ( ̃U α, ̃V α) remain in the plane or blade one generated by (U α, V α). In the same reference1 it was also proven that the transformation ∗Aα →∗Aα + ∗Λ,α generates a "rotation" of the tetrad vectors (Zα, W α) into ( ̃Zα, ̃W α) such that these "rotated" vectors ( ̃Zα, ̃W α) remain in the plane or blade two generated by (Zα, W α). In manuscript1 it was demonstrated that the group of local electromagnetic gauge transformations is isomorphic to the group LB1 of boosts plus two discrete transformations on blade one, and independently to LB2, the group of spatial rotations on blade two. Equations like 6 U α (φ) = cosh(φ) U α + sinh(φ) V α (21) V α (φ) = sinh(φ) U α + cosh(φ) V α , (22) on the local plane one represent a local electromagnetic gauge transformation of the vectors (U α, V α). The transformation of the two vectors (U α, V α) on blade one, given in (16-17) by the "angle" φ in (21-22) is a proper transformation, that is, a boost. For discrete improper transformations the result follows the same lines, see reference1. Analogously on the local plane two, Zα (φ) = cos(φ) Zα -sin(φ) W α (23) W α (φ) = sin(φ) Zα + cos(φ) W α . (24) Equations (23-24) represent a local electromagnetic gauge transformation of the vectors (Zα, W α), the transformation of the two tetrad vectors (Zα, W α) on blade two, given in (1819), by the "angle" φ. It is straightforward to check that the equalities U[α (φ) V β] (φ) = U [α V β] and Z[α (φ) W β] (φ) = Z[α W β] are true. These equalities mean that these antisymmetric tetrad objects are gauge invariant. In the Abelian case it was proved that the local group of electromagnetic gauge transformations is isomorphic to both the local groups LB1 and LB2, separately, independently. LB1 on the local plane one is a group composed by the tetrad boosts SO(1, 1) and two different kinds of discrete transformations. One of the discrete transformations is the full inversion or minus the identity two by two. The other discrete transformation is not Lorentzian1 because it is a reflection or flip, a two by two matrix with zeroes in the diagonal and ones off-diagonal. LB2 on plane two is the group of spatial tetrad rotations on this plane, that is SO(2). It is worth reminding ourselves about a point in mathematical language that could be loose or inaccurate but nonetheless immediate to understand. With the purpose of exemplifying we can mention the isomorphisms in the Abelian case1 between the local group of electromagnetic gauge transformations and the local groups of tetrad transformations LB1 and LB2, separately, independently. The isomorphisms strictly speaking are homomorphisms between the local algebra of scalars associated to the local group of electromagnetic gauge transformations and the local groups LB1 and LB2, independently. We know that between the local algebra of scalars and the local group of 7 electromagnetic gauge transformations there is a homomorphism, a homomorphism between the real numbers R that is, the algebra of local scalars associated to the local group of electromagnetic gauge transformations and U(1), that is, the local group of electromagnetic gauge transformations. We give this relationship as implicitly understood even though we talk about isomorphisms between the local group of electromagnetic gauge transformations and the local groups of tetrad transformations LB1 and LB2, separately, independently. We must also stress that the local transformations (21-22) are not imposed local boosts on the vectors that define the local plane one. They are the result of local gauge transformations of the vectors (U α, V α). For example, from reference1 a particular boost after the gauge transformation would look like, ̃V α (1) q - ̃V β (1) ̃V(1)β = (1 + C) q (1 + C)2 -D2 V α (1) q -V β (1) V(1)β + D q (1 + C)2 -D2 V α (2) q V β (2) V(2)β (25) ̃V α (2) q ̃V β (2) ̃V(2)β = D q (1 + C)2 -D2 V α (1) q -V β (1) V(1)β + (1 + C) q (1 + C)2 -D2 V α (2) q V β (2) V(2)β . (26) In equations (25-26) the following notation has been used, C = (-Q/2)V(1)σΛσ/(V(2)βV β (2)), D = (-Q/2)V(2)σ Λσ/(V(1)β V β (1) ) and [(1+C)2 -D2] > 0 must be satisfied. The notation Λα has been used for Λ,α where Λ is the local scalar generating the local gauge transformation. U α = V α (1) q -V β (1) V(1)β and V α = V α (2) q V β (2) V(2)β according to the notation used in paper1, V α (1) = ξαλ ξρλ Aρ (27) V α (2) = q -Q/2 ξαλ Aλ (28) V α (3) = q -Q/2 ∗ξαλ ∗Aλ (29) V α (4) = ∗ξαλ ∗ξρλ ∗Aρ . (30) For the particular case when 1 + C > 0, the transformations (25-26) manifest that an electromagnetic gauge transformation on the vector field Aα →Aα + Λα, that leaves invariant the electromagnetic field fμν, generates a boost transformation on the normalized tetrad vector fields V α (1) q -V β (1) V(1)β , V α (2) q V β (2) V(2)β . In this case cosh(φ) = (1+C) √ (1+C)2-D2, see also equation (21). This was just one of the possible cases in LB1. Similar analysis for the vector 8 transformations (23-24) in the local plane two generated by (Zα, W α). See reference1 for the detailed analysis of all possible cases. Back to our main line of work we can write the electromagnetic field in terms of these tetrad vectors, fαβ = -2 q -Q/2 cos α U[α Vβ] + 2 q -Q/2 sin α Z[α Wβ] . (31) Equation (31) entails the maximum simplification in the expression of the electromagnetic field. The true degrees of freedom are the local scalars q -Q/2 and α. We can also present both local degrees of freedom as q -Q/2 cos α and q -Q/2 sin α. The object U[αVβ] remains invariant1 under a "rotation" of the tetrad vectors U α and V α by a scalar angle φ like in (21-22) on blade one. This is the way in which local gauge invariance is manifested explicitly on this local plane. Analogous for discrete transformations on blade one. Similar analysis on blade two. A spatial "rotation" generated by a gauge transformation of the tetrad vectors Zα and W α through an "angle" φ as in (23-24), such that the object Z[α Wβ] remains invariant1. All this formalism clearly provides a technique to maximally simplify the expression for the electromagnetic field strength as in equation (31). It is block diagonalized automatically by the tetrad (16-19). This is not the case for the non-Abelian SU(2) field strength. We do not have an automatic block diagonalization. To this purpose a new algorithm was developed in reference11. In the next section we study the construction of tetrads similar to the Abelian case but in the non-Abelian environment. III. OVERVIEW OF NEW TETRADS AND SYMMETRIES FOR THE NON-ABELIAN CASE For the non-Abelian case we first would like to present a set of examples on how to build extremal fields that are locally invariant under SU(2) gauge transformations. Let us remember that in the Abelian case in the tetrads (12-15) the only dependence on gauge came through the gauge vectors Xμ and Y μ. The tetrad skeletons as it was mentioned previously are local gauge invariants. The advantage of this method is that when we introduce local gauge transformations, the vectors that span the local plane one, do not leave this plane after the transformation and analogous in the local plane two. This fact implies in turn that the metric tensor, be non-flat or flat will not change and will remain invariant. It is then 9 important to show explicitly that we can construct extremal fields invariant under both the Abelian and the non-Abelian gauge transformations. One example could be for instance given by, ζμν = cos β fμν -sin β ∗fμν , (32) Following the Abelian pattern we can define the new complexion β, and to this end we will impose the new SU(2) local invariant condition, Tr[ζμν ∗ζμν] = ζk μν ∗ζkμν = 0 , (33) where the summation convention is applied on the internal index k. We are just using a generalized duality transformation, and defining through it, this new local scalar complexion β. Therefore, the complexion condition (33) is not an additional condition for the field strength. We simply introduced a possible generalization of the definition for the Abelian complexion, found through a new duality transformation as well. Then, we find the local SU(2) invariant complexion β to be, tan(2β) = -f k μν ∗f kμν/f p λρ f pλρ , (34) where once again the summation convention was applied on both k and p. We can also consider gauge covariant derivatives since they will become useful in the ensuing analysis. For example, the gauge covariant derivatives of the three extremal field internal components, ζkμν|ρ = ζkμν ; ρ + g εklp Alρ ζpμν . (35) where εklp is the completely skew-symmetric tensor in three dimensions with ε123 = 1, with g the coupling constant. As in the previous section II the symbol ";" stands for the usual covariant derivative associated with the metric tensor gμν. Next we consider the Einstein-Maxwell-Yang-Mills vacuum field equations, Rμν = T (ym) μν + T (em) μν (36) 10 f μν ;ν = 0 (37) ∗f μν ;ν = 0 (38) f kμν |ν = 0 (39) ∗f kμν |ν = 0 . (40) The field equations (37-38) provide two electromagnetic potentials9, not independent from each other, but due to the symmetry of the equations, available for our construction. Aμ and ∗Aμ are the two electromagnetic potentials, see the comments made about the Abelian potentials and the star nomenclature ∗Aμ in section II. Similar for the two NonAbelian equations (39-40). The Non-Abelian potential Akμ is available for our construction as well12-19. With all these elements put together, we can proceed to define the auxiliary antisymmetric field, ωμν = Tr(∗ζ στ ζμν + ζ στ ∗ζμν) Tr(ζσρ |ρ ∗ζτλ |λ) . (41) This particular antisymmetric auxiliary field in our construction could also be alternatively chosen to be, ωμν = Tr(ζ στ ζμν) Tr(ζσρ |ρ ∗ζτλ |λ) . (42) We can choose this antisymmetric auxiliary field ωμν in many different ways, we just show two examples. It is clear that (41) or (42) are invariant under SU(2) local gauge transformations. Expressions (41) or (42) are nothing but explicit examples among many, see for example reference3. Once our choice is made we perform a local duality rotation in order to obtain the new extremal field. We remind ourselves through the algorithm created in section II and reference1 that extremal fields are found through local duality rotations of second rank antisymmetric tensors like in equation (1) because then we can use the equations analogous to (4-5) to define an orthogonal tetrad. That is the core of this algorithm. εμν = cos θ ωμν -sin θ ∗ωμν . (43) As always we choose this complexion θ to be defined by the condition, 11 εμν ∗εμν = 0 . (44) Thus we find the new local scalar complexion analogously to section II to be, tan(2θ) = -ωμν ∗ωμν/ωλρ ωλρ . (45) We used our new algorithm to find a new kind of local SU(2) gauge invariant extremal tensor εμν, that enables the construction of the new tetrad, Sμ (1) = εμλ ερλ Xρ (46) Sμ (2) = q -Qym/2 εμλ Xλ (47) Sμ (3) = q -Qym/2 ∗εμλ Yλ (48) Sμ (4) = ∗εμλ ∗ερλ Y ρ , (49) where Qym = εμν εμν and we assume this local scalar not to be zero. With the help of identity (3), when applied to the case Aμα = εμα and Bνα = ∗ενα we obtain as in section II the equivalent condition to (44), εαν ∗εμν = 0 , (50) It is a simple excercise using (3) for Aμα = εμα and Bνα = ενα, and (50), to prove that the vectors (46-49) are orthogonal. As we did before in section II we will call for future reference εμλερλ the skeleton of the tetrad vector Sμ (1), and Xρ the gauge vector. In the case of Sμ (3), the skeleton is ∗εμλ, and Yλ the gauge vector. It is clear now that skeletons are gauge invariant under SU(2) × U(1) as we announced at the start of this section. This property guarantees that the vectors under local U(1) or SU(2) gauge transformations will not leave their original planes or blades, keeping therefore the metric tensor explicitly invariant. Our final task in this construction will be to define the gauge vectors Xσ and Y σ for the tetrad (46-49). A nontrivial although useful choice that we can make is Xσ = Y σ = Tr[Σαβ E ρ α E λ β ∗ξ σ ρ ∗ξλτ Aτ]. The nature of the object Σαβ is explained in section VI, Appendix II in reference3 and also 12 section VI. The object Σαβ is basically built with the Pauli matrices and the identity two by two. The tetrad vectors E ρ α inside the expression Tr[Σαβ E ρ α E λ β ∗ξ σ ρ ∗ξλτ Aτ] can be chosen to be the tetrad vectors that we already know from manuscript1 and section II for electromagnetic fields in curved space-times. Following the same notation as in1 and equations (16-19), we call E ρ (o) = U ρ, E ρ (1) = V ρ, E ρ (2) = Zρ, E ρ (3) = W ρ. The electromagnetic extremal tensor ξρσ, and its dual ∗ξρσ are also already known from reference1 and section II. We make use of the already defined tetrads built for Einstein-Maxwell spacetimes in order to enable the use of the object Σαβ which is key in our construction. The key lies in the translating property of this object between SU(2) local gauge transformations S and local Lorentz transformations Λα γ, see reference3 and notice from section VI that S-1 Σαβ S = Λα γ Λβ δ Σγδ. We would like to study one more property of these chosen gauge vector fields Xσ = Y σ = Tr[Σαβ E ρ α E λ β ∗ξ σ ρ ∗ξλτ Aτ]. The structure E [ρ α E λ] β ∗ξρσ ∗ξλτ is invariant under U(1) local gauge transformations. The electromagnetic extremal field property1,8, ξμσ ∗ξμτ = 0 is useful in the contractions E ρ α E λ β ∗ξ σ ρ ∗ξλτ. Because it is leaving in the contraction of E ρ α E λ β with ∗ξρσ ∗ξλτ only the antisymmetric object E [ρ 2 E λ] 3 , which is locally U(1) gauge invariant. Precisely because of property (5). Let us remember that the object Σαβ is antisymmetric and contracted with the electromagnetic tetrads as Σαβ E ρ α E λ β inside the local gauge vector, see section III. In the first paper1 we proved that the group U(1) is isomorphic to the local group of boosts plus discrete transformations on blade one that we called LB1. The same group U(1) is isomorphic to SO(2), that we also called LB2 since it is related to local tetrad rotations on blade two. This is a fundamental result in group theory alone, let alone in physics. We proved in references3,4 that the local group of SU(2) gauge transformations is isomorphic to the tensor product of three LB1 groups. Second, the local group of SU(2) gauge transformations is isomorphic to the tensor product of three LB2 or SO(2) groups. All the local gauge groups of the Standard Model have been proven to be isomorphic to local groups of tetrad transformations in four-dimensional Lorentzian curved or flat spacetimes. The no-go theorems of the sixties20-22 have been proven to be incorrect. Not because of their internal logic but for the assumptions made at the outset of these theorems. We read in reference22 "S (the scattering matrix) is said to be Lorentz-invariant if it possesses a symmetry group locally isomorphic to the Poincar`e group P.. . . A symmetry transformation is said to be an internal symmetry transformation if it commutes with P. This implies that it 13 acts only on particle-type indices, and has no matrix elements between particles of different four-momentum or different spin. A group composed of such transformations is called an internal