Table Of ContentAn Alternative Approach to Unification of Gauge and Geometric Interactions
∗
James Lindesay
Computational Physics Laboratory
HowardUniversity
7
1
0
2
Abstract
n
a ThealgebraofthegeneratorsforinfinitesimaltransformationsoftheΓ= 1 representationofcausal
J 2
spinor fields (Dirac fields) explicitly constructs theMinkowski metric within the internal group space
3
asaconsequenceofnon-vanishingcommutationrelationsbetweengeneratorsthatcarryasinglespace-
2
timeindex. This representation isa subgroup of theset of all of thegenerators that transform under
the group GL(4). The sixteen hermitian generators of GL(4) include the three angular momentum
]
h spin matrices, a matrix proportional to the Dirac matrix γ0, and 12 additional matrices that have
p the same number of degrees of freedom as SU(3)×SU(2)×U(1). In this paper, the construction of
-
linearly independent internal SU(3) and SU(2) local symmetry groups for the causal spinor fields is
n
e demonstrated tonecessarily involvetheCKMmixingof threesets oftheSU(3)eigenstates as related
g tothe SU(2) eigenstates.
.
s
c
i 1 Introduction
s
y
h
Perhaps the most fundamental characteristic of the standard model implementation of fundamental
p
[
micro-physical interactions is their relation to the gauge symmetry product group SU(3) SU(2) U(1).
× ×
2
Localunitary gauge transformationsgenerally refer to an invarianceof physicalmeasurables under trans-
v
2 formations of the particle fields of the form Ψ(x) Ψ˜(x) = U (α(x))Ψ(x), where U is a unitary
(a) (a)
3 →
3 representationof a group of transformations , and the group parameters α=α(x) can take on an ar-
(a)
G
1
bitrary functional dependency on the space-time location x. In order to implement this symmetry, vector
0
.
1 gauge potentials As(x) are introduced using “minimal coupling”, with the requirement that they trans-
µ
0
7 formunder the gaugetransformationaccordingto U−(a1)(α)h¯i ∂∂xµU(a)(α)= qca sAsµU−(a1)(α)G(sa)U(a)(α),
:1 where Gs(a) is the generator of infinitesimal translations of group parameterPαs. The parameter qa is
v
the charge coupling the gauge fields As(x) to particle currents, making these vector gauge potentials the
i µ
X
carriers of micro-physicalinteractions between the particles.
r
a
General relativity has been consistently successful in describing the macro-physical phenomena of
gravitation,aswellasthegravitationofcoherentquantumstates[1,2,3]. Thetheoryformulatesequations
of motion for systems on curvilinear space-times, where the local curvature is generated by the local
energy-momentumdensity. Allmeasurablescanbe derivedfroma localandsymmetric space-timemetric
functiong (x)thatthroughthe principleofequivalencecanbe transformedintoalocallyflatMinkowski
µν
metric form η = ∂xµg ∂xν on coordinates ξ(x). The metric η cannot be explicitly generatedby the
αβ ∂ξα µν∂ξβ αβ
∗ Address: DepartmentofPhysics,HowardUniversity,Washington, DC20059, email: [email protected]
Lorentz group algebra through a Casimir-like construction, since there are no non-abelian generators in
that algebra that carry a single space-time index.
The Dirac equation[4] for spin 1 fermions utilizes a matrix algebrato construct a relativistic equation
2
forparticlewavefunctionsψthatislinearinthequantumoperatorsfor4-momentum,generatesapositive
semi-definite probability density ψ ψ, and maintains the expected non-relativistic correspondence with
†
the Schrodinger equation[5]. In particular, the evolution dynamics describing equations that are linear in
energy-momentum operators is mathematically straightforward[6]. The development of Dirac’s equation
required the introduction of 4 4 matrices that each carry a single space-time index.
