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Quantum Hamiltonian for gravitational collapse

Using a Hamiltonian formulation of the spherically symmetric gravity-scalar field theory adapted to flat spatial slicing, we give a construction of the reduced Hamiltonian operator. This Hamiltonian, together with the null expansion operators presented in an earlier work, form a framework for studying gravitational collapse in quantum gravity. We describe a setting for its numerical implementation, and discuss some conceptual issues associated with quantum dynamics in a partial gauge fixing.


I. INTRODUCTION
A complete understanding of the many puzzles related to black holes, such as entropy [1], Hawking radiation [2], and the final state question [3], will require a quantum theory describing gravitational collapse, where both matter and geometry are fully dynamical. The ideal would be the ability to follow the evolution of an initial matter-geometry quantum state to "black hole formation" and beyond. This goal has been implicit in many works on black hole physics beginning with Unruh thirty years ago [4], followed a few years later by work of Hajicek [6]. More recently it has been emphasized by Isham as one of the motivations for studying quantum gravity [5]. To date, however, no complete quantum framework is available for studying gravitational collapse of a scalar field. This paper continues a recent line of work aimed at developing the tools necessary to realize this ideal. So far, we have developed the quantum kinematics for spherically symmetric gravitational systems, in a setting which leads to a result concerning the resolution of the classical singularity at the quantum level [7]. Furthermore, we have proposed how to define a black hole at the quantum level, without fixing any classical "horizon boundaries" [8]. Finally, we formulated the classical dynamics of black holes in a framework that results in a true Hamiltonian, rather than a system with a Hamiltonian constraint [9].
The current paper is concerned with the development of the quantum dynamics of the gravity-scalar field system in spherical symmetry. It consists of two main parts. The first one deals with the conceptual issues connected with the notion of true dynamics. This is necessitated by the fact that our treatment of the dynamics is based neither on Dirac's constraint quantization, nor on a complete gauge-fixing, but lies in between those two extremes by employing a partial gauge-fixing only of time. As this type of approach has -to the best of our knowledge -not been discussed before in any detail, we include a careful discussion. The second part is of a more technical nature and describes the construction of the Hamiltonian operator starting from the reduced Hamiltonian derived in [9].
The work is an attempt to complete a proposal for the quantum theory of the gravityscalar field system in spherical symmetry. It provides a calculational framework for studying gravitational collapse in quantum gravity, which we hope will lead to a better understanding of Hawking radiation, and of the final state of gravitational collapse.
The next section discusses some conceptual issues concerning quantum evolution in a setting where only a partial (time) gauge fixing is utilized. This is followed in Section III by a review of the kinematical framework for quantization of the 1+1 dimensional field theory that describes the gravity-scalar field system. Section IV contains a construction of the Hamiltonian operator. Since the classical Hamiltonian contains a square root, we introduce here a new method of defining the corresponding operator. In Section V we give the action of the Hamiltonian on basis states, describe how to implement unitary evolution, and discuss the issue of dynamical singularity resolution. We conclude in Section V with a summary and outlook for numerical implementation.

II. CONCEPTUAL SETTING
There are at least two methods to introduce a notion of time into a theory with time reparameterization invariance. The most obvious way is by means of gauge-fixing, where a suitable function on the classical phase space is taken to be the time function. Another is the closely related method of using partial and complete observables [10,11], where a degree of freedom is chosen as the reference clock, and the evolution of the remaining degrees of freedom is measured with respect to that clock.
We have chosen the former since it is closest to the setting in which the well-known semiclassical results about black holes have been derived. In earlier work [9] on the classical theory, we derived a reduced Hamiltonian for the gravity-scalar field theory in spherical symmetry by performing only a time gauge fixing, and leaving the remaining spatial coordinate freedom untouched. The Hamiltonian constraint was solved as a strong condition, and a reduced spatial diffeomorphism generator remained as the only (first class) constraint.
Together with the surface term, this constraint forms the total Hamiltonian. If the surface term is written as a bulk integral and combined with the other bulk terms, one can identify the true local Hamiltonian density and a diffeomorphism generator for the remaining degrees of freedom.
