Geometry and symmetries in lattice spinor gravity

0.59 MB·English

by  C. Wetterich

Geometry and symmetries in lattice spinor gravity
C. Wetterich
Institut fu¨r Theoretische Physik
Universita¨t Heidelberg
Philosophenweg 16, D-69120 Heidelberg
Latticespinorgravityisaproposalforregularizedquantumgravitybasedonfermionicdegreesof
freedom. Inourlattice modelthelocal Lorentzsymmetryisgeneralized tocomplex transformation
parameters. The difference between space and time is not put in a priori, and the euclidean and
Minkowskiquantumfieldtheoryareunifiedinonefunctionalintegral. Themetricanditssignature
arise as a result of the dynamics, corresponding to a given ground state or cosmological solution.
Geometricalobjectsasthevierbein,spinconnectionorthemetricareexpectationvaluesofcollective
fieldsbuiltfromanevennumberoffermions. Thequantumeffectiveactionforthemetricisinvariant
2 undergeneral coordinatetransformations inthecontinuumlimit. Theaction ofourmodel isfound
1 to be also invariant under gauge transformations. We observe a “geometrical entanglement” of
0 gauge- and Lorentz-transformations due to geometrical objects transforming non-trivially under
2 both typesof symmetry transformations.

n
a
J I. INTRODUCTION the sixthcriterionnew methods fora reliablecomputation
1 of the quantum effective action need to be developed, as
3 Lattice spinor gravity [1] has been proposed as a regu- sketchedbrieflyinthe conclusions. We emphasizethatthe
larized model for quantum gravity. It is based on a Grass- derivative expansion of a diffeomorphism symmetric effec-
h]mann functional integral for fermions which is mathemat- tive action for the metric permits only few invariants with
tically well defined for a finite number of lattice sites. For a low number of derivatives. The two leading ones are
p-a realistic model of quantum gravity the decisive feature a “cosmological constant term” with zero derivatives, and
eis the invariance of the quantum effective action under an “Einstein-Hilbert term” proportional to the curvature
hgeneral coordinate transformations (diffeomorphism sym- scalar with two derivatives. The coefficients of both terms
[metry). The basic degrees of freedom used for the for- maydependonadditionalscalarfields. Ifthe cosmological
1 mulation of the functional integral are less important. In constantterm is smallenoughthe gravitationalfield equa-
vour fermionic formulation the metric and vierbein, as well tions are close to Einstein’s equations of general relativity
5as other geometrical objects, arise as expectation values and therefore to a realistic theory of gravity.
0of suitable collective fields built from an even number of The use of fermions as basic variables has several ad-
5
fermions. vantages: (i) Fermions transform as scalars with respect
6
For a realistic lattice quantum field theory for quantum to diffeomorphisms. This facilitates the formulation of a
.
1gravity we require the following six criteria: lattice diffeomorphism invariant functional measure which
0 would be much harder (and has never been achieved so
2
(1) The functional integral is well defined for a finite far) for fundamental metric degrees of freedom. (ii) For
1
number of lattice points. a Grassmann functional integral there is no problem of
:
v “boundedness” of the action. Criterion (1) is obeyed au-
Xi (2) The functional measure and lattice actionare lattice tomatically. (iii) Fermions need to be included anyhow in
diffeomorphism invariant.
anyrealisticmodelofparticlephysics. Bosonsasthegravi-
r
a (3) Latticediffeomorphisminvarianceturnstodiffeomor- ton,photon, W-andZ-boson,gluonsandHiggsscalarcan
arise as collective states. Thus an extension of the present
phism symmetry for the quantum effective action in
model of lattice spinor gravity can be a candidate for a
the continuum limit.
unifieddescriptionofallinteractions. Forthesereasonswe
(4) For a model with fermions the functional measure stick here to a purely fermionic functional integral and do
and lattice action are invariant under local Lorentz not introduce bosonic lattice variables as in ref. [2]. The
transformations. fermionic formulation and the implementation of lattice
diffeomorphism invariance distinguish our approach from
(5) Thecontinuumlimitincludesmassless(orverylight)
other lattice proposals for quantum gravity [3–5].
degrees of freedom with gravitationalinteractions.
Diffeomorphism symmetry of the continuum action can
(6) Aderivativeexpansiongivesareasonableapproxima- be achieved rather easily for a purely fermionic model. It
tion for the quantum effective action for the metric is underlying earlier versions of spinor gravity [6–8] and
at long wavelength. has been pioneered very early [9–11]. The work in [6–8]
and [9–11] does not implement local Lorentz symmetry,
We will present a model that obeys the first four cri- however. If the action is invariant only with respect to
teria. The fifth criterion is not yet shown, but likely to global Lorentz symmetry additional torsion-type massless
hold in view of the diffeomorphismsymmetry of the quan- degrees of freedom are present. Their phenomenology is
tum effective actionfor the metric. For an investigationof discussed in ref. [7]. In order to avoid such complications
2
we stick here to the criterion (4) and formulate an action The lattice action [1] realizing the criteria (1)-(4) in-
that is invariantunder localLorentz transformations. Our volves two species of Dirac fermions. It is found to
formulation of lattice spinor gravity differs therefore from be invariant under global chiral SU(2) SU(2) gauge
L R
×
ref. [6–11]. We investigate the lattice action proposed in transformations acting in “flavor-space”. These symme-
ref. [1]forwhichlocalLorentzsymmetryismanifest. This tries are actually extended to their complexified versions
model resembles in several aspects the higher-dimensional SU(2, ) SU(2, ) . In the continuum limit the ac-
L R
C × C
continuum action with local Lorentz symmetry proposed tion exhibits even local gauge symmetry. The presence
in ref. [12]. of additional gauge symmetries has the interesting conse-
Inref. [13]wehaveformulatedtheconceptoflatticedif- quence that some of the collective geometrical degrees of
feomorphism invariance. In this paper it was shown that freedomtransformnon-triviallyunderboththegeneralized
a lattice diffeomorphism invariant action and functional Lorentz transformations SO(4, ) and the gauge trans-
C
measure imlpy diffeomorphism symmetry for the quantum formations. This new “geometric entanglement” between
effective action, including the quantum effective action for Lorentz- and gauge-transformations results in interesting
the metric as expectation value of a collective field. Our aspects of “gauge-gravityunification”.
latticemodelislatticediffeomorphisminvariantandthere- In the present paper we work out several aspects of lat-
fore also obeys criteria (2) and (3). tice spinor gravity that are crucial for progress towards a
Our approachhas atthe basic levelneither a metric not realistic theory of gravity in this setting. We investigate
a vierbein. We also do not employ any objects of lattice the fermion-bilinears that can play the role of a vierbein,
geometrythatreplacethesefieldsinadiscreteformulation. both for the continuum limit and the lattice version. We
The absence of a metric contrasts with “induced gravity” discusstheirbehaviorundersymmetrytransformations,in-
[14] where a metric is present, while its kinetic term in cluding gauge symmetries. We further establish the con-
the effective action is induced by matter fluctuations. In nectionbetweenthevierbeinbilinearsandacollectivefield
ourmodelthe metric andthe vierbeinariseas expectation for the metric. We investigate various continuous and dis-
values of suitable collective fields. In this respect there is crete symmetries of the action, including an extension of
someresemblancetotheappearanceofavierbeinoramet- the model where the Lorentz- and gauge transformations
ricascondensatesororderparametersincertaincondensed are unified within a larger group SO(8, ). We give a de-
C
matter systems [15], or other ideas that the vierbein may tailed account of the lattice formulation.