symmetry group". The local electromagnetic gauge group of transformations U(1) has been proven to be isomorphic to local groups of tetrad transformations LB1 and LB2 on both the orthogonal planes one and two. These local groups of transformations LB1 and LB2= SO(2) are composed of Lorentz transformations except in LB1 for an improper discrete reflection, see reference1. Therefore the local Lorentz group of spacetime transformations cannot commute with LB1 or LB2 since Lorentz transformations on a local plane do not necessarily commute with Lorentz transformations on another local plane at the same point in spacetime. The local internal groups of transformations do not necessarily commute with the local Lorentz transformations, because they are isomorphic to local groups of tetrad transformations. Analogous results were proven for the non-Abelian cases SU(2)×U(1) and SU(3) × SU(2) × U(1) Yang-Mills, see references3-5. IV. SPACETIME FEYNMAN CALCULUS It is of fundamental importance to understand the geometry of spacetime when particle interactions are taking place. Using the accumulated analysis for different kinds of gauge theories carried out in1-5,11,23, we will show explicitly how to assign to different Feynman diagrams in weakly interacting processes, different sets of tetrad vectors. The massive weak interactions boson mediators have an associated gravitational field as well as electrons, muons and neutrinos and even though these gravitational fields might be weak, they possess the necessary geometrical structure that enables the local symmetries of the standard model to be realized in an explicit fashion as it was analyzed in previous manuscripts1-5. We judge relevant to understand that the transformations of colliding particles into the same or other emerging particles can occur through the local transformation properties of gravitational fields which will differ for different settings even though they exhibit analogous manifest local invariance under the internal symmetries of the standard model. We remind ourselves that in manuscripts1-5,11,23 all the local internal gauge symmetries of the standard model have been proved isomorphic to local groups of tetrad transformations in four-dimensional curved Lorentz spacetimes. However, in order not to get foundational contradictions with quantum field theories at this stage of analysis we will assume that the kind of tetrads introduced in 14 sections II and III are defined in Minkowski spacetime. This choice of spacetime involves no contradiction since the basic operation of local duality field transformation can be performed in flat Minkowski spacetimes as well. We can define the tetrads in Minkowski spacetime and prove that these tetrads define locally two orthogonal planes of stress-energy diagonalization. These geometrical structures exist not only in curved spacetimes but also in flat spacetimes. In essence local tetrad gauge states of spacetime would represent different microparticles in their spacetime manifestation, that can transform through local gauge transformations into other microparticles or other local tetrad gauge states of spacetime. The notation is a replica of the notation in24, so we refer the reader to this reference. We also refer the reader to24-28 for abundant literature citation, specially in the field of particle physics. A. Weak interactions The existence of mediators as it was shown in1-4 is irreplaceable as far as we are concerned with the construction of these kind of tetrads in weak interactions. In this case it is the existence of local SU(2) "extremal" fields that allow us to build tetrads in weak processes. There are interactions involving the massive mediators where any virtual effect is negligible. For instance the W -as the mediator in inverse Muon decay. The Zo mediator in elastic Neutrino-Electron scattering. This is important because the existence of virtual processes would require a different approach. We will analyze these processes through the use of appropriately defined tetrads. 1. Inverse Muon decay Let us consider the process e-(1) + νμ(2) →νe(3) + μ-(4). There are two vertices. We invoke then the existence of the SU(2) tetrads introduced in3,4, specially the general tetrad structure presented in the section "Gauge geometry" and also section III. We called these general SU(2) tetrad vectors Sμ (1) · · · Sμ (4) and the structure of these latter tetrads was introduced in equations (46-49) in the section III dedicated to the overview of these objects. There was a remaining freedom in the choice of two vector fields, Xρ and Y ρ. It is exactly through an appropriate choice for these two vector fields that we can identify a tetrad set for each vertex at the same spacetime point. In addition to the previously introduced notation 15 and structures, let us call the non-null electromagnetic tetrads, following again the notation in references1-4 and sections II-III, E ρ α . There are local non-null electromagnetic tetrads in both vertices at the same spacetime point since in one vertex we have an electron and in the other vertex a muon. The indices α and β are reserved for locally inertial coordinate systems. Then, we can proceed to define for the first vertex the two gauge vector fields, Xρ = Y ρ = u(3) γα (1 -γ5) u(1) Eρ α . (51) We are basically associating to the first vertex a current24 jα -= u(3) γα (1 -γ5) u(1). This current describes the process e-→νe + W -. For the second vertex we can choose for instance, Xρ = Y ρ = u(4) γα (1 -γ5) u(2) Eρ α . (52) Again, we are assigning to the second vertex a current24 jα -= u(4) γα (1 -γ5) u(2) describing the process νμ + W -→μ-. It is evident from all the analysis in3,4 that the geometrical transition from vertex one to vertex two and vice-versa, is an SU(2) generated local gauge transformation. That is only allowed through the existence of massive mediators. Following the ideas in3 we can start by choosing for instance, Xρ = Y ρ = Tr[Σαβ E σ α E λ β ∗ξ ρ σ ∗ξλτ Aτ] (53) The Σαβ objects are analyzed in appendix II in reference3 and sections III-VI in this present paper, ξσρ are the electromagnetic "extremal" fields introduced in reference1 and section II in this present paper, etc. Through a local SU(2) gauge transformation on blade one, we can "rotate" the normalized version of vectors (46-47) on blade one, until Xρ in (53) becomes Xρ in (51). We can also "rotate" the normalized version of vectors (48-49) on blade two, until Y ρ in (53) becomes Y ρ in (51). Let us remember that the tetrad skeletons are locally gauge invariant under U(1)×SU(2). This is just a sample of local gauge transformations of the normalized version of vectors (46-49). We proved in references1,3,4 that the maps that both in the local plane one and two send the tetrad vectors that generate these planes from an initial gauge vector into another gauge vector are injective and surjective maps. The 16 map that assigns local groups of gauge transformations into local groups of tetrad transformations on either local orthogonal planes of stress-energy symmetry are isomorphisms. Again we can start with (53) and appropriately "rotate" the tetrad vectors on blade one, until they become the ones corresponding to Xρ given in (52). Similar for Y ρ in this second case (52). It is evident then that (51) and (52) are connected through local SU(2) gauge transformations on blades one and two, that in turn, leave invariant the metric tensor. That is, these local gauge transformations exist because of transitivity. The local groups of gauge transformations have been proven to be isomorphic to the local groups of tetrad transformations on the local orthogonal planes of symmetry. Given two sets of tetrads on the local plane one, then there is a local gauge transformation that sends one set into the other and vice-versa. Similar in the local othogonal plane two. These local orthogonal planes we remind ourselves are the local planes of covariant diagonalization of the stress-energy tensor, for the Abelian case and also for the non-Abelian case, see references1-5,11. We can also notice that the vector fields (51-52) are not strictly vectors but pseudovectors under local parity transformations, see reference24. But the metric tensor remains unaltered under these local parity transformations. It is as if the geometry associated to the e-(1) and νe(3) can be transformed through the existence of a massive mediator into the geometry associated to the νμ(2) and μ-(4) without altering the spacetime. The vertices are local tetrad gauge states of the same flat spacetime. 2. Elastic Neutrino-Electron scattering Now, we are considering neutral currents. In particular the interaction process νμ(1) + e-(2) →νμ(3) + e-(4). As before we can assign to the first vertex the choice, Xρ = Y ρ = u(3) γα (1 -γ5) u(1) Zρ α . (54) The current jα -= u(3) γα (1 -γ5) u(1), represents the process νμ(1) →νμ(3) + Zo. The tetrad Zρ α is built as follows. Following again the notation in24 we know we have available a local vector field Zμ that results from the Weinberg rotation through the angle θw, in addition to the standard electromagnetic local vector field Aμ. The rotation can be written, 17 Aμ = Bμ cos θw + W 3 μ sin θw (55) Zμ = -Bμ sin θw + W 3 μ cos θw . (56) The local tetrad field Zρ α is present in both vertices at the same spacetime point, since the massive neutral mediator is present in both local vertices and Zρ α is a local tetrad associated to this flat spacetime. The electro-weak mixing involves a weak isotriplet of intermediate vector bosons W coupled to three weak isospin currents, and an isosinglet intermediate vector boson Bμ coupled to the weak hypercharge current. If we follow all the steps in reference1 and the method developed in section II in the present paper, we can build out of the curl Zμ,ν -Zν,μ a new tetrad. This auxiliary local tetrad Zρ α present in the definition of gauge-vectors (54) would once more in its own construction involve the choice of two gaugevector fields, see reference1. We can choose for instance Zμ and Bμ as these two vector fields needed in turn in the definition and construction of the local auxiliary tetrad Zρ α. Then, the tetrad that couples to the neutrino current is associated to the massive Zo. The second vertex could be assigned a choice, Xρ = Y ρ = u(4) γα (cV -cA γ5) u(2) Eρ α , (57) representing e-(2) + Zo →e-(4). For this particular interaction cV = -1 2 + 2 sin θw, and cA = -1 2, where θw is again the Weinberg angle24. The massive mediator allows again for a SU(2) local gauge transformation between the tetrad vectors chosen for vertex one and the ones chosen for vertex two at the same spacetime point. The neutral current works as a geometry mediator between the scattered particles keeping the spacetime invariant at the same spacetime point. The vertices function as local tetrad gauge states of the same spacetime. V. CONCLUSIONS We have explored the possibility of assigning tetrads to Feynman diagrams. In interactions where we can assume the existence of particles with associated local fields like Abelian or non-Abelian gauge fields. At no step of analysis we have specified the tetrad themselves 18 making all these geometrical properties outstanding since they can be put forward with all generality without the need to study case by case as long as the gauge fields are non-null for example. We can think the massive weak interactions boson mediators to have associated gravitational fields as well as electrons, muons and neutrinos do and even though these gravitational fields might be weak, they possess the necessary geometrical structure that enables the local symmetries of the standard model to be realized in an explicit fashion as it was studied thoroughly in previous manuscripts1-5. However we decided to consider a background flat spacetime since gravitational fields would entail foundational contradictions with standard quantum field theories. New concepts would have to be introduced and we do not want to do this at this stage in analysis. We deem fundamental to understand that the transformations of colliding particles into the same or other emerging particles through elastic or inelastic processes can occur through the local transformation properties of spacetime which will differ for different settings through the new notion of tetrad gauge states of spacetime. Having done this explicitly, a number of questions naturally arise. We want these concluding remarks to be a summary of these open questions. • The order of the formulations24-31. We have worked out the low-order diagrams. Then, what happens with higher order diagrams ?. The tetrads admit the choice of two gauge vector fields Xρ and Y ρ, and the higher orders are additive exactly as in the quantum theories in these vector fields available as a choice. But there is more to understand. Do the higher order diagrams represent contributions coming from higher order perturbative theories of a full relativistic formulation of these interactions23 involving perturbations of the electromagnetic field, etc, for instance, or just perturbative expansions in the gauge vector fields?. As an example of this kind of situation we might want to qualitatively consider the quark decays b →s γ, see chapter XIV-7 in reference32 for instance. There are several possibilities but there are certainly higher order contributions to these kind of processes. Let us focus on the contribution that involves a W -boson mediator in rare decays. There is the b →W -+ c vertex and the subsequent c + W -→s vertex for example. Each one of these has an associated current vector, let us call them for short jα [bc] and jα [cs]. Both vertices have particles with electric charge so there is at each vertex an associated electromagnetic tetrad Eρ [bc]α and Eρ [cs]α respectively. Then we can associate to each vertex in this higher order diagram 19 gauge vectors Xρ [bc] = Y ρ [bc] = jα [bc] Eρ [bc]α and Xρ [cs] = Y ρ [cs] = jα [cs] Eρ [cs]α. The jα [bc] Eρ [bc]α contribution can then be added to the non-Abelian tetrad gauge vectors for vertex [bc] and similar for jα [cs] Eρ [cs]α at the vertex [cs]. In this contribution there is also a photon involved. This photon emission will change the stress-energy tensor and therefore will be associated to the perturbed extremal field which is in turn a local duality rotation of the perturbed electromagnetic field. These perturbation in the extremal field will in turn perturb the tetrad skeletons. Therefore, the vertices in the diagram will be associated to tetrad gauge states of the spacetime and the photon emission to tetrad skeleton perturbations. The b →s γ decays might involve contributions with top quarks and the analysis will be similar, the b →s γ decays could include a loop of the kind cc for example and we will proceed similarly as well. We will add to the gauge vectors associated to the corresponding vertex, higher and higher contributions with a corresponding expansion parameter. The currents at the corresponding vertex are the key objects necessary to produce gauge vectors associated to vertices. There could be that both, the