From the Einstein-Cartan to the Ashtekar-Barbero canonical constraints, passing through the Nieh-Yan functional

0.15 MB·English

by  Simone Mercuri

From the Einstein-Cartan to the Ashtekar-Barbero canonical constraints, passing
through the Nieh-Yan functional
Simone Mercuri1,2,∗
1Centre de Physique Th´eorique de Luminy, Universit´e de la M´editerran´ee, F-13288, Marseilles, EU;
2Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, P.le Aldo Moro 5, I-00185, Rome, EU.
The Ashtekar-Barbero constraints for General Relativity with fermions are derived from the
Einstein-Cartancanonicaltheoryrescalingthestatefunctionalofthegravity-spinorcoupledsystem
by the exponential of the Nieh-Yan functional. A one parameter quantization ambiguity naturally
appears and can be associated with theImmirzi parameter.
PACSnumbers: 04.20.Fy,04.20.Gz
Keywords: Nieh-Yantopological term,canonical formulationofgravity
8
0
0 I. INTRODUCTION can be derived starting from the canonical Einstein-
2 Cartan theory, by considering a topologically suggested
rescaling of the wave function describing the quantum
n The growinginterest that the Ashtekar formulationof
a General Relativity (GR) [1] has gained over the years states of the system. This demonstration provides us
J with an interesting hint for interpreting the Immirzi pa-
is a consequence of the remarkable simplification it in-
7 rameterwhich,asthesocalledθ-angleofQCD,isaquan-
troduces in the structure of the canonical constraints.
1 Specifically,byusingtheself-dualSL(2,C)Ashtekarcon- tization ambiguity connected with the non-trivial struc-
ture of the quantum configuration space [21].
nections asfundamentalvariables,the constraintsofGR
]
c reduce to a polynomial form. On the other hand, the
q
complex character of the self-dual connections forces to
-
r introduce suitable reality conditions, which, until now,
g
have prevented the construction of a complete quantum The paper is organized as follows: in Section II we
[
theory. This led to the adoption of the real Ashtekar- briefly recall the canonical formulation of gravity and

3 Barbero SU(2) or SO(3) valued connections as funda- writetheconstraintsoftheEinstein-Cartanaction. Then
v mental variables in a partially gauge fixed version of we recallthe expressionofthe non-minimalactionintro-
7 tetrads GR [2, 3]. By using the new variables Ashtekar, ducedin[22](seealsotheinterestingpaper[24]). Analo-
3
RovelliandSmolinconstructedanon-perturbative(back- gously to the Holst action, the non-minimal action devi-
0
ground independent) quantum theory of gravity known atesfromtheEinstein-Cartan,specificallyitischaracter-
0
. as Loop Quantum Gravity (LQG) [4, 5, 6, 7], which, izedby two modifications,which, as soonas the solution
8
at least in symmetric physical systems, allows to cure ofthe secondCartanstructure equationis takeninto ac-
0
the inevitable singular behavior of General Relativity count, reduce to the Nieh-Yan topological term [22, 25].
7
[8, 9, 10, 11]. The results recently obtained about the In Section III we are going to demonstrate that this fact
0
: graviton propagator have further strengthened the the- makesit possible to calculate the Ashtekar-Barberocon-
v
ory, by providing other evidences on its non-singularbe- straintsforGRstartingfromthecanonicalconstraintsof
i
X havior [12, 13, 14]. theEinstein-Cartantheoryinthetimegaugebyusingan
original and interesting method, which sheds some light
r The Ashtekar-Barbero connections contain a free pa-
a on the underlying topological structure of the theory.