originate from a fermion condensate [16]. One of the important aspects of this work concerns the
The formulation of a functional integral for quantum observation that a given lattice action can be seen from
gravity without basic metric degrees of freedom opens the different geometrical perspectives. This is related to the
doorfortheinterestingpossibilitythatthesignatureofthe different possibilities to group fermions into collective bi-
metric which distinguishes between space and time is not linear fields. From one point of view the proposed action
introducedapriori. Thedifferencebetweentimeandspace for lattice spinor gravity appears a type of kinetic term
can therefore arise dynamically, as induced by a particu- for scalar bilinears, involving four derivatives contracted
lar expectation value of a collective field. For fermions, by an ǫ-tensor. These scalars are invariant under general-
such a setting requires that we do not fix the choice be- ized local Lorentz transformation such that the symmetry
tween a euclidean rotation group SO(4) or the Lorentz of the action is manifest. From a different point of view
group SO(1,3) a priori. This can be realized [12] if the theactioninvolvesvierbeinbilinearswhichtransformnon-
modelisinvariantunderthecomplexifiedorthogonalgroup triviallyunderLorentzandgaugetransformations. Finally,
SO(4, ). This group obtains if the real transformation the action can also be seen from a purely fermionic point
paramCeters of SO(4) are generalized to arbitrary complex of view where eight spinors without derivatives and four
transformation parameters. The group SO(4, ) contains derivatives of spinors are grouped into invariants.
C
both SO(4) and SO(1,3) as subgroups. Which one is re- This paper is organized as follows: In sect. II we for-
alized depends again on the ground state (or cosmological mulate the functional integral and discuss the symmetry
solution for an appropriate Lorentz-type signature). transformations SO(4, ) as well as the continuum limit
C
ModelswithSO(4, )symmetryhavetheimportantad- of the action. This section has substantial overlap with
vantagethatboththeCeuclideanandtheMinkowskisetting the shorter presentation in the letter of ref. [1] and per-
are realized by one and the same functional integral. On mits a systematic and self-consistent presentation for the
the level of the basic theory there are no longer two dif- present paper. In sect. III we introduce the geometrical
ferentfunctionalintegralsthatarerelatedby anoperation objects as the vierbein bilinears and the collective metric
of analytic continuation. Now “analytic continuation” ap- field in a continuum version. They will later be related
pears within one given functional integral and relates pos- to corresponding lattice objects. Sect. IV discusses the
sible expectation values of objects that correspond to the symmetries of the action.
vierbein. Such a realization of analytic continuation has In sect. V we turn to the detailed lattice formulation.
been discussed previously in ref. [17]. The requirement Basic building blocks are invariant hyperlinks that corre-
of local SO(4, ) symmetry restricts the possible form of spondtotheplaquettesinlatticegaugetheories. Weintro-
C
the lattice action. In short, the signature tensor η can duce lattice derivatives and compute the continuum limit,
mn
no longer be used as a basic object for the construction of showing that it is diffeomorphism symmetric. In sect. VI
invariants. we introduce fermionbilinears that actas links, somewhat
3
similar to the link variables in lattice gauge theories. We relate d4xtoasumoverthelatticepoints. Inthecontin-
present an equivalent expression for the lattice action in uumliRmitxlabelspointsofaregionof 4,∂µϕ(x)becomes
terms of links. The lattice vierbeins are closely connected apartialderivativeofaGrassmannfieldR,and d4xdenotes
to these links. Their continuum limit yield the vierbein the integration over the region in 4. R
R
bilinears of sect. III. In sect. VII is devoted to the lattice In the continuum limit the invariance of the action un-
metriccollectivefield. Insect. VIIIwefinallyestablishlat- der general coordinate transformations follows from the
tice diffeomorphism invariance of the proposed functional use of the totally antisymmetric product of four deriva-
integral. Our conclusions are drawn in sect. IX. In order tives∂ =∂/∂xµ. Indeed,withrespecttodiffeomorphisms
µ
to keep the main lines of the presentation clear we display ϕ(x) transforms as a scalar, and ∂ ϕ(x) as a vector. The
µ
the moretechnicalaspects ofour argumentsin variousap- particular contraction with the totally antisymmetric ten-
pendices. sor ǫµ1µ2µ3µ4,ǫ0123 = 1, allows for a realization of diffeo-
morphism symmetry without the use of a metric. In sect.
VIIIweintroduceforthediscretelatticetheconceptoflat-
II. ACTION ticediffeomorphisminvariance: Thispropertyoftheaction
(2)guaranteesdiffeomorphismsymmetryinthecontinuum
1. Functional integral limit. Finally,theobjectJ willbechosensuchthattheac-
tion is invariant under local Lorentz transformations and
It is our aim to formulate a quantum field theory for
their generalization to SO(4, ).
gravity based on the standard functional integral formal-
C
The partition function Z is defined as
ism. Our basic degrees of freedom are fermions, and the
functional integral will therefore be a Grassmann func-
tional integral. We also want all operations for this func- Z = ψg exp( S)g ,
f in
Z D −
tional to be mathematically well defined. We therefore
implement spinor gravity with a lattice regularization. 2
ψ = dψa(x)... dψa(x) . (3)
Let us explore a setting with 16 Grassmann variables Z D Z 1 Z 8
ψa at every spacetime point x, γ = 1...8, a = 1,2. The Yx aY=1(cid:8) (cid:9)
γ
coordinates x parametrize the discrete points of a four di-
Forafinitenumberofdiscretespacetimepointsonalattice
mensional lattice, i.e. xµ = (x0,x1,x2,x3). We will later
theGrassmannfunctionalintegral(3)iswelldefinedmath-
associatet=x0withatimecoordinate,andxk, k =1,2,3,
ematically. We assume that the time coordinates x0 = t
with space coordinates. There is, however, a priori no dif-
obey t t t . The boundary term g is a Grass-
in f in
ference between time and space coordinates. The spinor ≤ ≤
mann element constructed from ψ (t ,~x), while g in-
γ in f
index γ denotes the eight real Grassmann variables that
volvestermswithpowersofψ (t ,~x),were~x=(x1,x2,x3).
