A realist interpretation of unitarity in quantum gravity

Unitarity is a difficult concept to implement in canonical quantum gravity because of state non-normalisability and the problem of time. We take a realist approach based on pilot-wave theory to address this issue in the Ashtekar formulation of the Wheeler–DeWitt equation. We use the postulate of a definite configuration in the theory to define a global time for the gravitational-fermionic system recently discussed in Alexander et al (2022 Phys. Rev. D 106 106012), by parameterising a variation of a Weyl-spinor that depends on the Kodama state. The total Hamiltonian constraint yields a time-dependent Schrodinger equation, without semi-classical approximations, which we use to derive a local continuity equation over the configuration space. We implement the reality conditions at the level of the guidance equation, and obtain a real spin-connection, extrinsic curvature and triad along the system trajectory. We obtain quantum corrections to deSitter spacetime from the guidance equation. The non-normalisable Kodama state is naturally factored out of the full quantum state in the conserved current density, opening the possibility for quantum-mechanical unitarity. We also give a pilot-wave generalisation of the notion of unitarity applicable to non-normalisable states, and show the existence of equilibrium density for our system. Lastly, we find unitary states in mini-superspace by finding an approximate solution to the Hamiltonian constraint.


I. INTRODUCTION
Quantum gravity effects may become important in regimes where quantum fluctuations of the gravitational field and high curvature coincide, such as close to the classical big bang and black hole singularities.In such situations and given the perturbative nonrenormalizibility of quantum gravity, a non-perturbative treatment is a desired option.
A conservative approach to quantization is a Schrodinger quantization, such as the Wheeler-DeWitt equation (WDW) [1].The WDW equation is non-polynomial in the metric variables and is difficult to solve.Progress was made with the Ashtekar variables which rendered the WDW equation polynomial in the configuration variables [2].
A major leap in progress was found by Kodama by solving the WDW equation of general relativity in terms of the Ashtekar connection [3].This solution, called the Chern-Simons Kodama (CSK) is an exact wavefunction that solves the quantum WDW equation for a positive cosmological constant.It was shown that this Chern-Simons Kodama state consistently reduces to the Hartle-Hawking-Vilenkin state of de-Sitter space, and contains a multitude of other solutions, including black-hole quantum spacetimes [4,5].Recently, an exact CSK state was found with the inclusion of fermions [6].
Despite this success, the CSK state as well as other formulations of the WDW equation, is fraught with conceptual and technical problems that all approaches to the WDW suffer [7][8][9].Since time evolution is a gauge redundancy, the CSK state is timeless.Also, the Lorentzian CSK state is non-normalizable for the naive-inner product, although a recent proposal for a new non-perturbative inner product was proposed [10].The twin problems of time and non-normalizability make the definition of unitarity murky.Another issue is that the non-normalizable part of the Kodama state, when linearized yields gravitons with negative energy in its spectrum.These problems are to be expected since the CSK state is background independent and a proposed ground state.In this work, we address the interconnected problems of time, normalizability and unitarity by recasting the WDW equation in the Ashtekar formalism using pilot-wave theory [11][12][13][14][15], which is a realist formulation of quantum mechanics.
Our approach introduces three new ideas to attack these problems.First, we use the postulate of a definite configuration in pilot-wave to define for the first time a real, relational time in terms of variation of massless fermionic field.This allows us to discuss time evolution of the quantum state, which is shown to follow a Schrodinger equation, without using semi-classical approximations.Second, we approach the question of unitarity by deriving a continuity equation over the configuration space, instead of using operator valued reality conditions.This enables us to find a locally conserved current density on the configuration space and thereby discuss unitarity from quantum-mechanical perspective.
Third, we also generalize the notion of unitarity from pilot-wave perspective, which allows us to discuss unitarity without imposing normalizability.
The article is structured as follows.In section II we give an introduction to the Ashtekar formalism and the Kodama state.We give an introduction to field-theoretic pilot-wave theory with a brief discussion of complex massive scalar field in section III.In section IV, we develop a pilot-wave formulation of the gravitational-fermionic system in [6], making use of some of the ideas developed in [16].We first introduce a global time that parameterizes a particular variation of the fermionic field and then derive the continuity equation and corresponding current density in IV A. We discuss the physical interpretation in IV B, including normalizability in IV B 1, guidance equation for the Ashtekar connection in IV B 2, and reality conditions in IV B 3. We discuss the notion of unitarity in our approach in section V. We discuss quantum-mechanical unitarity using our continuity equation in V A, a generalized notion of unitarity using pilot-wave in V B, and the existence of pilot-wave unitary states in mini-superspace in V C 2. We discuss our results and future directions in VI.