rameterin their definition, calledthe Immirzi parameter
and here denoted as β. The presence of the Immirzi pa- Namely,afterhavingcanonicallyquantizedthe Einstein-
Cartantheory, we rescale the state functional by the ex-
rameterdoesnotaffecttheclassicaltheory,butitappears
ponential of the Nieh-Yan functional, which here plays a
in the spectra of geometrical non-perturbative quantum
role analogous to the one that the Chern-Simons func-
operators as, for instance, the Area and Volume opera-
tionals play in Yang-Mills gauge theories (the Immirzi
tors. Even though many attempts have been made to
parameterbeinganalogoustotheangularparameterusu-
understand the physical meaning of the Immirzi param-
ally indicated by θ) and demonstrate that the canonical
eter[15,16,17,18,19,20],itsroleincanonicalquantum
constraints of the non-minimal action can be easily de-
gravity has not completely clarified yet [21].
rived once the modification to the conjugated variables
In this brief paper, motivated by the interesting anal-
are calculated. Finally, in Section IV we discuss the re-
ysis proposed by Gambini, Obregon and Pullin [15] and
sults obtained.
by using the results of previous works [22, 23], we will
be demonstrating that the Ashtekar-Barberoconstraints
The signature throughoutthe paper is +,−,−,−. We
∗Electronicaddress: [email protected] assume 8πG=1.
2
II. CANONICAL GR WITH FERMION FIELDS the two pairs of fermion conjugate fields: either ψ,Π
or Ω,ψ . (cid:0) (cid:1)
The interaction between gravity and fermion fields is O(cid:0)nce (cid:1)the phase space is equipped with the following
described by the Einstein-Cartan action, symplectic structure
S e,ω,ψ,ψ = 1 e ∧e ∧⋆Rab (1) Kαi(t,x),Ekβ(t,x′) =δαβδkiδ(3)(x−x′) (4a)
2Z a b n o
(cid:0) (cid:1) ψA(t,x),Π (t,x′) =δAδ(3)(x−x′), (4b)
+ 2i Z ⋆ea∧ ψγaDψ−Dψγaψ . (cid:8)ΩB(t,x′),ψB (t,x)(cid:9)+ =δBBδ(3)(x−x′), (4c)
(cid:2) (cid:3) A + A
(cid:8) (cid:9)
ThecovariantderivativescontaintheLorentzvaluedspin we can verify that the constraints (3) are first class.
connections according to the following definitions: Theyareconnectedwiththegaugefreedomofthetheory,
namelyinvarianceunderspatialrotationsandspace-time
i i
Dψ =dψ− ωabΣ ψ and Dψ =dψ+ ψΣ ωab, diffeomorphisms2.
ab ab
4 4
In [22] we demonstrated that the non-minimal action
(2)
with a,b,c··· = 0,...,3. Assuming that space-time is 1 1
globally hyperbolic, we can extractthe 3+1 form of the S e,ω,ψ,ψ = ea∧eb∧ ⋆Rab− Rab (5)
2Z (cid:18) β (cid:19)
above action and, once the temporal gauge is fixed and (cid:0) (cid:1)
i i
the second class constraints solved [26] (further details + ⋆e ∧ ψγa 1− γ Dψ+h.c.
about this procedure in the presence of spinor fields will 2Z a (cid:20) (cid:18) β 5(cid:19) (cid:21)
be given in [21]), we can extract the following canonical
has the following characteristics: i) it has the Einstein-
constraints:1
Cartanactionwithoutanymodificationaseffectivelimit,
i i consequentlytheImmirziparameterdisappearsfromthe
R :=ǫ kKjEα− ΠΣ ψ+ ψΣ Ω≈0, (3a)
i ij α k 4 i 4 i classical equations as in the original (purely gravita-
C :=2EβD Ki +ΠD ψ+D ψΩ≈0, (3b) tional)Holstapproach;ii)itcanbeextendedtoarbitrary
α i [α β] α α complex values of the Immirzi parameter, in particular
1
C :=− EαEβ (3)R ik−2Ki Kk (3c) for β = ±i it reduces to the Ashtekar-Romano-Tate ac-
2 i k (cid:16) αβ [α β](cid:17) tion [27]; iii) it is suitable for a geometrical description,
−iEiα ΠΣ0iDαψ−DαψΣ0iΩ since the non-minimal term in the fermionic sector to-
i (cid:0) (cid:1) getherwiththeHolstmodificationcanbereducedtothe
+ 2ǫiklEiαKαk ΠΣlψ−ψΣlΩ Nieh-Yaninvariant[25]oncethesecondCartanstructure
(cid:0) (cid:1) equation is solved. In other words, the modifications in-
1
+ 2EkαKαk Πψ+ψΩ ≈0, troduced in (5) with respect to the Einstein-Cartan ac-
(cid:0) (cid:1) tion (1) can be reduced to the Nieh-Yan invariant, i.e.
