A first-order approach to conformal gravity

0.4 MB·

by  T. G. Zlosnik

A first-order approach to conformal gravity
T.G. Zl o´snik1∗and H.F. Westman2†
(1) Imperial College Theoretical Physics, Huxley Building, London, SW7 2AZ
(2) Instituto de F´ısica Fundamental, CSIC, Serrano 113-B, 28006 Madrid, Spain
January 5, 2016
6
1
0 Abstract
2
We investigate whether a spontaneously-brokengauge theory of the group SU(2,2) may be
n
a a genuine competitorto GeneralRelativity. The basicingredients ofthe theoryare anSU(2,2)
J gaugefieldA andaHiggsfieldW intheadjointrepresentationofthegroupwiththeHiggsfield
µ
4 producing the symmetry breaking SU(2,2) SO(1,3) SO(1,1). The action for gravity is
polynomial in Aµ,W and the field equation→s are first-o×rderin derivatives of these fields. The
] { }
c new SO(1,1) symmetry in the gravitationalsector is interpreted in terms of an emergent scale
q symmetryandtherecoveryofconformalizedGeneralRelativityandfourth-orderWeylconformal
- gravityaslimits ofthe theory-followingimpositionofLagrangianconstraints-isdemonstrated.
r
g Maximallysymmetricspacetimesolutionstothefulltheoryarefoundandstabilityofthetheory
[
around these solutions is investigated; it is shown that regions of the theory’s parameter space

1 describe perturbations identical to that of General Relativity coupled to a massive scalar field
v and a massless one-form field. The coupling of gravity to matter is considered and it is shown
7 that actions for all fields are naturally gauge-invariant, polynomial in fields and yield first-
6
order field equations; no auxiliary fields are introduced. Familiar Yang-Mills and Klein-Gordon
5
type Lagrangians are recovered on-shell in the General-Relativistic limit of the theory. In this
0
0 formalism, the General-Relativistic limit and the breaking of scale invariance appear as two
. sides ofthe same coinanditis shownthat the latter generatesmass terms forHiggsandspinor
1
0 fields.
6
1
1 Introduction
:
v
i
X The prevailing classical theory of gravity remains Einstein’s General Relativity. In General
r Relativity, the gravitational field is described solely by a field eI eIdxµ, the co-tetrad. The
a ≡ µ
index I denotes that the one-formeI is in the fundamental representationof the Lorentz group
SO(1,3); the actionfor GeneralRelativity- the Einstein-Hilbert action-possessesan invariance
under local Lorentz transformationsrepresentedby matrices ΛI (x) with eI ΛI eJ [1]. This
J → J
field is sufficient to describe the inherent dynamics of gravity and the coupling of gravity to
known matter fields: scalar fields, gauge fields, and fermionic fields; the latter are spinorial
representationsofthegroupSL(2,C)whichisthedouble-coverofSO(1,3)andsothe inclusion
of fermions into gravitationaltheory implies that most correctly the localsymmetry of General
Relativity is that of SL(2,C). In General Relativity the coupling of gravity to each of these
fields requires use of the tetrad (e−1) eµ∂ , where eµ is the matrix inverse of eI. Therefore
I ≡ I µ I µ
the coupling of gravity to all matter in General Relativity is non-polynomial. In the absence
of fermionic fields, it suffices to use a composite object called the metric tensor g , which is
µν
defined as follows:
∗[email protected]
†[email protected]
1
2
g = η eIeJ (1)
µν IJ µ ν
where η = diag( 1,1,1,1) is the invariant matrix of SO(1,3). Coupling to scalar fields
IJ
−
and gauge fields then requires use of the inverse metric gµν. As was discovered by Cartan,
an elegant reformulation of General Relativity is provided by introducing a new, independent
field ωI ω I dxµ- the spin connection- alongside eI in the description of gravity. In these
J ≡ µ J
variables, the action for gravity is given by the Palatini action:
S [ω,e] = ǫ eIeJ dωKL+ωK ωML (2)
P IJKL M
Z
(cid:0) (cid:1)
where multiplication of differential forms with one another is via the wedge product. Gravity
fromthis perspective is knownas Einstein-Cartangravity1. This actionis invariantunder local
SO(1,3) transformations if ωI transforms as a gauge field for those transformations. Indeed
J
we can see that the terms containing ωI in (2) combine to form the curvature two-form RI
J J
of a non-Abelian gauge field. The equation of motion for ωI is as follows:
J
deI +ωI eJ = 0 (3)
J
Unlike the equations of motion of gauge fields in particle physics, this equation is algebraic
in ωI . Using the solution for ωI (e) in the eI equation of motion yields the Einstein field
J J
equations. Alternatively,the solutionfor ωI (e)may be insertedback into the action,resulting
J
in the Einstein-Hilbert action of General Relativity, which- upon the addition of a boundary
term- yields the Einstein field equations upon variation. Similarly, ωI as an independent field
J
may be used to write the action of a fermionic field coupledto gravityin a polynomialmanner.
