arXiv:2012.08663v2 [gr-qc] 11 Mar 2021 otx ftepolmo ieb oel 5 ]adPavˇsiˇc and 8]. 6] [7, [5, be Rovelli has by system time n of This fixed problem theory. a the field for of gravity quantum context quantum consider of to study having the for without allows This field. scalar eacutdfr hr r aiu tepst nwrti que [1–4]. this approach answer right to the attempts on various consensus ho are a of There from question the for. to t accounted leads and Ψ be Hamiltonian of the time-independence functional: The wave constraint. the time-independ on be constraints to merely out turns are functional wave the that is feature ucinlas safnto fasaa field scalar a of function a is also functional eed na rirr parameter arbitrary an on depends fspace-time of h h tt sdsrbdb aefntoa Ψ( functional wave a by Wheeler-DeW the described to is leads state gravity the Einstein of quantization Canonical Introduction 1 ij 1 ∗ † ( lsial,tehsoyo on-atcei ecie yisworld-line its by described is point-particle a of history the Classically, nti ae,w osdrteculn fgaiyt point-particles to gravity of coupling the consider we paper, this In nt r sue uhthat such assumed are Units etu orLgc nFlsfi a eWtncapn ULeuven KU Wetenschappen, de van Filosofie en Belgium Logica Leuven, voor KU Centrum Fysica, Theoretische voor Instituut x hee-eitqatzto o point-particles for quantization Wheeler-DeWitt 1.Acmo a oitouemte struhasaa ed oth so field, scalar a through is matter introduce to way common A [1]. ) rn rvt,w osdrteculn fapril ihel with simpler. particle but a similarly, of treated coupling a is the is consider derivative we gravity, time ering the where This equation, time-independent. Klein-Gordon-type is function wave A Hamiltonia the theory, equation. particular DeWitt Klein-Gordon-type a a satisfies with It together constraint, 3-metric. function a the Whe is and `a la functional nates quantization wave po the canonical theory, relativistic its quantum a resulting and of gravity Einstein formulation to Hamiltonian the present We x ( = t, x ). 1 h atceadfil r ec aaeeie differently, parameterized hence are field and particle The ~ = c 1. = ac 5 2021 15, March adStruyve Ward λ hl h pc-iemetric space-time the while , Abstract 1 φ ( h x ij ,ie,Ψ=Ψ( = Ψ i.e., ), ,wihi ucino 3-metric a of function a is which ), ∗ † crmgeim which ectromagnetism, sa nteWheeler- the in usual s n diffeomorphism and n ftepril coordi- particle the of sn.Bfr consid- Before bsent. sas eetdi the in reflected also is lrDWt.I the In eler-DeWitt. Belgium , n.Tewv equations wave The ent. n-atcecoupled int-particle h to,btteei far is there but stion, ieeouincan evolution time w ij g φ , µν ediffeomorphism he me fparticles of umber nsuidi the in studied en .Aremarkable A ). t hoywhere theory itt ( x safunction a is ) X nta fa of instead ttewave the at µ ( λ ,which ), with X0(λ) = t. To pass to the Hamiltonian picture, a common temporal parameter for the field and the particle needs to be chosen. This is already familiar from the case of the free particle in Minkowski space-time. In that case, the disadvantage of choosing the parameter t is that one needs to take a square root to express the Hamiltonian in terms of the momenta, resulting in the Hamiltonian H = p2 + m2. Choosing the ± parameter λ instead leads to a zero Hamiltonian, but with the constraint p pµ +m2 = 0. p µ Quantization of the latter leads to the Klein-Gordon equation, while the former leads to the positive or negative frequency part of the Klein-Gordon equation. The situation in the case of gravity is similar. Rovelli [5,6] considered the parameter t and ended up taking a square root. Pavˇsiˇcseems to use t for the field, while using λ for the particle. This does not amount to a standard Hamiltonian formulation and hence makes this quantization of the theory questionable. In this paper, the Hamiltonian formulation is presented using λ as the common temporal parameter, together with the corresponding quantum theory. This avoids needing to take the square root and as such generalizes Rovelli’s treatment. The wave function is again time-independent; It does not depend on X0, just as in Rovelli’s treatment, but contrary to the findings of Pavˇsiˇc. So X0 does not appear as a possible candidate for the evolution parameter (contrary to for example the treatment of dust [9]). It will appear however that Pavˇsiˇc’s theory (when suitably interpreted) can be derived from ours in a mixed Schr¨odinger-Heisenberg picture. To start with, the treatment of point-particles coupled to an electromagnetic field is considered, in Minkowski space-time, which shares many features with the gravitational case.