wherewehaveintroducedthefollowingnotations: Greek
1
indexes indicate the spatial components of space-time Rab∧e ∧e +⋆e ∧ ψγ5γaDψ−Dψγaγ5ψ
a b a
tensorsα,β,···=1,2,3,while Latinindexes referto the 2β Z
(cid:2) (cid:0) (cid:1)(cid:3)
internal degrees of freedom i,j,···=1,2,3; Eα =−eeα 1
i i = Rab∧e ∧e −Ta∧T − d ⋆e Ja
andKβj =ωβ0j formacanonicalpairandrepresentrespec- 2β (cid:20)Z (cid:0) a b a(cid:1) Z (cid:16) a (A)(cid:17)(cid:21)
tivelythedensitizedtriadandthetheextrinsiccurvature; 1
Π = i eψγ0 and Ω = −i eγ0ψ are respectively the mo- = d(ea∧Ta) , (6)
2 2 2β Z
mentaconjugatetoψ andψ;Σ = 1ǫ jkΣ arethegen-
i 2 i jk
eratorsofspatialrotations. Moreoverthe new derivative where Jd = ψγdγ5ψ. Passing from the first to the
(A)
symbol, D, representing the connections of the rotations secondline, we usedthe solutionofthe Cartanstructure
group,hasbeenintroduced. We wouldliketostressthat equation, namely
the apparent doubling of the fermionic degrees of free-
1
dom is a consequence of the specific form of the action ωab =ea ∇ eρbdxµ+ ǫab ecJd , (7)
westartedfrom. Therightnumberofdegreesoffreedom ρ µ 4 cd (A)
can be restored simply imposing a suitable set of second
resulting from the variation of the action (5) with re-
class constraints, which make it possible to refer the dy-
spect to the spin connection, which, once the factor
namics of the fermion sector of the theory to only one of
2 The theory was initiallyinvariant under the fullLorentz group,
1 ItisworthrecallingthattheadditionalstrongequationDαEiβ = but, as well known, the temporal gauge fixes the boost sector,
0 follows from the solution of the second class constraints, rep- leaving a remnant invariance under the local spatial rotations
resentingthesocalledcompatibility condition. only.
3
1ǫab −1 δ[aδb] isdropped,isequivalenttotheonechar-
2 cd β c d
acterizing the usual Einstein-Cartan theory. Finally, the
result in the third line can be easily obtained using the ψAΦ(E,ψ)=ψAΦ(E,ψ) , (13a)
following identity Ta ∧e = −1 ⋆ e Ja . So, the ac-
a 2 a (A) δ
tion (5) is dynamically equivalent to the usual Einstein- Πb Φ(E,ψ)=−i Φ(E,ψ) . (13b)
B δψB
Cartan action, since the additional terms introduced in
b
action (5) with respect to (1) can be reduced to a total
divergence containing torsion. The fact that the effec-
Suppose now to rescale the state functional Φ(E,ψ)
tive action derived from (1) contains a topological mod- by the exponential of the Nieh-Yan functional3
ification with respect to (5) suggests that the two sets
of canonical constraints resulting from them have to be
i
connectedbyatopologicallymotivatedredefinitionofthe Φ′(E,ψ,ψ)=exp Y(E,ψ,ψ) Φ(E,ψ,ψ), (14)
conjugate momenta. In fact, in the next section we are (cid:26)β (cid:27)
going to derive the constraints of the non-minimal the-
ory (5) by redefining the state functional of the formally
where the parameter β will result to be associated with
quantizedEinstein-Cartantheory,namelybyrescalingit
the Immirzi parameter. It is worth stressing that the
by the exponential of the Nieh-Yan functional
study of large gauge transformations in temporal gauge
fixed gravity leads precisely to the rescaling (14) of the
Y(e,ψ,ψ)= e ∧Ti(e,ψ,ψ), (8) state functional, so a consistent geometrical interpreta-
i
Z tion of the entire procedure can be provided, but this
discussionisbeyondthescopeofthisbriefletterandwill
where we have to take into account the solution of the
be presented in a longer, forthcoming paper [21].