×
TheDiracformulationcanbeextendedbydevelopinggeneraloperatorsΓˆµ thatrequiretheformΓˆµPˆ
µ
to be a Lorentzscalarwith aneigenvaluelinear inthe particle mass. The field equationthatresults takes
the form
¯h ∂ q
Γµ a AsG Ψˆ(Γ)(~x)= γmcΨˆ(Γ)(~x), (1.1)
· i ∂xµ − c µ s! γ − γ
s
X
where m is positive semi-definite for all particle types (both particles and antiparticles), and the Γµ are
finite dimensional matrix representations of the operators Γˆµ. In this equation, γ is the eigenvalue of
the operator Γˆ0 on the standard (rest) state particle spinor, and this eigenvalue takes differing signs for
particles vs. antiparticles. Using this formulation, energies are always non-negative. This form is seen
to be invariant under local gauge transformations of the form previously introduced. The Dirac matrices
result from the Γ = 1 finite dimensional representation of the operators Γˆµ. A representation for the
2
Γ = 1 matrices can be found in Appendix D.2.1 of reference [6]. In what follows, the group structure of
the algebra will be briefly discussed, and internal symmetries of the causal fields Ψˆ(21) will be explored.
γ
2 Properties of a Closed Group Inclusive of Γµ
The closedalgebrathat includes the operatorsΓˆµ addthe followingcommutationrelationsto those of
the Lorentz group:
Γ0, Γk = i K , Γ0, J = 0, Γ0, K = i¯hΓk,
h¯ k k k − (2.2)
Γj, Γk = i ǫ J , Γj, J = i¯hǫ Γm, Γj, K = i¯hδ Γ0.
(cid:2) (cid:3) −h¯ jkm m (cid:2) k(cid:3) jkm (cid:2) k(cid:3) − jk
Eigenvalues Γ(cid:2)of the(cid:3)Casimir operator C(cid:2) = (cid:3)1 (J J K K(cid:2))/¯h2 +(cid:3)Γ0Γ0 Γ Γ will label the
Γ 6 · − · − ·
irreducible representations of this group. Addition(cid:0)al labels can be attributed to the eig(cid:1)envalues of the
mutuallycommutingoperatorsC ,Γ0,J2,andJ ,givenby 2Γ(Γ+2),γ,J(J+1)¯h2,,andM¯h,respectively.
Γ 3 6
Finite dimensionalrepresentationsofthe algebrahavedimensionN = 1(Γ+1)(2Γ+1)(2Γ+3),and
Γ 3
general representation eigenstates χΓ,J can be constructed using the spinor form
γ,M
J+M
(J M)! ∂ ∂
χΓγ,,MJ =s(2J)!(−J +M)![x−y]Γ−Jχ+(+)M+γχ(+−)M−γ (cid:20)∂x + ∂y(cid:21) xJ−γyJ+γ(cid:12)(cid:12)(cid:12)(cid:12) yx==χχ(−(+++))χχ+(−(−−)) (,2.3)
(cid:12)
2
where the χ(±) are Γ = 1 representation spinors with (γ) = ( 1) and M = 1. The signatures (half-
± 2 ±2 ±2
integral vs integral) of Γ, J, γ, and M are necessarily the same, with 0 J Γ, J γ J, and
≤ ≤ − ≤ ≤
J M J. The spinor forms of the operatorssatisfying the commutation relations (2.2) can be found
− ≤ ≤
in[6]. The matrices correspondingto Γ= 1 havedimensionalityN =4,with Γ0 = 1γ0 andJ both
2 21 2 Dirac 3
diagonal. This representation forms a subgroup of GL(4).
2.1 Development of a group metric on space-time indexes
For a general group algebra Gˆ , Gˆ = i (c )m Gˆ with appropriately non-vanishing structure
r s − m s r m
constants csrm = crsm, thehJacobi iidentity dPefines a representation in terms of the structure constants
−
which can be used to construct a group metric η [7],
ab
η (c )s (c )r, ηab ((η) 1) . (2.4)
ab ≡ a r b s ≡ − ab
sr
X
This group metric on the generators defines invariants on products of group generators, such as the
CasimiroperatorCˆG ≡ rsηGrGsGˆrGˆs. The non-commutingoperatorsΓˆµ thatcarrya space-timeindex
conjugate to the 4-momPentum define a group metric of Lorentz sub-group invariants given by
1
ηΓµΓν = η (2.5)
µν
−6
where η is the usual Minkowski metric. Thus, the Minkowski metric defining invariant products of
µν
groupgeneratorsis explicitly generatedwithin this closedalgebra,beyond the Lorentz invarianceimplicit
in Lorentz transformations. This group theoretic metric can be used to develop Lorentz invariants of
any operators carrying the group indexes of Γµ. Once space-time translations are incorporated, since
the generators of space-time translations P transform as covariant 4-vectors under arbitrary coordinate
µ
transformations(of whichgrouptransformationsarea specialsubset[8]), this groupstructure is explicitly
tied to curvilinear space-time dynamics expressing the principle of equivalence.