Due to the coupling to ωI , fermionic currents can act as a source in (3); one may still solve
J
for ωI algebraically but it now depends on eI and the fermionic fields. Inserting the solution
J
back into the action yields terms quartic in the fermionic fields, over and above terms usually
present when describing gravity coupled to fermions in General Relativity [2, 3, 4, 5].
Thus, the Einstein-Cartanapproachresults in a simplification of some actions involvingthe
gravitational field and introduces structure (the SO(1,3) gauge field ωI ) reminiscent of the
J
Yang-Millsfieldsofparticlephysics. However,eI hasnocounterpartamongstnon-gravitational
fields. It possesses a spacetime index but does not transform as a gauge field. Intriguingly
though,thereexistsare-writingofthePalatiniactionwithappearsastepclosertocommonality
withtheingredientsofparticlephysics. Theidea,originallyduetoMacDowellandMansouri[6]
(closelyresemblingearlierworkduetoCartan[7]),istoenlargethegaugegroupofgravityfrom
SO(1,3) to SO(2,3) or SO(1,4) (henceforth collectively referred to as SO(1,4)SO(2,3)). The
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trick is to imagine there there exists structurein a hypothetical SO(1,4)SO(2,3)gaugetheory
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to break the symmetry down to the SO(1,3) of the Einstein-Cartan theory. For example, this
could be accomplished via a gravitational Higgs field VA in the fundamental representation of
SO(1,4)SO(2,3) achieving a non-vanishing expectation value for its norm V2 η VAVB.
AB
| ≡
Thisnormshouldbepositive/spacelikeforSO(1,4)andnegative/timelikeforSO(2,3). Thenwe
maychooseagaugewhereVA =ℓδA (whereℓisaconstant);theresidualgaugetransformations
4
that leave this explicit form of VA invariant are those of SO(1,3), and we may decompose the
SO(1,4)SO(2,3) gauge field AA A A dxµ as follows:
| B ≡ µ B
AI AI
AA = J 4 (4)
B A4 0
(cid:18) I (cid:19)
We see then that there is a new field in the formalism: AI . This is a one-form field that
4
will transform homogeneously under the residual SO(1,3) transformations, precisely like eI!
1Within Einstein-Cartan gravity one may additionally consider polynomial Lagrangians ǫIJKLeIeJeKeL and
eIeJ(dωIJ +ωIKωKJ) which correspond to the cosmological constant term and the Holst term respectively. The
Holsttermonlyproducesanon-zerocontributiontofieldequationswhenthespinconnectioncouplestootherfields.
3
SO(1,4)SO(2,3) Scalar tensor theory
| V unconstrained
gauge theory with Euclidean
S[A,V] and Lorentzian phases
Constrain V
Einstein-Cartan
General Relativity
theory
S[e]
Solve for ω(e)
S[ω,e]
Figure 1: Diagramdepicting known results of SO(1,4)SO(2,3) gravity and relationto General
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Relativity.
There exists a simple actionprinciple due to Stelle andWest [8] that- postsymmetry breaking-
containsthePalatiniaction. Forconcretenesswelookatthe caseofthegroupSO(1,4)andthe
action is given by:
S [A,V,λ] = αǫ VEFABFCD+λ(V VA ℓ2) (5)
SW ABCDE A
−
Z
Where α is a constant and the four-form field λ is introduced entirely to enforce the constraint
that V VA is constant. To illustrate the relation of this action to Einstein-Cartan theory,
A
we may enforce the fixed-norm constraint at the level of the action, choosing a gauge where
VA =ℓδA, and identifying AI =ωI , AI =eI/ℓ we have:
4 J J 4
AAB =∗ ωIJ eℓI (6)
eI 0 !
