Quantization of the massive gravitino on FRW spacetimes

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by  Alexander Schenkel

Quantization of the massive gravitino on FRW spacetimes
Alexander Schenkel1,2,∗ and Christoph F. Uhlemann2,†
1Fachgruppe Mathematik
Bergische Universita¨t Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany
2Institut fu¨r Theoretische Physik und Astrophysik
Universita¨t Wu¨rzburg, Am Hubland, 97074 Wu¨rzburg, Germany
(Dated: January 13, 2012)
In this article we study the quantization and causal properties of a massive spin 3/2 Rarita-
Schwinger field on spatially flat Friedmann-Robertson-Walker (FRW) spacetimes. We construct
Zuckerman’suniversalconservedcurrentandprovethatitleadstoapositivedefiniteinnerproduct
on solutions of the field equation. Based on this inner product, we quantize the Rarita-Schwinger
fieldintermsofaCAR-algebra. Thetransversalandlongitudinalpartsconstitutingtheindependent
on-shelldegreesoffreedomdecouple. WefindaDirac-typeequationforthetransversalpolarizations,
2 ensuringacausalpropagation. TheequationofmotionforthelongitudinalpartisalsoofDirac-type,
1 butwithrespecttoan‘effectivemetric’. Weobtainthatforallfour-dimensionalFRWsolutionswith
0 a matter equation of state p = ωρ and ω ∈ (−1,1] the light cones of the effective metric are more
2
narrow than the standard cones, which are recovered for the de Sitter case ω =−1. In particular,

n this shows that the propagation of the longitudinal part, although non-standard for ω (cid:54)= −1, is
a completely causal in cosmological constant, dust and radiation dominated universes.
J

2 PACSnumbers: 04.65.+e,04.62.+v
1

] I. INTRODUCTION imposing canonical anticommutation relations (CAR).
h
t In this work we improve on this point and show
p- Supergravity is a well-motivated extension of Ein- that a consistent quantization of the gravitino on FRW
e stein’s theory of general relativity and may have inter- spacetimes is indeed possible. We consider the Rarita-
h esting consequences for cosmology and particle physics. Schwinger field in d dimensions without assuming a spe-
[ Specificissueslikeproductionmechanismsandproperties cificmodel,butwithpropertiesgeneralenoughtoinclude

2 of gravitino dark matter or the analysis of scattering ex- the relevant supergravity cases. In particular, we allow
v periments are best addressed in terms of effective quan- for a spacetime-dependent mass as it arises in lineariza-
1 tum field theory for the fluctuations around appropri- tionsofsupergravityaroundnon-trivialbackgroundslike
5 ate solutions of classical supergravity. While this means FRW [1–3], but do not fix the dependence a priori. We
9 Minkowski-space for collider physics, the less symmetric show that there is a canonical conserved current for the
2 FRW backgrounds are of particular interest for studies Rarita-Schwinger field on all spacetimes of dimension
.
9 of the early universe. A consistent quantum field the- d 3. Specializingtothecaseofd-dimensionalspatially
0 ory for the gravitino linearized around such supergravity fla≥tFRWspacetimes,weprovethattheinnerproductde-
1 solutionsisthereforeofgreatimportanceforphysicalap- rivedfromthiscurrentispositivedefiniteonsolutionsof
1
plications. the Rarita-Schwinger equation. In particular, it satisfies
:
v A matter-coupled supergravity with FRW solutions non-negativity, the necessary condition for a consistent
Xi has been proposed and analyzed in detail at the clas- implementationofCARemphasizedin[4]. Weconstruct
sical level by Kallosh, Kofman, Linde and Van Proeyen the CAR-algebra and discuss causality and the role of
r
a [1–3]. It has been shown that due to the special form supergravity in that respect. We find that the propaga-
of the scalar field potential, the massive gravitino equa- tion is in general non-standard, yet completely causal on
tions are consistent and the propagation is causal on the awideclassofFRWspacetimesincludingdust,radiation
FRW solutions. However, the quantization of the mas- and cosmological constant dominated universes. Specifi-
sive gravitino on spacetimes which are not necessarily cally,thedomainsofdependenceonthesespacetimesare
Einstein has been discussed only recently [4], arriving at in general more narrow than na¨ıvely expected. Time-
theconclusionthataconsistentquantizationisonlypos- variations of the mass stretch these domains, eventually
sible on Einstein spaces, i.e. when the Einstein tensor arriving at the standard light cones in the supergravity
is proportional to the metric. This statement is based model of [1–3].