3 Construction of Local Invariance Groups
In this section, the invariance groups of Γ= 1 representation eigenstates will be explored.
2
3.1 A complete set of linearly independent hermitian generators in GL(4)
As previously mentioned, the Γ = 1 representation is a subgroup of GL(4). There are 16 hermitian
2
matrices in GL(4), including that associated with the U(1) transformation proportional to the identity.
The subgroup connected to particle representations already includes the 4 hermitian generators Γ0 and
the angular momenta J . This leaves an additional 11 matrices, which will be constructed as T = iΓj
k j
and T =iK for j =1 3, two matrices given by
j+3 j
→
i 0 1 1 0 1
T = , T = , (3.6)
7 2 1 0 8 2 1 0
(cid:18) − (cid:19) (cid:18) (cid:19)
3
where 1 and 0 are the 2 2 identity and zero matrices respectively, and a final set of three generators
×
T , T , and T forming a closed representation of SU(2) on the lower components given by
9 10 11
1 0 0
T = , (3.7)
j+8 2 0 σj
(cid:18) (cid:19)
where the σ are Pauli spin matrices.
j
It is intriguing that the number of additional hermitian generators in GL(4) is precisely the same as
the numberofhermitiangeneratorsinSU(3) SU(2) U(1). However,the setof8Hermitiangenerators
× ×
T for s:1 8 do not form a closed algebra.
s
→
3.2 Transformation properties of causal spinor fields
CausalspinorfieldsΨˆ(x)eithercommuteoranti-commute Ψˆ(x),Ψˆ(y) =0forspace-likeseparationsof
the space-timecoordinates(~y ~x)ofthosefields accordinghtowhetherith∓e spinisintegralorhalf-integral.
−
Thus, microscopic causality compels a well defined local relationshipbetween components of spinor fields
in configuration space. The form of a causal linear spinor field that has the expected properties under
parity, time reversal,and charge conjugation is given by[6]
Ψˆ(Γ)(~x) mc2d3p eh¯i(p·x−ǫ(p)t) u(Γ)(p~,m,J,s )aˆ(Γ)(p~,m,J,s )+
γ ≡ ǫ(p) " (2π¯h)3/2 γ z γ z
XJ,szZ
(−)J+sz e−h¯(i2(pπ·¯hx−)3ǫ/(2p)t) u(−Γγ)(p~,m,J,−sz)aˆ(−Γγ)†(p~,m,J,sz)#, (3.8)
where the normalization has been chosen to have non-relativistic correspondence, mc2d3p d3p for
ǫ(p) →
p<<mc,ands specifiesthez-componentofinternalangularmomentumtoavoidconfusionwiththemass
z
m. Inthisexpression,theaˆ(Γ)(p~,m,J,s )areannihilationoperatorsofparticlestatesofthegivenquantum
γ z
numbers, ǫ(p) = p2c2+m2c4, and the spinors satisfy Γµp u(Γ)(p~,m,J,s ) = γmcu(Γ)(p~,m,J,s ).
µ γ z γ z
| | −
Inthis paper,thepfocuswillbe onthe Γ= 12 representation,forwhichthe u(γ12) have4spinorcomponents.
Foramorecompletetreatmentofthealgebra,symmetries,andcausalitypropertiesofcausalspinorfields,
the reader is invited to examine sections 4.3 and 4.4 in reference [6].