γ f
correspondtoacomplexfour-componentDiracspinor. We
The product g g can be generalizedto a “boundary ma-
f in
consider two flavors of fermions, labeled by a, similar to
trix” ρ that depends on ψ(t ,~x) and ψ(t ,~x). If g g
fi f in f in
the electron and neutrino, or up- and down-quark (with-
or ρ are elements of a real Grassmann algebra the par-
out color). The “real” Grassmann variables ψa can be fi
γ tition function is real. We may restrict the range of the
combined to complex Grassmann variables ϕa,α=1...4,
α spacecoordinatesoruse periodicorantiperiodicboundary
conditions for ϕ(x) with xk discrete points on a torus T3.
ϕa(x)=ψa(x)+iψa (x), (1)
α α α+4 The Grassmann integration involves then a finite number
of Grassmann variables. (For suitable ρ also the time-
with α the “Dirac index”. fi
coordinatecanbe putona torus.) The boundarytermg
We concentrate on an action which involves twelve in
specifies the initial values of a pure quantum state, while
Grassmann variables and realizes diffeomorphism invari-
ρ can be associated with a type of density matrix. The
ance and extended local Lorentz symmetry of the group fi
particular form of the boundary terms play no role in the
SO(4, )
C discussion of this paper and we may, for simplicity, put
both t and ~x on a torus T4.
S = αZ d4xϕaα11...ϕaα88ǫµ1µ2µ3µ4 (2) Observables A will be represented as Grassmann ele-
ments constructed from ψ (x). We will consider only
×Jαa11......αa88bβ11......bβ44∂µ1ϕbβ11∂µ2ϕbβ22∂µ3ϕbβ33∂µ4ϕbβ44 +c.c.. bosonic observablesthat invγolveanevennumber of Grass-
mann variables. Their expectation value is defined as
We sum over repeated indices. The complex conjugation
c.c. replaces α α∗,J J∗ and ϕ (x) ϕ∗(x) =
ψ (x) iψ (x→), such th→at S∗ = S.α(Occ→asionαally we =Z−1 ψgf exp( S)gin. (4)
α α+4 hAi Z D A −
−
use a notation where the flavor index a is not written ex-
plicitly, such that each ϕ should be interpreted as a two- “Realobservables” are elements of a realGrassmannalge-
α
component complex vector.) In terms of the Grassmann bra, i.e. they are sums of powers of ψ (x) with real coef-
γ
variables ψ (x) the action S as well as exp( S) are ele- ficients. For real g and g all real observables have real
γ in f
−
ments of a real Grassmann algebra. expectation values. We will take the continuum limit of
Theprecisemeaningofthederivatives∂ ϕ(x)inthelat- vanishing lattice distance at the end. Physicalobservables
µ
ticeformulationwillbeexplainedinsect. V.Therewealso are those that have a finite continuum limit.
4
In the remainder of this section and sections III-IV we with Σmn = Σmn , γm = γm. The first “homogeneous
E E
will discuss the properties of the action (2) in the contin- term” ∂ ϕ transforms as ϕ . Contributions of the sec-
µ β
∼
uumlimitwhere∂ denotespartialderivatives. Mostprop- ond “inhomogeneous term” to the variation of the action
µ
ertiescanbe directlyextendedtothe discreteformulation. δS involveatleastnine spinorsat the same position x, i.e.
Sects. V-VIII will then provide an explicit discussion of (Σmnϕ)b(x)ϕa1(x)...ϕa8(x). Thisinhomogeneouscontri-
β α1 α8
the discrete setting on a lattice. bution to δS vanishes due to the identity ϕ (x)ϕ (x) =0
α α
2. Generalized Lorentz transformations (nosumhere). ThisinvarianceofS underglobalSO(4, )
C
transformationsentailstheinvarianceunderlocalSO(4, )
We do not want to introduce a difference between time
C
transformations. Wehaveconstructedinref. [12]asixteen
and space from the beginning. In consequence, we do not
dimensional spinor gravity with local SO(16, ) symme-
want to fix the signature of the Lorentz rotations a pri-
C
try. The present four-dimensional model shows analogies
ori. This is achievedby extending the euclideanSO(4) ro-
to this.
tations to complex transformations SO(4, ). Depending
C It is important that all invariants appearing in the ac-
on the choice of parameters both the euclidean rotations
tion (2) involve either only factors of ϕ =ψ +iψ or
SO(4) and the Lorentz group SO(1,3) are subgroups of α α α+4
only factors of ϕ∗ = ψ iψ . It is possible to con-
SO(4,C). struct SO(1,3) inαvarianαts−whicαh+4involve both ϕ and ϕ∗.
Ouraimistheconstructionofanaction(2)thatisinvari-
Those will not be invariant under SO(4, ), however. We
ant under local SO(4, ) transformations. It is therefore
necessary that the tensCor Ja1...a8b1...b4 is invariant under canalsoconstructinvariantsinvolvingϕCandϕ∗ whichare
α1...α8β1...β4 invariant under euclidean SO(4) rotations. They will not
globalSO(4, )transformations. We willoftenusedouble
C be invariant under SO(1,3). The only types of invariants
indices ǫ = (α,a) or η = (β,b), ǫ,η = 1...8. The tensor invariant under both SO(4) and SO(1,3), and more gen-
J is totally antisymmetric in the first eight in-
ǫ1...ǫ8η1...η4 erally SO(4, ), are those constructed from ϕ alone or ϕ∗
dicesǫ1...ǫ8,andtotallysymmetricinthelastfourindices alone,orprodCuctsofsuchinvariants. (Invariantsinvolving
η ...η . This follows from the anticommuting properties
1 4 bothϕandϕ∗ canbeconstructedasproductsofinvariants
of the Grassmann variables ϕ ϕ = ϕ ϕ . We will see
ǫ η − η ǫ involving only ϕ with invariants involving only ϕ∗.)
that for an invariant J the action (2) is also invariant un-
We conclude that for a suitable invariant tensor J the
der local SO(4, ) transformations.
C action has the symmetries required for a realistic theory
Local SO(4, ) transformations act infinitesimally as
C of gravity for fermions, namely diffeomorphism symmetry
1 and local SO(1,3) Lorentz symmetry. No signature and
δϕa(x)= ǫ (x)(Σmn) ϕa(x), (5)
α −2 mn E αβ β no metric are introduced at this stage, such that there is
no difference between time and space [12]. As discussed
with arbitrary complex parameters ǫ (x) =
mn extensively in ref. [7], global Lorentz symmetry may be
ǫ (x),m = 0,1,2,3. The complex 4 4 matrices
− nm × sufficient for a realistic theory of gravity. Nevertheless,
Σmn are associated to the generators of SO(4) in the
E models with local SO(1,3)-symmetry may be preferable,
(reducible) four-component spinor representation. They
since they contain only the metric as massless composite
can be obtained from the euclidean Dirac matrices
bosonic degree of freedom.