2 Electrodynamics
To formulate the quantum theory of a point-particle interacting with an electromagnetic field, we start from the classical theory, pass to the Hamiltonian picture, and apply the usual canonical quantization methods [10–14]. Consider a classical point-particle with worldline Xµ(λ) interacting with an electro- µ magnetic field with vector potential A =(A0, A). Derivatives with respect to λ and t will be denoted by respectively by primes and dots throughout, e.g. X′µ = dXµ/dλ. It is also assumed throughout that X′0 > 0. The action is
= + , (1) S SM SF with = m dλ X′µ(λ)X′ (λ) e dλX′µ(λ)A (X(λ)) (2) SM − µ − µ Z q Z and 1 = d4xF (x)F µν (x), (3) SF −4 µν Z where F µν = ∂µAν ∂ν Aµ is the electromagnetic field tensor. The temporal parameter in the matter action and− the field action are different. In the matter action, the parameter
2 λ acts as temporal parameter, while t does it for the electromagnetic field. In order to pass to a Hamiltonian formulation, a common temporal coordinate is needed. We will first use t and then λ. To write the matter action in terms of t, the identity 1 = dtδ(t X0(λ)) is inserted − in the action and an integration over λ is performed. Defining Xµ(t)= Xµ(λ(t)), where R X0(λ(t)) 1, leads to ≡ e ˙ ˙ = m dt 1 X(t)2 + e dtX(t) A(t, X(t)) e dtA (t, X(t)). (4) SM − − · − 0 Z q Z Z ˙ e e e e Writing = dtL(X, X, A, A˙), with L the Lagrangian, the canonical momenta are S R e e ∂L X P = = m + eA(X), (5) ˙ 2 ∂X ˙ 1 e X e q − e δL e δL Π0(x)= =0, Π(x)= e = A˙ (x)+ ∇A0(x). (6) δA˙ 0(x) δA˙ (x) The canonical Hamiltonian is ˙ H = P X + d3x(π A˙ + Π A˙ ) L · 0 0 · − m Z = e + eA0(X)+ HF , (7) ˙ 2 1 X q − e where HF is the usual Hamiltoniane for the free electromagnetic field: 1 1 H = d3x Π2 + (∇ A)2 + A ∇ Π . (8) F 2 2 × 0 · Z As is well known, there are two constraints: the primary constraint
Π0 =0, (9) and the secondary constraint (which is the Gauss law)
∇ Π + eδ(x X)=0. (10) · − The matter part of the Hamiltonian (7) is not yete expressed in terms of the momenta. Using m2 2 = m2 + P eA(X) , (11) ˙ 1 X(t)2 − − we obtain e e e 2 H = m2 + P eA(X) + eA (X)+ H , (12) ± − 0 F r e 3 e e which involves the choice of a square root. Quantization in the Schr¨odinger picture (and dropping the tildes) leads to the fol- lowing equations for Ψ(X, A, t):
i∂ Ψ= m2 (∇ ieA(X))2Ψ+ eA (X)Ψ+ H Ψ, (13) t ± − − 0 F q δΨ δΨ =0, i∇ eδ(x X)Ψ=0,b (14) δA0 · δA(x) − − where 1 δ2 H = d3x + [∇ A(x)]2 . (15) F 2 −δA(x)2 × Z The equations (14) areb the constraints (9) and (10) that are imposed as operator con- straints. The square root could be eliminated by considering the square of (13) to obtain a Klein-Gordon-like equation. If λ rather than t is taken as a common temporal parameter then this is indeed the equation that will be obtained, as we will now show. To use λ as the temporal parameter, 1 = dλδ(λ λ(t)) is inserted in the field − action. Defining A(λ, x) = A(X0(λ), x), so that A′ = AX˙ ′0, the field action can be R written as e = dtL (A, A˙)= dλLe ∗ (A, A′), (16) SF F F Z Z with e e ∗ ′ ′0 ′ ′0 LF (A, A )= X LF A, A /X . (17)