Since the spinor components of a causal 4-spinor field Ψ(21) maintain local spatial relationships de-
scribed in (3.8), a particular local Euclidean rotationR (x) will be constructed to define a spinor with a
E
single component. If the components of the causal 4-spinor are written in the form
φ1(x)eiω1(x) φ1(x)eiω1(x)
Ψ(x)= φ2(x)eiω2(x) −φ1(x)tan(ζ12)sec(ζ13)sec(ζ14)eiω2(x) , (3.9)
φ3(x)eiω3(x) ≡ φ1(x)tan(ζ13)sec(ζ14)eiω3(x)
−
φ4(x)eiω4(x) −φ1(x)tan(ζ14)eiω4(x)
thenthecausalspinorcanbeexpressedbytheactionofaEuclideanrotationonasinglecomponentspinor
4
as follows:
cos(ζ14) 0 0 sin(ζ14)eiω14 cos(ζ13) 0 sin(ζ13)eiω13 0
0 1 0 0 0 1 0 0
R14 = 0 0 1 0 , R13 = sin(ζ13)e−iω13 0 cos(ζ13) 0 ,
−
sin(ζ14)e−iω14 0 0 cos(ζ14) 0 0 0 1
−
cos(ζ12) sin(ζ12)eiω12 0 0
R12 = −sin(ζ102)e−iω12 cos(0ζ12) 10 00 , ω1s ≡ω1−ωs, RE ≡R14R13R12, Ψ(x)=RE(x)Φ¯(x).
0 0 0 1
(3.10)
The single component spinor takes the form
(φ1(x))2+(φ2(x))2+(φ3(x))2+(φ4(x))2eiω1(x)
Φ¯(x)= p 00 =R−E1(x)Ψ(x). (3.11)
0
It is important to note that this transformation must be made uniquely at each location x. This imposes
a local nature to any internal symmetries that utilize this construction.
3.3 SU(2) and SU(3) invariance transformations on Φ¯(x)
Thepreviouslydefinedgeneratorsτ T withj :1 3formaclosedmatrixgroupalgebraM(2) that
j j+8
≡ →
transform components in the subspace S(2):
1 0
M(2)(θ)= =ei θsτs. (3.12)
0 S(2)(θ)
(cid:18) (cid:19) P
In this expression, the matrix S(2) is a unitary unimodular transformation matrix in SU(2), and 1 is the
2 2 identity matrix. Since τ Φ¯(x) = 0, the field Φ¯(x) from Eq. 3.11 is invariant under transformations
j
×
involving M(2). The generators τ share this basis simultaneously with Γ0 and J .
j 3
As was previously mentioned, the remaining hermitian generators in this basis T ,...,T do not
1 8
{ }
form a closed algebra. However, a basis of SU(3) states can be constructed through mixing of the her-
mitian generators in this basis. This mixing between precisely three sets of the basis states will next be
demonstrated.
Consider the set of SU(3) transformations of the form
T
M(3)(α)= 1 0 =ei αbtb, (3.13)
0 S(3)(α) !
P
whereS(3) isaunitaryunimodulartransformationmatrixinSU(3),and0isa1 3zerovector. Thesetof
×
eight generators t ,...,t form a closed algebra of SU(3). Since t Φ¯(x) =0, the field Φ¯(x) is invariant
1 8 b
{ }
under transformations involving M(3). The SU(3) eigenstates will be defined using this basis. A set of
SU(3) generators (including 2 diagonal generators)can be found in [9].
5
TheconstructionofarepresentationofSU(3)thatisinclusiveofthehermitiangenerators Γ0,J ,J ,J
1 2 3
{ }
in GL(4) necessarily involves the mixing of the three basis states of the transformation (3.13). A conve-
nientmechanismformixingthreeeigenstatesisprovidedthroughgeneralCKMmatrices[10,11]embedded
withinGL(4). Theparticularchoiceformixingwillbethesetofalltransformationsonthe3 3subspace
×
that leaves the second spin component of Φ¯(x) invariant as demonstrated below:
1 0 0 0 cos(θ31) 0 0 sin(θ31)e−iδ31
0 1 0 0 0 1 0 0
U23 = 0 0 cos(θ ) sin(θ ) , U31 = 0 0 1 0 ,
23 23
0 0 −sin(θ23) cos(θ23) −sin(θ31)eiδ31 0 0 cos(θ31)
cos(θ ) 0 sin(θ ) 0
12 12
0 1 0 0
U12 = sin(θ12) 0 cos(θ12) 0 , UCKM ≡U23U31U12. (3.14)
−
0 0 0 1
Most CKM transformations of this type on the generators ˜ts ≡ UCKMtsU−C1KM will produce a set of
transformed generators ˜t that both satisfy the algebra of SU(3) as well as can be included among a set
s
of 11 linearly independent traceless hermitian generators that complete the set Γ0,J ,J ,J .