Fixing only a time gauge raises the question of the extent to which unambiguous evolution can be achieved in the quantum dynamics, since evolution still contains a gauge part manifested in the freedom in the shift function. This may be seen schematically through the requirement that a gauge condition such as f (p, q) = 0 be preserved in time. It leads to the equation which fixes the lapse function N in terms of N a . The latter remains arbitrary, and must be specified to compute evolution classically or quantum mechanically. This situation is similar to Yang-Mills theory where the total Hamiltonian density before gauge fixing and solving the Gauss law is where Λ i is the Lagrange multiplier and G i is the Gauss law expression.
In contrast the Hamiltonian for our problem (in a fixed time gauge) contains an additional twist (see below for the details). Schematically, it is of the form where C r is the radial diffeomorphism generator, q, π denote the collection of canonical phase space variables, and N r is a lagrange multiplier (the remaining radial component of the shift function in spherical symmetry). The similarity between the two cases (YM and gravity) is the clear separation of the gauge generators from the "true" Hamiltonians. The difference is that the gravity case contains Lagrange multiplier dependence also in the first term, but only through its radial derivative.
This poses the conceptual issue of obtaining unambiguous evolution, since the gravity reduced Hamiltonian generates a family of time evolutions for observers specified by N r .
This freedom is limited to the spatial reference system only. Once it is fixed so is the clock.
This situation lies in the middle of the two common scenarios in dealing with Hamiltonian gravity, where either no gauges are fixed, or all are fully fixed.
We will address this issue in two steps. Firstly, we obtain an Hamiltonian operator that depends on N r , determine its action on basis states, and thereby obtain a prescription for evolution with any N r (with the prescribed asymptotic conditions). This gives a family of evolved states parameterized by N r . A unique evolution is then obtained by specifying a specific functional form for this function. This may be viewed as the quantum analog of evolving with a fixed shift function.
Secondly, we will not impose the diffeomorphism generator as a constraint on states a'la Dirac. Rather we will give prescriptions for obtaining information pertaining to the collapse problem in a manner which is manifestly invariant under this symmetry. This means looking at the dynamics of certain phase space observables, such as null expansion, as functions of other phase space variables. Suitable semiclassical states that are peaked on classical solutions of the radial diffeormorphism constraint will be used as the physical states.
Taken together, the implementation of these ideas, provided in the following sections, appears to provide a tractable approach to the gravitational collapse problem in quantum gravity.

III. REVIEW OF THE KINEMATICAL SETTING
Before addressing the issue of quantum dynamics, we briefly review the quantum kinematical setting, as introduced in [7]. The starting point is a classical field theory in 1+1 dimensions, characterized by the canonical pairs (R, P R ), describing the geometry degrees of freedom, and (φ, P φ ), describing a scalar field. The basic variables that are turned into quantum operators are the smeared fields and where f and g are suitable test functions. These configuration variables, together with the translation generators and form a closed Poisson algebra, and are taken as the basic variables to be converted into operators. All other operators are be constructed in terms of these.
The Hilbert space for the quantum theory is spanned by basis states of the form where a k , b l are real numbers, and N 1 and N 2 are positive integers. The factors of L in the exponents reflect the length dimensions of the respective field variables. The intuitive picture is that each basis state 'tests' the quantum scalar field at N points in space and the basis of the 'excitation space' at each point consists of (generalized) plane waves. The inner product is if the states contain the same number of sampled points, and is zero otherwise. The action of the basic operators isR and is not part of the original basis state. In this case the action creates a new 'excitation' at the point x j (resp. y k ) with 'mode' −λ j (resp. −µ k ). These definitions give the commutator Comparing with the classical Poisson bracket relation, and using the Poisson bracket commutator correspondence, it turns out that L = √ 2l P , where l P is the Planck length. A similar commutator relation holds for the matter scalar field.