− ℓ
2α 1 ℓ2
S [ω,e]=∗ ǫ eIeJRKL eIeJeKeL RIJRKL (7)
SW − ℓ IJKL − 2ℓ2 − 2
Z (cid:18) (cid:19)
∗
where the notation = means that something holds in a specified gauge (here the gauge where
VA = ℓδA. We see then that the Palatini action (plus a cosmological constant term and a
4
topologicaltermquadraticinRIJ)canberecoveredfromaspontaneously-brokengaugetheory2.
Thefour-formλisasimplewaytoachieveanon-vanishingnormforVA butisnotnecessary
as an ingredient of the theory. Indeed, it has been found that there exist polynomial actions
solely in terms of the set AA ,VA that dynamically yield a non-vanishing expectation value
{ B }
of V2 [10, 11]. These theories possess a rich phenomenology with the dynamics of the scalar
V2 actingas a potentialsourceofcosmicinflationor quintessence,evenfacilitating moreexotic
behaviour such as cosmologicalchanges of the signature of the four-dimensional metric [12].
Therefore General Relativity (and scalar-tensor extensions thereof) can arise as a limit of
a spontaneously-broken gauge theory based on SO(1,4)SO(2,3). However, the presence of a
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new degree of freedom V2 in gravitation means the theory is more general than a re-casting of
the Einstein-Cartantheory. This situation is summarizedbriefly in Figure 1. Researchinto the
linkbetweenSO(1,4)SO(2,3)groupsandgravityandmatterisongoing[13,14,15,16,17,18].
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Thereare,however,issueswithadescriptionofgravitybasedonSO(1,4)SO(2,3). Although
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it is possible to couple matter fields to gravity in a fashion consistent with the gauge principle
[19], we nevertheless encounter a somewhatunnaturalstructure when coupling gravityto other
Yang-Mills fields. In this paper we consider an alternative to the above SO(1,4)SO(2,3),
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approachwherein these issues are resolved. Specifically we will develop a description of gravity
2See [9]for a more detailed discussion of these steps towards regarding gravity as a gauge theory
4
as a spontaneously-broken gauge theory based on the group SU(2,2), a group which contains
both SO(1,4)andSO(2,3)as sub-groups. This grouphas a matrixrepresentationas the setof
all 4 4 complex matrices Uα of unit determinant that satisfy
× β
hαα′ =hββ′UβαU∗βα′′, hαα′ = 0I I0 (8)
(cid:18) (cid:19)
where I is the 2 2 identity matrix.
×
The group SU(2,2) is the double cover of the orthogonal group SO(2,4) which itself is
the double coverof the conformal group C(1,3) of coordinate transformationswhich preservea
metricofsignature( ,+,+,+)uptoanoverallmultiplicativefunction. Insteadofbreakingthe
−
symmetry down to the SO(1,3) symmetry of General Relativity, our Higgs sector will recover
a symmetry SO(1,3) SO(1,1). The additional SO(1,1) symmetry now presentingravitation
×
sector will be found to be related to Weyl/scale invariance.
Let us illustrate this more concretely. If there exists structure in the theory to break
SU(2,2) SO(1,3) SO(1,1)then we may decompose the SO(2,4) SU(2,2)connectionas
→ × ≃
follows:
0 1(eI +fI) c
AAB =∗ 1(eI +fI) 2 ωIJ 1(eI fI) (9)
 −2 −2 − 
c 1(eI fI) 0
− 2 −
 
where we now use indices A,B,C,... to denote indices in the fundamental representation of
SO(2,4). Here we identify ωIJ as the spin-connection, c as a gauge field for the groupSO(1,1)
andwhatappeartobetwo framefieldseI andfI. UnderanSO(1,1) SO(2,4)transformation
⊂
we have the following transformation properties:
eI eα(x)eI, fI e−α(x)fI, c c+dα (10)
→ → →
By analogy with (6) and (7) we may immediately speculate as to the kind of theory that may
result following symmetry breaking SU(2,2) SO(1,3) SO(1,1). We will construct the
→ ×
SU(2,2) gauge theory from an action first-orderand polynomial in fields. We may then expect
that in the limit where all dynamics due to the Higgs field is ‘frozen’ (much like V2 = ℓ2 in
the Stelle-Westcase),thatwewillrecoverthe mostgeneralSO(1,3) SO(1,1)invariantaction
×
that can be built from the ingredients (9). It is straightforward to write down such an action,
it takes the following form:
S[ω,c,e,f]= αǫ eIfJRKL+βe f RIJ +γǫ eIfJeKfL+µe fIe fJ +ξe fIdc(11)
IJKL I J IJKL I J I
Z
Where α,β,µ,ξ are constants and we have not included possible boundary terms. Due to
{ }
the symmetries (10), this action defines a scale-invariant theory of gravity that we may call
Conformal Einstein-Cartan theory. Remarkably- as we shall see in more detail later in the
paper- in the absence of matter its solution space contains that of General Relativity with
positive/negative cosmological constant, corresponding to cases where dynamically the frames
eI and fI become aligned or anti-aligned i.e. fI = Ω2(x)eI.