second Cartan structure equation, which for spatial in-
ternal indexes gives The price paid for this rescaling is a modification in
thedefinitionsofthecanonicalconjugatemomentumop-
Ti(e,ψ,ψ)= 1ǫi ej ∧ekJ0 . (9) erators Kαi, ΠB and ΩB. Specifically we get:
2 jk (A)
b b b
K′αiΦ′(E,ψ,ψ)=eβi Y(E,ψ,ψ)Kiαe−βi Y(E,ψ,ψ)Φ′(E,ψ,ψ)
III. FROM EINSTEIN-CARTAN TO 1
ASHTEKAR-BARBERO CONSTRAINTS b =(cid:18)Kαi − 2β ǫαβγTβiγ(cid:19)Φ′b(E,ψ,ψ), (15a)
Weak equations (3) represent a set of first class con- Π′BΦ′(E,bψ,ψ)=eβi Y(bE,ψ,ψ)ΠBe−βi Y(E,ψ,ψ)Φ′(E,ψ,ψ)
straints, so we can quantize the system by adopting the b = Π − iΠ γ5 A bΦ′(E,ψ,ψ), (15b)
Dirac procedure, i.e. the constraints are directly imple- (cid:18) B β A B(cid:19)
(cid:0) (cid:1)
mentedinthequantumtheorybyrequiringthatthewave b b
functional be annihilated by their operator representa- Ω′BΦ′(E,ψ,ψ)=eβi Y(E,ψ,ψ)ΩBe−βi Y(E,ψ,ψ)Φ′(E,ψ,ψ)
tfiigounr.aLtieotnmspeaacsesutmheefaosllcoowoirndginfiaetldess:onEitαh,eψqAuaanntdumψBc,osno- b =(cid:18)ΩB− βi γ5 BAΩA(cid:19)Φb′(E,ψ,ψ), (15c)
that the wave function is b (cid:0) (cid:1) b
where we have formally calculatedthe functional deriva-
Φ=Φ E,ψ,ψ . (10)
tives,takingintoaccountexpression(9)andthefollowing
(cid:0) (cid:1)
useful form of the Nieh-Yan functional
The quantum gravitational equations are formally ex-
pressed as
1
Y(E,ψ,ψ)=− d3xǫ αβEγTi . (16)
R Φ E,ψ,ψ =0, (11a) 2Z γ i αβ
i
(cid:0) (cid:1)
CbαΦ E,ψ,ψ =0, (11b)
Themodificationsoftheconjugatemomentumoperators
(cid:0) (cid:1)
bCΦ E,ψ,ψ =0, (11c) correspond to canonical modifications of the respective
(cid:0) (cid:1) classical conjugate momenta, which can be easily evalu-
b
where the operatorial translation of the canonical con- ated. So that, reintroducing the new classical conjugate
straints relies on the following prescriptions momentaintothecanonicalconstraints(3)weobtainthe
EαΦ(E,ψ)=EαΦ(E,ψ) , (12a)
i i
δ
KbβkΦ(E,ψ)=iδEβΦ(E,ψ) , (12b) 3 ForaconciseexplanationofthisprocedureforSU(N)Yang-Mills
k gaugetheoriessee[28].
