Wheeler-DeWitt quantization for point-particles ∗† Ward Struyve∗ March 15, 2021 Abstract We present the Hamiltonian formulation of a relativistic point-particle coupled to Einstein gravity and its canonical quantization `ala Wheeler-DeWitt. In the resulting quantum theory, the wave functional is a function of the particle coordi- nates and the 3-metric. It satisfies a particular Hamiltonian and diffeomorphism constraint, together with a Klein-Gordon-type equation. As usual in the Wheeler- DeWitt theory, the wave function is time-independent. This is also reflected in the Klein-Gordon-type equation, where the time derivative is absent. Before consid- ering gravity, we consider the coupling of a particle with electromagnetism, which is treated similarly, but simpler. 1 Introduction Canonical quantization of Einstein gravity leads to the Wheeler-DeWitt theory where the state is described by a wave functional Ψ(hij), which is a function of a 3-metric hij(x) [1]. A common way to introduce matter is through a scalar field, so that the wave functional also is a function of a scalar field φ(x), i.e., Ψ = Ψ(hij,φ). A remarkable feature is that the wave functional turns out to be time-independent. The wave equations are merely constraints on the wave functional: the Hamiltonian and the diffeomorphism arXiv:2012.08663v2 [gr-qc] 11 Mar 2021 constraint. The time-independence of Ψ leads to the question of how time evolution can be accounted for. There are various attempts to answer this question, but there is far from a consensus on the right approach [1–4]. In this paper, we consider the coupling of gravity to point-particles instead of a scalar field. This allows for the study of quantum gravity for a fixed number of particles without having to consider quantum field theory. This system has been studied in the context of the problem of time by Rovelli [5, 6] and Pavˇsiˇc[7, 8]. Classically, the history of a point-particle is described by its world-line Xµ(λ), which depends on an arbitrary parameter λ, while the space-time metric gµν(x) is a function of space-time x = (t, x).1 The particle and field are hence parameterized differently, ∗Instituut voor Theoretische Fysica, KU Leuven, Belgium †Centrum voor Logica en Filosofie van de Wetenschappen, KU Leuven, Belgium 1Units are assumed such that ~ = c = 1. 1 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.