perturbations in the skeletons and the perturbations in gauge vectors proceed simultaneously as in the b →s γ case. On one hand the local orthogonal planes of symmetry will tilt and on the other hand the tetrad vectors that span these local planes will rotate inside them. • A point outside the scope of this manuscript. The issue of "gauge gravity". Since in references1,3 it was explicitly proved that the Abelian and non-Abelian gauge theories represent special symmetries of the gravitational field, we can ask about the meaning of "gauge gravity". The electromagnetic field is associated to the LB1 and LB2 symmetries of the gravitational field, see reference1. The SU(2) group of local gauge transformations is associated to the symmetries of the tensor product of three LB1 or three LB2 groups of transformations, see references3,4. Analogous for SU(3), see reference5. Then, it is not obvious to understand what is the meaning of a statement like, "casting the theory of gravity into a Yang-Mills formulation". We have reason to believe that we can truly cast the theory of gravity into a Yang-Mills formulation but as said, it is not obvious and requires a whole new work. • Another point outside the scope of this manuscript. The issue of quantum gravity. It has been proved explicitly that metric tensors can be associated with microparticle in20 teractions. These constructions are possible by means of non-null Abelian tetrad fields, and by means of SU(2) local non-Abelian tetrad fields. Perturbative formulations of these tetrad field structures as in reference23 can take care of quantum fluctuations as well. The quantum is connected through the existence of these tetrad fields to gravity. A treatment for a curved spacetime where a gravitational field is present would entail several new notions that we would like not to introduce at this stage of analysis. Nonetheless we can advance that in curved spacetimes there will be local orthogonal planes one and two and the isomorphisms between local gauge groups Abelian and non-Abelian and local groups of tetrad transformations LB1 and LB2 would also apply in a similar fashion as to flat spacetimes1-5. The quantization will be reflected through interactions that alter the local plane-symmetry structure by tilting these local planes of symmetry. Continuously for continuous perturbations and discretely in quantum settings situations. The main idea behind these quantum formulations in curved spacetimes is that the local planes of stress-energy diagonalization are always and during quantum interactions local orthogonal planes of symmetry, because quantum problems are basically confronted through perturbations analysis and these perturbations lead to continuous or discrete evolution of the local planes of symmetry either in flat or curved spacetimes. Continuous or discrete tilt symmetry evolution, see reference23. We also established that during interactions of microparticles the tetrad vectors that span the local orthogonal planes one and two might rotate inside them. The tetrads of different nature that we were able to build in1-5 and the present work, establish a link between the standard locally inertial flat field environment of the traditional standard quantum theories in weak interactions on one hand, and the curved spacetime of gravity on the other hand. The point is the following, why are we using in quantum gravity similar conceptual foundations to theories that are not formulated in curved spacetimes but flat spacetimes ?. • Another point outside the limited scope of this paper. The issue of the Higgs mechanism. It is a device conceived in its relationship with the nature of mass, for instance of the mass mediators. In the present tetrad environment we can ask if it is necessary, or the mass comes into existence due to the presence of gravity ?. Is it possible that the local Higgs field and its quantum fluctuations are related to the perturbations of 21 the local gravitational weak field scalar approximation on a flat Minkowski background associated to the asymptotically curved spacetimes that in turn we can associate to elementary microparticles ?. • The issue of symmetry-breaking. It was proved in the general manuscripts1-4 that the gravitational field when built with tetrads along the lines of expressions (46-49) are manifestly invariant under local electromagnetic gauge transformations, and local SU(2) gauge transformations as well. But when assigning a tetrad set to a vertex in a low-energy weak process diagram in Minkowski spacetime, we make a particular choice for the two gauge vectors Xρ and Y ρ. For instance, through associated currents we choose a particular gauge, and a different one for each vertex, like in inverse Muon decay or elastic Neutrino-Electron scattering. Then, we wonder if this gauge fixing procedure could be the geometrical form of the standard symmetry-breaking process. Hereby, we can see that it is the tetrad fields that bridge the two gauges associated to the two vertices, through a local SU(2) gauge transformation, that in turn, leaves invariant the metric tensor. VI. APPENDIX I This appendix is introducing the object Σαβ that according to the matrix definitions introduced in the references is Hermitic. The use of this object in the construction of tetrads in section III enables the local SU(2) gauge transformations S, to get in turn transformed into purely geometrical transformations. That is, local rotations of the U(1) electromagnetic tetrads. The object σαβ is defined6,7 as σαβ = σα + σβ --σβ + σα -. The object σα ± arises when building the Weyl representation for left handed and right handed spinors. According to reference7, it is defined as σα ± = (1, ±σi), where σi are the Pauli matrices for i = 1 · · · 3. Under the (1 2, 0) and (0, 1 2) spinor representations of the Lorentz group this object transforms as, S-1 (1/2) σα ± S(1/2) = Λα γ σγ ± . (58) Equation (58) means that under the spinor representation of the Lorentz group, σα ± transform as vectors. In (58), the matrices S(1/2) are local objects, as well as7 Λα γ. The 22 SU(2) elements can be considered to belong to the Weyl spinor representation of the Lorentz group. Since the group SU(2) is homomorphic to SO(3), they just represent local space rotations. It is also possible to define the object σ†αβ = σα -σβ + -σβ -σα + in a similar fashion. Therefore, we can write, ı σαβ + σ†αβ = 0 if α = 0 and β = i 4 εijk σk if α = i and β = j , σαβ -σ†αβ = -4 σi if α = 0 and β = i 0 if α = i and β = j . We then may call Σαβ ROT = ı σαβ + σ†αβ , and Σαβ BOOST = ı σαβ -σ†αβ and a possible choice for the object Σαβ could be for instance Σαβ = Σαβ ROT +Σαβ BOOST. This is a good choice when we consider proper Lorentz transformations of the tetrad vectors nested within the structure of the gauge vectors Xμ and Y μ. For spatial rotations of the U(1) electromagnetic tetrad vectors which in turn are nested within the structure of the two gauge vectors Xμ and Y μ, as is the case under study in section III, we can simply consider Σαβ = Σαβ ROT. These possible choices also make sure the Hermiticity of gauge vectors. Since when defining the gauge vectors Xμ and Y μ we are taking the trace, then Xμ and Y μ are real. REFERENCES 1A. Garat, Tetrads in geometrodynamics, J. Math. Phys. 46, 102502 (2005). A. Garat, Erratum: Tetrads in geometrodynamics, J. Math. Phys. 55, 019902 (2014). 2A. Garat, New tetrads in Riemannian geometry and new ensuing results in group theory, gauge theory and fundamental physics in particle physics, general relativity and astrophysics, Int. J. Mod. Phys. Conf. Ser., Vol. 45, (2017), 1760004. 3A. Garat, Tetrads in Yang-Mills geometrodynamics, Gravitation and Cosmology, (2014) Vol. 20 No. 1, pp. 116-126. Pleiades Publishing Ltd. arXiv:gr-qc/0602049. 4A. Garat, The new electromagnetic tetrads, infinite tetrad nesting and the non-trivial emergence of complex numbers in real theories of gravitation, Int. J. Geom. Methods Mod. Phys., Vol. 14, No. 9 (2017), 1750132. 5A. Garat, Tetrads in SU(3) × SU(2) × U(1) Yang-Mills geometrodynamics, Int. J. Geom. Methods Mod. Phys., Vol. 15 no. 3 (2018), 1850045. . 23 6M. Kaku, Quantum Field Theory: A Modern Introduction (Oxford University Press, 1993). 7L. ́Alvarez-Gaum ́e and M. A. V ́azquez-Mozo, Introductory Lectures on Quantum Field Theory (arXiv:hep-th/0510040). 8C. Misner and J. A. Wheeler, Classical physics as geometry, Annals of Physics 2, 525 (1957). 9N. Cabibbo and E. Ferrari, Nuovo Cim. 23, 1147 (1962). 10J. A. Schouten, Ricci Calculus: An Introduction to Tensor Calculus and Its Geometrical Applications (Springer, Berlin, 1954). 11A. Garat, "Gauge invariant method for maximum simplification of the field strength in non-Abelian Yang-Mills theories", Int. J. Geom. Methods Mod. Phys., Vol. 12, No. 10 (2015), 1550104. . 12R. Gilmore, Lie Groups, Physics and Geometry (Cambridge University Press, 2008). 13R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (John Wiley & Sons, 1974). 14J. Stillwell, Naive Lie Theory (Springer Science + Business Media, L.L.C., 2010). 15N. Carter, Visual Group Theory (The Mathematical Association of America, Inc, 2009). 16M. Carmeli, Classical Fields: General Relativity and Gauge Theory (J. Wiley & Sons, New York, 1982). 17C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954). 18R. Utiyama, Phys. Rev., 101, 1597 (1956). 19T. W. B. Kibble, J. Math. Phys., 2, 212 (1961). 20S. Weinberg, Comments on relativistic supermultiplet theories, Phys. Rev. 139, B597 (1965). 21L. O'Raifeartagh, Lorentz invariance and internal symmetry, Phys. Rev. 139, B1052 (1965). 22S. Coleman and J. Mandula, All possible symmetries of the S matrix, Phys. Rev. 159, N5 1251 (1967). 23A. Garat, "Dynamical symmetry breaking in geometrodynamics", TMF, 195:2, (2018), 313-328; Theoret. and Math. Phys., 195:2, (2018), 764-776. . 24D. Griffiths , Introduction to elementary particles (John Wiley & Sons, Inc. , 1987). 25T. P. Cheng and L. F. Li , Gauge Theory of Elementary Particle Physics (Oxford University Press, 1989). 24 26W. Greiner and B. Mueller, Gauge Theory of Weak Interactions (Springer Verlag Gmbh, 1996). 27W. Greiner and B. Mueller, Quantum Mechanics, Symmetries (Springer Verlag, 1989). 28G. ,t Hooft, Renormalization of Gauge Theories (Lecture notes Erice, 1998). 29M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus Books Publishing L.L.C., 1995). 30M. Srednicki, Quantum Field Theory (Cambridge University Press, New York, 2007). 31R. Jackiw, Fifty Years of Yang-Mills Theory and my Contribution to it (arXiv:physics/0403109, 2004). 32J. F. Donoghue, E. Golowich and B. R. Holstein, Dynamics of the Standard Model (Cambridge monographs on particle physics, nuclear physics and cosmology) (Cambridge University Press, Cambridge, 2014). 25
|
math/0104241
|
arXiv:math/0104241v1 [math.CO] 25 Apr 2001
THE LAURENT PHENOMENON
SERGEY FOMIN AND ANDREI ZELEVINSKY
Abstract. A composition of birational maps given by Laurent polynomi-
als need not be given by Laurent polynomials; however, sometimes—quite
unexpectedly—it does. We suggest a unified treatment of this phenomenon,
which covers a large class of applications. In particular, we settle in the affir-
mative a conjecture of D. Gale and R. Robinson on integrality of generalized
Somos sequences, and prove the Laurent property for several multidimensional
recurrences, confirming conjectures by J. Propp, N. Elkies, and M. Kleber.
Contents
1.
Introduction
1
2.
The Caterpillar Lemma
4
3.
One-dimensional recurrences
8
4.
Two- and three-dimensional recurrences
11
5.
Homogeneous exchange patterns
20
References
21
1. Introduction
In this paper, we suggest a unified explanation for a number of instances in which
certain recursively defined rational functions prove, unexpectedly, to be Laurent
polynomials. We begin by presenting several instances of this Laurent phenomenon
established in the paper.
Example 1.1. (The cube recurrence) Consider a 3-dimensional array
(yijk : (i, j, k) ∈H)
whose elements satisfy the recurrence
yi,j,k = αyi−1,j,kyi,j−1,k−1 + βyi,j−1,kyi−1,j,k−1 + γyi,j,k−1yi−1,j−1,k
yi−1,j−1,k−1
.
(1.1)
Here H can be any non-empty subset of Z3 satisfying the following conditions:
if (i, j, k) ∈H, then (i′, j′, k′) ∈H whenever i ≤i′, j ≤j′, k ≤k′;
(1.2)
for any (i′, j′, k′) ∈H, the set {(i, j, k) ∈H : i ≤i′, j ≤j′, k ≤k′} is finite.
(1.3)
Date: April 25, 2001.
1991 Mathematics Subject Classification. Primary 14E05, Secondary 05E99, 11B83.
Key words and phrases. Laurent phenomenon, Somos sequence, Gale-Robinson conjecture.
The authors were supported in part by NSF grants #DMS-0049063, #DMS-0070685 (S.F.),
and #DMS-9971362 (A.Z.).
1
2
SERGEY FOMIN AND ANDREI ZELEVINSKY
Theorem 1.2. Let Hinit = {(a, b, c) ∈H : (a −1, b −1, c −1) /∈H}. For every
(i, j, k) ∈H, the entry yi,j,k is a Laurent polynomial with coefficients in Z[α, β, γ]
in the initial entries ya,b,c, for (a, b, c) ∈Hinit.
The cube recurrence (with α = β = γ = 1) was introduced by James Propp [10],
who was also the one to conjecture Laurentness in the case when H ⊂Z3 is given
by the condition i + j + k ≥0; in this case Hinit consists of all (a, b, c) ∈H such
that a + b + c ∈{0, 1, 2}. Another natural choice of H was suggested by Michael
Kleber: H = Z3
≥0, in which case Hinit = {(a, b, c) ∈Z3
≥0 : abc = 0}.
Example 1.3. (The Gale-Robinson sequence) Let p, q, and r be distinct positive
integers, let n = p + q + r, and let the sequence y0, y1, . . . satisfy the recurrence
yk+n = αyk+pyk+n−p + βyk+qyk+n−q + γyk+ryk+n−r
yk
.
(1.4)
David Gale and Raphael Robinson conjectured (see [7] and [8, E15]) that every
term of such a sequence is an integer provided y0 = · · · = yn−1 = 1 and α, β, γ are
positive integers. Using Theorem 1.2, we prove the following stronger statement.
Theorem 1.4. As a function of the initial terms y0, . . . , yn−1, every term of the
Gale-Robinson sequence is a Laurent polynomial with coefficients in Z[α, β, γ].
We note that the special case α = β = γ = 1, p = 1, q = 2, r = 3, n = 6 (resp.,
r = 4, n = 7) of the recurrence (1.4) is the Somos-6 (resp., Somos-7) recurrence [7].
Example 1.5. (Octahedron recurrence) Consider the 3-dimensional recurrence
yi,j,k = αyi+1,j,k−1yi−1,j,k−1 + βyi,j+1,k−1yi,j−1,k−1
yi,j,k−2
(1.5)
for an array (yijk)(i,j,k)∈H whose indexing set H is contained in the lattice
L = {(i, j, k) ∈Z3 : i + j + k ≡0 mod 2}
(1.6)
and satisfies the following analogues of conditions (1.2)–(1.3):
if (i, j, k) ∈H, then (i′, j′, k′) ∈H whenever |i′ −i| + |j′ −j| ≤k′ −k;
(1.7)
for any (i′, j′, k′) ∈H, the set {(i, j, k) ∈H : |i′ −i| + |j′ −j| ≤k′ −k}
(1.8)
is finite.
Theorem 1.6. Let Hinit = {(a, b, c) ∈H : (a, b, c −2) /∈H}. For every (i, j, k) ∈
H, the entry yi,j,k is a Laurent polynomial with coefficients in Z[α, β] in the initial
entries ya,b,c, for (a, b, c) ∈Hinit.
The octahedron recurrence on the half-lattice
H = {(i, j, k) ∈L : k ≥0}
(1.9)
was studied by W. H. Mills, D. P. Robbins, and H. Rumsey in their pioneering
work [9] on the Alternating Sign Matrix Conjecture (cf. [1] and [10, Section 10] for
further references); in particular, they proved the special case of Theorem 1.6 for
this choice of H.