II. THE ASHTEKAR FORMALISM AND THE CSK STATE
In pursuit of a Wheeler-DeWitt quantization of gravity, it is instructive to understand how the Ashtekar connection and the resulting Hamiltonian, diffeomorphism, and gauge constraints emerge from a manifestly covariant 4D theory of gravity.In what follows we closely follow the derivation of the Ashtekar variables in the work of [6].In the Ashtekar formalism [2,17], gravitational dynamics on a four-dimensional manifold M 4 is not described by a metric g µν but, 1 rather, a real-valued gravitational field e I µ (x), mapping a vector v µ in the tangent space of M 4 at the point x into Minkowski spacetime M 4 [with metric η IJ = diag(−1, 1, 1, 1) IJ ].Locally, the metric on M 4 is g µν = η IJ e I µ e J ν .The Lorentz connection ω J µI is ω J I ≡ ω J µI dx µ , dω J I ≡ ∂ µ ω J νI dx µ ∧ dx ν is the exterior derivative, and the curvature of ω is R J I = dω J I +ω K I ∧ω J K .The action of self-dual gravity is (up to the gravitational constant 8πG) where * is the Hodge dual, the first term is the Hilbert-Palatini action and the second is the Holst term (proportional to the first Bianchi identities in the absence of torsion).
Here we are interested in the Hamiltonian formulation in Ashtekar variables [2,19].In the gauge choice e 0 µ = 0, it is convenient to define the densitized triad E a i = ǫ ijk ǫ abc e j b e k c , which is conjugate to the self-dual connection As the Lorentz connection (and, in particular, the spin connection Γ i a ≡ − 1 2 ǫ ij k ω k aj ) is real, A is complex-valued and obeys the reality conditions (for a discussion, see e.g., [20]) where X denotes complex conjugate of X and the spin connection solves the equation The Poisson bracket of the elementary variables A and E is 1 We use the mostly plus metric signature, i.e. η µν = (−, +, +, +) in units of c = 1.We use boldface letters x to indicate 3-vectors, and we use x to denote 4-vectors.Conventions for curvature tensors, covariant and Lie derivatives are all taken from Carroll [18].Greek indices (µ, ν, . ..) denote spacetime indices, Latin indices (a, b, . ..) denote spatial indices, and Latin indices (I, J, . ..) and (i, j, . ..) denote indices for the internal space ranging from 0, . . . 3 for the former and 1, . . . 3 for the latter.
Introducing the "magnetic" field and the gauge field strength Now, let us construct the CSK state by solving the Wheeler-DeWitt equation.Given the Hamiltonian which acts on some wave function ψ[A], and we want to find the form of ψ[A] that is annihilated by (7).Applying the regular canonical quantization procedure, i.e.
the annihilation of the quantum state becomes Putting the expression inside the brackets to zero, we get Contracting both sides with ǫ dab gives us where ℓ 2 Pl = 8πG is the Planck length.Recognizing the term multiplying the wave function to be the Chern-Simons functional, we can write down the exact solution to the Wheeler-DeWitt equation as being where N is some normalization constant independent of the gauge field and is the Chern-Simons functional, with the trace taken in the Lie algebra.It can be said that the WKB semiclassical limit of the CSK state is de Sitter spacetime [21], 2 with Now that we have the CSK state solely in terms of the gravitational connection and the cosmological constant, we would like to explore a full nonperturbative state that also includes the fermionic Hamiltonian.