on the non-conservation of the specific gravitino current
The outline is as follows: In Section II we review the
proposedin[4],whichwouldleadtoinconsistencieswhen
action, equation of motion and constraints for a massive
Rarita-Schwingerfieldonad-dimensionalspacetime. We
derive a current for this field in Section III using varia-
∗ [email protected] tional bicomplex methods [5–7] and show that this cur-
† [email protected] rent is conserved. For d-dimensional spatially flat FRW
2
spacetimesweproveinSectionIVthatthederivedinner III. CONSERVED CURRENT
product is positive when evaluated on solutions of the
equation of motion. The quantization is outlined in Sec- WeconstructZuckerman’suniversalconservedcurrent
tionVandadiscussionofthecausalpropagationandthe [5–7] for the Rarita-Schwinger field using the variational
role of supersymmetry is given in Section VI. Examples bicomplex. We will verify its conservation explicitly, so
ofcosmologicalspacetimesallowingforacausalpropaga- the reader may also pass directly to (11).
tion are studied in Section VII. We conclude in Section The basic idea of the variational bicomplex is to con-
VIII. Our notation and conventions are summarized in sider functions and differential forms on the product
Appendix A. space , with being spacetime and the space
M×S M S
of field configurations (this can be made precise by us-
ing -jet bundles [7]). The differential forms on
II. EQUATION OF MOTION AND ∞ M×S
can be decomposed into subspaces of a definite horizon-
CONSTRAINTS
tal (i.e. spacetime) and vertical (i.e. field space) degree.
Likewise, the exterior differential on splits into
We consider a Dirac Rarita-Schwinger field ψ on a M×S
µ a horizontal differential d and a vertical differential δ,
spacetime of dimension d 3 with metric signature
increasing the horizontal/vertical degree by one.
≥
mostly minus. The action reads
The starting point of the construction is a Lagrangian
(cid:90) described by a (d,0)-form, i.e. of maximal horizontal de-
S = ddxe ψ µ[ψ] , (1)
µR gree, on . The Lagrangian form corresponding to
M×S
the Dirac Rarita-Schwinger action (1) reads
with the Rarita-Schwinger operator
µ[ψ]:=iγµνρ ψ +mγµνψ . (2) L=iψ (cid:63)V3 ψ+( 1)dmψ (cid:63)V2 ψ , (6)
R Dν ρ ν ∧ ∧D − ∧ ∧
TΓfigρµhuνeψrcaρot.vioanr,iWanie.te.daetsrhsiuveamtceiovneanisectDotiµrosψnioνns:-y=fmre∂beµoψlbsνa+cakr14geωrµosauybmnγdambψectoνrn−ic- wDanhψdere:t=h(cid:63)edDenµnoψortνmedsaxltiµzhee∧dHdnoxd-νfg.oeldoFppuerrrotahdtoeurrcm,tψories:,=dVψenµ:o=dtxeγµdµdanxbdµy
Γbuρµtν =isΓaρνsµsu.mTehdetmoabssemreamlaayndbepsopsaitcievtei.meT-dheipsenadcteinotn, tVhne:L=agn1r!aVng∧ia.n.∧adVm.itTshaedveceortmicpaolseitxitoenrior derivative of
with d=4 is the quadratic gravitino part of the matter-
coupled =1supergravitydiscussedin[1–3], uptomet- δL=E+dΘ , (7)
N
ric conventions and the Majorana condition. The mass
min[1–3]isrelatedtotheK¨ahlerandsuperpotentialvia with a unique source form E of degree (d,1) yielding