1 2 3
{ }
For clarity, consider anexample CKM transformationfrom the SU(3) eigenbasis to that including the
generators T ,...,T parameterizedusing data consistentwith knownmixing between the generations
1 11
{ }
in quark phenomenology, (θ 0.227, θ 0.446, θ 0.0035, δ 1.96):
12 23 31 31
→ → → →
0.974 0 0.225 0.0013 0.0032i
− −
0 1 0 0
UCKM = 0.225+0.00014i 0 0.973+0.000033i 0.0446 . (3.15)
− −
0.00875 0.00316i 0 0.0437 0.00073i 0.999
− − −
The set of 15 linearly independent hermitian generators Γ0,J ,J ,J ,T ,...,T can be directly de-
1 2 3 1 11
composed in terms of an alternative set of 15 linearly inde(cid:8)pendent hermitian genera(cid:9)tors given by
Γ0,J ,J ,J ,˜t ,...,˜t ,∆ (T T )/2,∆ (T +T )/2,∆ (T +T )/2 .
1 2 3 1 8 1 1 5 2 2 4 3 3 7
≡ − ≡ ≡
(cid:8) (cid:9)
Although the set of matrices ˜t ,...,˜t now generate a closed SU(3) algebra in this basis, the remaining
1 8
{ }
generators ∆ , ∆ , and ∆ do not form a closed algebra. The transformed SU(3) generators ˜t are
1 2 3 b
decomposed in (3.16).
Γ0 J1 J2 J3 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11
˜˜˜˜˜˜˜˜tttttttt87631254 −−−−2..90000275793.5350 −−231000020..52341 −321.0000202.5341 −−−120..0000275293.5350 −−−−−411..2181000.2.72751.71463 −−−−4215...2410002..82517637861 −−−−−..00.214503761..065369.05988 −−−.44105.2.810012.2727566.183 −−−1.410.2.1800.752177114.263 −−−−−.13215141061..090565.99985 −−.2..530041100667...035599688 −−−−241.15110913...09065699585 −−−−319.81..2070237.206941.5517 −−−−.173..2900713.2712115414 −−−.10890000006453.581
(3.16)
Foreachofthetransformedgenerators˜t = kjG ,thecoefficientskj aretobeobtainedbymultiplying
b j b j b
the number in the column under the previoPusly defined generator Gj by 10−3 in (3.16). It is clear that
6
the set of independent generators Γ0,J ,J ,J ,˜t ,...,˜t ,∆ ,∆ ,∆ that includes a closed SU(3) sub-
1 2 3 1 8 1 2 3
{ }
algebra ˜t ,...,˜t necessarily mixes in an invertible manner the basis Γ0,J ,J ,J ,T ,...,T that
1 8 1 2 3 1 11
{ } { }
includes an internal closed SU(2) sub-algebra T ,T ,T on Φ¯(x). The independent group of SU(3)
9 10 11
{ }
transformations on the 4-spinors generated by ˜t ,...,˜t will be denoted U(3)(α(x)), so that
1 8
{ }
M(3) ≡U−C1KMU(3)UCKM, (3.17)
where M(3) is of the form defined in Eq. 3.13.
3.4 Local gauge symmetries of spinor fields
A local internal SU(3) symmetry on the causal spinor field Ψˆ(12)(~x) from (3.8) can be constructed using
γ
the internalSU(3)symmetry transformationM(3) onΦ¯(x)from(3.17),alongwiththe relationshipofthe
transformed spinor Φ¯(x) to the general causal spinor field expressed in (3.11), via
U(3)(x)=RE(x)M(3)(α(x))R−E1(x)=RE(x)U−C1KMU(3)(α(x))UCKMR−E1(x), (3.18)
whereαs(x)aretheeightlocalgroupparametersofSU(3). Similarly,thetransformationM(2) from(3.12)
onΦ¯(x)definesalocalinternalSU(2)symmetryonthecausalspinorfieldΨˆ(12)(~x)undertransformations
γ
U(2)(x)=RE(x)M(2)(θ(x))R−E1(x), (3.19)
where θj(x) are the three local group parameters of SU(2). Although both (3.18) and (3.19) are local
internal symmetries on the causal spinor field (x)Ψ(x) = Ψ(x), the eigenbases of the internal SU(2)
U
and SU(3) symmetries are related via CKM mixing of the three SU(3) eigenstates in the enlarged unified
group GL(4) via U .