III. PILOT-WAVE FORMULATION OF MASSIVE COMPLEX SCALAR FIELD
It is helpful to begin with a discussion of pilot-wave theory with field ontology as an example (for further discussion, see [11,14,23,24]).Consider a massive complex scalar field φ( x, t) on flat space-time with the Lagrangian density where m labels the mass, φ labels the complex conjugate of φ and the space-time metric is η = (1, −1, −1, −1).The conjugate momenta are π = δL/δ∂ 0 φ = ∂ 0 φ and To quantize the system, the canonical commutation relations are imposed [25].Working in the φ, φ representation, the conjugate momenta are represented by the operators π → −i δ/δφ, π → i δ/δφ and the Schrodinger equation becomes where M labels the spatial manifold and Ψ[φ, φ, t] is a functional of φ and φ.Using (16) 2 See [22] for criticisms of this view.
and its complex conjugate, we can prove the following continuity equation where ∇ φ = M δ/δφ and Here Ψ = Re iS/ and R, S are real time-dependent functionals of φ, φ.The evolution of the field is given by the guidance equation Equation (19) implies that the evolution of the scalar field is spatially nonlocal (over M).
This is because S[φ, φ, t] is a functional of φ( x), φ( x), and thus depends on values of φ( x), φ( x) at all x in M in general.Also note that ( 19) is a local guidance equation in the configuration space (φ, φ).That is, the evolution of a particular field φ( x) does not depend on other field configurations as S[φ, φ, t] in ( 19) is evaluated at a particular point on the configuration space.

SYSTEM
We wish to quantize general relativity with a positive cosmological constant and a twocomponent Weyl Spinor.As we will see, the corresponding total Hamiltonian constraint, which was discovered in [6], becomes equivalent to a time dependent Schrodinger equation, where the first order spinor Hamiltonian will play exactly the role of a relational clock.This approach has advantages over scalar-field relational clocks since the latter may introduce negative norm states due to being second order in time derivative, as opposed to the Dirac equation, which is first order.
The Gravitational-Spinor action is: where the covariant derivative is: upon performing a 3 + 1 decomposition as discussed in section II, the total Hamiltonian density for the combined fermionic gravitational system [6] is and N > 0, N a are the lapse function and shift vectors respectively.Here ξ A ( x) is a two-component Weyl spinor and the corresponding conjugate momentum is labelled by Π B ( x) (A, B ∈ {+, −}) [6].We have chosen the Ashtekar ordering [2,3,21,26] for the purely gravitation part of the constraints.For the interaction terms between gravity and fermion, we have Weyl ordered Êai and ordered Π B to directly operate on the quantum state.We remove divergent terms throughout in our calculations.
The total Hamiltonian constraint is where M labels the spatial manifold.We use the quantization scheme (using commutators [27,28]) where we have used natural units.Equation (25) implies Let us define where That is, we choose a particular variation of the fermionic field ξ, suggested by the form of the Hamiltonian, to implicitly define a real, formal time variable t ′ that parameterizes this variation.Note that equation ( 29) is not a semi-classical background as ξ is piloted by Ψ[A, ξ] via its dependence on A ai (see equation ( 43) below).Equation ( 29) is naturally consistent with a pilot-wave interpretation as the latter posits a definite system configuration (A, ξ).It is also consistent with any other interpretation where a definite configuration of the quantum system is physically meaningful.
We can use (29) to rewrite equation (27) as which resembles the time-dependent Schrodinger equation.

A. Continuity equation and current density
Using the complex conjugates of equations ( 28) and ( 30), we define (suppressing the labels A, ξ in Ψ for brevity) We know from [16] that the current density is generally of the form |Ψ| 2 Ω, where Ω depends on the configuration variables and their conjugates, is independent of time, and is real and positive semi-definite.We define Ω ≡ Ω[A, A] as we have used ξ to define our time variable t ′ .We can then show that where c.c denotes complex conjugate of the term in square bracket, and we have used δΨ/δA ai = δΨ/δA ai = 0 ∀a, i as Ψ is a holomorphic functional of A. The right-hand side of equation ( 32) can be written as We require that Ω be such that all the source-like terms vanish.This will be true if Equation ( 34) supplies the Ω needed to define the current density.We observe that solves (34), where Ψ K [A] is the Kodama state.As the weight factor Ω is unique and does not depend on the Hamiltonian or the state [29], we take (35) henceforth.Equation (32) can then be written as where ∇ ai ≡ M δ/δA ai , ∇ B ≡ M δ/δξ B and Note that equation (36) is not yet a satisfactory continuity equation, as there are 'temporal flux' terms J B , J B corresponding to ξ B , ξ B .We can absorb them into the current density term by redefining the time parameter t ′ → t such that Equation ( 36) can then be written as the continuity equation where J ai is given by (37).