THE LAURENT PHENOMENON
3
Example 1.7. (Two-term version of the Gale-Robinson sequence) Let p, q, and n
be positive integers such that p < q ≤n/2, and let the sequence y0, y1, . . . satisfy
the recurrence
yk+n = αyk+pyk+n−p + βyk+qyk+n−q
yk
.
(1.10)
Using Theorem 1.6, one can prove that this sequence also exhibits the Laurent
phenomenon.
Theorem 1.8. As a function of the initial terms y0, . . . , yn−1, every term ym is
a Laurent polynomial with coefficients in Z[α, β].
We note that in the special case α = β = 1, p = 1, q = 2, n = 5 (resp., n = 4),
(1.10) becomes the Somos-5 (resp., Somos-4) recurrence [7].
The last example of the Laurent phenomenon presented in this section is of a
somewhat different kind; it is inspired by [2].
Example 1.9. Let n ≥3 be an integer, and consider a quadratic form
P(x1, . . . , xn) = x2
1 + · · · + x2
n +
X
i<j
αijxixj .
Define the rational transformations F1, . . . , Fn by
Fi : (x1, . . . , xn) 7→(x1, . . . , xi−1,
P
xi=0
xi
, xi+1, . . . , xn).
(1.11)
Theorem 1.10. For any sequence of indices i1, . . . , im, the composition map G =
Fi1 ◦· · · ◦Fim is given by
G : x = (x1, . . . , xn) 7→(G1(x), . . . , Gn(x)),
where G1, . . . , Gn are Laurent polynomials with coefficients in Z[αij : i < j].
This paper is an outgrowth of [6], where we initiated the study of a new class
of commutative algebras, called cluster algebras, and established the Laurent phe-
nomenon in that context. Here we prove the theorems stated above, along with a
number of related results, using an approach inspired by [6]. The first step is to
reformulate the problem in terms of generalized exchange patterns (cf. [6, Defini-
tion 2.1]), which consist of clusters and exchanges among them. The clusters are
distinguished finite sets of variables, each of the same cardinality n. An exchange
operation on a cluster x replaces a variable x ∈x by a new variable x′ = P
x , where
P is a polynomial in the n−1 variables x−{x}. Each of the above theorems can be
restated as saying that any member of the cluster obtained from an initial cluster
x0 by a particular sequence of exchanges is a Laurent polynomial in the variables
from x0. Theorem 1.10 is explicitly stated in this way; in the rest of examples
above, the rephrasing is less straightforward.
Our main technical tool is “The Caterpillar Lemma” (Theorem 2.1), which es-
tablishes the Laurent phenomenon for a particular class of exchange patterns (see
Figure 1).
This is a modification of the namesake statement [6, Theorem 3.2],
and its proof closely follows the argument in [6]. (We note that none of the two
statements is a formal consequence of another.)
4
SERGEY FOMIN AND ANDREI ZELEVINSKY
In most applications, including Theorems 1.2 and 1.6 above, the “caterpillar”
patterns to which Theorem 2.1 applies, are not manifestly present within the origi-
nal setup. Thus, we first complete it by creating additional clusters and exchanges,
and then apply the Caterpillar Lemma.
The paper is organized as follows. The Caterpillar Lemma is proved in Sec-
tion 2. Subsequent sections contain its applications. In particular, Theorems 1.2,
1.4, 1.6, and 1.8 are proved in Section 4, while Theorem 1.10 is proved in Sec-
tion 5. Other instances of the Laurent phenomenon treated in this paper include
generalizations of each of the following: Somos-4 sequences (Example 3.3), Elkies’s
“knight recurrence” (Example 4.1), frieze patterns (Example 4.3) and number walls
(Example 4.4).
We conjecture that in all instances of the Laurent phenomenon established in this
paper, the Laurent polynomials in question have nonnegative integer coefficients.
In other contexts, similar nonnegativity conjectures were made earlier in [4, 5, 6].
Acknowledgments.
We thank Jim Propp for introducing us to a number of
beautiful examples of the Laurent phenomenon, and for very helpful comments on
the first draft of the paper. In particular, it was he who showed us how to deduce
Theorem 1.8 from Theorem 1.6.
This paper was completed during our stay at the Isaac Newton Institute for
Mathematical Sciences (Cambridge, UK), whose support and hospitality are grate-
fully acknowledged.
2. The Caterpillar Lemma
Let us fix an integer n ≥2, and let T be a tree whose edges are labeled by the
elements of the set [n] = {1, 2, . . ., n}, so that the edges emanating from each vertex
receive different labels. By a common abuse of notation, we will sometimes denote
by T the set of the graph’s vertices. We will write t
k
−−−t′ if vertices t, t′ ∈T are
joined by an edge labeled by k.
From now on, let A be a unique factorization domain (the ring of integers Z
or a suitable polynomial ring would suffice for most applications). Assume that
a nonzero polynomial P ∈A[x1, . . . , xn], not depending on xk , is associated with
every edge t
k
−−−t′ in T . We will write
t −−−
P
t′
or
t
k
−−−
P
t′ , and call P the
exchange polynomial associated with the given edge. The entire collection of these
polynomials is called a generalized exchange pattern on T . (In [6], we introduced a
much narrower notion of an exchange pattern; hence the terminology.)
We fix a root vertex t0 ∈T , and introduce the initial cluster x(t0) of n in-
dependent variables x1(t0), . . . , xn(t0). To each vertex t ∈T , we then associate a
cluster x(t) consisting of n elements x1(t), . . . , xn(t) of the field of rational functions
A(x1(t0), . . . , xn(t0)). The elements xi(t) are uniquely determined by the following
exchange relations, for every edge t
k
−−−
P
t′:
xi(t) = xi(t′)
for any i ̸= k;
(2.1)
xk(t) xk(t′) = P(x(t)).
(2.2)
(One can recursively compute the xi(t)’s, moving away from the root. Since the
exchange polynomial P does not depend on xk, the exchange relation (2.2) does
not change if we apply it in the opposite direction.)
THE LAURENT PHENOMENON
5
We next introduce a special class of “caterpillar” patterns, and state conditions
on their exchange polynomials that will imply Laurentness.
For m ≥1, let Tn,m be the tree of the form shown in Figure 1.
✲
✲
✲
✲
✲
✲
✲
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
❆
✁
✁
✁
✁
✁
✁
✁
✁
✁
✁
✁
✁
✁
✁
✁
✁
❍
❍
✟✟
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
t0
t1
thead
Figure 1. The “caterpillar” tree Tn,m, for n = 4, m = 8
The tree Tn,m has m vertices of degree n in its “spine” and m(n−2)+2 vertices
of degree 1. We label every edge of the tree by an element of [n], so that the n
edges emanating from each vertex on the spine receive different labels. We let the
root t0 be a vertex in Tn,m that does not belong to the spine but is connected to
one of its ends. This gives rise to the orientation of the spine, with all the arrows
pointing away from t0 (see Figure 1). We assign a nonzero exchange polynomial
P ∈A[x1, . . . , xn] to every edge t −−−t′ of Tn,m, thus obtaining an exchange
pattern.
For a rational function F = F(x, y, . . . ), we will denote by F|x←g(x,y,... ) the
result of substituting g(x, y, . . . ) for x into F. To illustrate, if F(x, y) = xy, then
F|x←y
x = y2
x .
Theorem 2.1. (The Caterpillar Lemma) Assume that a generalized exchange pat-
tern on Tn,m satisfies the following conditions:
For any edge •
k
−−−
P
•, the polynomial P does not depend on xk, and is not
(2.3)
divisible by any xi, i ∈[n].
For any two edges •
i
−−−
P
•
j
−−→
Q
•, the polynomials P and Q0 =Q|xi=0
(2.4)
are coprime elements of A[x1, . . . , xn].
For any three edges •
i
−−−
P
•
j
−−→
Q
•
i
−−−
R
• labeled i, j, i, we have
(2.5)
L · Qb
0 · P = R
xj←Q0
xj ,
where b is a nonnegative integer, Q0 =Q|xi=0 , and L is a Laurent
monomial whose coefficient lies in A and is coprime with P.
Then each element xi(t), for i ∈[n], t ∈Tn,m , is a Laurent polynomial in
x1(t0), . . . , xn(t0), with coefficients in A.
(Note the orientation of edges in (2.4)–(2.5).)
Proof. Our argument is essentially the same as in [6, Theorem 3.2]. For t ∈Tn,m,
let
L(t) = A[x1(t)±1, . . . , xn(t)±1]
6
SERGEY FOMIN AND ANDREI ZELEVINSKY
denote the Laurent polynomial ring in the cluster x(t) with coefficients in A. We
view each L(t) as a subring of the ambient field of rational functions A(x(t0)).
In this notation, our goal is to show that every cluster x(t) is contained in L(t0).
We abbreviate L0 = L(t0).
Note that L0 is a unique factorization domain, so
any two elements x, y ∈L0 have a well-defined greatest common divisor gcd(x, y)
which is an element of L0 defined up to a multiple from the group L×
0 of invertible
elements in L0; the group L×
0 consists of Laurent monomials in x1(t0), . . . , xn(t0)
whose coefficient belongs to A×, the group of invertible elements of A.
To prove that all x(t) are contained in L0, we proceed by induction on m, the
size of the spine. The claim is trivial for m = 1, so let us assume that m ≥2, and
furthermore assume that our statement is true for all “caterpillars” with smaller
spine. It is thus enough to prove that x(thead) ⊂L0 , where thead is one of the
vertices most distant from t0 (see Figure 1).
We assume that the path from t0 to thead starts with the following two edges:
t0
i
−−−
P
t1
j
−−→
Q
t2. Let t3 ∈Tn,m be the vertex such that t2
i
−−−
R
t3. The following
lemma plays a crucial role in our proof.
Lemma 2.2. The clusters x(t1), x(t2), and x(t3) are contained in L0. Further-
more, gcd(xi(t3), xi(t1)) = gcd(xj(t2), xi(t1)) = 1.
Proof. The only element in the clusters x(t1), x(t2), and x(t3) whose inclusion
in L0 is not immediate from (2.1)–(2.2) is xi(t3). To simplify the notation, let us
denote x = xi(t0), y = xj(t0) = xj(t1), z = xi(t1) = xi(t2), u = xj(t2) = xj(t3),
and v = xi(t3), so that these variables appear in the clusters at t0, . . . , t3, as shown
below:
y,x
•
t0
i
−−−−−−−
P
z,y
•
t1
j
−−−−−−→
Q
u,z
•
t2
i
−−−−−−−
R
v,u
•
t3
.
Note that the variables xk, for k /∈{i, j}, do not change as we move among the
four clusters under consideration. The lemma is then restated as saying that
v ∈L0;
(2.6)
gcd(z, u) = 1 ;
(2.7)
gcd(z, v) = 1 .
(2.8)
Another notational convention will be based on the fact that each of the polynomials
P, Q, R has a distinguished variable on which it depends, namely xj for P and R,
and xi for Q.
(In view of (2.3), P and R do not depend on xi, while Q does
not depend on xj.) With this in mind, we will routinely write P, Q, and R as
polynomials in one (distinguished) variable. For example, we rewrite the formula
in (2.5) as
R
Q(0)
y
= L(y)Q(0)bP(y),
(2.9)
where we denote L(y) = L|xj←y. In the same spirit, the notation Q′, R′, etc., will
refer to the partial derivatives with respect to the distinguished variable.
THE LAURENT PHENOMENON
7
We will prove the statements (2.6), (2.7), and (2.8) one by one, in this order.
We have:
z = P(y)
x
;
u = Q(z)
y
=
Q
P (y)
x
y
;
v = R(u)
z
=
R
Q(z)
y
z
=
R
Q(z)
y
−R
Q(0)
y
z
+
R
Q(0)
y
z
.
Since
R
Q(z)
y
−R
Q(0)
y
z
∈L0
and
R
Q(0)
y
z
= L(y)Q(0)bP(y)
z
= L(y)Q(0)bx ∈L0 ,
(2.6) follows.
We next prove (2.7). We have
u = Q(z)
y
≡Q(0)
y
mod z .
Since x and y are invertible in L0, we conclude that gcd(z, u) = gcd(P(y), Q(0)) = 1
(using (2.4)).
It remains to prove (2.8). Let
f(z) = R
Q(z)
y
.
Then
v = f(z) −f(0)
z
+ L(y)Q(0)bx .
Working modz, we obtain:
f(z) −f(0)
z
≡f ′(0) = R′
Q(0)
y
· Q′(0)
y
.
Hence
v ≡R′
Q(0)
y
· Q′(0)
y
+ L(y)Q(0)bx mod z .
Note that the right-hand side is a polynomial of degree 1 in x whose coefficients
are Laurent polynomials in the rest of the variables of the cluster x(t0). Thus (2.8)
follows from gcd
L(y)Q(0)b, P(y)
= 1, which is a consequence of (2.4)–(2.5).
□
We can now complete the proof of Theorem 2.1. We need to show that any
variable X = xk(thead) belongs to L0.
Since both t1 and t3 are closer to thead
than t0, we can use the inductive assumption to conclude that X belongs to both
L(t1) and L(t3). Since X ∈L(t1), it follows from (2.1) that X can be written as
X = f/xi(t1)a for some f ∈L0 and a ∈Z≥0 . On the other hand, since X ∈L(t3),
it follows from (2.1) and from the inclusion xi(t3) ∈L0 provided by Lemma 2.2
that X has the form X = g/xj(t2)bxi(t3)c for some g ∈L0 and some b, c ∈Z≥0 .
The inclusion X ∈L0 now follows from the fact that, by the last statement in
8
SERGEY FOMIN AND ANDREI ZELEVINSKY
Lemma 2.2, the denominators in the two obtained expressions for X are coprime
in L0.
□
3. One-dimensional recurrences
In this section, we apply Theorem 2.1 to study the Laurent phenomenon for
sequences y0, y1, . . . given by recursions of the form
ym+nym = F(ym+1, . . . , ym+n−1),
(3.1)
where F ∈A[x1, . . . , xn−1].