1
Σmn = [γm,γn] , γm,γn =2δmn. (6) For an action of the type (2) local SO(4, ) symme-
E −4 E E { E E} try is realized for every invariant tensor J. WeCdefine the
Subgroups of SO(4, ) with different signatures obtain by SO(4, ) variation of arbitrary tensors with Dirac indices
appropriate choicesCof ǫmn. Real parameters ǫmn corre- α1...αCN as
spond to euclidean rotations SO(4). Taking ǫ ,k,l =
kl
δT =T Σ + +T Σ , (10)
1,2,3 real, and ǫ = iǫ(M) with real ǫ(M), realizes the α1...αN α˜α2...αN α˜α1 ··· α1...α˜ α˜αN
0k − 0k 0k
Lorentz transformations SO(1,3). The Lorentz transfor- with
mations can be written equivalently with six real trans-
fgoernmeraattioonrspΣamranmaentderssigǫn(mMant)ur,eǫη(kmMln)==dǫiaklg,(us1i,n1g,1L,o1r)e,ntz- Σαβ =−21ǫmnΣmαβn. (11)
M −
1 WecanexpressglobalSO(4, )-transformations(withǫmn
δϕ=−2ǫ(mMn)ΣmMnϕ, (7) independent of x) of the actiCon equivalently by a transfor-
mation (5) of the spinors ϕ with fixed J, or by a transfor-
with
mation (10) of J with fixed ϕ. For δJ = 0 the action is
1 invariant under global SO(4, )-transformations.
ΣmMn =−4[γMm,γMn ] , {γMm,γMn }=ηmn. (8) C
3. Action with local Lorentz symmetry
The euclidean and Minkowski Dirac matrices are related We willcompose the invariantJ froma totallysymmet-
by γ0 = iγ0,γk =γk. ricfour-indexinvariantL andatotallyantisymmetric
M − E M E η1...η4
The transformation of a derivative involves an inhomo- eight-index invariant A(8) . For two flavors we can con-
geneous part ǫ1...ǫ8
struct two real symmetric SO(4, ) invariants
C
1 1
δ∂µϕβ =−2ǫmn(Σmn∂µϕ)β − 2∂µǫmn(Σmnϕ)β, (9) Sη±1η2 =−(τ2)β1β2(τ2)b1b2, (12)
5
where (β ,β ) are restricted to the values (1,2) for S+, D+ D− , such that again D is invariant. (For a
and (3,4)1 fo2r S−, and τk are the Pauli matrices. These suµit1aµb2le↔choiµc1eµ2γ0 = τ1 1 the transformation ϕ γ0ϕ
invariants and their relation to Weyl spinors are described actually corresponds to ϕ⊗ ϕ , cf. app. A.) W→e can
+,η −,η
in more detail in appendix A. We work here in a basis (cf. also decompose ↔
app. A) where γ¯ = diag(1,1, 1, 1) such that the Weyl
spinors ϕ =(1 γ¯)ϕ/2 corre−spon−d to the upper or lower A(8) =A+A−, (19)
±
±
two components of ϕ.
with
A totally symmetric four-index invariant can be con-
structedbysymmetrizingaproductoftwo-indexinvariants
A+ =ϕ1 ϕ1 ϕ2 ϕ2 (20)
+1 +2 +1 +2
1
Lη1η2η3η4 = 6(Sη+1η2Sη−3η4 +Sη+1η3Sη−2η4 +Sη+1η4Sη−2η3 involving four Weyl spinors ϕ+, and similarly for A−.
The invariants A± can be expressed in terms of the
+Sη+3η4Sη−1η2 +Sη+2η4Sη−1η3 +Sη+2η3Sη−1η4). (13) Lorentz invariant bilinears H±
k
MultiplicationofLwithfourspinorderivatives∂ ϕ yields
µ η H± = ϕa (τ ) (τ τ )abϕb , (21)
an expression D that is invariant under global SO(4, ) k ± ±,α 2 αβ 2 k ±,β
C
transformations
(with ϕ two-component Weyl spinors). These scalar
±,α
fields are simple building blocks for SO(4, ) invariantac-
D =ǫµ1µ2µ3µ4∂µ1ϕη1∂µ2ϕη2∂µ3ϕη3∂µ4ϕη4Lη1η2η3η4. (14) tions. One finds (cf. sect. VI), C
This invariantinvolvestwoWeylspinorsϕ andtwoWeyl
+ 1
spinors ϕ . A± = H±H±. (22)
− −24 k k
Furthermore, an invariant with eight factors of ϕ (with-
out derivatives) involves the totally antisymmetric tensor The combinations
for the eight values of the double-index ǫ
F± =A±D± (23)
1 µ1µ2 µ1µ2
A(8) = ǫ ϕ ...ϕ
8! ǫ1ǫ2...ǫ8 ǫ1 ǫ8 aretherefore composedofsix Weylspinors ϕ+ or six Weyl
= (214)2ǫα1α2α3α4ϕ1α1...ϕ1α4ǫβ1β2β3β4ϕ2β1...ϕ2β4 Fsp+inaonrsdϕF−−,,respectively. The action involves products of
= ϕ1ϕ1ϕ1ϕ1ϕ2ϕ2ϕ2ϕ2. (15)
1 2 3 4 1 2 3 4 S =α d4xǫµ1µ2µ3µ4F+ F− +c.c. (24)
Z µ1µ2 µ3µ4
It is easy to verify that δ(ǫ ) = 0 in the sense of eq.
ǫ1...ǫ8
(10). AninvariantJ ineq. (2)canthereforebeconstructed Since the tensor S involves (τ )ab all contributions
η1η2 2
by multiplying Lη1...η4 with ǫǫ1...ǫ8. to Dµ+ν have one spinor of each flavor, i.e. only terms
Inconclusionofthisdiscussionwewillconsideranaction ∂ ϕ1∂ ϕ2 appear. This implies that D+ is odd under
with local SO(4, ) symmetry which takes the form µ α ν β µν
C the transformation ϕ2α → −ϕ2α. On the other hand, A+
is even under this transformation, such that F+ is odd.
S =α d4xA(8)D+c.c. (16) Since F− is invariant, the action changes sign. Sµiνmilar in-
Z µν
volutionsthatchangethe signofS canbe foundbynoting
Indeed, the inhomogeneous contribution (9) to the vari- that each term in S contains exactly three spinors of each
ation of D(x) contains factors (Σmnϕb) (x). It vanishes of the four sorts ϕ1,ϕ2,ϕ1 and ϕ2.