±
As a theory in and of itself, the Conformal Einstein-Cartan theory provides an elegant
and simple scale-invariant generalization of the Einstein-Cartan theory (and hence of General
Relativity). However, we note again that the fields eI and fI do not have analogues within
Yang-Mills theory: they are one-forms but do not transformas Yang-Mills gauge fields. In this
paper we shall focus on a different description in which gravity- at the fundamental level- is
describedinthe samelanguageasYang-Millstheory: aSO(2,4) SU(2,2)=Spin(2,4)gauge
≃
connection AA A A dxµ and a Higgs field WA in the adjoint representation of SO(2,4).
B ≡ µ B B
In contrast to prior attempts to construct theories of gravity based on the group SU(2,2), our
5
approach will be to place no constraints on the fields of the theory- no object is required to be
‘invertible’, constant, or non-zero. In addition, no metric tensor is needed to write down our
actions, nor do our actions make use of the Hodge dual operator. Instead, all actions will be
requiredtobegauge-invariantandpolynomialinthefieldsandtheirexteriorderivativesandall
(classical)behaviourofthefieldswillresultfromtheirfieldequations. Asasimpleconsequence,
allfieldequationsinthe Lagrangianformalismwillfundamentally first-orderpartialdifferential
equations3. ThismightseemalarmingbutwewillseethattheHodge-dualstructureofstandard
Yang-Mills fields and Higgs fields pops up naturally on-shell for the simple actions we study in
this paper. This marks a departure for conventional formulations in which both gravity and
bosonic fields are governed in the Lagrangianformalism by second-order differential equations.
Allphysicalfieldsaretakentobedifferentialformsonafour-dimensionalmanifold,thoughthey
may have internal indices in representations of non-gravitational groups. No non-dynamical
fields will allowed in the action: we only allow invariants of the groups such as the invariants
hα′α and ǫαβδγ of SU(2,2).
By comparison, the gravitational actions based on the Einstein-Cartan fields ωI ,eI or
{ J }
SO(1,4)SO(2,3)gaugegravityfields AA ,VA arepolynomialintheLagrangianformulation
| { B }
and produce first-order field equations; these properties persist when including coupling to
fermionic matter. This is rather different from the conventional Lagrangians describing other
Yang-Millsfields andHiggsfields whichproducesecond-orderequationsofmotione.g. consider
the action for a Yang-Mills field B = B dxν with curvature F = dB +BB = 1F dxµdxν
µ 2 µν
(internal indices suppressed):
1
S = gαγgβδTr(F F )√ gdx4 (12)
YM αβ γδ
−4 −
Z
where Tr denotes contractionof internaladjoint indices using the Killing metric. This action
G
is clearly polynomial in B but non-polynomial in gravitational fields because we must use the
inverse metric gµν
4ǫαβγµǫǫζηνg g g
gµν = αǫ βζ γη (13)
ǫ̺χκλǫξρστg g g g
̺ξ χρ κσ λτ
When one couples these matter fields to gravityinEinstein-Cartantheory,this non-polynomial
coupling appears in the stress-energy tensor of matter and so the gravitational field equations
become fundamentally non-polynomial in fields unless one introduces auxiliary non-dynamical
fields which have the sole purpose of re-writing (12) as a first-order theory. Indeed, a popular
modificationtotheEinstein-Cartangravitytheoryhasbeentoallowtermsnon-polynomialineI
into the action of pure-gravity, much as they are present in the Einstein-Cartan matter sector.
This allowsone to constructnon-topologicalterms quadraticinthe SO(1,3)curvatureRIJ(ω).
This approachtypically is referredto as Poincar´e gauge theory and it presents a wealth of new
phenomenology in the gravitational sector, notably the propagation of ωIJ itself via its own
field equations (as opposedto simply being solvable for ωIJ =ωIJ(e) as in the Einstein-Cartan
case) [20, 21, 22, 23, 24].