In this formalism we can also define other operators of interest. For example for the Hamiltonian, we need operator analogs of the radial derivatives R ′ and φ ′ , and the square of the momentum P 2 φ . Definitions for the operators corresponding to R ′ and other derivatives of fields are obtained by implementing the idea of finite differencing using the operatorR f (4). We use narrow Gaussians with variance proportional to the Planck scale, peaked at coordinate points r k + ǫl P , where 0 < ǫ ≪ 1 is a parameter designed to sample neighbouring points [24]: Denoting R fǫ by R ǫ for this class of test functions we definê A further motivation of this form is that in the gauge R = r the corresponding classical expression gives unity in the limit ǫ → 0. This definition captures the simplest finite difference approximation to the derivative at the operator level. Second derivatives may be similarly defined by converting a finite differencing scheme into the corresponding operator.
The quantization defined above does not give definitions for operators corresponding to momenta, since only translation generators are directly realized as operators at the first step. This of course is similar to any quantum theory on a lattice. Operators for momenta can however be realized using the translation generators, for example by expressions such These λ dependent expressions will be utilized below for the Hamiltonian operator.
Finally, let us note that the background independent (ie. metric free) quantization outlined here is not the same as the one for the scalar field reported in [12,13], where the basic variables used are the integral of the scalar field momentum over space Σ P φ , and the exponential of the scalar configuration e iλφ . The present approach has the advantage that it is fairly straightforward to write a local Hamiltonian density operator using the translation and φ ′ operators. A further technical point of difference, apart from the choice of basic variables, arises from the fact that R and φ are scalars, and P R and P φ are scalar densities of weight 1. This means that the functions f in (4) and the λ in (6) are scalar densities of weights 1 and −1 respectively. These combine to give eigenvalues ofR f andφ f that are scalars. In contrast, for the quantization studied in [12,13], it is the space-integrated momentum density that is diagonal; this is the direct analog of the surface integrated densitized dreibein in loop quantum gravity.
This ends our review of the kinematical setup. Our goal in the next section is to define the Hamiltonian operator for the collapse problem on this Hilbert space.

IV. HAMILTONIAN
The classical Hamiltonian in a time gauge fixing corresponding to flat spatial slicing given was derived in [9]. The classical phase space before gauge fixing has an extra pair of canonical variables (Λ, P Λ ), with the spatial metric The condition Λ = 1 is second class with the Hamiltonian constraint, which is solved classically for the conjugate momentum P Λ . This leads to the reduced Hamiltonian where N r is the remaining (and still arbitrary) non-zero component of the shift function N a after imposing spherical symmetry, and The Hamiltonian (18) is obtained by writing the surface term in the reduced action as a bulk integral and combining terms. It gives the time-gauge fixed evolution equations for the fields R(r, t) and φ(r, t), and their canonical conjugates. Our main goal in this paper is to construct the corresponding operator.
Let us write the first part of the Hamiltonian density (excluding the radial diffeomorphism term) as where we have defined There are two ways to construct an operator corresponding to a square root. One approach is to find the eigenvalues of its argument and work with the corresponding basis of eigenvectors. The operator can then be defined as the square root of the eigenvalues if the spectrum is positive semi-definite. A potential alternative is to see if the square root operator can be defined using Dirac's idea.
In this procedure an operator for the classical function H = p 2 i + m 2 (i = 1 · · · 3) is constructed by writingĤ = α ipi + βm using anticommuting matrices α i , β, such that H 2 = (p 2 i + m 2 )I, where H 2 defined by matrix multiplication. This works because the momentum componentsp i commute. This is not true for the elements h i in the argument of our Hamiltonian, so Dirac's trick cannot be used, at least in its basic form.