m=eK/2W/M2. Theaction(1)isrealuptoaboundary the equations of motion and Θ of degree (d 1,1), which
P −
term and the Rarita-Schwinger operator (2) is formally is unique up to horizontally exact parts. For the La-
self-adjoint with respect to (cid:0)ψ ,ψ (cid:1) := (cid:82) ddxe ψµψ . grangian (6) we find
1 2 1 2µ
Thatis,forallψ andψ withsupportsofcompactover-
1 2
lap we have Θ= iψ (cid:63)V3 δψ . (8)
− ∧ ∧
(cid:0) (cid:1) (cid:0) (cid:1)
ψ , [ψ ] = [ψ ],ψ . (3)
1 R 2 R 1 2 Zuckerman’s universal current is defined as the contrac-
Contracting the equation of motion µ[ψ] = 0 with γ tion of the (d 1,2)-form u:=δΘ with two Jacobi fields,
leads to the on-shell constraint R µ i.e. solutions o−f the linearized equations of motion. Since
we are considering a linear theory, the Jacobi fields co-
d 1
i / γ ψ i µψµ+ − m γ ψ =0 . (4a) incide with solutions of the Rarita-Schwinger equation
D · − D d 2 ·
− µ[ψ]=0. From (8) we find
Acting with on µ[ψ] = 0 and using (4a) yields the R
µ
D R
second constraint u= iδψ (cid:63)V3 δψ , (9)
− ∧ ∧
i d 1
Gµνγ ψ +(∂ m)γµνψ + − im2γ ψ =0 , (4b)
2 µ ν µ ν d 2 · and contracting with the two Jacobi fields ψ1 and ψ2 we
− obtain the (d 1,0)-form current
whereGµν := µν 1gµν istheEinsteintensor. Using −
(4a), the RariRta-Sc−hw2ingeRr equation µ[ψ] = 0 can be u[ψ ,ψ ]=i( 1)dψ (cid:63)V3 ψ . (10)
written as R 1 2 − 1∧ ∧ 2
(cid:0)i / m(cid:1)ψ (cid:0)i + m γ (cid:1)γ ψ =0 . (5) Notethat(10)doesnotdependonthefieldspacecoordi-
D− µ− Dµ d 2 µ · nates. Wepullback(10)to toobtainad 1-formcur-
− M −
Due to the derivative in the second term, (5) is not of rent (denoted by the same symbol) on spacetime. From
Dirac-type [8] and the causal propagation of the Rarita- that current on we define the more familiar one-form
M
Schwinger field on a generic spacetime is not guaranteed current j[ψ1,ψ2]:=i(cid:63)u[ψ1,ψ2], which reads explicitly
a priori. We will discuss this point further in Section VI
j [ψ ,ψ ]= ψνγ ψρ . (11)
and VII. µ 1 2 − 1 νµρ 2
3
Conservation of the d 1-form current u[ψ ,ψ ], i.e. whereforthesecondequationwehaveusedFriedmann’s
1 2
du[ψ ,ψ ] = 0, is equiv−alent to jµ[ψ ,ψ ] = 0, with equations G0 = ρ and Gn = pδn (in units M = 1).
1 2 ∇µ 1 2 0 m − m P
being the covariant derivative on vector fields. We Theseexpressionsind=4havebeenobtainedin[1–3],up
µ
∇
obtain to metric conventions. Combining the µ=0 component
of (5) with (15a) yields
jµ[ψ ,ψ ]= ψ γνµρψ +ψ (cid:0)γνµρψ (cid:1)
−∇µ 1 2 Dµ 1ν 2ρ 1νDµ 2ρ (cid:16) i a(cid:48) (cid:17)
=γρµν ψ ψ +ψ γνµρ ψ iγmn∂ ψ = m+ (d 2) γ0 γmψ . (16)
Dµ 1ν 2ρ 1ν Dµ 2ρ m n − 2 − a m
=i ρ[ψ ]ψ iψ ν[ψ ] . (12) (cid:124) (cid:123)(cid:122) (cid:125)
R 1 2ρ− 1νR 2 =:B
In the first line we have used the Leibniz rule for the Due to the constraints (16), (15b), only (d 2) 2(cid:98)d/2(cid:99)
covariant derivative and µψν = µψν, and in line two of the d 2(cid:98)d/2(cid:99) complex degrees of freed−om ·of the