For gravity based on SO(1,4)SO(2,3), it becomes possible to naturally base the dynamics
|
of Higgs fields in a first order formalism. Essentially a Higgs field is taken to be a field in
the fundamental representation of the gravity group as well as whatever representation it may
belong to for the internal gauge group (e.g. SU(2) U(1) for the electroweak field). The
G ×
fundamental representationof SO(1,4)SO(2,3) is five-dimensionaland the Higgs field has five
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‘gravity’ components- four of these are found on-shell to be related to derivatives of the Higgs
fieldwhilstthefinalcomponentcontainsthefamiliarHiggsfield. However,asmentionedabove,
coupling to other Yang-Mills either still requires the introduction of auxiliary fields or of exotic
transformation properties of B that depend on the dynamics of the gravitational Higgs field
3The generation of higher-order partial derivatives is made impossible by a polynomial Lagrangian and Bianchi
identitiesDF =0andDDV =FV whereF isacurvaturetwo-formofsomegaugeconnectionA;Distheassociated
gauge covariant exterior derivative,and V is a field in some representation of thegauge group
6
[19]. We see it as a comparative advantage of the SU(2,2) approach that it- as we shall see
later- allows for a simple first order formalism for all matter fields.
Perturbed action
SU(2,2) SO(2,4) Maximally symmetric
≃ around maximally
gauge theory solutions
symmetric solutions
S[A,W] E[A¯,W¯ ]= 0
S[δA,δW]
Constrain W
Constrain
Conformal Einstein-Cartan
ω = ω(e,c)
theory S[e,f,c]
S[A = e,f,ω,c ]
{ }
Solve for f = f(e,c)
2
If e= Ω f
±
Fourth-order Weyl gravity
Conformalized General Relativity
S[g,c]
in first-order form
S[e,Ω,ω] Constrain dc = 0
Solve for ω = ω(e) Fourth-order conformal gravity
S[g]
Conformalized General Relativity
in second-order form
S[g,Ω]
Gauge fix Ω = 1
General Relativity
in second-order form
S[g]
Figure 2: Diagram depicting known results of SU(2,2) SO(2,4) gravity and relation to other
≃
gravitational models. Dotted paths denote those taken with WAB entirely unconstrained.
The outline of the paper is as follows: In Section 2 we present our proposed actions and
discuss the field content of the theory. In Section 3 we consider the analog of the Stelle-
West approach for these theories, that is: we see what theory emerges by constraining all the
degreesoffreedompresentinthe Higgsfields. We willshowthatthis constrainedtheoryindeed
corresponds to Conformal Einstein-Cartan theory and we shall show that this theory possesses
a limit where it corresponds to ‘conformalized General Relativity’ i.e. a scale-invariant theory
ofascalarfield(‘dilaton’)conformallycoupledtoametrictensorg -inthe‘scalegauge’where
µν
this field is a constant, the theory is identical to General Relativity. Furthermore we will show
how by further constraining the theory, we recover fourth-order conformal gravity i.e. gravity
with action
S [g]= α αβδγ√ gd4x (14)
C αβδγ
− C C −
Z
whereαisaconstantand istheWeyltensorbuiltwithmetricg andChristoffelsymbols
αβδγ µν
C
Γα .
βδ
In Section 4 we return to the full theory and show that there exist maximally symmetric
solutions to the theory i.e. solutions interpretable as spacetimes possessing ten Killing vectors
each. In Section 5 we examine the nature of small perturbations around these solutions, estab-
lishing conditions for linear stability. To aidthe reader,the flow chartFigure 2 summarizesthe
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structureofthese sectionsandsomeresultstherein. InSection6wediscusshowonecancouple
thegravitationalfieldstomatterfieldsinasimpleandelegantfashion. Thisconstructionmakes
essential use of the conformal group. All actions are polynomial and yield first order partial
differentialequationsbutyieldon-shellthe familiarsecondorderequationsforYang-Millsfields
and Higgs fields. It is suggested that Yang-Mills fields for a symmetry group are necessarily
G
accompanied by scalar fields in the adjoint representation of . In Section 7 we discuss the
G
relation of the work presented in this paper to previous approaches in the literature and in
Section 8 we discuss the paper’s results and present conclusions.