There is the additional problem that the term R, which is the Ricci scalar, is not a squared quantity, and so can be negative. (In the gauge R = r this term vanishes since we are using the flat slice asymptotic conditions [9].)R is however diagonal in the basis since the operator analogs of the fields R, R ′ and R ′′ are all diagonal [8]. Therefore at least for the subset of states where its eigenvalues λ R are positive we can define an operatorĥ 4 whose eigenvalues are √ λ R . We will return to this issue below after describing an approach to address the square root problem. For now, and the following discussion, we define The idea is to obtain a definition of the square root partÂ of the Hamiltonian operator by working in a larger Hilbert space for some V to be specified, and writingÂ where theĥ k act in H Kin , and theē k act in V . The key requirement is that the action ofÂ in H must be such thatÂ where η kl = diag(+ − −+) and I is the identity operator in the space V . All other operatorŝ O acting in H Kin are extended to H by the identity action in V . Since the h k do not commute, this means that the operatorsē k must satisfȳ e kēl = η kl I.
That no such operators exist for any space V may be seen by the following argument. Let |ψ k be a complete set of normalisable states, and let us assume that operators satisfying (28) exist. Then we are led to contradictions such as There is a way out of this situation if we demand the weaker condition that for every |Ψ ∈ H that is of a form specified below. The basic idea is to introduce a four-dimensional Euclidean vector space V and consider its decomposition into orthogonal k = 1 · · · 4. Let us denote by P k the projection operators onto these subspaces. By definition the P k satisfy P k P l = P k δ kl . Now define operatorsē k bȳ for k = 1, 4, andē for k = 2, 3. These giveÂ Consider now the action ofÂ 2 on states |Ψ of the form where |ψ ∈ H Kin , and the |ρ k denote the basis of V corresponding to the decomposition into the V k . The result isÂ From this it is evident that (30) holds. Finally, we still need a refinement to get a self-adjoint A. To do this we define the operatorsĥ 2 andĥ 3 such that they have the propertŷ These ensure that the square root operatorÂ in the extended Hilbert space is self-adjoint.
The last issue to address is the definition of the operator corresponding to R (the Ricci scalar) in Eqn. (22). This contains derivatives of R which are defined by the finite difference operators introduced in [8]. Since these are diagonal on basis states, so isR . Now, in the gauge R = r, which fixes the shift vector N r to be proportional to 1/ √ r everywhere, the Ricci scalar vanishes. Our Hamiltonian is not in this coordinate gauge (which is still free), but as alluded to in Sec. II, the idea is to evolve quantum states with this choice of lapse function, with initial states for which the values a k of the radial field R are distributed linearly with the graph points r k , ie. a k ∼ r k . This guarantees that the eigenvalue ofR vanishes, at least initially. However it will not remain so because the Hamiltonian contains terms with P R , which is represented by the operator The translation operators on the r.h.s. act to shift excitations at the point r k which result in the eigenvalue λ R ofR being moved from zero at the selected point. However this move is very small since 0 < λ << 1. Because of this we define the action ofĥ 4 bŷ where |ψ is a basis state. This definition makes the assumption that, starting from a basis state with λ R = 0, the deviation from zero after the action ofP R is positive.
This completes our prescription for the Hamiltonian operator for the scalar field collapse problem. With each of its constituents well defined, it is straightforward to compute its action on a basis state. The fact that this can be done in contrast to the case for full gauge fixing [3] represents some progress which we anticipate will lead to concrete calculations for quantum collapse. A strategy to do this is presented in the next section.
We close with a discussion of the quantization choices, or ambiguities, inherent in the steps outlined here. Firstly, let us note that the general procedure used in obtaining the square root operator is new. The only other possible approach is to seek the spectrum of the argument of the square root, which given its form appears a formidable task. If this could

V. QUANTUM EVOLUTION
In classical numerical simulations of scalar field collapse the general approach is to start with initial data representing a shell of scalar field, and to evolve it using some choice of lapse and shift functions [14]. At each step of the simulation an "apparent horizon" check is made as a criterion for black hole formation. This is a null geodesic trapping condition at each step or leaf of the evolution. In spherical symmetry the goal is to find the outermost radial location on each leaf where the condition is satisfied. The evolution of this location is taken to represent the dynamical boundary or horizon of the black hole.