B. Physical interpretation
Let us consider the physical interpretation given the weight factor (35) and the continuity equation (40).Let us first take the question of normalizability of the quantum state.

Normalizability
It was shown by the authors of [6], that Ψ ] is a good ansatz for the gravitational-fermionic WDW equation.The continuity equation ( 40) can be rewritten as which makes it evident that the non-normalizable Chern-Simons-Kodama state Ψ K is factored out of the current density.Therefore, to interpret (41) as a probability conservation equation, the normalizability condition is imposed on Φ[A, ξ] -not the full quantum state Ψ[A, ξ].Using (30), we can show that Φ[A, ξ] follows the Schrodinger equation with respect to the time parameter t in (39).We further discuss probabilities in section (V).

Guidance equations
The conceptual role of the continuity equation derived from the quantum state, in pilotwave theory, is to define the guidance equation.Using (37) and the standard pilot-wave prescription for the ansatz Ψ[A, ξ] = Ψ K [A]Φ[A, ξ], the guidance equation determines the evolution of the Ashtekar connection with respect to the fermionic time t.
We note that the first line of ( 43) is the classical equation of motion for the connection with E bj substituted by δ ln Ψ K /δA bj .This form of E bj can be shown to give the classical deSitter solution [21].The first term in the second line of ( 43) contains the quantum corrections to the deSitter solution, whereas the second term contains the quantum contribution from the fermionic interaction with Π B given by δ ln Φ/δξ B .Also note that equation ( 43) is nonlocal in the sense that the evolution of the connection at a particular point in physical space generally depends upon the value of the connection at other points in physical space, similar to equation (19).
The guidance equation for the fermion is given by equation (39).We note that (39) resembles the classical equation of motion with E ai substituted by δ ln Ψ K /δA ai .However, as A ai is guided by the full quantum state Ψ[A, ξ] in (43), the evolution of the fermion is implicitly state dependent and shows quantum behaviour.

Reality conditions
We impose the reality conditions at the level of the guidance equation (43).We first note that, in the Ashtekar formulation of classical general relativity, the following conditions have to be imposed to recover the real sector (with real metric), where Γ ai is the 3D spin connection.In the orthodox quantum formulation of canonical quantum gravity, the reality conditions are generalised to the operator conditions We pursue here an approach based on pilot-wave theory to generalizing the classical reality conditions (44), (45).We demand that these conditions be met for the configuration-space trajectory determined by the guidance equation (43).This allows us to extract the real sector for an arbitrary solution to the Schrodinger-like equation ( 30), regardless of normalizability issues.
It is clear from (43) that the first reality condition (44) will be trivially satisfied for any arbitrary solution Ψ if we define at all points of the system trajectory.Let us next consider the second reality condition (45).
Using the definition where E ≡ det(E), we can solve for E ai (t) given Γ ai (t) along the system trajectory from (48).Since the Γ ai is real, (49) admits real solutions and the second reality condition (45) is thereby satisfied.
Lastly, we can obtain the extrinsic curvature along the system trajectory from the imaginary part of the connection as Equations ( 41) and ( 52 of the configuration space has been defined to be in pilot-wave equilibrium [34], which is a generalization of the notion of quantum equilibrium [30][31][32][33].For example, an initial density (up to normalization factor) where Ω 0 ≡ {(A, ξ)|ρ[A, A, ξ, ξ, 0] > 0} is a compact support on the configuration space C, will evolve to where Ω t ≡ {(A, ξ)|ρ[A, A, ξ, ξ, t] > 0} is the time evolved support on the configuration space.The behaviour of such densities has been explored for the case of harmonic oscillators in [34].
Let us next consider the notion of unitarity from a pilot-wave perspective.We define Φ[A, ξ] to be a unitary state if and only if lim for any initially normalized ρ[A, A, ξ, ξ, 0] evolving via (52) at any finite t > 0, and where δA ck /δt is determined from (43).As δξ B /δt ∝ ξ B from (39), the condition (57) implies that ρ[A, A, ξ, ξ, t] remains normalized with time.Clearly, the pilot-wave notion of unitarity (57) is applicable regardless of the normalizability of Φ[A, ξ].Note that a unitary non-normalizable state is not identical to a bound non-normalizable state [34].
We now explore the behaviour of solutions to the Hamiltonian constraint in the context of this discussion.This is a technically challenging question to investigate in full generality, so we address this here in the mini-superspace (FRW) approximation, which is relevant for quantum cosmology.