∗ ′ ′ With = dλL (X,X , A, A ),e thee conjugate momentae e are S R ∗ e e δL δL Πµ(x)= = = Πµ(x), (18) δA′µ(x) δA˙ µ(x)
e∗ ′ ∂L X e µ X 0 Pµ = ′µ = m ′ν ′ eAµ( ) δµHF (A, Π). (19) ∂X − X Xν − − So the expressions for the momentap for the field aree just the sameease before, cf. (6). The particle momentum P0 gets a contribution from field Lagrangian which is just minus the field Hamiltonian (8). The canonical Hamiltonian is
H∗ = P X′µ + d3xΠ (x)A′µ(x) L∗ =0 (20) µ µ − Z and is zero because of the reparameterizatione invariancee of the action. There are two primary constraints: χ1 = Π0 = 0 (as before) and
χ = P + eA (X)+ δ0H (A, Π) P µ + eAµ(X)+ δµH (A, Π) m2 =0. (21) 2 µ µ µ F 0 F − h i h i e e e e e e 4 There is a secondary constraint which follows from the requirement that the Poisson bracket of the two primary constraints with the total Hamiltonian
∗ ∗ 3 HT = H + d xλ1(x)χ1(x)+ λ2χ2 (22) Z vanishes, which results in [χ1(x), χ2]P =0 or
δH (A, Π) F + eδ(x X)=0, (23) δA0(x) − e e which amounts to the Gauss constrainte (10). Quantization (again dropping the tildes) leads to the following equation for Ψ(X, A):
0 µ µ µ 2 ∂µ + ieAµ(X) + iδµHF ∂ + ieA (X) + iδ0 HF Ψ+ m Ψ=0, (24) with HF as before in (15), togetherb with the constraints (14).b This Klein-Gordon-type equation is just the square of (13), provided X0 is identified with t. Sob far we have dealt with just a single particle. How to extend the theory to many particles? The theory with the square root Hamiltonian (13) is directly generalized to many particles (13), with now ψ = ψ(X1,..., Xn, A, t):
i∂ Ψ= m2 (∇ ieA(X ))2 + eA (X ) Ψ+ H Ψ, (25) t − k − k 0 k F k X q b δΨ δΨ =0, i∇ e δ(x X )Ψ=0. (26) δA · δA(x) − − k 0 k X We have chosen the positive square root Hamiltonian for each particle. The Klein- Gordon-type equation (24) is not that straightforwardly generalizable to many particles because it is not of the Schr¨odinger form like (13). One option is to run through the quantization procedure again for many particles, but this seems rather complicated. Another option is to recast the Klein-Gordon-type equation (24) in a Schr¨odinger form, which could be done using the Kemmer formulation [15] (which actually concerns a Dirac-like equation for spin-0), and which at least in the case of external field is directly extendable to many particles [16]. Yet another option is to use the multi-time picture where the wave function depends on the time-component of each of the particles [17]. We will not pursue this further here.