b
4
following weak equations: canonical theory. The Dirac quantization of such a the-
ory requires that the state functional be annihilated by
1
R′ = D Eα+ǫ kKjEα (17a) thequantumoperatorconstraint,implyingthatthestate
i β α i ij α k functional has to be invariant under small gauge trans-
i i formations, i.e. gauge transformations in the connected
− Πf5 Σ ψ+ ψΣ f5 Ω≈0,
4 (β) i 4 i (β) componentoftheidentity. Butthecanonicaltheorydoes
1 not provide any suggestion on how the state functional
C′ =2EβD Ki + ǫij EβR k (17b)
α i [α β] 2β k i αβj behaves under large gauge transformations. The behav-
ior of the wave function under large gauge transforma-
+Πf5 D ψ+D ψf5 Ω≈0,
(β) α α (β) tions can be studied by using the Chern-Simons func-
tionals. More specifically, by rescaling the wave func-
tion by the exponential of the Chern-Simons we can au-
1
C′ = EαEβ ǫikRl +2Ki Kk (17c) tomatically diagonalize the large gauge transformations
2 i k l αβ [α β]
(cid:16) (cid:17) operator [15]. The result is that the modifications intro-
−iEiα Πf(5β)Σ0iDαψ−DαψΣ0if(5β)Ω duced by the rescaling affect the Gauss constraint and
(cid:16) (cid:17)
the Hamiltonianof the theory,providingus with exactly
i
+ ǫi EαKk Πf5 Σlψ−ψΣlf5 Ω theexpressionwewouldhaveobtainedifwestartedfrom
2 kl i α (β) (β)
(cid:16) (cid:17) the well known topological modification of the standard
1
+ EαKk Πf5 ψ+ψf5 Ω ≈0, action.
2 k α (β) (β)
(cid:16) (cid:17) Here we have demonstrated that the same happens in
wheref5 =1− iγ5,andRl = 1ǫl R ik. Itisworth gravity and, even though the details regarding the large
(β) β αβ 2 ik αβ gauge transformations have been omitted to limit the
noting that the gravitational contributions in the above
length of the paper (and will be described in [21]), their
expressionsareexactlythoseobtainablestartingfromthe
absence does not prevent one from drawing the conclu-
Holst action. In fact, the above modified canonical con-
sion that in temporal gauge fixed gravity the Nieh-Yan
straints (17) can be extracted from a 3+1 splitting of
functionalplaysthesameroleastheChern-Simonsfunc-
the non-minimal action (5), so that an interesting topo-
tionals,allowingus to shedsomelightonthe topological
logical new aspect of the Ashtekar-Barbero formulation
aspectsoftheAshtekar-Barberoformulationofcanonical
of canonical gravity has been extracted and will be fur-
gravity.
ther reinforced by studying the role of the large gauge
And, finally, we would like to stress that the Immirzi
transformations in this framework [21].
parameterβisintroducedinthisframeworkintherescal-
ing (14), where it plays the same role as the so called
IV. DISCUSSION θ-angle plays in QCD.
In this brief paper we have demonstrated that a non-
Acknowledgments
minimal actionexists which is dynamically equivalentto
theEinstein-Cartanaction,sinceitdeviatesfromthelat-
ter due to the presence of two modifications which can The author expresses his gratitude to C. Rovelli for
easily be reduced to a topological term. This fact sug- stimulatingdiscussionsandsuggestionsandtotheQuan-
gests a simple analogy with the extension of Yang-Mills tum Gravity group in CPT Marseilles for hospitality.
gauge theories to contain topological terms. It is well This work is financially supported by “Fondazione A.
known that in Yang-Mills gauge theories the presence Della Riccia”. Shelly Kittleson is also acknowledged for
of a local symmetry generates a Gauss constraint in the her reading of the manuscript.
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