β + + − −
when multiplied with A(8)(x), since the Pauli principle The action (16) is not the only possible action with
ϕa(x) 2 = 0 admits at most eight factors ϕ for a given twelve spinors and SO(4, ) symmetry. The tensor
α J in eq. (2) mustCbe totally antisymmetric in
(cid:0)x. In co(cid:1)nsequence, the inhomogeneous variation of the ac- ǫ1...ǫ8η1...η4
tion (16) vanishes andS is invariantunder local SO(4, ) the first eight indices ǫ1...ǫ8, implying J = ǫ(8)L˜. Since
transformations. In contrast to d4xD(x) the action SCis thetotallyantisymmetrictensorwitheightindicesǫ1...ǫ8
not a total derivative. R is a singlet with respect to SO(4, ), the remaining piece
The derivative-invariantD can be written in the form L˜η1...η4 must be a SO(4,C)-singleCt which is totally sym-
metric in the four indices η ...η . Besides L in eq. (13)
1 4
D =ǫµ1µ2µ3µ4D+ D− , (17) we can also construct a tensor L where S− is replaced
µ1µ2 µ3µ4 + η1η2
byS+ ,andsimilarlyforatensorL . The threepossible
where η1η2 −
terms in L˜ =αL+βL +γL alllead to local SU(2, )
+ − F
C
D± =∂ ϕ S± ∂ ϕ , (18) gauge symmetry of the action, cf. sect. IV. The action (2)
µν µ η1 η1η2 ν η2 becomes unique, however, if we require an equal number
involves two Weyl spinors ϕ or two Weyl spinors ϕ , of six Weyl spinors ϕ and six Weyl spinors ϕ . One can
+ − + −
respectively. This shows that D is invariant under an also find contributions to an action with local SO(4, )
C
exchange ϕ ϕ of the Weyl spinors. The trans- symmetry that involve only eight or ten spinors. They are
+ −
formation ϕ ↔γ0ϕ maps S+ S− and therefore discussed in a separate publication.
→ η1η2 ↔ η1η2
6
4. Minkowski action appearastheexpectationvalueofasuitablefermionbilin-
Defining the Minkowski action by ear. Bilinears of the type
S = iS , e−S =eiSM, (25) (E˜m )ab =ϕa(C γm) ∂ ϕb = ∂ ϕb(C γm) ϕa
− M 1,µ α 1 M αβ µ β − µ α 1 M αβ β
(29)
one finds the usual “phase factor” for the functional inte- transform as a vector under general coordinate transfor-
gral written in terms of SM. We emphasize that one and mations and as a vector under global SO(4, ) variations
thesamefunctionalintegral(3)describestheeuclideanand (5) of ϕ, i.e. for ǫ independent of x, C
mn
the Minkowski setting. The use of the euclidean action S
or the Minkowski action SM is purely a matter of conve- δ(E˜m )ab = (E˜n )abǫ(M)ηpm. (30)
nience. There is no “analytic continuation” between the 1,µ − 1,µ np
euclidean and the Minkowski setting. They are the same, The same holds for E˜m for which we replace C C
and different signatures arise only from different expecta- 2,µ 1 → 2
in eq. (29). In this case the sign of the second term in
tionvaluesforcollectivebosonicfieldsdescribingthemetric
eq. (29) is positive. The antisymmetric invariant 4 4-
or the vierbein. Our setting realizes a version of analytic ×
matricesC andC aredisplayedexplicitly inapp. A.Un-
1 2
continuation in terms of the continuation of the possible
der local generalized Lorentz transformation the vierbein
values of the vierbein [17].
bilinear acquires an inhomogeneous piece that we discuss
We candefine the operationofatranspositionasa total
in appendix C.
reorderingofallGrassmannvariables. The resultoftrans-
Acandidatefora vierbeincanbe obtainedby asuitable
positionforaproductofGrassmannvariablesdependsonly
contraction with a 2 2 matrix Vab,
on the number of factors N . For N = 2,3 mod 4 the ×
ϕ ϕ
transposition results in a minus sign, while for Nϕ = 4,5 E˜m =Vab(E˜m )ab. (31)
µ 1,µ
mod 4 the product is invariant. In consequence, one finds
for the action (2) with 12 spinors We observe that for antisymmetric Vab = Vba the vier-
−
bein is the derivative of a vector
ST =S. (26)
1
The hermiteanconjugationhc is the combinationoftrans- E˜µm = 2∂µ(ϕaC1γMmϕbVab), (32)
position with the complex conjugation c.c, such that
while for symmetric Vab =Vba one has
hc(S)=S. (27)
E˜m = ϕaC γm∂ ϕbVab
With respect to the complex conjugation c.c. used in eq. µ 1 M µ
(2) the Minkowski action is therefore antihermitean = −∂µϕaC1γMmϕbVab. (33)
hc(S )= S . (28) FurtherobjectsVab(E˜m )ab withthetransformationprop-
M − M 2,µ
erty of a vierbein can be found by replacing C C =
1 2
There exists a different complex structure for which SM C1γ¯. In this case a symmetric V leads to the deri→vative of
is hermitean. This is discussed in appendix B. One can a vector.
use the complex structure (1) in order to show that we We conclude that for real ǫ(M) the vierbein bilinear E˜m
µ
deal with a real Grassmann algebra, while the different has almost the transformation properties of the vierbein
complexstructurewithhermiteanSM canbeemployedfor in general relativity with a Minkowski signature. The
establishing a unitary time evolution. only difference concerns the inhomogeneous piece for lo-
cal Lorentz transformations. It is also not clear at this
stage which one of the possible candidates for E˜m should
µ
III. GEOMETRY be selected.
2. Absence of cosmological constant invariant
The action (2), (16) or (24) has all required symmetries
We next investigate if the action (16) can be written in
for a quantum field theory of gravity. The geometric con-
tent is not very apparent, however,in this formulation. In terms of a suitable vierbein bilinear E˜µm and its deriva-
this section we will discuss collective bosonic fields whose tives, plus suitable collective bosonic fields that transform
expectation values correspond to usual geometric objects asscalarsunderdiffeomorphismsandglobalLorentztrans-
as the vierbein or metric. We will express the action in formations. We will see that this is indeed the case.
terms of such fields. One may first ask if an invariant action of the type (2)
can be written in the intuitive form
1. Vierbein bilinear
Theactionofspinorgravityhasthesymmetriesofathe- S =α d4xW det(E˜m)+c.c., (34)
ory for gravity, namely invariance under diffeomorphisms Z µ
and local Lorentz transformations. One may therefore ex-
pect that a geometrical formulation in terms of a vierbein withE˜m givenbyeqs. (29),(31)forsomeparticularchoice
µ
and metric should be possible. Indeed, the vierbein may of V , or by similar expressions with C instead of C .
ab 2 1
7
In eq. (34) E˜m is interpreted as a matrix with first in- metric, cf. app. A.
µ
dex µ and second index m. The invariant W must in- The vector with respect to SU(2, ) which is not a
F
C
volve two Weyl spinors ϕ and two Weyl spinors ϕ . The total derivative involves the matrix C ,
+ − 1
invariance of the action (34) under diffeomorphisms and
SO(4 ) transformations would be particularly transpar- E¯1m(k)µ =ϕa(τ2τk)abC1γMm∂µϕb. (36)
C
ent in this language. The transformation (30) implies the
invariance of S under global Lorentz transformations in a Now the matrices C1γMm are symmetric, cf. app. A, and
simple way. With respect to diffeomorphisms the determi- also (τ2τk) are symmetric 2 2 matrices. The three “com-
×
nantE˜ =det(E˜m)hasthesametransformationproperties ponents”ofE1 labeledby(k)transformindeedasavector
µ
asthedeterminantofthevierbeiningeneralrelativity. The with respect to the gauge symmetry SU(2, )F.