that due to the vielbeinDpostulateD µγνµρ = 0. Thus, Rarita-Sch·winger field are independent. It is convenient
D
the current is conserved when evaluated on solutions. to transform to spatial Fourier space via ψ (τ,x) =
m
(2π)1−d(cid:82) dd−1k eiknxnψ(cid:101) (τ,k). As in [1–3, 9] we sep-
m
IV. POSITIVITY OF THE INNER PRODUCT arate the spatial part of the Rarita-Schwinger field ψ(cid:101)m
into the γm and km traceless part ψ(cid:101)mT and the traces
As noted in [4], non-negativity of the inner product
constructed from the current (11) is a necessary condi- χ(cid:101):=γnψ(cid:101)n , ζ(cid:101):=knψ(cid:101)n . (17)
tionforaconsistentimplementationofCAR.Thisisdue With kˆ := k /k , k := √ k kn and kˆ/ := kˆ γm we
to the anticommutator of the smeared quantum fields m m | | | | − n m
Ψ (f¯µ) and Ψµ(f ) being an expression of the form find
µ µ
A†A+AA†, which has a non-negative expectation value γ +kˆ kˆ/ γ kˆ/ (d 1)kˆ
in any normalized state, see also Section V for more de- ψ(cid:101)m =ψ(cid:101)mT+ md 2m χ(cid:101)+ m (d− 2−)k m ζ(cid:101). (18)
tails. − − | |
We define the inner product associated to (11) by The constraint (16) in k-space yields the following rela-
(cid:90) tion between the traces
(cid:104)ψ1,ψ2(cid:105):= Σnµjµ[ψ1,ψ2] , (13) ζ(cid:101)=(cid:0)k/−B(cid:1)χ(cid:101) . (19)
where Σ is a Cauchy surface with future-directed unit With (19) the decomposition (18) becomes
normal vector field nµ. Splitting µ = (0,m) we choose
coordinates such that ds2 = g dτ2 +g dxmdxn and (cid:18) (d 1)kˆ γ kˆ/ (cid:19)
likewise fix ea0 = √g00δ0a. With00a choicemonf Σ such that ψ(cid:101)m =ψ(cid:101)mT− kˆmkˆ/− −(d m2)−k m B χ(cid:101) . (20)
(cid:112) − | |
n= g00∂ the integrand evaluates to
τ
The traceless part ΨT comprises (d 3) 2(cid:98)d/2(cid:99) degrees
nµjµ[ψ1,ψ2]=−(cid:16)ψ1†mψ2m+(cid:0)γmψ1m(cid:1)†(cid:0)γnψ2n(cid:1)(cid:17) . oUfsfirnegedtohmis aonnd-shtheleltdrmeaccoempparotsiγtinoΨnn, tt−hheerien·mnearinpinrogd2u(cid:98)cdt/2i(cid:99)n.
(14)
Fourier space evaluates to
Weverifynon-negativityoftheinnerproduct(13)eval- (cid:90) dd−1k (cid:16) (cid:17)
uated on solutions of the Rarita-Schwinger equation for (cid:104)ψ1,ψ2(cid:105)= (2π)d−1 ad−1 −ψ(cid:101)1Tm†ψ(cid:101)2Tm+Cχ(cid:101)1†χ(cid:101)2 ,
d-dimensional FRW spacetimes, ea = a(τ)δa. For com-
µ µ (21)
patibility with the FRW symmetries we assume that the
mass depends on time only, m = m(τ). The spin con- where
nection is given by ω = 2a(cid:48)a−1e e0, where prime
µab µ[a b] d 1 (cid:16) 1 a(cid:48)2(cid:17)
denotesthederivativewithrespecttoτ. Theconstraints = − m2+ (d 2)2 (22)
(4) read for the FRW background C (d 2) k 2 4 − a4
− | |
(cid:104)i a(cid:48) d 1 (cid:105) is positive. The integrand is pointwise (in k-space) non-
iγµν∂µψν + 2(d−2)aγ0+ d−2m γ·ψ negative, since in our conventions the spatial metric is
− (15a) negative definite. Thus, for any nonzero solution ψ 0
(cid:16) 3(cid:17)a(cid:48) (cid:54)≡
i d ψ0 =0 , the norm is positive, ψ,ψ >0.