2 Gravitational action
ThoughwewilltakegravitytobeagaugetheoryofSU(2,2),forcalculationpurposesitisrather
convenient in the gravitationalsector to work with representations of SO(2,4). The machinery
to move from representations of SU(2,2) to SO(2,4) is discussed in detail in section 6.3 when
we discuss the coupling of gravity to fermions, which we take to be fields in the fundamental
representation of SU(2,2).
A group element of SO(2,4) may be represented as a matrix ΛA with unit-determinant
B
and satisfying
η =η ΛC ΛD , η =diag( 1, 1,1,1,1,1) (15)
AB CD A B AB − −
A field in the fundamental representation of SO(2,4) is a six-component vector UA and a field
intheadjointrepresentationtakesoftheformofanantisymmetricmatrixYAB = YBA,where
−
indices are lowered and raised with η and its inverse.
AB
WeseektobuildamodeldescribinggravitationfromalocallySO(2,4)-invariantpolynomial
action. ThegravitationalfieldswillbeanSO(2,4)gaugefieldAAB =A ABdxµandafieldWAB
µ
intheadjointrepresentation. ThefieldWAB canalwaysbeputinthefollowing‘block-diagonal’
form by appropriate SO(2,4) transformations:
t Σ 0 0
1 0 1
WAB = 0 t Σ 0 , Σ (16)
 2  ≡ 1 0
0 0 t3Σ (cid:18) − (cid:19)
 
If t = t = 0 (labeling indices on their rows and columns by I,J,K, = 0,1,2,3) and
1 2
···
t = 0 (labeling indices on its rows and columns by a,b,c, = 1,4) then this form of WAB
3
6 ··· −
will be invariant under the SO(2,4) transformations ΛI and Λa . If sign(η ) = sign(η )
J b −1−1 6 44
then- by implication- Λa represent hyperbolic rotations/boostsin the ( 1,4) plane whilst ΛI
b − J
are transformations generated by the Lorentz group subgroup of SO(2,4). Thus with t = 0,
3
6
t ,t =0, the residual symmetry is SO(1,3) SO(1,1).
1 2
×
Usefulquantitiesarethecurvaturetwo-formFAB andcovariantderivativeofWAB,DWAB,
given as follows:
FAB = dAAB +AACA B (17)
C
DWAB = dWAB+AACW B+ABCWA (18)
C C
WewilllooktoconsiderthemostgenerallocallySO(2,4)-invariantanddiffeomorphism-invariant
action polynomial in AAB,WAB . To do this we may build differential four-forms from FAB
{ }
and DWAB, thus guaranteeing that the Lagrangian is coordinate independent. To further en-
force local SO(2,4) invariance,we will look to contract awayall free SO(2,4) indices for which
wecaninprincipleadditionallyusethescalarWAB andtheSO(2,4)invariantsthematrixη
AB
and the completely antisymmetric symbol ǫ :
ABCDEF
S[A,W] = a FABFCD+b DWA DWEBFCD
ABCD ABCD E
Z
+c DWA DWEBDWC DWFD (19)
ABCD E F
8
where:
a a ǫ WEF +a W WE η +a W W
ABCD ≡ 1 ABCDEF 2 AE D BC 3 AB CD
+a η η +a η W (20)
4 AC BD 5 AC BD
b b ǫ WEF (21)
ABCD 1 ABCDEF
≡
c c ǫ WEF (22)
ABCD 1 ABCDEF
≡
The coefficients a ,b ,c may in general depend on SO(2,4) invariants built from WAB and
i i 1
{ }
thegroupinvariantsη ,ǫ . Inthispaper,asafirstapproachtothetheory,wewilltake
AB ABCDEF
these coefficients to be constant numbers but it is conceivable that functional dependences on
such invariants cannot be consistently neglected. Though the action contains terms quadratic
in FAB and quartic in DWAB, the wedge-product structure guarantees that components of
these fields appear at most linearly in the action. As mentioned in the previous section, the
generation of higher-order partial derivatives in the equations of motion is made impossible by
a polynomial Lagrangian and Bianchi identities DFAB = 0 and ‘DDV = FV’. Therefore, as
theLagrangianisatmostlinearinderivativesofanycomponent,theequationsofmotionareat
most first order in derivatives. The action may look very unfamiliar and so in the next section
we will initially look at a simpler theory emerges when we ‘freeze’ all the degrees of freedom in
the field WAB.
