New commutation relations for quantum gravity

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by  Chopin Soo

New commutation relations for quantum gravity
Chopin Soo
∗
Department of Physics, National Cheng Kung University, Taiwan
Hoi-Lai Yu
†
Institute of Physics, Academia Sinica, Taiwan
A new set of fundamental commutation relations for quantum gravity is presented. The basic
variables are the eight components of the unimodular part of the spatial dreibein and eight SU(3)
generators which correspond to Klauder’s momentric variables. The commutation relations are
not canonical, but they have well defined group theoretical meanings. All fundamental entities
are dimensionless; and quantum wave functionals are preferentially selected to be in the dreibein
representation.
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Publication details:
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New Commutation Relations for Quantum Gravity,
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Chopin Soo and Hoi-Lai Yu,Chin. J. Phys. 53, 110106 (2015),

n ChineseJournalofPhysics(Volume53,Number6(November2015)) SpecialIssueOntheoccasion
a of 100 years since thebirth of Einstein’s General Relativity
J

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1

]
c
I. COMMUTATION RELATIONS OF GEOMETRODYNAMICS AND MOMENTRIC VARIABLES
q
-
r
Successful quantization of the gravitational field remains the preeminent challenge a century after the completion
g
[ ofEinstein’sgeneralrelativity(GR).Inquantumfieldtheories,‘equaltime’fundamentalcommutationrelations(CR)
- part and parcel of causality requirements - are predicated on the existence of spacelike hypersurfaces. Geometro-
1
dynamics bequeathed with positive-definite spatial metric is the simplest consistent framework to implement them.
v
Expressedintermsofdreibein, e ,thespatialmetric,q :=δabe e isautomaticallypositive-definiteifthedreibein
4 ai ij ai bj
2 is realandnon-vanishing,moduloSO(3,C) gaugerotationswhichleaveqij invariantunder these localLorentztrans-
8 formations. To incorporate fermions, it is also necessary to introduce the dreibein which is more fundamental than
3 the metric.
0 Inusualquantumtheories,the fundamental canonical CR,[Q(x),P(y)]=i~δ(x−y), implies P is the generatorof
~
1. translations of Q which are also symmetries of the ‘free theories’ in the limit of vanishing interaction potentials. For
0 geometrodynamics, the corresponding canonical CR is [qij(x),π˜kl(y)] = i~21(δikδjl +δilδjk)δ(x−y). However, neither
6 positivityofthespatialmetricispreservedunderarbitrarytranslationsgeneratedbyconjugatemomentum;noristhe
1 ‘free theory’ invariant under translations when interactions are suppressed, because G π˜klπ˜mn in the kinetic part
klmn
:
v of the Hamiltonianalso contains the DeWitt supermetric[1], Gijkl, whichis dependent upon qij. In quantum gravity,
i states which are infinitely peaked at the flat metric, or for that matter any particular metric with its corresponding
X
isometries, cannot be postulated ad hoc; consequently, the underlying symmetry of even the ‘free theory’ is obscure.
ar Decomposition of the geometrodynamic degrees of freedom, (qij,πij), singles out the canonical pair (lnq13, π)
whichcommutes with the remainingunimodular qij :=q−13qij, andtraecelessπij :=q13 πij−31qijπ variables. Hodege
decomposition for compact manifolds yields δlnq31 = δT +∇iδYi, wherein the spatia(cid:0)lely-indepened(cid:1)ent δT is a three-
dimensionaldiffeomorphisminvariant(3dDI)quantitywhichservesastheintrinsictimeinterval,whereas∇ δYicanbe
i
gaugedawaysince Lδ→−N lnq31 = 32∇iδNi. A theory of quantumgeometrodynamicsdictated by first-orderSchr¨odinger
evolution in intrinsic time, i~δΨ = H Ψ, and equipped with diffeomorphism-invariant physical Hamiltonian and
δT Phys.
time-orderinghas been discussedand advocatedin Refs.[2–5]. The Hamiltonian, H = H¯(x)d3x, andorderingof
Phys β
the time development operator, U(T,T )=T{exp[−i T H (T )δT ]}, are 3dDI providRed
0 ~ T0 Phys ′ ′
R
H¯ = π¯ijG¯ π¯kl+V[q ] (1)
ijkl ij
q