For an integer m, let ⟨m⟩denote the unique element of [n] = {1, . . . , n} satisfying
m ≡⟨m⟩mod n. We define the polynomials F1, . . . , Fn ∈A[x1, . . . , xn] by
Fm = F(x⟨m+1⟩, x⟨m+2⟩, . . . , x⟨m−1⟩);
(3.2)
thus Fm does not depend on the variable xm. We introduce the infinite “cyclic
exchange pattern”
t0
⟨0⟩
−−−−−−−
F⟨0⟩
t1
⟨1⟩
−−−−−−−
F⟨1⟩
t2
⟨2⟩
−−−−−−−
F⟨2⟩
t3
⟨3⟩
−−−−−−−
F⟨3⟩
t4 −−−· · · ,
(3.3)
and let the cluster at each point tm consist of the variables ym, . . . , ym+n−1, labeled
within the cluster according to the rule ys = x⟨s⟩(tm). Then equations (3.1) become
the exchange relations associated with this pattern.
To illustrate, let n = 4. Then the clusters will look like this:
y1,y2,y3,y0
•
t0
4
−−−−−−−
y1,y2,y3,y4
•
t1
1
−−−−−−−
y5,y2,y3,y4
•
t2
2
−−−−−−−
y5,y6,y3,y4
•
t3
3
−−−−−−−
y5,y6,y7,y4
•
t4
4
−−−−−−−· · · .
In order to include this situation into the setup of Section 2 (cf. Figure 1), we create
an infinite “caterpillar tree” whose “spine” is formed by the vertices tm, m > 0.
We thus attach the missing n−2 “legs” with labels in [n]−{⟨m −1⟩, ⟨m⟩}, to each
vertex tm.
Our next goal is to state conditions on the polynomial F which make it possible
to assign exchange polynomials satisfying (2.3)–(2.5) to the newly constructed legs.
The first requirement (cf. (2.3)) is:
The polynomial F is not divisible by any xi, i ∈[n −1].
(3.4)
For m ∈[n −1], we set
Qm = Fm|xn←0 = F(xm+1, . . . , xn−1, 0, x1, . . . , xm−1).
(3.5)
Our second requirement is
Each Qm is an irreducible element of A[x±1
1 , . . . , x±1
n−1].
(3.6)
To state our most substantial requirement, we recursively define a sequence of
polynomials Gn−1, . . . , G1, G0 ∈A[x1, . . . , xn−1]; more precisely, each Gm will be
defined up to a multiple in A×. (Later, G1, . . . , Gn−2 will become the exchange
polynomials assigned to the “legs” of the caterpillar labeled by n = ⟨0⟩; see Fig-
ure 2.)
We set Gn−1 = F, and obtain each Gm−1 from Gm, as follows. Let
∼
Gm−1= Gm
xm←Qm
xm .
(3.7)
THE LAURENT PHENOMENON
9
•
⟨0⟩
−−−−−−−
F⟨0⟩
•
⟨1⟩
−−−−−−−
F⟨1⟩
•
⟨0⟩
G1
•
⟨2⟩
−−−−−−−
F⟨2⟩
•
⟨0⟩
G2
•
⟨3⟩
−−−−−−−
F⟨3⟩
•
⟨0⟩
−−−−−−−
F⟨0⟩
• −−−· · · .
Figure 2. Constructing a caterpillar; n = 4.
Let L be a Laurent monomial in x1, . . . , xn−1, with coefficient in A, such that
≈
Gm−1=
∼
Gm−1
L
(3.8)
is a polynomial in A[x1, . . . , xn−1] not divisible by any xi or by any non-invertible
scalar in A. Such an L is unique up to a multiple in A×. Finally, we set
Gm−1 =
≈
Gm−1
Qbm
,
(3.9)
where Qb
m is the maximal power of Qm that divides
≈
Gm−1. With all this notation,
our final requirement is:
G0 = F.
(3.10)
Theorem 3.1. Let F be a polynomial in the variables x1, . . . , xn−1 with coefficients
in a unique factorization domain A satisfying conditions (3.4), (3.6), and (3.10).
Then every term of the sequence (yi) defined by the recurrence
ym+n = F(ym+1, . . . , ym+n−1)
ym
is a Laurent polynomial in the initial n terms, with coefficients in A.
Proof. To prove the Laurentness of some yN, we will apply Theorem 2.1 to the
caterpillar tree constructed as follows. We set thead = tN−n+1; this corresponds to
the first cluster containing yN. As a path from t0 to thead, we take a finite segment
of (3.3):
t0
⟨0⟩
−−−−−−−
F⟨0⟩
t1
⟨1⟩
−−−−−−−
F⟨1⟩
t2
⟨2⟩
−−−−−−−
F⟨2⟩
· · ·
⟨N−1⟩
−−−−−−−
F⟨N−1⟩
tN−n
⟨N⟩
−−−−−−−
F⟨N⟩
tN−n+1 .
(3.11)
We then define the exchange polynomial Gj,k−1 associated with the leg labeled j
attached to a vertex tk on the spine (see Figure 3) by
Gj,k−1 = G⟨k−j−1⟩(x⟨j+1⟩, . . . , xn, x1, . . . , x⟨j−1⟩),
where in the right-hand side, we use the polynomials G1, . . . , Gn−2 constructed in
(3.7)–(3.9) above.
It remains to verify that this exchange pattern satisfies (2.3), (2.4), and (2.5).
Condition (2.3) for the edges appearing in (3.11) is immediate from (3.4), while for
the rest of the edges, it follows from the definition of
≈
Gm−1 in (3.8).
10
SERGEY FOMIN AND ANDREI ZELEVINSKY
•
⟨k−1⟩
−−−−−−−
F⟨k−1⟩
tk
j
Gj,k−1
•
⟨k⟩
−−−−−−−
F⟨k⟩
•
Figure 3.
Turning to (2.4), we first note that we may assume i = ⟨0⟩= n (otherwise apply
a cyclic shift of indices). Under this assumption, we can identify the polynomials P
and Q0 in (2.4) with the polynomials Gm−1 and Qm in (3.9), for some value of m.
(The special case of P attached to one of the edges in (3.11) corresponds to m = 1,
and its validity requires (3.10).) Then the condition gcd(Gm−1, Qm) = 1 follows
from (3.6) and the choice of the exponent b in (3.9).
Finally, (2.5) is ensured by the construction (3.7)–(3.9), which was designed
expressly for this purpose. As before, the special case of P attached to one of the
edges in (3.11) holds due to (3.10).
□
In the rest of this section, we give a few applications of Theorem 3.1. In all of
them, conditions (3.4) and (3.6) are immediate, so we concentrate on the verification
of (3.10).
Example 3.2. Let a and b be positive integers, and let the sequence y0, y1, . . .
satisfy the recurrence
yk = ya
k−2yb
k−1 + 1
yk−3
.
We claim that every term of the sequence is a Laurent polynomial over Z in y0, y1,
and y2. To prove this, we set n = 3 and construct the polynomials G2, G1, and G0
using (3.7)–(3.9). Initializing G2 = F(x1, x2) = xa
1xb
2 + 1, we obtain:
Q2 = F(0, x1) = 1,
∼
G1= F
x2←Q2
x2 = xa
1x−b
2
+ 1,
G1 =
≈
G1= xa
1 + xb
2,
Q1 = F(x2, 0) = 1,
∼
G0= G1
x1←Q1
x1 = x−a
1
+ xb
2,
G0 =
≈
G0= 1 + xa
1xb
2 = F,
as desired.
Example 3.3. (Generalized Somos-4 sequence) Let a, b, and c be positive integers,
and let the sequence y0, y1, . . . satisfy the recurrence
yk = ya
k−3yc
k−1 + yb
k−2
yk−4
.
(The Somos-4 sequence [7], introduced by Michael Somos, is the special case a =
c = 1, b = 2.) Again, each yi is a Laurent polynomial in the initial terms y0, y1,
y2, and y3. To prove this, we set n = 4 and compute G3, . . . , G0 using (3.7)–(3.9)
and beginning with G3 = F = xa
1xc
3 + xb
2:
Q3 =F(0, x1, x2)=xb
1,
G3
x3←Q3
x3=xa+bc
1
x−c
3
+ xb
2,
G2 =xa+bc
1
+ xb
2xc
3,
Q2 =F(x3, 0, x1)=xc
1xa
3,
G2
x2←Q2
x2=xa+bc
1
+xbc
1 x−b
2 xab+c
3
,
G1 =xa
1xb
2 + xab+c
3
,
Q1 =F(x2, x3, 0)=xb
3,
G1
x1←Q1
x1=x−a
1 xb
2xab
3 + xab+c
3
,
G0 =xb
2+xa
1xc
3 =F,
THE LAURENT PHENOMENON
11
and the claim follows.
Remark 3.4. The Laurent phenomena in Theorems 1.4 and 1.8 can also be proved
by applying Theorem 3.1: in the former (resp., latter) case, the polynomial F is
given by F = αxpxn−p + βxqxn−q + γxrxn−r (resp., F = αxpxn−p + βxqxn−q).
The proofs are straightforward but rather long. Shorter proofs, based on J. Propp’s
idea of viewing one-dimensional recurrences as “projections” of multi-dimensional
ones, are given in Section 4 below.
4. Two- and three-dimensional recurrences
In this section, we use the strategy of Section 3 to establish the Laurent phenom-
enon for several recurrences involving two- and three-dimensional arrays. Our first
example generalizes a construction (and the corresponding Laurentness conjecture)
suggested by Noam Elkies and communicated by James Propp. Even though the
Laurent phenomenon in this example can be deduced from Theorem 1.6, we choose
to give a self-contained treatment, for the sake of exposition.
Example 4.1. (The knight recurrence) Consider a two-dimensional array (yij)i,j≥0
whose entries satisfy the recurrence
yi,jyi−2,j−1 = αyi,j−1yi−2,j + βyi−1,jyi−1,j−1 .
(4.1)
We will prove that every yij is a Laurent polynomial in the initial entries
Yinit = {yab : a < 2 or b < 1},
with coefficients in the ring A = Z[α, β].
We will refer to Yinit as the initial cluster, even though it is an infinite set.
Notice, however, that each individual yij only depends on finitely many variables
{yab ∈Yinit : a ≤i, b ≤j}.
Similarly to Section 3, we will use the exchange relations (4.1) to create a se-
quence of clusters satisfying the Caterpillar Lemma (Theorem 2.1).
This is done in the following way. Let us denote by H = Z2
≥0 the underlying set
of indices; for h = (i, j) ∈H, we will write yh = yij . The variables of the initial
cluster have labels in the set
Hinit = {(i, j) ∈H : i < 2 or j < 1}.
In Figure 4, the elements of Hinit are marked by •’s.
We introduce the product partial order on H:
(i1, j1) ≤(i2, j2)
def
⇔(i1 ≤i2) and (j1 ≤j2).
(4.2)
For an element h = (i, j) ∈H −Hinit , let us denote h−= (i −2, j −1); in this
notation, the exchange relation (4.1) expresses the product yh ·yh−as a polynomial
in the variables yh′ , for h−< h′ < h.
We write h−∼h, and extend this to an equivalence relation ∼on H.
The
equivalence class of h is denoted by ⟨h⟩. These classes are shown as slanted lines in
Figure 4. All our exchange polynomials will belong to the ring A[xa : a ∈H/∼].
Note that Hinit has exactly one representative from each equivalence class. We
will now construct a sequence of subsets H0 = Hinit, H1, H2, . . . , each having this
12
SERGEY FOMIN AND ANDREI ZELEVINSKY
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟
✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟
✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟
✟✟✟✟✟✟
✟✟✟✟
✟✟
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
❝
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
0
H
0
Figure 4. The initial cluster and the equivalence classes ⟨h⟩
property, using the following recursive rule. Let us fix a particular linear extension
of the partial order (4.2), say,
(i1, j1) ⪯(i2, j2)
def
⇔(i1 + j1 < i2 + j2) or (i1 + j1 = i2 + j2 and i1 ≤i2).
Restricting this linear ordering to the complement H −Hinit of the initial cluster,
we obtain a numbering of the elements of this complement by positive integers:
h0 = (2, 1), h1 = (2, 2), h2 = (3, 1), h3 = (2, 3), h4 = (3, 2),
h5 = (4, 1), h6 = (2, 4), h7 = (3, 3), h8 = (4, 2),
and so on.
Having constructed Hm, we let Hm+1 = Hm ∪{hm} −{h−
m}.
To
illustrate, the set H9 is shown in Figure 5.
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
0
0
Figure 5. Indexing set H9
We next create the infinite exchange pattern
t0
⟨h0⟩
−−−−−−−
P⟨h0⟩
t1
⟨h1⟩
−−−−−−−
P⟨h1⟩
t2
⟨h2⟩
−−−−−−−
P⟨h2⟩
t3
⟨h3⟩
−−−−−−−
P⟨h3⟩
t4 −−−· · · .
(4.3)
(cf. (3.3)) The cluster at each point tm is given by x(tm) = {yh : h ∈Hm}; as
before, each cluster variable yh corresponds to the variable x⟨h⟩. The exchange
THE LAURENT PHENOMENON
13
polynomial P⟨h⟩for an edge •
⟨h⟩
−−−• with h = (i, j) is given by
P⟨h⟩= αx⟨(i,j−1)⟩x⟨(i−2,j)⟩+ βx⟨(i−1,j)⟩x⟨(i−1,j−1)⟩.
(4.4)
Then equations (4.1) become the exchange relations associated with this pattern.
To establish the Laurent phenomenon, we will complete the caterpillar pattern by
attaching “legs” to each vertex tm and assigning exchange polynomials to these legs
so that the appropriate analogues of conditions (3.4), (3.6) and (3.10) are satisfied.
Since we now work over the polynomial ring A[xa : a ∈H/∼] in infinitely many
indeterminates, the number of legs attached to every vertex tm will also be infinite
(one for every label a different from ⟨hm−1⟩and ⟨hm⟩). This will not matter much
for our argument though: to prove the Laurentness for any yhm, we will simply
restrict our attention to the finite part of the infinite caterpillar tree lying between
t0 and thead = tm+1, and to the legs labeled by ⟨hk⟩for 0 ≤k ≤m.
The role of conditions (3.4) and (3.6) is now played by the observation that each
exchange polynomial P⟨h⟩is not divisible by any variable xa , and furthermore every
specialization P⟨h⟩
xa←0 is an irreducible element of the Laurent polynomial ring.