− − 2 a (cid:104) (cid:105)
p 2m2d−1 +2im(cid:48)γ0 V. QUANTIZATION
γ0ψ = − d−2 γmψ , (15b)
0 ρ+2m2d−1 m
d−2 Using the inner product (13), we can quantize the
(cid:124) (cid:123)(cid:122) (cid:125)
=:A Dirac Rarita-Schwinger field on d-dimensional FRW
4
spacetimes analogously to the spin 1/2 Dirac field [10]. to the Majorana solutions hermitian smeared field oper-
We briefly outline the construction of the CAR-algebra atorsviatheR-linearmapf Ψµ (f ). Theself-dual
µ (cid:55)→ maj µ
and refer for details on fermionic quantization to [8, 10, CAR-algebra is defined as the -algebra with unit 1 gen-
∗
11] and references therein. erated by these operators, subject to the relations
We denote by Sol the space of spinor solutions of the
Rarita-Schwingerequationwhichareofcompactsupport (cid:8)Ψµmaj(fµ),Ψνmaj(hν)(cid:9)=(cid:104)f,h(cid:105)1 . (29)
when restricted to any Cauchy surface. The space Sol of
cospinorsolutionsisdefinedastheimageofSolunderthe
map Sol f fµ. To the spinor/cospinor solutions VI. CAUSALITY AND THE ROLE OF
µ
we associ(cid:51)ate sm(cid:55)→eared field operators via C-linear maps SUPERGRAVITY
f Ψµ(f ) and fµ Ψ (fµ). The CAR-algebra is
µ µ µ
defi(cid:55)→ned as the -algebr(cid:55)→a with unit 1 generated by these In this section we discuss the propagation of the
operators, subj∗ect to the relations transversal and longitudinal parts of the Rarita-
Schwinger field on d-dimensional FRW spacetimes [13].
Ψ (fµ)† =Ψµ(f ) , (23a) The relevant equations are the constraints (15) and the
µ µ
(cid:8)Ψ (fµ),Ψν(h )(cid:9)= f,h 1 , (23b) equation of motion (5), or equivalently (5), (15b) and
µ ν (cid:104) (cid:105) (16). Thenon-dynamicalψ canbeeliminatedbysolving
(cid:8)Ψµ(fµ),Ψν(hν)(cid:9)=(cid:8)Ψµ(fµ),Ψν(hν)(cid:9)=0 , (23c) (15b), ψ0 =γ0Aχ, and (106) is manifestly implemented
in the decomposition (20). It thus remains to solve (5).
with denoting the involution in the algebra. The µ=0 component yields the equation of motion for χ
†
Aspointedoutin[4],non-negativityoftheinnerprod-
uspcatciesreespsreensteianltaftoironthoef CthAeRalg(2e3b)r:a Aabsosuvme.eLaentyfµHilbSerotl i(cid:0)γ0∂0+γmA∂m(cid:1)χ+2Bd −2mχ+dd−12B†Aχ=0.
∈
be arbitrary and define A := Ψµ(fµ), then (23a) and − − (30)
(23b) imply
The spatial components of (5) give – after using (30) –
A†A+AA† = f,f 1 . (24) the equation for the transversal polarizations
(cid:104) (cid:105)
From the expectation value in any normalized Hilbert (cid:18)ia(cid:48) (cid:19)
iγν∂ ψT + (d 3)γ0 m ψT =0 . (31)
space state ϕ one concludes ν m 2a − − m
| (cid:105)
f,f = AϕAϕ + A†ϕA†ϕ 0 , (25) Thus, the transversal and longitudinal parts decouple.
(cid:104) (cid:105) (cid:104) | (cid:105) (cid:104) | (cid:105)≥
Note that (31) is a Dirac-type operator and therefore
completing the argument. As we have shown in Sec- ψT propagates causally, see e.g. [8]. In order to under-
m
tion IV, for d-dimensional FRW spacetimes the inner standthecausalpropertiesofthelongitudinalpartχ,we
product (13) indeed satisfies this necessary condition for define the ‘effective gamma matrices’
a consistent quantization of the Rarita-Schwinger field.
The Dirac Rarita-Schwinger field as discussed above γ0 :=γ0 , γm :=γm . (32)
eff eff A
amounts to the generic case in d dimensions without im-
They form a Clifford algebra γµ ,γν = 2gµν with
posing restrictions on d. However, if a Majorana condi- { eff eff} eff
tion ψµ =ψµTC is available and used to reduce the Dirac an ‘effective metric’ geff with components ge0ffµ = g0µ =
spinor (e.g. in d=4 minimal supergravity), the quantiza- a−2δ0µ and
jtoiornanparosocleuetdisonisnSaolsmimajislaartiswfyayin:gWfµe =resftrµTicCt.STolhteoinMnear- gemffn =(cid:0)A21+a−2A22(cid:1)gmn =:c2eff(τ)gmn . (33)
product(13)forMajoranasolutionsf ,h Sol reads
µ µ ∈ maj The numerical coefficients 1 and 2 are defined by
A A
(cid:90) =: 1+i 2γ0, see (15b). Thus, (30) is a Dirac-type
f,h = nµfνTCγ hρ . (26) AoperaAtorontAhespacetimewith‘effectivemetric’gµν and
(cid:104) (cid:105) − Σ νµρ χ propagates causally with respect to gµν. Interepffreted
eff
with respect to the original metric it propagates with a
ItissymmetricsinceCγ isanti-symmetricinthecases
νµρ time-dependentspeedoflightc2 (τ),ascanbeseenfrom
where a Majorana condition is available, such that eff
(33). Thepropagationisthereforecausalwithrespectto
hνTCγνµρfρ =fρT(cid:0)Cγνµρ(cid:1)Thν =fρTCγρµνhν , (27) gµν as long as c2eff(τ) ≤ 1 for all times τ. Note that any
time-dependence of the mass leads to a positive contri-
bution to the effective speed of light since 2 (m(cid:48))2.