C
latter equals the usual volume factor √g = det(gµν)1/2, Objects with the transformation properties of the vier-
andwerecoverthe generalcoordinateinvaria|nceofthe|ac- beinunderdiffeomorphismsandLorentzrotations,butalso
tion(34). OnemaycallW detE˜ma“cosmologicalconstant transforming non-trivially with respect to gauge symme-
µ
invariant” due to its resemblance to a cosmological con- tries acting on flavor, may be called “flavored vierbeins”.
stant in standard general relativity for W =const. How- Suchobjectsarenotcommoninusualformulationsofgen-
ever, we show in appendix D that suchh ai “cosmological eral relativity. They give a first glance on a more intrinsic
constant invariant” is not possible in the present formula- unification of gravity and gauge symmetries that may be
tion of spinor gravity. realized in our scenario. Any nonzero expectation value
This observation may have interesting consequences for hE¯1m(k)µi would lead to spontaneous breaking of the gauge
the issue of a cosmological constant. For ∆ E˜m = em a symmetry.
h µ i µ
term in the effective action ∆−4 W detem could be as- 4. Dimension of vierbein
h i µ
sociated with a cosmological constant W /∆4. Earlier A third difference between E˜m and the usual vierbein
h i µ
proposalsforspinor gravity[6, 7] orsimilar theories[9–11] does not concern the transformation property, but rather
have based the action on a “cosmological constant invari- the dimension. In fact, the spinors ϕ are dimensionless,
ant” (with W = 1). The absence of such an invariant in such that the presence of a derivative in eq. (29) implies
thepresentformulationisadistinctivefeature. Anexpres- that E˜m has dimension of mass or inverse length. The
µ
sion of the action (16) in terms of the vierbein bilinears discrete formulation in sect. V will introduce the lattice
E˜m must involve derivatives of those bilinears. distance ∆ with dimension of length. One may therefore
µ
consider the dimensionless vierbein bilinears
3. Flavored vierbein
Besidesthe inhomogeneoustransformationpropertyun- e˜m =∆E¯m , e˜m =∆E¯m . (37)
der localgeneralizedLorentztransformationsa secondim- 1(k)µ 1(k)µ 2,µ 2,µ
portant difference between the vierbein bilinears and the
Other dimensionless fields transforming as scalars with
usual vierbein concerns the nontrivial transformation of
respect to general coordinate transformations and vectors
E˜m withrespecttogaugetransformations. Wewilldiscuss
µ with respectto globalgeneralizedLorentztransformations
in the next section that the action (16) is invariant under
are
chiralgaugetransformationsSU(2,C) SU(2, ) . Here
L R
× C
the first factor SU(2,C)L acts on the indices of the Weyl A¯m =ϕa(τ2)abC1γMmϕb (38)
spinor ϕ , while the second factor SU(2, ) acts on ϕ .
+ R −
C
SincethevierbeinbilinearinvolvesoneWeylspinorϕ+ and and
one Weyl spinor ϕ it transforms in the (2,2) representa-
−
tionofthisgaugegroup. Thenon-trivialtransformationof S¯m =ϕa(τ τ )abC γmϕb. (39)
(k) 2 k 2 M
geometricalobjectsundergaugetransformationsisanovel
feature of our approach. Possible interesting observational These objects transform as scalars or vectors with respect
consequences of this new type of “gauge-gravity unifica- tothevectorlikegaugetransformationofSU(2, )F. With
C
tion” will be postponed to future investigations. We only respecttoSU(2, )L SU(2, )R theybelongagaintothe
describeheresomefeaturesthatwillbeneededlaterinthis representation (2C,2),×similar tCo e˜m and e˜m .
2,µ 1(k)µ
work. We also discuss in appendix E some other collective The inhomogeneous part of the Lorentz transformation
fields which show this entanglement between geometrical (9) mixes the bilinears e˜m ,S¯m and e˜m ,A¯m, cf. app.
1(k)µ (k) 2,µ
and gauge aspects. C,
With respect to the vectorlike gauge transformations
SU(2, )F which consist of the diagonal subgroup of δ e˜m = 1∆∂ ˜ǫ(M)mA¯n,
SU(2,C) SU(2, ) the vierbein bilinears transform inh 2,µ 2 µ n
L R
as singClets a×nd threeCcomponent vectors. The singlet that δ e˜m = 1∆∂ ǫ˜(M)mS¯n . (40)
is not a pure derivative is given by inh 1(k)µ 2 µ n (k)
E¯m = ϕa(τ )abC γm∂ ϕb Oneobservesthatthisinhomogeneouspartvanishesinthe
2,µ 2 2 M µ
limit ∆ 0. We will discuss this important property
= −∂µϕa(τ2)abC2γMmϕb, (35) in more →detail later. If the inhomogeneous part can be
whereweobservethatthematrices(C γm) areantisym- neglected the expectation values e˜m and e˜m have
2 M αβ h 2,µi h 1(k)µi
8
preciselythelocalLorentz-transformationpropertiesofthe Other forms of the action can be obtained by further
vierbein in Cartan’s formulation [18] of general relativity. reordering of the Grassmann variables. For example, we
may use
5. Connection bilinear
We will now proceed to an expression of the action (16) 1
A(8) = H+H−H+H−, (46)
in terms of the vierbein bilinears e˜m2,µ and e˜m1(k)µ. For this 192 k k l l
purpose we will introduce a collective field related to the
which follows from squaring the relation
spin connection. More generally, the expectation values
of suitable bosonic collective fields can be used to define
H+H− = 8 ϕ1 ϕ1 ϕ2 ϕ2 +ϕ2 ϕ2 ϕ1 ϕ1
geometrical objects transforming as vierbein, metric, spin k k { +1 +2 −1 −2 +1 +2 −1 −2}
connection, curvature, tensor etc.. If we can find geomet- − 4{ϕ1+1ϕ2+2ϕ1−1ϕ2−2+ϕ1+1ϕ2+2ϕ2−1ϕ1−2
ricfieldswiththestandardtransformationproperties,they +ϕ2 ϕ1 ϕ1 ϕ2 +ϕ2 ϕ1 ϕ2 ϕ1 .(47)
can be used to construct diffeomorphism and Lorentz in- +1 +2 −1 −2 +1 +2 −1 −2}
variantobjectsinthestandardway. Oneonlyhastoverify A reordering can now be performed in the factor
thatsuchobjectsdonotvanishidenticallyduetothePauli H+H−D+ D− .
k k µν ρσ
principle for spinors. We will see, however, that not all
6. Inverse vierbein
standard geometrical objects can be implemented in this
Whatisnotavailableonthe levelofmulti-fermionfields
way. Inparticular,theinversevierbeincannotbeobtained
isthe inversevierbein. Anygivenchoiceofthe vierbeinbi-
as a polynomial of spinors.