C. Mini-superspace
Assuming homogenity and isotropy, we take A ck ( x) = iAδ ck and ξ B ( x) = ξ.This implies that The Hamiltonian constraint ĤΨ = 0 simplifies to 3i As such, equation (60) does not have separable solutions in A, ξ.

Approximately separable solutions
Let us make the simplifying assumption that the last term in (60) is small, which we will justify later.We then look for separable solutions Ψ(A, ξ) = χ(A)φ(ξ).For such solutions, Clearly, the first two terms depend only on A whereas the third term depends only on ξ.
Let us introduce a separation constant E (in general complex) such that 3i The differential equation for φ can be written as where E 0 = 2E/(σ + aA σ aA+ + σ − aA σ aA+ + σ + aA σ aA− + σ − aA σ aA− ) and we have used τ ≡ −iσ/2.The general solution to (64) is φ(ξ) = ce −iE 0 ln ξ , where c is an arbitrary constant.We note the resemblance of this solution to the time-dependent part e −iEt/ of an energy eigenstate corresponding to energy E.
The approximate solution to (62) for χ is where we have neglected the third-derivative term multiplied by Λ.Note that we can select E in (65) such that the last term in (60) is indeed small, as assumed.

Unitarity and Torsion
The current (37) can be rewritten as The guidance equation (43)  Equation (68) also implies that, in general, A(t) will have both real and imaginary parts.
This implies the presence of both normal and parity-violating torsion [35,36].

Evolution of the fermionic field
Lastly, the evolution of the fermionic field (39) becomes Equation (69) implies that ξ + and ξ − will quickly differ, even if ξ B = ξ at t = 0.

VI. DISCUSSION
We have described an interacting gravitational-fermionic system in Ashtekar formulation using the language of pilot-wave theory.We summarise here the key results of our work and potential directions for future research.
We have obtained a real time variable for the combined system, without semiclassical assumptions, by parameterizing variation of the fermionic field that depends on the Kodama state.In both classical and quantum canonical gravity, time disappears and the Hamiltonian becomes a constraint.Various approaches to define a natural time variable using a matter field as a clock have been discussed in the literature [37][38][39][40].Our approach analogously uses a fermionic field to define time, but also supplements it with the pilot-wave postulate of a definite joint configuration for both the clock and the Ashtekar connection.
Time is then defined to be a real variable that naturally parameterizes the variation of the connection naturally resolves into the classical equation of motion, giving us deSitter spacetime as a solution, plus quantum corrections.We have also shown the existence of pilot-wave equilibrium densities [34], which lead to Born-rule-like probabilities.It is interesting that we have used commutators to quantize the fermionic field [27,28], and this leads to considerable simplicity in interpretation for the guidance equations.It will be interesting to explore the violation of spin-statistics theorem in quantum gravity in the future, as this is closely related to the long-standing question of particle versus field ontology for fermions in pilot-wave theory [23,24,43].
It is important to extract testable cosmological predictions from our approach.We know that the connection in FRW is the co-moving Horizon, A ∼ Ha [35,36], so that the evolution of the Horizon may be obtained from the guidance equation and this may yield predictions in light of the Hubble tension.Such a link would connect non-local dynamics in pilot-wave theory to the evolution of the Hubble parameter, but this is still speculative.