To formulate the analogue of (3.10), let us fix an equivalence class a ∈H/∼
and concentrate on defining the exchange polynomials for the legs labeled by a and
attached to the vertices squeezed between two consecutive occurrences of the label
a on the spine:
•
a
−−−
Pa
•
a1
−−−•
a
G1
•
a2
−−−•
a
G2
•
−−−•
a
•
−−−•
a
•
aN−2
−−−•
a
GN−2
•
aN−1
−−−•
a
−−−
Pa
• .
(4.5)
We note that the labels a1, . . . , aN−1 ∈H/∼appearing on the spine between these
two occurrences of a are distinct. For m = N −2, N −3, . . . , 1, we denote by Gm
the exchange polynomial to be associated with the a-labeled leg attached between
the edges labeled am and am+1 (cf. (4.5)).
The polynomials Gm are defined with the help of a recursive procedure analogous
to (3.7)–(3.9). We initialize GN−1 = Pa, and obtain each Gm−1 from Gm, as follows.
The step (3.7) is replaced by
∼
Gm−1= Gm
xam←Qm
xam
with
Qm = Pam
xa←0 .
(4.6)
We then compute
≈
Gm−1 and Gm−1 exactly as in (3.8)–(3.9). By the argument
given in the proof of Theorem 3.1, the equality G0 = Pa would imply the desired
Laurentness (cf. (3.10)).
14
SERGEY FOMIN AND ANDREI ZELEVINSKY
To simplify computations, we denote the equivalence classes “surrounding” a, as
shown below:
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
q
p
f
c
a
· · ·
· · ·
f
c
a
e
b
· · ·
· · ·
a
e
b
g
d
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
.
(4.7)
In other words, if a = ⟨(i, j)⟩, then b = ⟨(i, j −1)⟩, c = ⟨(i −1, j)⟩, etc. With this
notation, we can redraw the pattern (4.5) as follows:
•
a
−−−•
g
−−−· · · •
a
Gk−1
•
f
−−−•
a
Gk
•
e
−−−•
d
−−−· · · •
a
Gℓ−1
•
c
−−−•
a
Gℓ
•
b
−−−• · · ·
a
−−−•,
(4.8)
for appropriate values of k and ℓ.
We will call a value of m essential if Gm−1 ̸= Gm . We are going to see that the
essential values of m are those for which am ∈{b, c, e, f}; in the notation of (4.8),
these values are ℓ+ 1, ℓ, k + 1, and k.
We initialize GN−1 = Pa = αxbxf + βxcxe . The values of m in the interval
ℓ< m < N are not essential since the variable xam does not enter Pa, which is
furthermore not divisible by Qm (because the latter involves variables absent in Pa).
The first essential value is m = ℓ+ 1, with am = b:
Qℓ+1 = Pb|xa←0 = (αxaxd + βxexg)|xa←0 = βxexg ,
∼
Gℓ= Pa
xb←
Qℓ+1
xb
= α βxexg
xb
xf + βxcxe ,
Gℓ= αxgxf + xbxc .
Step m = ℓ(here am = c):
Qℓ= Pc|xa←0 = (αxexp + βxaxf)|xa←0 = αxexp ,
∼
Gℓ−1= Gℓ
xc←
Qℓ
xc = αxgxf + xb
αxexp
xc
,
Gℓ−1 = xcxgxf + xbxexp .
Notice that Gℓ−1 does not involve xd, so the value m = k + 2 is not essential, as
are the rest of the values in the interval k + 1 < m < ℓ.
Step m = k + 1, with am = e:
Qk+1 = Pe|xa←0 = (αxcxg + βxaxb)|xa←0 = αxcxg ,
∼
Gk= xcxgxf + xbxp
αxcxg
xe
,
Gk = xfxe + αxbxp .
Step m = k, with am = f:
Qk = Pf|xa←0 = (αxaxq + βxcxp)|xa←0 = βxcxp ,
∼
Gk−1= βxcxp
xf
xe + αxbxp ,
Gk−1 = βxcxe + αxbxf .
THE LAURENT PHENOMENON
15
The values of m in the interval 0 < m < k are not essential since none of the
corresponding variables xam appears in Gk−1; in particular, m = 1 is not essential,
since Gk−1 does not involve xg . Hence
G0 = Gk−1 = βxcxe + αxbxf = Pa ,
as desired. The Laurentness is proved.
Remark 4.2. The Laurent phenomenon for the recurrence (4.1) actually holds in
greater generality. Specifically, one can replace H by any subset of Z2 which satisfies
the following analogues of conditions (1.2)–(1.3) and (1.7)–(1.8):
if h ∈H, then h′ ∈H whenever h ≤h′;
(4.9)
for any h′ ∈H, the set {h ∈H : h ≤h′} is finite.
(4.10)
Then take Hinit = {h ∈H : h−/∈H}.
The proof of Laurentness only needs one adjustment, concerning the choice of
a linear extension ≺.
Specifically, while proving that yh is given by a Laurent
polynomial, take a finite set H(h) ⊂H containing h and satisfying the conditions
if h′ ∈H(h), then h′′ ∈H(h) whenever h′′ ≤h′ and h′′ ∈H;
(4.11)
for any h′ ∈H such that h′ ≤h, there exists h′′ ∈H(h) such that
(4.12)
h′′ ≥h and h′′ ∼h.
(The existence of H(h) follows from (4.9)–(4.10).) Then define ⪯exactly as before
on the set H(h); set h′ ≺h′′ for any h′ ∈H(h) and h′′ ∈H −H(h); and define ⪯on
the complement H −H(h) by an arbitrary linear extension of ≤. These conditions
ensure that the sets Hm needed in the proof of Laurentness of the given yh are well
defined, and that the rest of the proof proceeds smoothly.
Armed with the techniques developed above in this section, we will now prove
the main theorems stated in the introduction.
Proof of Theorem 1.2. Our argument is parallel to that in Example 4.1, so we
skip the steps which are identical in both proofs. For simplicity of exposition, we
present the proof in the special case H = Z3
≥0; the case of general H requires the
same adjustments as those described in Remark 4.2.
We define the product partial order ≤and a compatible linear order ⪯on H by
(i1, j1, k1) ≤(i2, j2, k2)
def
⇔
(i1 ≤i2) and (j1 ≤j2) and (k1 ≤k2),
(i1, j1, k1) ⪯(i2, j2, k2)
def
⇔
(i1 + j1 + k1 < i2 + j2 + k2)
or (i1 + j1 + k1 = i2 + j2 + k2 and i1 + j1 < i2 + j2)
or (i1 + j1 = i2 + j2 and k1 = k2 and i1 ≤i2).
For h = (i, j, k), we set h−= (i −1, j −1, k −1); thus, the exchange relation (1.1)
expresses the product yh ·yh−as a polynomial in the variables yh′ , for h−< h′ < h.
All the steps in Example 4.1 leading to the creation of the infinite exchange
pattern (4.3) are repeated verbatim. Instead of (4.4), the exchange polynomials
P⟨h⟩along the spine are now given by
P⟨(i,j,k)⟩
= αx⟨(i−1,j,k)⟩x⟨(i,j−1,k−1)⟩+βx⟨(i,j−1,k)⟩x⟨(i−1,j,k−1)⟩+γx⟨(i,j,k−1)⟩x⟨(i−1,j−1,k)⟩.
16
SERGEY FOMIN AND ANDREI ZELEVINSKY
The role of (4.7) is now played by Figure 6, which shows the “vicinity” of an
equivalence class a.
This figure displays the orthogonal projection of H along
the vector (1, 1, 1). Thus the vertices represent equivalence classes in H/∼. For
example, if a = ⟨(i, j, k)⟩, then
b = ⟨(i, j, k −1)⟩,
c = ⟨(i, j −1, k)⟩,
d = ⟨(i −1, j, k)⟩,
e = ⟨(i, j −1, k −1)⟩,
f = ⟨(i −1, j, k −1)⟩,
g = ⟨(i −1, j −1, k)⟩.
With this notation, we have:
Pa = αxdxe + βxcxf + γxbxg .
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟✟✟✟✟
✟✟✟✟✟✟✟✟
✟✟✟✟
✟✟✟✟
❍❍❍❍❍❍❍❍
❍❍❍❍❍❍❍❍❍❍❍❍
❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍
❍❍❍❍❍❍❍❍❍❍❍❍
❍❍❍❍❍❍❍❍✟✟
✯
❍
❍
❨
❄
v
u
d
f
p
g
a
b
s
c
e
q
r
1
2
3
Figure 6. The cube recurrence
With the polynomials G1, G2, . . . defined as in (4.5), the essential values of m
are now those for which am ∈{b, c, d, e, f, g}. (The verification that the rest of the
values are not essential is left to the reader.) We denote these values by m1, . . . , m6 ,
respectively.
The computation of the polynomials Gm begins by initializing
GN−1 = Pa = αxdxe + βxcxf + γxbxg .
Step m = m1, am = b:
Qm1 = Pb|xa←0 = αxfxq + βxexp ;
∼
Gm1−1= Gm1
xb←
Qm1
b
= αxdxe + βxcxf + γ αxfxq+βxexp
xb
xg ;
Gm1−1 = αxbxdxe + βxbxcxf + αγxfxgxq + βγxexgxp .
Step m = m2, am = c:
Qm2 = Pc|xa←0 = αxgxr + γxexs;
∼
Gm2−1= αxbxdxe + βxb
αxgxr+γxexs
xc
xf + αγxfxgxq + βγxexgxp ;
Gm2−1 = αxbxcxdxe+αβxbxfxgxr+βγxbxexfxs+αγxcxfxgxq+βγxcxexgxp .
THE LAURENT PHENOMENON
17
Step m = m3, am = d:
Qm3 = Pd|xa←0 = βxgxv + γxfxu ;
∼
Gm3−1= αxbxc
βxgxv+γxfxu
xd
xe
+αβxbxfxgxr + βγxbxexfxs + αγxcxfxgxq + βγxcxexgxp ;
Gm3−1 = αβxbxcxexgxv + αγxbxcxexfxu + βγxbxdxexfxs + βγxcxdxexgxp
+αβxbxdxfxgxr + αγxcxdxfxgxq .
Step m = m4, am = e:
Qm4 = Pe|xa←0 = βxbxr + γxcxq ;
∼
Gm4−1=
Qm4
xe (αβxbxcxgxv + αγxbxcxfxu + βγxbxdxfxs + βγxcxdxgxp)
+αxdxfxgQm4 ;
Gm4−1 = αγxbxcxfxu + βγxbxdxfxs + αxdxexfxg
+αβxbxcxgxv + βγxcxdxgxp .
Step m = m5, am = f:
Qm5 = Pf|xa←0 = αxbxv + γxdxp ;
∼
Gm5−1=
Qm5
xf (αγxbxcxu + βγxbxdxs + αxdxexg) + βxcxgQm5 ;
Gm5−1 = αxdxexg + βxcxfxg + αγxbxcxu + βγxbxdxs .
Step m = m6, am = g:
Qm6 = Pg|xa←0 = αxcxu + βxdxs ;
∼
Gm6−1=
Qm6
xg (αxdxe + βxcxf) + γxbQm6 ;
Gm6−1 = αxdxe + βxcxf + γxbxg = Pa ,
completing the proof.
□
We will now deduce the Gale-Robinson conjecture from Theorem 1.2.
Proof of Theorem 1.4.
To prove the Laurentness of a given element yN of
the Gale-Robinson sequence (ym), we define the array (zijk)(i,j,k)∈H by setting
zijk = yN+pi+qj+rk , with the indexing set
H = H(N) = {(i, j, k) ∈Z3 : N + pi + qj + rk ≥0} .
Then (1.4) implies that the zijk satisfy the cube recurrence (1.1). Note that H
satisfies the conditions (1.2)–(1.3). Thus Theorem 1.2 applies to (zijk), with Hinit =
{(a, b, c) ∈Z3 : 0 ≤N + pa + qb + rc < n}. It remains to note that yN = z000 ,
while for any (a, b, c) ∈Hinit , we have zabc = ym with 0 ≤m < n.
□
Proof of Theorem 1.6. This theorem is proved by the same argument as The-
orem 1.2. We treat the Mills-Robbins-Rumsey special case (1.9) (cf. also (1.6));
similarly to Theorem 1.2, the case of general H requires the standard adjustments
described in Remark 4.2. We use the partial order on the lattice L defined by
(i, j, k) ≤(i′, j′, k′) : |i′ −i| + |j′ −j| ≤k′ −k .
18
SERGEY FOMIN AND ANDREI ZELEVINSKY
For h = (i, j, k) ∈L, we set h−= (i, j, k −2), and define the equivalence relation ∼
accordingly. Figure 7 shows equivalence classes “surrounding” a given class a (cf.
Figure 6).
s
❝
s
❝
s
❝
s
❝
s
✻
✲
d
a
c
e
p
b
q
s
r
j
i
Figure 7.
The initialization polynomial GN−1 = Pa is given by Pa = αxcxd + βxbxe . The
table below displays am, Qm,
∼
Gm−1, and Gm−1 for all essential values of m.
am
Qm
∼
Gm−1
Gm−1
b
αxpxq
αxcxd + αβ xpxq
xb xe
xbxcxd + βxexpxq
c
βxqxr
β xqxr
xc xbxd + βxexpxq
xbxdxr + xcxexp
d
βxpxs
β xpxs
xd xbxr + xcxexp
βxbxrxs + xcxdxe
e
αxrxs
βxbxrxs + α xrxs
xe xcxd
βxbxe + αxcxd
We see that G0 = Ge−1 = Pa , completing the proof.
□
Proof of Theorem 1.8. The proof mimics the above proof of Theorem 1.4. To
prove the Laurentness of an element yN of the sequence (ym) satisfying (1.10),
we define the array (zijk)(i,j,k)∈H by setting zijk = yN+ℓ(i,j,k) , where ℓ(i, j, k) =
n i+j+k
2
−pi −qj. The indexing set H is now given by
H = H(N) = {(i, j, k) ∈Z3 : N + ℓ(i, j, k) ≥0} .
Then (1.10) implies that the zijk satisfy the octahedron recurrence (1.5). It is easy
to check that H satisfies the conditions (1.7)–(1.8). Thus Theorem 1.6 applies to
(zijk), with Hinit = {(a, b, c) ∈L : 0 ≤N +ℓ(a, b, c) < n}, and the theorem follows.