There are at least two concrete versions of what is a dynamical horizon. The first was formulated by Hayward [15]. In addition to the usual criteria for null geodesic expansions, this work consists of additional conditions designed to distinguish future, past, inner and outer local horizons. The second [16] lifts some of these conditions, and points out that the resulting definition of dynamical horizon allows a nice formulation of local flux laws. The common and minimal feature of both definitions are the conditions where θ ± are the in(out)going null geodesic expansions. Equivalent information is captured in the observable θ + θ − , which goes from negative to positive as a dynamical black hole boundary is crossed.
With the Hamiltonian operator defined in the last section, and the operator analogs of the null expansion operators given in [8], we are in a position to give a procedure for a quantum collapse calculation. This involves specifying (i) initial states, (ii) an evolution procedure, and (iii) a quantum test for black hole formation.
The first question is what are suitable initial states. The basis states represent values of the scalar field φ(r, t) and the radial field R(r, t) at a set of discrete coordinate points r 1 · · · r k , which is a sample of the half line. As such these states may be compared with the discrete data for a classical numerical simulation. This suggests the use of "profile" states where we take for example the scalar field excitations to have a gaussian profile, and the radial excitations to be linearly distributed. Another possibility is the use of suitable coherent [18] or other form of semi-classical states that are peaked at classical configurations.
These would be infinite linear combinations of basis states, and so computation with them would be more involved, especially in the present field theory setting.
The second step is the implementation of evolution. Rather than constructing a finite evolution operator by exponentiation of the Hamiltonian, which is very cumbersome, it is more suitable to implement repeated infinitesimal evolution using a suitable scheme. The simplest possibility is not unitary. However there are unitary schemes available for this purpose. One example is based on the well known Crank-Nicholson method, where the Schrodinger equation in discrete time (labelled by n) is written as This leads to the manifestly unitary (but implicit) evolution scheme given by SinceĤ depends also on the free (classical) function N r , so does the evolved state |ψ n+1 .
This function must be fixed (with the fall off condition N r ∼ 1/ √ r) to get a unique evolution [9]. The simplest choice is to take this form for all points r k in the chosen initial state.
The third step is to implement a test for black hole formation at each time step of the evolution. As mentioned above, the minimum requirement for this is that we must have operators corresponding to the null expansions. A prescription for constructing these were given in [8]. In addition we require this test to be invariant under the remaining radial diffeomorphism constraint. This is achieved, for example, by looking at the quantities θ + (r k , t)θ − (r k , t) as functions of < R(r k , t) >, where the expectation values are in the state arrived at by stepwise evolution from some initial state. The resulting curve is radial diffeomorphism invariant, and its intersection with the R axis gives the location and size of the evolving horizon. It is the dynamics of this curve which is of interest for black hole formation and subsequent evolution at the quantum level.
There are many other useful diffeomorphism invariant quantities of interest that can be computed at each time step. Two examples are the scalar field configuration φ(r k , t) , and a curvature measure such as π(r k , t) (the trace of the ADM momentum) [7], both viewed as functions of R (r k , t) .
The second and third steps are to be repeated for multiple time steps to extract time dependent profiles of the functions of interest, such as the ones just mentioned. It is in this manner that physical information about collapse may be obtained in the kinematical Hilbert space after fixing the time gauge classically. Although evolution is with respect to a fixed N r so that the evolved states depend on it, the physical information contained in the suggested functions is independent of the coordinate points r k : radial diffeormorphisms act to shift the points r k to r ′ k without changing the values of the field variables.

VI. DYNAMICAL AVOIDANCE OF THE SINGULARITY
In earlier work we gave a construction of an operator corresponding to the classical variable 1/R that is bounded on the Hilbert space we are using for the quantum theory [7].
This result has the direct consequence that curvature singularities in spherical symmetry are avoided. This is because phase space variables that classically diverge at the singularity do so as positive powers of 1/R. The corresponding quantum operators are constructed as products of the 1/R operators, and so are also bounded. This does not involve the Hamiltonian in any way so the result may be viewed as kinematical singularity avoidance.
In this section we show, using the construction of the Hamiltonian operator given above, that inclusion of dynamics does not alter this result.