CKM
Furthermore, the independent sets of hermitian generators Γ0,J ,J ,J ,˜t ,...,˜t ,∆ ,∆ ,∆ and
1 2 3 1 8 1 2 3
{ }
Γ0,J ,J ,J ,T ,...,T include local gauge invariance transformations under SU(3) and SU(2) for
1 2 3 1 11
{ }
interacting fields satisfying (1.1) that connect the two basis sets via CKM mixing, where
U(3)(α)=ei bαb˜tb and U(2)(θ)=ei sθsτs. (3.20)
P P
Intheseexpressions,α=α(x)arelocalgroupparametersofSU(3)andθ =θ(x)arelocalgroupparameters
ofSU(2)undergaugetransformationsofthe fieldΨ(x) Ψ˜(x)=U(3,2)(β(x))Ψ(x). Thetopologyofthe
→
local mapping of gauge group parameters β(x) in space-time determines the monopole structure of the
sourcesofthegaugeinteractions(seesection4.2of[6]),aswellasthepropertiesofanyhigherdimensional
structures (e.g., strings, etc.) in the formulation.
4 Symmetric Tensors and Conservation Properties
Itisofinteresttodeterminewhetherasymmetricenergy-momentumtensorsuitableforgeometrodynamics
can be generated from a Lagrangian that produces the spinor field equations. It should be noted that
7
the equations that define the properties of the spinors given in (1.1) generally need not be the same as
the equations used to construct the energy momentum tensor, and one can always construct symmetric
energy momentum tensors from a Lagrangianof the form
1 1 ¯h ∂ q † ¯h ∂ q
L= 2m" i ∂xµ − ca AsµGs!Ψ# gµν i ∂xν − ca AsνGs!Ψ+ mc2Ψ†Ψ+
Xs Xs
1 c4
Tr(FµνF ) , (4.21)
µν
−16π − 16πG R
N
where g is the determinant of the space-time metric tensor, F = ∂ A ∂ A iqa[A ,A ], A
µν µ ν − ν µ − h¯c µ ν µ ≡
AsG , and is the Ricci scalar. Gravitational field equations result from (4.21) by requiring that
s µ s R
P2 δ (√ g )=0. Causalfields (3.8) that satisfy the spinor field equations (1.1) will also satisfy the
√−gδgµν − L
Klein-Gordon equation resulting from this Lagrangian form, which directly produces symmetric energy-
momentum tensors.
To determine if the symmetric, locally conserved tensors can also be constructed from a Lagrangian
form that generates the spinor field equations, it suffices to examine non-interacting fields. Spinor field
equations (1.1) will result from a Lagrangianof the form
1 ¯h ∂ 1 ∂xβ ¯h ∂
= Ψ Γµ Ψ+γmcΨ +cc= Ψ Γµ Ψ+γmcΨ +cc. (4.22)
Ls 2 · i ∂ξµ 2 ∂ξµ · i ∂xβ
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:18) (cid:19) (cid:21)
where Ψ is the Dirac conjugate of field Ψ, cc refers to the complex conjugate of the previous term, and
ξµ are locally flat coordinates. A conserved second rank tensor can be constructed consistent with the
Euler-Lagrangeequations of the form
∂
Πµ L , µν =Πµgνβ∂ Ψ+cc gµν , (4.23)
≡ ∂(∂ Ψ) T β − L
µ
However,in this case the tensor µν is notsymmetric. Following Belinfante[12][13], the tensor describing
T
internal transformations under general Lorentz transformations given by
i¯h
βµν = Πβ[Γµ,Γν]Ψ Πµ[Γβ,Γν]Ψ Πν[Γβ,Γµ]Ψ +cc (4.24)
S 2 − −
(cid:0) (cid:1)
can be used to construct a symmetric second rank tensor Tνµ = Tµν µν +∂ βµν that satisfies the
β
≡ T S
conservation equation ∂ Tµν = 0 in flat space-time (since βµν is antisymmetric under the interchange
µ
S
β µ). However,using curvilinear coordinates, the results of the construction are as follows:
↔
1
Tµν µν +D βµν, D Tµν = Rν βµλ. (4.25)
≡T βS µ −2 λβµS
Thus,thecovariantdivergenceoftheBelinfanteconstructdoesnotgenerallyvanishincurvedspace-times.