∗Electronicaddress: [e-mail]
†Electronicaddress: [e-mail]
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is a scalar density of weight one; and Einstein’s GR (with β = 1 and V = − q [R − 2Λ ]) is a particular
√6 (2κ)2 eff
realization of this wider class of theories. Natural extensions of GR and the precise quantum Hamiltonian are
discussed in Refs.[2, 4].
The Poissonbrackets for the barred variables are,
{q¯ (x),q¯ (y)}=0, {q¯ (x),π¯ij(y)}=Pijδ(x−y), {π¯ij(x),π¯kl(y)}= 1(q¯klπ¯ij −q¯ijπ¯kl)δ(x−y); (2)
ij kl kl kl 3
with Pij := 1(δiδj +δiδj)− 1q¯ijq¯ denoting the traceless projection operator. This set is not strictly canonical. In
kl 2 k l l k 3 kl
themetricrepresentation,theimplementationofπ¯kl astraceless,symmetric,andself-adjointoperatorsisproblematic.
Remarkably, these difficulties can be cured by passing to the ‘momentric variable’ (first introduced by Klauder[6])
whichis classicallyπ¯i =q¯ π¯im. Intermsofspatialmetricandmomentricvariables,the fundamentalCRpostulated
j jm
(from which the classical Poisson brackets corresponding to (2) can be recovered)are then[4]
i~
[q¯ (x),q¯ (y)]=0, [q¯ (x),π¯k(y)]=i~E¯k δ(x−y), [π¯i(x),π¯k(y)]= δkπ¯i−δiπ¯k δ(x−y); (3)
ij kl ij l l(ij) j l 2 j l l j
(cid:0) (cid:1)
wherein E¯i := 1 δi q +δiq − 1δiq (with properties δjE¯i = E¯i q¯mn = 0;E¯i = E¯i = 5q ) is
j(mn) 2 m jn n jm 3 j mn i j(mn) j(mn) jil jli 3 jl
the vielbein for the su(cid:0)permetric G¯ (cid:1)= E¯m E¯n . Quantum mechanically, the momentric operators and CR can
ijkl n(ij) m(kl)
be explicitly realized in the metric representation by
~ δ ~ δ
π¯ji(x):= iE¯ji(mn)(x)δq¯ (x) = i δq¯ (x)E¯ji(mn)(x)=πˆ¯j†i(x) (4)
mn mn
which are self-adjoint on account of [ δ ,E¯i (x)]=0.
δq¯mn(x) j(mn)
II. NEW COMMUTATION RELATIONS
These momentric variables generate SL(3,R) transformations of q¯ = δabe¯ e¯ which preserve its positivity and
ij ai bj
unimodularity. Moreover, they generate at each spatial point, an SU(3) algebra. In fact, with 3×3 Gell-Mann
matrices λA=1,...,8, it can be checked that TA(x) := 1 (λA)jπˆ¯i(x) generates the SU(3) algebra with structure
~δ(0) i j
constantsfAB [7]. In(3)thereisanasymmetryinthatthereareonlyfiveindependentcomponentsinthesymmetric
C
unimodularq¯ (likewiseforthesymmetrictracelessπ¯ij in(2)),whereasthemixed-indextracelessmomentricvariable,
ij
π¯i, contains eight components. This asymmetry is rectified by unimodular dreibein-traceless momentric variables,
j
(e¯ai := e−13eai,TA), each having eight independent components, and they obey the new fundamental CR advocated
in this work,
λA δ(x−y) δ(x−y)
[e¯ (x),e¯ (y)]=0, [e¯ (x),TA(y)]=i( )ke¯ , [TA(x),TB(y)]=ifAB TC . (5)
ai bj ai 2 i ak δ(0) C δ(0)
Anumberofremarkableandintriguingfeaturesareencodedinthis setofCR.Itisnoteworthythatin(5)allentities,
including(e¯ai,TA)and δ(δx(−0)y),aredimensionless;andneitherthegravitationalcouplingconstantnorPlanck’sconstant
make their appearance. δ(0) := limx→yδ(x−y) denotes the coincident limit; so there are no divergences in δ(δx(−0)y)
which is unity in the coincident limit and vanishing otherwise. The second CR in (5) implies
α (x)λA j
exp i α TBδ(0)d3y e¯ (x)exp −i α TAδ(0)d3y = exp( A ) e¯ (x), (6)
(cid:16) Z B ′(cid:17) ai (cid:16) Z A (cid:17) (cid:16) 2 (cid:17)i ja
with exp(αA(x)λA) being a local SL(3,R) transformation; while the final CR in (5) is the statement that TA(x)
2
generates, at each spatial point, a separate SU(3) algebra.
Local Lorentz transformations are generated by the Gauss Law constraint Ga(x) := ǫabcebiπ˜ci − iπ˜ψτ2aψ = 0,
wherein π˜ai (which is classically π˜ai =2π˜ijea) is the canonical conjugate momentum of the dreibein, (ψ,π˜ ) denotes
j ψ
the conjugate pair of Weyl fermion, and τa are the Pauli matrices. However, e:=det(eai) and (lnq13, π) are Lorentz
singlets which commute with the Gauss Law constraint, and the latter can be reexpressed with traceless momentric,
ratherthanmomentum,variableasGa =ǫabce¯bie¯cjπ¯ji−iπ˜ψτ2aψ =0. ThatitgeneratesSO(3,C)localLoreentzrotations
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of the dreibein can be verified from [e¯ (x), i ηbG d3y] = ǫ ηb(x)e¯c(x), with ηb as gauge parameter; in addition,