The fundamental question here is whether quantum evolution remains well-defined through the region of highest curvature, or whether it stalls or breaks down there. Classically dynamics is encoded either in (i) the Hamiltonian constraint, or (ii) a Hamiltonian derived from a partial (time) gauge-fixing, or (iii) a Hamiltonian derived from a complete gauge-fixing as in [4]. Although our work is concerned with the second case, we will look at the issue of singularity avoidance from all three points of view. Let us consider first the Hamiltonian constraint. Its classical expression is Due to the 1/R factors, it is divergent at the points r where R = 0. Upon quantization these factors turn into bounded operators as shown in [7]. Consider next the fully reduced Hamiltonian that was found in [4] This is clearly divergent at r = 0. As r is a parameter rather than a configuration field variable, quantization of the totally reduced theory cannot resolve the singularity at the quantum level.
Finally, let us consider the Hamiltonian whose quantization is the subject of this paper.
Firstly, an inspection of the classical reduced Hamiltonian (18) shows that, unlike the Hamil- in [19,20], and the Schwarzschild black hole studied in [21]. In both these cases the systems are finite dimensional, unlike the model in this paper. In the first case, the action of the Hamiltonian constraint on a basis state gives a finite difference equation with coefficients such that action on the state of zero volume gives a bounce. In the second case the interior (Kantowksi-Sachs) and exterior of the Schwarzschild spacetimes are quantized with appropriate matching at the event horizon, which is taken as the fundamental classical dividing line. Our perspective is that these cases are quite different from the matter coupled case we treat here. In particular, with non-vanishing matter fields, the extended Schwarzschild spacetime does not arise, so quantizing it to discuss singularity resolution is not relevant for the quantum treatment of the collapse problem.

VII. DISCUSSION
We have constructed the quantum Hamiltonian for the spherically symmetric system of gravity coupled to a scalar field in a fixed time gauge. Together with earlier papers [7,8,9], this work completes the construction of a quantum theory for studying the gravitational collapse of a scalar field in spherical symmetry. We are now in a position to address the questions surrounding black holes that have been generally acknowledged to find a resolution only within a full quantum treatment.
Any application of our formalism to a given physical situation will require a choice of initial state. As all the black holes that have been detected so far have macroscopic size, the first task is to determine how to represent what we know as a classical black hole in the quantum theory. This obviously calls for a construction of the semiclassical sector, including the search for suitable semiclassical states. This will be the subject of a future publication [22]. Once the issue of what initial state to take has been settled, one can then investigate the quantum time evolution of the system.
Among the problems of physical interest is the evolution of a quantum state that satisfies the quantum horizon conditions. One would like to know how this evolution depends on the matter part of the state, and how the horizon grows or shrinks. An indispensable tool for such questions will be the null expansion operators which were constructed in [8], which serve as horizon finders. It should be pointed out that, even if one does not subscribe to the exact definition of the horizon of a black hole proposed in [8], these operators would still play an important role in any other approach to a quantum description of black hole horizons.
Perhaps the most interesting questions to address in this framework are whether and how Hawking radiation arises, and what is the end point of collapse. A preliminary investigation [17], indicates that the end result of gravitational collapse is a Planck size remnant. This appears to be intimately connected with the existence of an upper bound on curvature, which in turn has an appealing intuitive analogy with Fermi pressure; whereas the latter can be ultimately overcome by gravitational forces, the former cannot since it is a fundamental quantum gravity effect. It represents the ultimate limit to which matter can be compactified.
Finally, as time evolution in our setting is unitary, one can surmise already at this point that the solution to the so-called information loss paradox is that there has never been a paradox in the first place.
Apart from applications, another potential line of investigation is to improve on the framework developed here. One of the questions here concerns time gauge fixing. There are of course many other choices, so it would be useful to see if there are others that might lead to simpler reduced Hamiltonians. One possibility is to look at only the time gauge fixing used in [4], without fixing the radial gauge R = r. Another approach to the same issues using the connection-triad variables is being developed by Bojowald and collaborators [23].
This work offers a parallel approach in the loop quantum gravity programme.