□
We conclude this section by a couple of examples in which the Laurent phenom-
enon is established by the same technique as above. In each case, we provide:
• a picture of the equivalence classes “surrounding” a given class a, which plays
the role of (4.7) in Example 4.1;
• the initialization polynomial GN−1 = Pa;
• a table showing am, Qm,
∼
Gm−1, and Gm−1 for all essential values of m.
Example 4.3. (Frieze patterns) The generalized frieze pattern recurrence (cf., e.g.,
[3, 11]) is
yijyi−1,j−1 = ε yi,j−1yi−1,j + β ,
(4.13)
THE LAURENT PHENOMENON
19
where ε ∈{1, −1}. To prove Laurentness (over Z[β]), refer to Figure 8. Then
Pa = ε xb xc + β, and the essential steps are:
am
Qm
∼
Gm−1
Gm−1
b
β
ε β xc
xb
+ β
ε xc + xb
c
β
ε β
xc + xb
β + ε−1xb xc
s
s
s
s
❅
❅
❅
❅
❅
❅
■
✒
c
b
a
a
i
j
Figure 8.
Example 4.4. (Number walls) Consider the 2-dimensional recurrence
yijyi,j−2 = yp
i−1,j−1yr
i+1,j−1 + yq
i,j−1 ,
(4.14)
where p, q, and r are nonnegative integers. To prove Laurentness, refer to Figure 9.
Then Pa = xp
dxr
b + xq
c, and the essential steps are:
am
Qm
∼
Gm−1
Gm−1
b
xq
f
xp
d
xq
f
xb
r + xq
c
xp
dxqr
f + xq
cxr
b
c
xp
gxr
f
xp
dxqr
f +
xp
gxr
f
xc
qxr
b
xp
dxq
c + xpq
g xr
b
d
xq
g
xq
g
xd
pxq
c + xpq
g xr
b
xq
c + xr
bxp
d
s
s
s
s
s
s
s
✻
✲
d
c
b
a
a
g
f
j
i
Figure 9.
Remark 4.5. As pointed out by J. Propp, the Laurent phenomenon for certain
special cases of Examples 4.3 and 4.4 can be obtained by specialization of Exam-
ple 1.5.
20
SERGEY FOMIN AND ANDREI ZELEVINSKY
5. Homogeneous exchange patterns
In this section, we deduce Theorem 1.10 and a number of similar results from
the following corollary of Theorem 2.1.
Corollary 5.1. Let A be a unique factorization domain. Assume that a collection
of nonzero polynomials P1, . . . , Pn ∈A[x1, . . . , xn] satisfies the following conditions:
Each Pk does not depend on xk, and is not divisible by any xi, i ∈[n].
(5.1)
For any i ̸= j, the polynomials Pji
def
= (Pj)|xi=0 and Pi are coprime.
(5.2)
For any i ̸= j, we have
(5.3)
L · P b
ji · Pi =Pi
xj←
Pji
xj
,
where b is a nonnegative integer, and L is a Laurent monomial whose
coefficient lies in A and is coprime with Pi.
Let us define the rational transformations Fi, i ∈[n], by
Fi : (x1, . . . , xn) 7→(x1, . . . , xi−1, Pi
xi
, xi+1, . . . , xn).
Then any composition of the form Fi1 ◦· · · ◦Fim is given by Laurent polynomials
with coefficients in A.
Proof.
Let Tn denote a regular tree of degree n whose edges are labeled by
elements of [n] so that all edges incident to a given vertex have different labels.
Assigning Pi as an exchange polynomial for every edge of Tn labeled by i, we
obtain a “homogeneous” exchange pattern on Tn satisfying conditions (2.3)–(2.5)
in Theorem 2.1. This implies the desired Laurentness.
□
Example 5.2. Let n ≥3 be an integer, and let P be a quadratic form given by
P(x1, . . . , xn) = x2
1 + · · · + x2
n +
X
i<j
αijxixj .
Theorem 1.10 is a special case of Corollary 5.1 for Pi = P
xi=0 and A = Z[αij : i <
j]. Conditions (5.1)–(5.2) are clear. To verify (5.3), note that
Pi = Pji + x2
j + xj
X
k
αkjxk +
X
ℓ
αjℓxl
,
where k (resp. ℓ) runs over all indices such that k ̸= i and k < j (resp. ℓ̸= i and
ℓ> j). It follows that
Pi
xj←
Pji
xj
= Pji + P 2
ji
x2
j
+ Pji
xj
X
k
αkjxk +
X
ℓ
αjℓxl
= Pji
x2
j
Pi ,
verifying (5.3).
In the remainder of this section, we list a few more applications of Corollary 5.1.
In each case, the verification of its conditions is straightforward.
Example 5.3. Let P and Q be monic palindromic polynomials in one variable:
P(x) = (1 + xd) + α1(x + xd−1) + α2(x2 + xd−2) + . . . ;
Q(x) = (1 + xe) + β1(x + xe−1) + β2(x2 + xe−2) + . . . .
THE LAURENT PHENOMENON
21
Then every member of the sequence y0, y1, . . . defined by the recurrence
yk =
µ2P(yk−1/λ)
yk−2
if k is odd;
λ2Q(yk−1/µ)
yk−2
if k is even
is a Laurent polynomial in y0 and y1 with coefficients in A = Z[λ±1, µ±1, αi, βi].
This follows from Corollary 5.1 with n = 2, P1 = µ2P(x2/λ), and P2 = λ2Q(x1/µ).
Example 5.4. Consider the sequence y0, y1, . . . defined by the recurrence
yk = y2
k−1 + cyk−1 + d
yk−2
.
(5.4)
Every term of this sequence is a Laurent polynomial in y0 and y1 with coefficients
in Z[c, d].
Example 5.5. Define the rational transformations F1, F2, F3 by
F1 : (x1, x2, x3) 7→( x2 + x2
3 + x2
2x3
x1
,
x2,
x3
),
F2 : (x1, x2, x3) 7→(
x1,
x1 + x3
x2
,
x3
),
F3 : (x1, x2, x3) 7→(
x1,
x2,
x2 + x2
1 + x2
2x1
x3
).
(5.5)
Then any composition Fi1 ◦Fi2 ◦· · · is given by (x1, x2, x3) 7→(G1, G2, G3), where
G1, G2, G3 are Laurent polynomials in x1, x2, x3 over Z.
References
[1] D. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices
Amer. Math. Soc. 46 (1999), no. 6, 637–646.
[2] J. H. Conway and R. K. Guy, The book of numbers, Copernicus, New York, 1996.
[3] J. H. Conway and H. S. M. Coxeter, Triangulated polygons and frieze patterns, Math. Gaz.
57 (1973), no. 400, 87–94; and ibid., no. 401, 175–183.
[4] S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc.
12 (1999), 335–380.
[5] S. Fomin and A. Zelevinsky, Total positivity: tests and parametrizations, Math. Intelligencer
22 (2000), no. 1, 23–33.
[6] S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, preprint math.RT/0104151.
[7] D. Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13
(1991), no. 1, 40–43.
[8] R. K. Guy, Unsolved problems in number theory, 2nd edition, Springer-Verlag, New York,
1994.
[9] W. H. Mills, D. P. Robbins, and H. Rumsey, Alternating sign matrices and descending plane
partitions. J. Combin. Theory Ser. A 34 (1983), 340–359.
[10] J. Propp, The Many Faces of Alternating-Sign Matrices, Discrete Math. Theor. Comput.
Sci., to appear.
[11] R. P. Stanley, Enumerative combinatorics, vol. 2, Cambridge University Press, 1999.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
E-mail address: [email protected]
Department of Mathematics, Northeastern University, Boston, MA 02115
E-mail address: [email protected]
|
arXiv:math/0104241v1 [math.CO] 25 Apr 2001 THE LAURENT PHENOMENON SERGEY FOMIN AND ANDREI ZELEVINSKY Abstract. A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes-quite unexpectedly-it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of D. Gale and R. Robinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by J. Propp, N. Elkies, and M. Kleber. Contents 1. Introduction 1 2. The Caterpillar Lemma 4 3. One-dimensional recurrences 8 4. Two- and three-dimensional recurrences 11 5. Homogeneous exchange patterns 20 References 21 1. Introduction In this paper, we suggest a unified explanation for a number of instances in which certain recursively defined rational functions prove, unexpectedly, to be Laurent polynomials. We begin by presenting several instances of this Laurent phenomenon established in the paper. Example 1.1. (The cube recurrence) Consider a 3-dimensional array (yijk : (i, j, k) ∈H) whose elements satisfy the recurrence yi,j,k = αyi-1,j,kyi,j-1,k-1 + βyi,j-1,kyi-1,j,k-1 + γyi,j,k-1yi-1,j-1,k yi-1,j-1,k-1 . (1.1) Here H can be any non-empty subset of Z3 satisfying the following conditions: if (i, j, k) ∈H, then (i′, j′, k′) ∈H whenever i ≤i′, j ≤j′, k ≤k′; (1.2) for any (i′, j′, k′) ∈H, the set {(i, j, k) ∈H : i ≤i′, j ≤j′, k ≤k′} is finite. (1.3) Date: April 25, 2001. 1991 Mathematics Subject Classification. Primary 14E05, Secondary 05E99, 11B83. Key words and phrases. Laurent phenomenon, Somos sequence, Gale-Robinson conjecture. The authors were supported in part by NSF grants #DMS-0049063, #DMS-0070685 (S.F.), and #DMS-9971362 (A.Z.). 1 2 SERGEY FOMIN AND ANDREI ZELEVINSKY Theorem 1.2. Let Hinit = {(a, b, c) ∈H : (a -1, b -1, c -1) /∈H}. For every (i, j, k) ∈H, the entry yi,j,k is a Laurent polynomial with coefficients in Z[α, β, γ] in the initial entries ya,b,c, for (a, b, c) ∈Hinit. The cube recurrence (with α = β = γ = 1) was introduced by James Propp [10], who was also the one to conjecture Laurentness in the case when H ⊂Z3 is given by the condition i + j + k ≥0; in this case Hinit consists of all (a, b, c) ∈H such that a + b + c ∈{0, 1, 2}. Another natural choice of H was suggested by Michael Kleber: H = Z3 ≥0, in which case Hinit = {(a, b, c) ∈Z3 ≥0 : abc = 0}. Example 1.3. (The Gale-Robinson sequence) Let p, q, and r be distinct positive integers, let n = p + q + r, and let the sequence y0, y1, . . . satisfy the recurrence yk+n = αyk+pyk+n-p + βyk+qyk+n-q + γyk+ryk+n-r yk . (1.4) David Gale and Raphael Robinson conjectured (see [7] and [8, E15]) that every term of such a sequence is an integer provided y0 = · · · = yn-1 = 1 and α, β, γ are positive integers. Using Theorem 1.2, we prove the following stronger statement. Theorem 1.4. As a function of the initial terms y0, . . . , yn-1, every term of the Gale-Robinson sequence is a Laurent polynomial with coefficients in Z[α, β, γ]. We note that the special case α = β = γ = 1, p = 1, q = 2, r = 3, n = 6 (resp., r = 4, n = 7) of the recurrence (1.4) is the Somos-6 (resp., Somos-7) recurrence [7]. Example 1.5. (Octahedron recurrence) Consider the 3-dimensional recurrence yi,j,k = αyi+1,j,k-1yi-1,j,k-1 + βyi,j+1,k-1yi,j-1,k-1 yi,j,k-2 (1.5) for an array (yijk)(i,j,k)∈H whose indexing set H is contained in the lattice L = {(i, j, k) ∈Z3 : i + j + k ≡0 mod 2} (1.6) and satisfies the following analogues of conditions (1.2)-(1.3): if (i, j, k) ∈H, then (i′, j′, k′) ∈H whenever |i′ -i| + |j′ -j| ≤k′ -k; (1.7) for any (i′, j′, k′) ∈H, the set {(i, j, k) ∈H : |i′ -i| + |j′ -j| ≤k′ -k} (1.8) is finite. Theorem 1.6. Let Hinit = {(a, b, c) ∈H : (a, b, c -2) /∈H}. For every (i, j, k) ∈ H, the entry yi,j,k is a Laurent polynomial with coefficients in Z[α, β] in the initial entries ya,b,c, for (a, b, c) ∈Hinit. The octahedron recurrence on the half-lattice H = {(i, j, k) ∈L : k ≥0} (1.9) was studied by W. H. Mills, D. P. Robbins, and H. Rumsey in their pioneering work [9] on the Alternating Sign Matrix Conjecture (cf. [1] and [10, Section 10] for further references); in particular, they proved the special case of Theorem 1.6 for this choice of H. THE LAURENT PHENOMENON 3 Example 1.7. (Two-term version of the Gale-Robinson sequence) Let p, q, and n be positive integers such that p 0. We thus attach the missing n-2 "legs" with labels in [n]-{⟨m -1⟩, ⟨m⟩}, to each vertex tm. Our next goal is to state conditions on the polynomial F which make it possible to assign exchange polynomials satisfying (2.3)-(2.5) to the newly constructed legs. The first requirement (cf. (2.3)) is: The polynomial F is not divisible by any xi, i ∈[n -1]. (3.4) For m ∈[n -1], we set Qm = Fm|xn←0 = F(xm+1, . . . , xn-1, 0, x1, . . . , xm-1). (3.5) Our second requirement is Each Qm is an irreducible element of A[x±1 1 , . . . , x±1 n-1]. (3.6) To state our most substantial requirement, we recursively define a sequence of polynomials Gn-1, . . . , G1, G0 ∈A[x1, . . . , xn-1]; more precisely, each Gm will be defined up to a multiple in A×. (Later, G1, . . . , Gn-2 will become the exchange polynomials assigned to the "legs" of the caterpillar labeled by n = ⟨0⟩; see Figure 2.) We set Gn-1 = F, and obtain each Gm-1 from Gm, as follows. Let ∼ Gm-1= Gm xm←Qm xm . (3.7) THE LAURENT PHENOMENON 9 • ⟨0⟩ ------- F⟨0⟩ • ⟨1⟩ ------- F⟨1⟩ • ⟨0⟩ G1 • ⟨2⟩ ------- F⟨2⟩ • ⟨0⟩ G2 • ⟨3⟩ ------- F⟨3⟩ • ⟨0⟩ ------- F⟨0⟩ • ---· · · . Figure 2. Constructing a caterpillar; n = 4. Let L be a Laurent monomial in x1, . . . , xn-1, with coefficient in A, such that ≈ Gm-1= ∼ Gm-1 L (3.8) is a polynomial in A[x1, . . . , xn-1] not divisible by any xi or by any non-invertible scalar in A. Such an L is unique up to a multiple in A×. Finally, we set Gm-1 = ≈ Gm-1 Qbm , (3.9) where Qb m is the maximal power of Qm that divides ≈ Gm-1. With all this notation, our final requirement is: G0 = F. (3.10) Theorem 3.1. Let F be a polynomial in the variables x1, . . . , xn-1 with coefficients in a unique factorization domain A satisfying conditions (3.4), (3.6), and (3.10). Then every term of the sequence (yi) defined by the recurrence ym+n = F(ym+1, . . . , ym+n-1) ym is a Laurent polynomial in the initial n terms, with coefficients in A. Proof. To prove the Laurentness of some yN, we will apply Theorem 2.1 to the caterpillar tree constructed as follows. We set thead = tN-n+1; this corresponds to the first cluster containing yN. As a path from t0 to thead, we take a finite segment of (3.3): t0 ⟨0⟩ ------- F⟨0⟩ t1 ⟨1⟩ ------- F⟨1⟩ t2 ⟨2⟩ ------- F⟨2⟩ · · · ⟨N-1⟩ ------- F⟨N-1⟩ tN-n ⟨N⟩ ------- F⟨N⟩ tN-n+1 . (3.11) We then define the exchange polynomial Gj,k-1 associated with the leg labeled j attached to a vertex tk on the spine (see Figure 3) by Gj,k-1 = G⟨k-j-1⟩(x⟨j+1⟩, . . . , xn, x1, . . . , x⟨j-1⟩), where in the right-hand side, we use the polynomials G1, . . . , Gn-2 constructed in (3.7)-(3.9) above. It remains to verify that this exchange pattern satisfies (2.3), (2.4), and (2.5). Condition (2.3) for the edges appearing in (3.11) is immediate from (3.4), while for the rest of the edges, it follows from the definition of ≈ Gm-1 in (3.8). 