ai ~ b abc i
it generates the associated local SL(2,C) tranRsformations of the fermionic degrees of freedom (d.o.f.).
ThegeneratorsTAobeytheSU(3)algebraof (5)andthusdonottotallycommuteamongthemselves. Aremarkable
consequence is that quantum wave functions cannot be chosen as functionals of TA, but quantization in the dreibein
representationisviableandpreferentiallyselectedbythefundamentalCR.Thisdiffersstarklyfromtheusualcanonical
CR which allows wave functionals to be realized in either of the conjugate representations; and it may explain why
our universe seems fundamentally and intuitively ‘metric’ in nature, and not ‘conjugately realized’. The dreibein
(andmetric representation)consistentlyguaranteesthathypersurfacesonwhichthe fundamentalCRare definedwill
be spacelike. An explicit realization in the dreibein representation is TA(x) = − i (λA)je¯ (x) δ ; and when
δ(0) 2 k ja δe¯ka(x)
operating on wave functionals of the spatial metric, Ψ[q ], this action of TA agrees with that of (4).
ij
The symmetry of the free theory now becomes transparent. It is characterized by SU(3) invariance generated by
the momentric, because the Casimir invariant TATA is related to the kinetic operator through
~2[δ(0)]2
2 TATA =π¯ji†π¯ij =π¯jiπ¯ij =π¯ijG¯ijklπ¯kl. (7)
ThespectrumofthefreeHamiltoniancanthusbelabeledbyeigenvaluesofthecompletecommutingsetateachspatial
point comprising the two Casimirs L2 = TATA,C = d TATBTC ∝ det(π¯i) with d = 1Tr({λ ,λ }λ ),
ABC j ABC 4 A B C
CartansubalgebraT3,T8,andisospinI = 3 TBTB. Anunderlyinggroupstructurehastheadvantagetheaction
B=1
of momentric onwave functionals by functiPonal differentiation canbe traded for the welldefined action of generators
of SU(3) on states expanded in this basis. To wit,
λA e¯ (x) δ
( )j ja he¯ | |l2,C,I,m ,m i =he¯ |TA(x) |l2,C,I,m ,m i . (8)
2 k iδ(0) δe¯ (x) bl 3 8 y bl 3 8 y
ka Y Y
y y
Grouptheoreticalconsiderationsalsoleadto a succinctdescriptionofgravitond.o.f. andtheir associatedquantum
excitations. In geometrodynamics, local SL(3,R) transformations of q¯ are generated through U (α)q¯ (x)U(α) =
kl † kl
(eα(2x))mk q¯mn(x)(eα(2x))nl, wherein U(α) = e−~i Rαijπ¯ijd3y[6]; while the generator of spatial diffeomorphisms for the
momentric and unimodular spatial d.o.f. is effectively D = −2∇ π¯j, with smearing NiD d3x = (2∇ Ni)π¯jd3x
i j i i j i
after integration by parts. The action of spatial diffeomorphisms can thus be subsumRed by specialRization to αi =
j
2∇ Ni, with the upshot that SL(3,R) transformations which are not spatial diffeomorphisms are parametrized by
j
αij complement to 2∇jNi. Given a background metric qiBj =q31q¯iBj, this complement is precisely characterized by the
choice of transverse traceless (TT) parameter (α )i := q α(ik) , because the condition ∇j(α )i = 0 excludes
TT j jBk Phys TT j
non-trivial Ni through ∇2Ni =0 if (α )i were of the form 2∇ Ni. The TT conditions impoBse four restrictions on
TT j Bj
thesymmetricα(Pijh)ys(x),lBeavingexactlytwofreeparameters. TheactionofUPhys(αTT)=e−~i R(αTT)ijπ¯ijd3x (whichis
thuslocalSL(3,R)modulospatialdiffeomorphism)onany3dDIwavefunctionalwouldresultinaninequivalentstate.
TT conditions however require a particular background metric to be defined. In Ref.[8] a basis of infinitely squeezed
stateswasexplicitlyrealizedbyGaussianwavefunctionalsΨ[q¯]qB ∝exp[−12 f˜ǫ(q¯ij−q¯iBj)G¯ijkl(q¯kl−q¯kBl)d3x]. 3dDIis
recoveredinthe limit ofzeroGaussianwidth withdivergentlim f˜ →δ(0R). These localizBedNewton-Wigner states
ǫ 0 ǫ
are infinitely peaked at qB which can then be deployedto actuali→ze the TT conditions. The action of U (α ) on
ij Phys TT
these states would thus generate two infinitesimal local physical excitations at each spatial point.
Acknowledgments
This work was supported in part by the Ministry of Science and Technology (R.O.C.) under Grant Nos. NSC101-
2112-M-006-007-MY3 and MOST104-2112-M-006-003,and the Institute of Physics, Academia Sinica.
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