3 Gravity
The analysis of gravity proceeds completely analogously as that of electromagnetism. The action for a classical point-particle coupled to gravity is
= + , (27) S SM SG 5 with = m dλ g (X(λ))X′µ(λ)X′ν(λ) (28) SM − µν Z q and 1 = d4x√ gR, (29) SG −κ − Z with κ = 16πG, is the Einstein-Hilbert action. Before considering the Hamiltonian picture in terms of the temporal parameter λ, we recall the usual Hamiltonian formulation for the Einstein-Hilbert action in terms of t (following the conventions of [18]). It is supposed that space-time can be foliated in terms of space-like hypersurfaces such that the space-time manifold is diffeomorphic to R Σ, with Σ a 3-surface. Coordinates xµ = (t, x) are chosen such that the time × coordinate t labels the leaves of the foliation and x are the coordinates on Σ. In terms of these coordinates, the metric and its inverse are written as
2 i 1 −N i N NiN Ni µν N 2 N 2 gµν = − − , g = −N i N iN j ij , (30) Ni hij 2 2 h − − N N − i where N and N are respectively the lapse and the shift vector, and hij is the Riemannian i metric on Σ, with h its determinant. Spatial indices of hij and N and the corresponding momenta are raised and lowered with this spatial metric. With LG the gravitational Lagrangian, the canonically conjugate momenta are δL 1 πij = G = √h(Kij Khij) (31) δh˙ ij −κ −
(which is a tensor density of weight 1), with − 1 K = (D N + D N h˙ ), K = K hij, (32) ij 2N i j j i − ij ij the extrinsic curvature, and δL δL π = G =0, π = G =0. (33) δN i δN i The Hamiltonian is H = d3x N + N i , (34) G H Hi ZΣ with = κG πijπkl + , = 2h D πjk, (35) H ijkl V Hi − ik j where G = (h h + h h h h )/2√h is the DeWitt metric, D is the covariant ijkl ij jl il jk − ij kl i derivative corresponding to h , and = √hR(3)/κ is the gravitational potential den- ij V − sity. Apart from the primary constraints (33), there are also the secondary constraints
=0, =0. (36) H Hi 6 0 In order to use the temporal variable λ, we introduce gµν (λ, x)= gµν (X (λ), x) and similarly for other variables. The total action is e ∗ ′ ′ = dλL (X,X , g, g ). (37) S Z The momenta for the metric are the same as before,e e i.e., ∗ ∗ ∗ ij δL δLG ij δL δL π = = = π , π = =0, πi = =0, (38) ′ ˙ i δhij δhij δN δN e e µ e and the momentum fore the particle becomes (indicese of X and Pµ eare raised and lowered with the metric gµν (X))
∗ ′ ∂L Xµ 0 e Pµ = ′µ = m ′ν ′ δµHG, (39) ∂X − X Xν − where p e i ij HG = HG(N, N , gij, π ). (40) The canonical Hamiltonian is zero, i.e., e e e e e H∗ = P X′µ + d3xπij(x)g′ (x) L∗ =0. (41) µ ij − ZΣ There are three primary constraints: e e
π =0, πi =0, (42)
χ = gµν(X) P + δ0H P + δ0H m2 =0. (43) µe µ G eν ν G − h i h i There are two secondary constraints corresponding to δχ/δN = 0 and δχ/δN i = 0, e e e resulting in
0 1 e e (x)= δ(x X) N(X)p + HG (44) H − N(X) ! and e e e (x)= δ(x X)p ,e (45) Hi − i where (as in (40)) the tilde on and denotes the functions (35) evaluated for fields H e Hi with tildes. There are no further constraints. By multiplying (44) with N(ex) ande (45) with N i(x), integrating over all space, and 0 i 2 using p =(p0 N (X)pi)/N(X) , it follows that − e e e e p0 =0. (46) Hence, another interesting consequence is that
N(x) (x)+ N i(x) (x)= δ(x X)H . (47) H Hi − G e e e e7 e Summarizing, the Hamiltonian dynamics is completely determined by the constraints (42)-(45). Using (46), they can be simplified to (dropping the tildes):
π =0, πi =0, (48)
H N i(X)p 2 N(X)2 hij(X)p p + m2 =0, (49) G − i − i j 1 (x)= δ(x X) N i(X)p + H , (50) H − N(X) − i G (x)= δ(x X)p . (51) Hi − i 2 Quantization in the Schr¨odinger picture leads to the following equations for Ψ(X, hij):
2 iN i(X) + H Ψ N(X)2 2 + m2 Ψ=0, (52) ∇i G − −∇ h i b 1 (x)Ψ = δ(x X) iN i(X) + H Ψ, (53) H − N(X) ∇i G h i b (x)Ψ = iδ(x X) Ψ, b (54) Hi − − ∇i where is the covariant derivative with respect to the metric h (X), with ψ = ∂ ψ, i b ij i i 2 = ∇ i is the Laplacian, and ∇ ∇ ∇i∇