That this construct does not generate an energy-momentumtensor that can drive the geometrodynamics
should not be surprising, since for integer values of Γ, there are states satisfying (1.1) with finite mass,
yet vanishing eigenvalues of ΓµPˆ , for which γ = 0. These states must nonetheless carry non-vanishing
µ
energy-momentum.
8
5 Conclusions
The fundamental representation of causal spinor fields has been shown to unify a set of internal
local symmetries including a U(1) symmetry along with 11 additional hermitian generators that can
represent a linearly independent SU(2) symmetry or a linearly independent SU(3) symmetry, but not
bothsimultaneously. TheeigenbasisoftheSU(3)internalsymmetryhasbeenrelatedtothatoftheSU(2)
internal symmetry via a CKM transformation mixing the symmetries in GL(4) consistent with observed
phenomenology. The closed representation of SU(2) can share the same basis as that in which Γ0 and J
3
are diagonal, whereas the SU(3) eigenbasis with two diagonal generators cannot share this basis.
Group algebraic invariants explicitly generate the Minkowski metric using the non-abelian algebra of
generatorsthatcarrysinglespace-timeindexes,therebyextendinginteriorgroupstructuretothedynamics
ofgeneralcoordinatetransformations. Thegeometrodynamicsofgeneralrelativityfollowsdirectlyviathe
principle of equivalence.
On-goingeffortsexaminehowthegaugebosonsoftheinternalSU(2) U(1)pre-symmetryoftheΓ= 1
× 2
representationcanrelateto the quantaofthe massive Γ=1bosonrepresentationwithN =10,which
Γ=1
containsaself-adjointscalarparticle,aself-adjointvectorparticle,anothervectorparticle,anditsadjoint
vectorparticle. Theself-adjointγ =0statesaredegenerateintheireigenvalue,andcanfreelymixmassive
as well as massless vector states in a manner that preserves the degrees of freedom.
References
[1] E.J. Weinberg and D.L. Nordstrom, eds, “Review of Particle Physics” Phys. Rev. D 86 (2012).See
section 20, pp.259-263.
[2] R. Colella, A.W. Overhauser, and S.A. Werner, “Observation of Gravitationally Induced Quan-
tum Interference”, Phys.Rev.Lett. 34, 1472-1474 (1975). See also A.W.Overhauser and R.Colella,
Phys.Rev.Lett. 33, 1237 (1974).
[3] H. Muller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the
interference of matter waves”. Nature 463, 926-929 (2010), doi:10:1038/nature08776.
[4] P.A.M. Dirac, Proc. Roy. Soc. (London), A117, 610 (1928); ibid, A118, 351 (1928).
[5] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York (1964).
[6] J.Lindesay,Foundations of Quantum Gravity,CambridgeUniversityPress(2013),ISBN978-1-107-
00840-3.
[7] M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley (1962).
See eg. Equations. (8-90) and (8-114)
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[8] J. Lindesay, “Linear Spinor Fields in Relativistic Dynamics”, arXiv:1312.0541 [hep-th] (2013) 13
pages.
[9] J. Lindesay, “A Model of Unified Gauge Interactions”, arXiv:1604.01375[physics:gen-ph] (2016) 11
pages.
[10] N. Cabibbo, “Unitary Symmetry and Leptonic Decays”. Physical Review Letters 10 (12): 531-533
(1963). doi:10.1103/PhysRevLett.10.531.
[11] M.KobayashiandT.Maskawa,“CP-Violationinthe RenormalizableTheoryofWeak Interaction”.
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[12] F. Belinfante, Physica 6, 887 (1939).
[13] S. Weinberg, The Quantum Theory of Fields, Volume 1, (Cambridge University Press,1999),p316.
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