linear E˜m (givenchoice ofV ) is anelement of the Grass-
We define “spin connection bilinears” by µ ab
mann algebra. Inverse elements are not defined, however,
1 for a Grassmann algebra. Nevertheless, with
Ω˜m = (∂ e˜m ∂ e˜m ),
2,µν −2 µ 2,ν − ν 2,µ E˜ = det(E˜m) (48)
1 µ
Ω˜m1(j)µν = −2(∂µe˜m1(j)ν −∂νe˜m1(j)µ). (41) = 1 ǫµ1µ2µ3µ4ǫ E˜m1E˜m2E˜m3E˜m4,
24 m1m2m3m4 µ1 µ2 µ3 µ4
The transformations properties of these objects and their
we can define an object that transforms as the product of
relation to the usual spin connection are discussed in ap-
the inverse vierbein with the determinant of the vierbein
pendix F. There we also show that in terms of those bilin-
ears the action (16) can be written as 1
I˜µ = ǫµ1µ2µ3µ4ǫ E˜m2E˜m3E˜m4, (49)
m 6 m1m2m3m4 µ2 µ3 µ4
α
S = 16 242∆2 Z d4xǫµ1µ2µ3µ4ηmnHk+Hk+Hl−Hl− = E˜E˜mµ.
·
Ω˜m Ω˜n Ω˜m Ω˜n +c.c.. (42) It obeys b
{ 2,µ1µ2 2,µ3µ4 − 1(j)µ1µ2 1(j)µ3µ4}
ThisisobtainedbyasuitablereorderingoftheGrassmann I˜µE˜n =E˜δn , I˜µE˜m =E˜δµ, (50)
m µ m m ν ν
variables.
The expression(42) involvesfirst derivativesofthe vier- where we recall that E˜−1 is not defined.
bein, with the structure Similarly, the antisymmetrized product of two inverse
vierbeins, multiplied by E˜, can be defined as
1
D¯ = ǫµνρσΩ pΩ . (43)
16 µν ρσp I˜µν = 1ǫµνµ3µ4ǫ E˜m3E˜m4
mn 2 mnm3m4 µ3 µ4
From the transformation properties in app. F, eq. (F.8), = E˜(E˜ µE˜ ν E˜ νE˜ µ). (51)
onefindsthatD¯ isnotinvariantunderlocalLorentztrans- m n − m n
formations
It obeys b
1
δD¯ = ǫµνρσ∂ (e me n)∂ ǫ(M). (44) I˜µνE˜p = I˜µδp I˜µδp ,
16 µ ν ρ σ mn mn ν m n− n m
I˜µνE˜n = I˜µδν I˜νδµ. (52)
mn ρ m ρ − m ρ
Nevertheless, the action is invariant under local SO(4, )
C
transformations due to the Pauli principle. (Recall that One also has
also D in eq. (14) transforms inhomogeneously under lo-
1
cal SO(4, ).) The structure D¯ can be written as a total ǫµνρσǫ E˜q = E˜Aˆ E˜ µE˜ νE˜ ρ , (53)
C 6 mnpq σ { m n p }
derivative
b
1 where Aˆstands for totalantisymmetrizationinthe indices
D¯ = ∂µ(eνp∂ρeσq)ηpqǫµνρσ, (45) (mnp), or equivalently, in (µνρ).
16
andthe action cantherefore alsobe written with a deriva-
tive ∂ acting on H.
µ
9
7. Metric collective field The metric obtains then by a functional derivative with
On the level of the metric we can define an invariant respect to the sources
under local Lorentz transformations by use of the scalars
H± in eq. (21), δW[T˜]
k g (x)= , W(T˜)=lnZ[T˜]. (59)
µν δT˜µν(x)
2
g˜ = ∆2(∂ H+∂ H−+∂ H−∂ H+). (54)
µν 3 µ k ν k µ k ν k The quantum effective action is defined by a Legendre
Thisobjectinvolvesfourspinorsandtransformasasecond transform
rank symmetric tensor under general coordinate transfor-
mations. We may identify its expectation value with the Γ[g ]= W + g (x)T˜µν(x). (60)
µν µν
− Z
metric x
1 The metric obeys the exact field equation
g = ( g˜ + g˜ ∗). (55)
µν µν µν
2 h i h i
δΓ
We note, however, the particularity that g˜ is a singlet =T˜µν(x). (61)
µν δg (x)
µν
with respect to global vectorlike gauge transformations
SU(2, )F,whileδg˜µν acquiresaninhomogeneoustermfor We recognize the relation between the source T˜µν and the
C
local gauge transformations. Furthermore, for the chiral energy momentum tensor Tµν
gauge group SU(2, ) SU(2, ) the metric collective
L R
C × C
field g˜ is not a representation. (It is an element of the 1
µν T˜µν = √gTµν , g = detg . (62)
(3,3) representation.) Any nonvanishing metric therefore 2 | µν|
breaks this gauge symmetry.
One can express the collective metric field (55) in terms The effective action is diffeomorphism symmetric. This
of vierbein bilinears as isaconsequenceoflatticediffeomorphisminvarianceofthe
lattice action, as discussed in ref. [13] and briefly in sect.
1
g˜ = η e˜m e˜n + e˜m e˜n VIII. Diffeomorphism symmetry constitutes a strong re-
µν mn{ 2,µ 2,ν 3 1(k)µ 1(k)ν
striction for the possible form of the effective action. Re-
∆2∂ A¯m∂ A¯n ∆2∂ S¯m ∂ S¯n . (56) alistic gravity can be obtained if Γ admits a derivative ex-
− 4 µ ν − 12 µ (k) ν (k)} pansion for metrics with a long wavelength, for example
compared to ∆. In this case the leading terms are a cos-
ThiscanbeverifiedbyareorderingofGrassmannvariables
mologicalconstant(noderivatives)andanEinstein-Hilbert
and recombination to bilinears in the expression
terminvolvingthecurvaturescalar(twoderivatives). (The
g˜ = 8∆2 ∂ ϕa ϕb ∂ ϕc ϕd coefficientsofbothtermsmaydependonotherfieldsas,for
µν −3 { µ +α +β ν −γ −δ example, scalar fields.) If the cosmological constant term
(τ ) (τ ) (τ τ )ab(τ τ )cd+µ ν . (57) is smallenoughone wouldfindthe usualgeometricsetting
2 αβ 2 γδ 2 k 2 k
× ↔ } for a massless graviton. It remains to be seen if the effec-
For∆ 0the lasttwotermsineq. (56)canbe neglected. tiveactionforthemetriccanbecomputedinasatisfactory
If we a→ssociate, for example, the vierbein with e˜m2,µ we ob- approximationby using suitable methods, for example the
servea relationbetweenthe collective metric andthe vier- Schwinger-Dysonequation employed in ref. [8].
bein bilinear similar to the usual one between metric and
vierbein.