10 SERGEY FOMIN AND ANDREI ZELEVINSKY • ⟨k-1⟩ ------- F⟨k-1⟩ tk j Gj,k-1 • ⟨k⟩ ------- F⟨k⟩ • Figure 3. Turning to (2.4), we first note that we may assume i = ⟨0⟩= n (otherwise apply a cyclic shift of indices). Under this assumption, we can identify the polynomials P and Q0 in (2.4) with the polynomials Gm-1 and Qm in (3.9), for some value of m. (The special case of P attached to one of the edges in (3.11) corresponds to m = 1, and its validity requires (3.10).) Then the condition gcd(Gm-1, Qm) = 1 follows from (3.6) and the choice of the exponent b in (3.9). Finally, (2.5) is ensured by the construction (3.7)-(3.9), which was designed expressly for this purpose. As before, the special case of P attached to one of the edges in (3.11) holds due to (3.10). □ In the rest of this section, we give a few applications of Theorem 3.1. In all of them, conditions (3.4) and (3.6) are immediate, so we concentrate on the verification of (3.10). Example 3.2. Let a and b be positive integers, and let the sequence y0, y1, . . . satisfy the recurrence yk = ya k-2yb k-1 + 1 yk-3 . We claim that every term of the sequence is a Laurent polynomial over Z in y0, y1, and y2. To prove this, we set n = 3 and construct the polynomials G2, G1, and G0 using (3.7)-(3.9). Initializing G2 = F(x1, x2) = xa 1xb 2 + 1, we obtain: Q2 = F(0, x1) = 1, ∼ G1= F x2←Q2 x2 = xa 1x-b 2 + 1, G1 = ≈ G1= xa 1 + xb 2, Q1 = F(x2, 0) = 1, ∼ G0= G1 x1←Q1 x1 = x-a 1 + xb 2, G0 = ≈ G0= 1 + xa 1xb 2 = F, as desired. Example 3.3. (Generalized Somos-4 sequence) Let a, b, and c be positive integers, and let the sequence y0, y1, . . . satisfy the recurrence yk = ya k-3yc k-1 + yb k-2 yk-4 . (The Somos-4 sequence [7], introduced by Michael Somos, is the special case a = c = 1, b = 2.) Again, each yi is a Laurent polynomial in the initial terms y0, y1, y2, and y3. To prove this, we set n = 4 and compute G3, . . . , G0 using (3.7)-(3.9) and beginning with G3 = F = xa 1xc 3 + xb 2: Q3 =F(0, x1, x2)=xb 1, G3 x3←Q3 x3=xa+bc 1 x-c 3 + xb 2, G2 =xa+bc 1 + xb 2xc 3, Q2 =F(x3, 0, x1)=xc 1xa 3, G2 x2←Q2 x2=xa+bc 1 +xbc 1 x-b 2 xab+c 3 , G1 =xa 1xb 2 + xab+c 3 , Q1 =F(x2, x3, 0)=xb 3, G1 x1←Q1 x1=x-a 1 xb 2xab 3 + xab+c 3 , G0 =xb 2+xa 1xc 3 =F, THE LAURENT PHENOMENON 11 and the claim follows. Remark 3.4. The Laurent phenomena in Theorems 1.4 and 1.8 can also be proved by applying Theorem 3.1: in the former (resp., latter) case, the polynomial F is given by F = αxpxn-p + βxqxn-q + γxrxn-r (resp., F = αxpxn-p + βxqxn-q). The proofs are straightforward but rather long. Shorter proofs, based on J. Propp's idea of viewing one-dimensional recurrences as "projections" of multi-dimensional ones, are given in Section 4 below. 4. Two- and three-dimensional recurrences In this section, we use the strategy of Section 3 to establish the Laurent phenomenon for several recurrences involving two- and three-dimensional arrays. Our first example generalizes a construction (and the corresponding Laurentness conjecture) suggested by Noam Elkies and communicated by James Propp. Even though the Laurent phenomenon in this example can be deduced from Theorem 1.6, we choose to give a self-contained treatment, for the sake of exposition. Example 4.1. (The knight recurrence) Consider a two-dimensional array (yij)i,j≥0 whose entries satisfy the recurrence yi,jyi-2,j-1 = αyi,j-1yi-2,j + βyi-1,jyi-1,j-1 . (4.1) We will prove that every yij is a Laurent polynomial in the initial entries Yinit = {yab : a j). It follows that Pi xj← Pji xj = Pji + P 2 ji x2 j + Pji xj X k αkjxk + X l αjlxl = Pji x2 j Pi , verifying (5.3). In the remainder of this section, we list a few more applications of Corollary 5.1. In each case, the verification of its conditions is straightforward. Example 5.3. Let P and Q be monic palindromic polynomials in one variable: P(x) = (1 + xd) + α1(x + xd-1) + α2(x2 + xd-2) + . . . ; Q(x) = (1 + xe) + β1(x + xe-1) + β2(x2 + xe-2) + . . . . THE LAURENT PHENOMENON 21 Then every member of the sequence y0, y1, . . . defined by the recurrence yk = μ2P(yk-1/λ) yk-2 if k is odd; λ2Q(yk-1/μ) yk-2 if k is even is a Laurent polynomial in y0 and y1 with coefficients in A = Z[λ±1, μ±1, αi, βi]. This follows from Corollary 5.1 with n = 2, P1 = μ2P(x2/λ), and P2 = λ2Q(x1/μ). Example 5.4. Consider the sequence y0, y1, . . . defined by the recurrence yk = y2 k-1 + cyk-1 + d yk-2 . (5.4) Every term of this sequence is a Laurent polynomial in y0 and y1 with coefficients in Z[c, d]. Example 5.5. Define the rational transformations F1, F2, F3 by F1 : (x1, x2, x3) 7→( x2 + x2 3 + x2 2x3 x1 , x2, x3 ), F2 : (x1, x2, x3) 7→( x1, x1 + x3 x2 , x3 ), F3 : (x1, x2, x3) 7→( x1, x2, x2 + x2 1 + x2 2x1 x3 ). (5.5) Then any composition Fi1 ◦Fi2 ◦· · · is given by (x1, x2, x3) 7→(G1, G2, G3), where G1, G2, G3 are Laurent polynomials in x1, x2, x3 over Z. References [1] D. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc. 46 (1999), no. 6, 637-646. [2] J. H. Conway and R. K. Guy, The book of numbers, Copernicus, New York, 1996. [3] J. H. Conway and H. S. M. Coxeter, Triangulated polygons and frieze patterns, Math. Gaz. 57 (1973), no. 400, 87-94; and ibid., no. 401, 175-183. [4] S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), 335-380. [5] S. Fomin and A. Zelevinsky, Total positivity: tests and parametrizations, Math. Intelligencer 22 (2000), no. 1, 23-33. [6] S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, preprint math.RT/0104151. [7] D. Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13 (1991), no. 1, 40-43. [8] R. K. Guy, Unsolved problems in number theory, 2nd edition, Springer-Verlag, New York, 1994. [9] W. H. Mills, D. P. Robbins, and H. Rumsey, Alternating sign matrices and descending plane partitions. J. Combin. Theory Ser. A 34 (1983), 340-359. [10] J. Propp, The Many Faces of Alternating-Sign Matrices, Discrete Math. Theor. Comput. Sci., to appear. [11] R. P. Stanley, Enumerative combinatorics, vol. 2, Cambridge University Press, 1999. 48109, USA E-mail address: 02115 E-mail address:
|
2510.14979
| "FROM PIXELS TO WORDS – TOWARDS NATIVE VISION-\nLANGUAGE PRIMITIVES AT SCALE\nHaiwen Diao1\nMingxu(...TRUNCATED)
| "FROM PIXELS TO WORDS - TOWARDS NATIVE VISIONLANGUAGE PRIMITIVES AT SCALE Haiwen Diao1 Mingxuan Li2 (...TRUNCATED)
|
2510.14981
| "Preprint - Under Review\nCOUPLED DIFFUSION SAMPLING FOR TRAINING-FREE\nMULTI-VIEW IMAGE EDITING\nHa(...TRUNCATED)
| "Preprint - Under Review COUPLED DIFFUSION SAMPLING FOR TRAINING-FREE MULTI-VIEW IMAGE EDITING Hadi (...TRUNCATED)
|
2510.14978
| "Preprint\nLEARNING AN IMAGE EDITING MODEL\nWITHOUT IMAGE EDITING PAIRS\nNupur Kumari1\nSheng-Yu Wan(...TRUNCATED)
| "Preprint LEARNING AN IMAGE EDITING MODEL WITHOUT IMAGE EDITING PAIRS Nupur Kumari1 Sheng-Yu Wang1 N(...TRUNCATED)
|
2510.14976
| "Ponimator: Unfolding Interactive Pose for Versatile Human-human\nInteraction Animation\nShaowei Liu(...TRUNCATED)
| "Ponimator: Unfolding Interactive Pose for Versatile Human-human Interaction Animation Shaowei Liu(...TRUNCATED)
|
2510.14980
| "Technical Report\nAGENTIC DESIGN OF COMPOSITIONAL MACHINES\nWenqian Zhang1\nWeiyang Liu2,*\nZhen Li(...TRUNCATED)
| "Technical Report AGENTIC DESIGN OF COMPOSITIONAL MACHINES Wenqian Zhang1 Weiyang Liu2,* Zhen Liu1,*(...TRUNCATED)
|
2510.14971
| "ON THE INVARIANTS OF FINITE GROUPS ARISING IN A\nTOPOLOGICAL QUANTUM FIELD THEORY\nCHRISTOPHER A. S(...TRUNCATED)
| "ON THE INVARIANTS OF FINITE GROUPS ARISING IN A TOPOLOGICAL QUANTUM FIELD THEORY CHRISTOPHER A. SCH(...TRUNCATED)
|
2510.14973
| "ATTENTION IS ALL YOU NEED\nFOR KV CACHE IN DIFFUSION LLMS\nQuan Nguyen-Tri∗\nFPT AI Residency\nHa(...TRUNCATED)
| "ATTENTION IS ALL YOU NEED FOR KV CACHE IN DIFFUSION LLMS Quan Nguyen-Tri∗ FPT AI Residency Hanoi,(...TRUNCATED)
|
2510.14975
| "WithAnyone: Towards Controllable and ID Consistent Image Generation\nHengyuan Xu1,2\nWei Cheng2,†(...TRUNCATED)
| "WithAnyone: Towards Controllable and ID Consistent Image Generation Hengyuan Xu1,2 Wei Cheng2,† P(...TRUNCATED)
|
End of preview. Expand
in Data Studio
Arxiv Bulk Dataset
This dataset card aims to be a base template for new datasets. It has been generated using this raw template.
Dataset
ArXiv Bulk Dataset
This dataset is a bulk fetch of ArXiv articles, based on the official metadata dataset maintained and updated by Cornell https://www.kaggle.com/datasets/Cornell-University/arxiv/data.
This dataset was created to provide cross-domain academic training data, with existing datasets being domain-specific, and manual fetches being time-intensive.
Each article has the id, enabling lookup within the metadata dataset, the raw PDF content, and the content with cleaning to remove PDF, XML, and LaTeX artifacts.
Dataset Description
- Curated by: [More Information Needed]
- Funded by [optional]: [More Information Needed]
- Shared by [optional]: [More Information Needed]
- Language(s) (NLP): [More Information Needed]
- License: [More Information Needed]
Dataset Sources [optional]
- Repository: [More Information Needed]
- Paper [optional]: [More Information Needed]
- Demo [optional]: [More Information Needed]
Uses
Direct Use
[More Information Needed]
Out-of-Scope Use
[More Information Needed]
Dataset Structure
[More Information Needed]
Dataset Creation
Curation Rationale
[More Information Needed]
Source Data
Data Collection and Processing
[More Information Needed]
Who are the source data producers?
[More Information Needed]
Annotations [optional]
Annotation process
[More Information Needed]
Who are the annotators?
[More Information Needed]
Personal and Sensitive Information
[More Information Needed]
Bias, Risks, and Limitations
[More Information Needed]
Recommendations
Users should be made aware of the risks, biases and limitations of the dataset. More information needed for further recommendations.
Citation [optional]
BibTeX:
[More Information Needed]
APA:
[More Information Needed]
Glossary [optional]
[More Information Needed]
More Information [optional]
[More Information Needed]
Dataset Card Authors [optional]
[More Information Needed]
Dataset Card Contact
[More Information Needed]
- Downloads last month
- 113