FinallywenotethatthoughwewilluseWAB inthefollowingsections,wemayalternatively
(and equivalently) use an entirely antisymmetric field Y , where the two are related via
ABCD
1
Y = ǫ WEF (23)
ABCD ABCDEF
2
For example the equivalent of ‘a ’ term in the gravitational Lagrangian would then take the
1
form:
a Y FABFCD (24)
1 ABCD
Z
with the required symmetry breaking happening when Y = 0 (W = 0) and Y = 0
IJKL ab abKL
∗ 6 6
(W =0) whilst we may adopt Y (W =0) as a gauge choice.
IJ IJKa Ia
3 Conformal Einstein-Cartan theory
We first discuss a theory which emerges when the degrees of freedom of WAB in the model
(19) are completely frozen by means of constraints imposed at the level of the action. Recall
that the theory recovered by the same process of freezing the Higgs degree of freedom for the
SO(1,4)SO(2,3)gaugetheoriesresultedintheEinstein-Cartantheory(orequivalentlyGeneral
|
Relativity). We shall show that the corresponding theory for gravity based on the gauge group
SU(2,2) C(1,3) is in some senses a natural conformal generalization of the Einstein-Cartan
≃
theory.
Assuming the block-diagonal form of WAB from (31), degrees of freedom in WAB can be
characterized in terms of three SO(2,4)-invariant quantities:
= ǫ WABWCDWEF =6t t t (25)
1 ABCDEF 1 2 3
C
= W WAB = 2 η η (t )2+η η (t )2+η η (t )2 (26)
2 AB 00 11 1 22 33 2 −1−1 44 3
C −
C3 = WABWBCWCDW(cid:0)DA =2 (t1)4+(t2)4+(t3)4 (cid:1) (27)
If it is enforced = 0 and = ( )2, this im(cid:0)plies that two out o(cid:1)f t ,t ,t are zero. For
1 3 2 1 2 3
C C C { }
example, we can choose t =0 (the label ‘3’ is completely arbitrary at this point) from =0.
3 1
C
Subsequently, the condition ( )2 =0 takes the form:
2 3
C −C
9
8η η η η (t )2(t )2 =0 (28)
00 11 22 33 1 2
−
Hence we can choose t =0. Now if we further require >0 we have the condition:
2 2
C
2η η (t )2 >0 (29)
00 11 1
−
Therefore η η < 0 and so we have the breaking SO(1,1) SO(1,3). If we then add on the
00 11
×
following four-form Lagrange-multiplier constraints
S [WAB,λ ,λ ,λ ] = λ +λ φ¯2 +λ ( )2 (30)
λ 1 2 3 1 1 2 2 3 3 2
C C − C − C
Z
(cid:0) (cid:1) (cid:0) (cid:1)
then the constraints will be enforced via the field equations obtained by varying λ . We may
i
instead enforce the constraints at the level of the action, and so WAB may be assumed to take
the form at the level of the action:
0 0 0
WAB =∗ 0 0 0 (31)
 0 0 φ¯ǫab 
 
where we use indices a,b,c... as indices in the fundamental representation of SO(1,1) and we
use the convention ǫ = 1, ǫ−14 = 1 and φ¯ is a constant. Given the application of these
−14
−
constraints, there are no longer any degrees of freedom for WAB left in the action (19); the
action is now a functional only of AAB, a general ansatz for which is given in this gauge by:
ωIJ EIa
AAB = (32)
EIa cǫab
(cid:18) − (cid:19)
The one-form field ωIJ is the Lorentz-groupspin connection, while the one-form c is a connec-
tion for the group SO(1,1). The ‘off-diagonal’ components EIa look much less familiar; they
transform homogeneously under the remnant SO(1,3) SO(1,1) symmetry and appear in the
×
SO(2,4) covariantobject DWAB as follows:
0 φ¯EI ǫca
DWAB = c (33)
φ¯EI ǫca 0
(cid:18) − c (cid:19)
Now we write down the total constrained form of (19) in the ‘preferred gauge’, making use of
the following results:
RIJ EI EJa D(ω+c)EIa φ¯2EI EJd 0
FAB = − a , DWA DWCB = d (34)