and it is also real A2 ∝
Thus, time-variations in the mass can never reconcile an
f,h ∗ = h,f = f,h . (28) otherwise acausal propagation of the longitudinal part
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) with causality.
We quantize the Majorana Rarita-Schwinger field in Itisremarkablethatthesupergravitymodeldiscussed
terms of a self-dual CAR-algebra [8, 12]: We associate in [1–3] leads to c2 (τ) 1 for all d=4 FRW solutions.
eff ≡
5
This ensures causal propagation of the gravitino on the c2
eff
one hand, but on the other hand also means that the 1
time-varying mass m=eK/2W exactly compensates the
deficitintheeffectivespeedoflightfrombeingone, thus
leading to standard causal properties. As we will dis-
cuss in the next section, also for the Rarita-Schwinger
field alone, without the restrictions imposed by the su-
1
pergravity model, a causal propagation is possible on a 2
variety of FRW spacetimes. This, however, generically ω = 1
−
involves a time-dependent speed of light c2 (τ) 1. ω =0
eff ≤
ω = 1
3
VII. COSMOLOGICAL SPACETIMES 0
log(1+t)
Within the class of matter models described by the
equation of state p = ωρ, ω R, we identify the d=4 FIG.1. Cosmological-timedependenceoftheeffectivespeed
∈ oflightforthelongitudinalpart. Theplotshowswithincreas-
FRW spacetimes on which the Rarita-Schwinger field
ing dash length ω∈{−1,0,1}, corresponding to a cosmolog-
can propagate causally, and study the time-dependence 3
ical constant, dust and radiation dominated FRW universe,
of the effective speed of light. For a clearer physi-
respectively.
cal interpretation we work in cosmological time t de-
fined by dt = a(τ)dτ, such that the FRW metric reads
ds2 = dt2 a(t)2d(cid:126)x2. As discussed in the previous sec-
Figure 1. Note that for ω >0 the effective speed of light
−
tion, time-variations of the mass give a positive contri-
vanishes at t = 2√ω/(3m ω +1). This means that
bution to the effective speed of light and thus can only ± | |
the longitudinal part of the Rarita-Schwinger field effec-
tightentherestrictionsonthebackgroundspacetime. We
tivelydoesnotpropagateoverextendedspatialdistances
therefore focus on a constant mass m > 0 and identify
around these times.
the spacetimes for which c2 (t) 1 for all t.
eff ≤
Solving Friedmann’s equations for the Hubble rate H
yields VIII. CONCLUDING REMARKS
a˙(t) 2
H(t):= = (34)
We have investigated the massive spin 3/2 Rarita-
a(t) 3t(ω+1)+2α
Schwinger field focusing on the properties relevant for
with constant of integration α = H(0)−1. The energy quantization. Using the variational bicomplex we have
density is given by ρ = 3H2 and the effective speed of constructed for generic spacetimes of dimension d 3 a
≥
light (33) for the longitudinal part becomes current which is conserved on solutions of the equation
of motion. Furthermore, on d-dimensional FRW space-
(cid:18)m2 ωH(t)2(cid:19)2 times we have shown that the associated inner product
c2 (t)= − . (35)
eff m2+H(t)2 is positive definite and therefore allows for a consistent
quantization of the Rarita-Schwinger field in terms of
For the special case ω = 1, i.e. de Sitter space, we find
− a CAR-algebra. This shows that also on non-Einstein
c2 (t) 1 such that the Rarita-Schwinger field propa-
eff ≡ spaces a consistent quantization is possible, even with-
gateswiththestandardspeedoflight,asexpected. Con-
outemployingtheadditionalstructureofasupergravity.
sidernowthecaseω = 1,wherewesetα=0suchthat
(cid:54) − We then have studied the propagation of the transversal
the cosmological singularity is at t = 0. For t
→ ±∞ and longitudinal parts of the Rarita-Schwinger field and
the Hubble rate vanishes and the speed of light c2 ap-
eff foundthat,whilethetransversalpolarizationspropagate
proaches 1. Thus, we find standard causal properties at
causally on all FRW spacetimes, the propagation of the
latetimes. Ontheotherhand,fort 0theHubblerate
→ longitudinalparthasquitedistinctfeatures. Itspropaga-
diverges, such that c2 ω2. For a causal propagation
eff → tion is characterized by a time-dependent effective speed
at times close to t=0 we have to require ω [ 1,1]. In
∈ − oflight,anddemandingcausalityimposesrestrictionson
fact, from (35) this condition is necessary and sufficient
the background spacetime and on time-variations of the
for causal propagation
mass. This discussion offered an interesting perspective
c2 (t) 1 for all t ω [ 1,1] . (36) ontheroleofthetime-dependentmassinthesupergrav-
eff ≤ ⇐⇒ ∈ − ity model [1–3]. For a constant mass we have found that
Interestingly, the matter models used in standard cos- thepropagationiscausalford=4FRWspacetimeswitha
mology satisfy ω [ 1,1] and hence allow for a causal mattermodeldescribedbytheequationofstatep=ωρ,
∈ −
propagation of the Rarita-Schwinger field. We plot the ifandonlyifω [ 1,1]. Thisinparticularincludescos-
∈ −
effective speed of light for the cases ω = 1 (cosmolog- mological constant, dust and radiation dominated uni-
−
ical constant), ω =0 (dust) and ω =1/3 (radiation) in verses. Comparingthisresulttotheweak-fieldcondition
6
found for the electromagnetic background in [14] which products (of weight one) of gamma matrices are denoted
singles out preferred frames, we note that our condition by γµ1...µn := γ[µ1 γµn]. The contraction of γµ with
is invariant under the FRW isometries. the Rarita-Schwing·e·r·field is denoted by γ ψ := γµψ .
µ
·
The distinct features of the propagation of the longi- The covariant derivatives of the vielbein and Rarita-
tudinal modes with time-dependent speed of light may Schwinger fields are Dµeaν =∂µeaν +ωµabebµ−Γρµνeaρ and
also be relevant for models with explicit supersymme- ψ = ∂ ψ + 1ω γabψ Γρ ψ , ψ = ψ .
try breaking, e.g. the MSSM. Interesting physical con- TDhµeνvielbeµinνpos4tulµaatbe µea− =µν 0ρ reDlaµteµs theDµspiµn
Dµ ν
sequences may therefore be expected e.g. for bounds on connection to the Christoffel symbols. The co-
gravitino dark matter. variant derivative of the gamma matrices reads
γν = ∂ γν +Γν γρ + 1ω [γab,γν] and due to the
Dviµelbein poµstulate wµρe have4 µµaγbν1...νn =0.
D
Appendix A: Notation
ACKNOWLEDGMENTS
As usual, letters from the middle of the
greek/beginning of the latin alphabet denote space- We thank Thomas-Paul Hack, Thorsten Ohl and also
time/flat Lorentz indices. The metric signature is Julian Adamek and Florian Staub for useful discus-
(+, ,..., ) and ηab is the flat Minkowski metric. The sions and comments. CFU is supported by the Ger-
inve−rse vie−lbein is eµa, and e := det(eaµ) is the volume manNationalAcademicFoundation(Studienstiftungdes
element. The Hodge operator is defined by (cid:63)(dxµ1 deutschen Volkes). AS and CFU are supported by
··· ∧ dxµr) = (d−er)!(cid:15)µ1...µrνr+1...νddxνr+1 ∧ ··· ∧ dxνd∧. Deutsche Forschungsgemeinschaft through the Research
The (flat) Dirac matrices satisfy the Clifford algebra Training Group GRK1147 Theoretical Astrophysics and
γa,γb = 2ηab and γµ := eµγa. Antisymmetrized Particle Physics.
{ } a
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