3 i HG = d x N + N i , (55) Σ H H Z b δ2 b b δ = κGijkl + (h, φ), i = 2ihikDj . (56) H − δhijδhkl V H δhjk The wave functionb Ψ does not depend on N, N i andb X0, because of the operator constraints following from (46) and (48). The latter means that the wave functional does not depend on time as is familiar in Wheeler-DeWitt quantization. The wave functional does not depend on N and N i, but they still appear in the wave equations. Choosing N = 1 and N i = 0, results in
H2 + 2 m2 Ψ=0, (57) G ∇ − (bx)Ψ = δ(x X)H Ψ, (58) H − G (x)Ψ = iδ(x X) Ψ. (59) Hbi − − b∇i Applying the quantization recipe using t as temporal coordinate leads to the following b quantum theory [5, 6]:
(x)Ψ = δ(x X)√ 2 + m2Ψ, (x)Ψ = iδ(x X) Ψ, (60) H ± − −∇ Hi − − ∇i 2We have chosen the Laplace-Beltrami operator ordering for the particle but not for gravity [19]. b b
8 with Ψ(X, hij). This theory also follows from (48)-(51) if the square root is taken in (49). As in the case of electromagnetism, the extension to many particles is straightforward, leading to
(x)Ψ = δ(x X ) 2 + m2Ψ, (x)Ψ = i δ(x X ) Ψ, (61) H − k −∇k Hi − − k ∇ki k k X q X b b with k the particle label, Ψ(X1,..., Xn, hij). The extension of the Klein-Gordon form (57)-(59) to many particles requires a bit more effort (though apparently less than in the case of electromagnetism due to absence of the time derivative). The natural generalization seems to be O2 + 2 m2 Ψ=0, (62) k ∇k − (x)Ψb = δ(x Xk)OkΨ, (63) H − k X b b (x)Ψ = i δ(x X ) Ψ. (64) Hi − − k ∇ki k X b The operators Ok can be determined almost completely by self-consistency. Namely, for configurations (X1,..., XN ) with all the Xk different, we can integrate (63) over a test function f(bx) such that f(Xk) = 1 and f(Xl) = 0 for l = k, to obtain OkΨ = 3 6 d xf(x) (x)Ψ. If not all the Xk are different, the action of the Ok could be defined similarly byH requiring symmetry. For example, for a configuration with X = bX , we R k l can requireb that OkΨ = OlΨ. This generalization is consistent withb (61), in the sense that (61) follows by taking particular square roots in (62). The Wheeler-DeWittb b quantization of the relativistic particle was considered before by Pavˇsiˇc[7]. However, in the quantization procedure no common temporal parameter was used. In the definition of the particle momentum the parameter λ was used, while in the definition of the field momenta the parameter t was used. As such the quantization deviates from the usual recipe and the resulting quantum theory does not agree with the one presented here. However, the equations obtained by Pavˇsiˇcfollow from ours when interpreted in a partial Heisenberg picture (and as such is reminiscent of the Dirac-Fock- Podolsky formulation of a particle interacting with an electromagnetic field [20]). That is, defining
b 0 b 0 b 0 iHGX iHGX −iHGX Φ(X, hij) = e Ψ(X, hij), hij(X) = e hij(X)e , (65)
b 0 − b 0 b 0 − b 0 (x,X0) = eiHGX (x)e iHGX , (x,X0) = eiHGX (x)e iHGX , (66) H H Hi Hi the equations (57)-(59) reduce to b b b b ∂2 2 + m2 Φ=0, (67) 0 − ∇ (x,X0)Φe = iδ(x X)∂ Φ, (68) H − − 0 b 9 (x,X0)Φ = iδ(x X) Φ, (69) Hi − − ∇i 2 where and concern the covariant derivative with respect to the metric hij(X). (Pavˇsiˇcalso∇ chooses∇ a differentb operator ordering, bute that difference is not essential.) For Pavˇsiˇcthee e appearance of the time derivative in (68) is interesting the light of the problem of time. However, the proper quantum theory is time-independent.
4 Acknowledgments
This work is supported by the Research Foundation Flanders (Fonds Wetenschappelijk Onderzoek, FWO), Grant No. G066918N. It is a pleasure to thank Christian Maes and Kasper Meerts for useful discussions.
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