IV. SYMMETRIES
8. Emergent geometry
The task of determining the geometry for our model of
spinor gravity consists in evaluating the metric as the ex- SymmetriesconsistintransformationsoftheGrassmann
pectation value (55). We should do so in the presence of variables ψa(x) ψ′a(x) that leave the action and the
γ → γ
appropriate sources for the collective field, in order to ac- functional measure invariant. We note that symmetry
count for the response of the metric to an energy momen- transformations do not involve a complex conjugation of
tum tensor. The formalism of this program involves the parameters or other coefficients in the action, in contrast
quantum effective action Γ[g ] for the metric. In the reg- to hermitean conjugation or complex conjugation. Not
µν
ularized proposal for quantum gravity that we present in all symmetries must be compatible with a given complex
sect. Vallstepsforthedefinitionoftheeffectiveactionare structure.
mathematically well defined. We do not aim in this paper Besides the generalized Lorentz transformations
for a computation of Γ, but rather present here shortly its SO(4, ) the action (16) is also invariant under continu-
C
definition in a continuum language. ous gauge transformations. By the same argument as for
We first introduce sources T˜µν(x) for the collective field localSO(4, )symmetry,anyglobalcontinuoussymmetry
C
g˜ (x). The partition function (3) becomes then a func- of the action is also a local symmetry due to the Pauli
µν
tional of the sources principle.
1
Z[T˜]= exp S+ g˜ (x)+g˜∗ (x) T˜µν . (58)
Z D {− 2Z µν µν }
x(cid:0) (cid:1)
10
1. Vectorlike gauge symmetry and electric charge [17].) The global chiral U(1) symmetry (65) leaves E˜m
µ
The vectorlike gauge symmetry SU(2, ) transforms invariant.
F
C
A transformation which is compatible with the complex
i
δϕa(x)= α˜ (x)(τ )abϕb(x), (63) structure has to obey (for real α)
α 2 k k α
ϕ exp(iαγ¯)ϕ , ϕ∗ exp( iαγ¯∗)ϕ∗. (69)
with three complex parameters α˜k. For real α˜k these are → → −
standardgaugetransformationswithcompactgaugegroup
Defining
SU(2). Thebasicspinorsϕtransformasadoublet,andthe
left- and right-handed Weyl spinors ϕ+ and ϕ− have the 1
same transformation property with respect to this gauge ϕ∗± = 2(1±γ¯∗)ϕ∗ (70)
group. The spinors are therefore in a vectorlike represen-
tation, similar to quarks with respect to the color group yields
SU(3) , but different from the chiral representation of
C
quarks and leptons in the standard model of electroweak ϕ∗ e−iαϕ∗ , ϕ∗ =eiαϕ∗, (71)
+ → + − −
interaction. Wedonotaiminthispaperforrealisticgauge
symmetries of the standard model and are rather inter- and
ested in a consistent theory of gravity which is as simple
as possible. (E˜m)∗ =ϕ∗V∗C∗γm∗∂ ϕ∗ ϕ∗V∗C∗γm∗∂ ϕ∗ (72)
µ + + M µ −± − − M µ +
As onepossibilityonemayidentify the thirdcomponent
of the isospin with electric charge is invariant. We can extend this axial symmetry to
U(1, ) by using complex parameters α. Invariance un-
A
C
Q=2I =τ , (64) der this symmetry holds for all expressions containing an
3 3
equal number of factors ϕ and ϕ .
+ −
where τ acts in flavor space. Then our model describes
3
oneDiracspinorϕ1α withchargeQ=1,andanotherDirac 3. Chiral SU(2)L×SU(2)R gauge symmetry
spinor ϕ2α with charge Q=−1. At the present stage these The gauge symmetries SU(2)F ×U(1)A leave the vier-
are distinguished fermions. We will discuss elsewhere the bein bilinear (35) invariant. They are, however, not the
possibility to associate ϕ2 with the antiparticle of ϕ1. In only gauge symmetries of the action (16). We rather can
this case the two Dirac spinors are no longer independent extend the symmetry SU(2, )F to a chiral gauge sym-
C
and our model describes an electron, and its antiparticle, metry SU(2, )L SU(2, )R, where the first factor acts
C × C
the positron. Hereweconcentrate,however,onthesetting onlyontheWeylspinorsϕ+,whilethesecondactsonlyon
where the antiparticle of ϕ1 differs from ϕ2. ϕ−. Altogether, we have four SU(2, ) factors, and with
C
respect to G = SU(2, ) SU(2, ) SU(2, )
+ − L
2. Axial U(1)-symmetry C × C × C ×
SU(2, ) the Weyl spinors ϕ and ϕ transform as
R + −
Letusnextturntofurtherglobalcontinuoussymmetries C
(2,1,2,1) and (1,2,1,2), respectively.
that leave the action invariant. A global phase rotation of
In order to establish the extended chiral SU(2, )
ϕ is not a symmetry. We may, however,decompose ϕ into C L ×
SU(2, ) symmetry we write the invariant D in eq. (14)
R
irreducible representations of SO(4, ) and use different C
C in the form (18). Here
phase rotationsfor the different representations. Since the
action contains an equal number of Weyl spinors ϕ and
+ D+ = ∂ ϕb1(C ) (τ )b1b2∂ ϕb2 (73)
ϕ− it is invariant under global chiral U(1)A transforma- µ1µ2 − µ1 β1 + β1β2 2 µ2 β2
tions
involves only the Weyl spinor ϕ and is invariant under
+
ϕ+ eiαϕ+ , ϕ− e−iαϕ−. (65) the transformation SU(2,C)L,
→ →
i
We can express E˜m in terms of the Weyl spinors ϕ as δ ϕa = α˜ (τ )ab(1+γ¯) ϕb. (74)
µ ± L α 4 L,k k αβ β
E˜µm =ϕ+VC+γMm∂µϕ−±ϕ−VC−γMm∂µϕ+, (66) Similarly
where the + sign applies for C = C1, and the sign for D− =∂ ϕb1(C ) (τ )b1b2∂ ϕb2 (75)
C =C ,C =C γ¯. We employ here the matrice−s µ1µ2 µ1 β1 − β1β2 2 µ2 β2
2 2 1
involves only ϕ and is invariant under SU(2, ) , with
1 − R
C = (1 γ¯)C (67) C
± 1
2 ±
i
δ ϕa = α˜ (τ )ab(1 γ¯) ϕb. (76)
defined in app. A. We observe that the relations R α 4 R,k k − αβ β
{γ¯,γMm}=0 , Cγ¯ =γ¯TC (68) TvehcutosrDlikeissiunbvgarroiaunptSuUnd(e2r, S)U(2o,bCta)iLns×foSrUα˜(2,C=)Rα˜. (Th=e
F L,k R,k
C
hold both for C and C independently of the particular α˜ . We note that the total symmetrization in the indices
1 2 k
representation of the Dirac matrices. (For details cf. ref. (η ...η ) in eqs. (14), (13) results automatically from the
1 4