D(ω+c)EIa dcǫab E aEJb C 0 0
(cid:18) − − J (cid:19) (cid:18) (cid:19)
where RIJ dωIJ +ωI ωKJ is the SO(1,3) curvature two-form. The action then becomes:
≡ K
S[ω,c,E] = 2φ¯(2a φ¯2b )ǫ EI EJaRKL 2φ¯(a b φ¯2+c φ¯4)ǫ EI EJaEK ELb
1− 1 IJKL a − 1− 1 1 IJKL a b
Z
+a φ¯2R EJaEI +φ¯2(a +2a )dcǫcdE EJ
2 IJ a 2 3 Jc d
+a φ¯2E EJ E cEKd+a φ¯2ǫ ǫ E aEJbE cEKd
2 Jd c K 3 ab cd J K
+2φ¯2(a +2a )dcdc 2φ¯a ǫ RIJRKL (35)
2 3 1 IJKL
−
where we have used the result DDEIa = RI EJa +dcǫa EIb. To make further progress, we
J b
make the following general ansatz for EIa:
1 1
EIa = (eI +fI)Ua+ (eI fI)Va (36)
2 2 −
10
whereη UaUb =1andVa ǫa Ub. HereUa andVa shouldberegardedasanarbitrarychoice
ab ≡ b
of basis of the two-dimensional vector space and not a new fundamental field. As such, eI and
fI as defined in (36) are SO(1,1)-invariant (and so are not the same fields as the eI,fI of
{ }
(9)). Applying this ansatz to (35) yields:
1 Λ˜ 2
S[ω,c,e,f] = ǫ eIfJRKL eIfJeKfL eIfJR
32πG˜ IJKL − 6 !− γ IJ!
Z
1
+ξ e fIe fJ dceIf + ǫ RIJRKL+ dcdc (37)
I J I 1 IJKL 2
2 − C C
(cid:18) (cid:19)
where
1 6(a b φ¯2+c φ¯4) 4(2a b φ¯2)
(φ¯) = 4φ¯(2a b φ¯2), Λ˜(φ¯)= 1− 1 1 , γ(φ¯)= 1− 1
16πG˜ 1− 1 (2a1 φ¯2b1) a2φ¯
−
ξ(φ¯) = (a +2a )φ¯2, (φ¯)= 2φ¯a , (φ¯)=2(a +2a )φ¯2
2 3 1 1 2 2 3
C − C
The action (37)- up to boundary terms (those with coefficients)- indeed corresponds to the
i
C
action (11) i.e. Conformal Einstein-Cartan theory is indeed the theory that emerges in the
limit in which all degrees of freed of WAB are frozen out and WAB is constrained to break
SO(2,4) SO(1,3) SO(1,1). As expected, the actionis manifestly locally Lorentz invariant
→ ×
andinvariantunderlocalSO(1,1)gaugetransformationsc c+dθ(x). Theactionadditionally
→
possesses invariance under local dilations of eI and fI of opposite weight:
eI eα(x)eI, fI e−α(x)fI (38)
→ →
By appearance, the theory (37) resembles the Einstein-Cartan theory but instead has a pair
of frame fields eI,fI with a local scale symmetry under opposite rescalings. This theory
{ }
is indeed none other than the Conformal Einstein-Cartan theory proposed in equation (11).
Indeed, if there exist solutions when eI = fI then the first three terms in (37) become terms
±
familiar from the Einstein-Cartan theory: Palatini, cosmological,and Holst terms respectively.
The terms proportionalto the coefficient ξ vanish in this limit and so representnew behaviour,
whilst the final terms quadratic in curvature are boundary terms and do not contribute to the
equations of motion.
3.1 General-Relativistic limit
Byconductingsmallvariationsof(37)onemaystraightforwardlyobtaintheequationsofmotion.
Remarkably, solutions to these equations of motion exist for the ansatz fI eI i.e.
∝
fI = Ω2(x)eI (39)
±
We now showthat- assuggestedabove-suchsolutions constitute a General-Relativisticlimit of
the theory. In this limit, we find from combining the eI and fI field equations that
dc=0 (40)
and the field c disappears fromthe systemof field equations. The remaining field equationsare
equivalent to those obtained from the following action:
Ω2 Ω2Λ˜ 2
S[ω,e,Ω] = ǫ eIeJRKL eIeJeKeL R eIeJ
32πG˜ IJKL ± − 6 !∓ γ IJ !
Z
This action is invariant under local rescalings eI eα(x)eI, Ω e−α(x)Ω. Then, varying with
→ →
respect to ω and solving for ω(e,Ω), eliminating it from the action, we recover the following,
second-order action: