# On the Construction of Quantum Dirac Observables and the Emergence of WKB Time

###### Abstract

We describe a method of construction of gauge-invariant operators (Dirac observables or “evolving constants of motion”) from the knowledge of the eigenstates of the gauge generator of time-reparametrisation invariant mechanical systems. These invariant operators evolve unitarily with respect to an arbitrarily chosen time variable. We emphasise that the dynamics is relational, both in the classical and quantum theories. In this framework, we show how the “emergent WKB time” often employed in quantum cosmology arises from a weak-coupling expansion of invariant transition amplitudes, and we illustrate an example of singularity avoidance in a vacuum Bianchi I (Kasner) model.

^{†}

^{†}affiliationtext: Institute for Theoretical Physics, University of Cologne,

Zülpicher Straße 77, 50937 Köln, Germany

## 1 Introduction

One of the most important issues in any attempt to quantise the gravitational field is the proper understanding of the gauge symmetry of the theory (“general covariance”) at the quantum level. Presumably, a *complete* quantum theory of gravity would involve a Hilbert space of physical states and a set of linear operators which would represent the observables of the theory. Both states and observables should transform appropriately under gauge transformations. There is no general agreement on how such a theory should be constructed or even if a Hilbert space is really necessary [1], although there are multiple approaches being actively pursued [2].

Classically, the diffeomorphism symmetry induces a group of transformations in phase space (the Bergmann-Komar group [3, 4]) and it is associated to a set of first class constraints [5, 6, 7, 8, 9, 10, 11, 12]. The gauge transformations (Lie derivatives) are generated in phase space by a combination of these constraints [5, 4, 13, 14]. In this way, phase-space functions which are gauge invariant must have vanishing Lie derivatives and, if these functions do not depend explicitly on the spacetime coordinates, then they Poisson-commute with the constraint functions. In this case, they are often called Dirac observables.

An important class of such observables is given by “evolving constants of motion” [15], which are phase-space functions that encode the relational evolution between tensor fields according to the appropriate field equations (e.g., the Einstein field equations in general relativity). The “evolving constants” can be understood as gauge-invariant extensions of non-invariant quantities given in a particular frame [16, 17, 12, 18, 19, 20] and have been contemplated in the literature in the context of canonical gravity both in (quantum) geometrodynamics and loop quantum gravity [21, 22, 23].

If one takes the view that the physical content of a generally covariant theory is entirely encoded in such relational phase-space functions, it is indeed reasonable to construct a canonical quantum theory based on operators which represent Dirac observables and physical states that are superpositions of eigenstates of the “evolving constants”. In this framework, the physical Hilbert space is the vector space of wave functions that are annihilated by the constraint operators. This apparently leads to a “problem of time” [24, 25]: physical states seem to be time independent and one has the impression that the dynamics is “frozen”.

While there are many possible solutions to this “problem” (see, for instance, [26] and references therein), it is arguably sufficient to note that the quantum dynamics has to be relational, as it is in the classical theory. The dynamics is not “frozen”, but rather encoded in the relational evolution of Dirac observables. Variations of this argument have been explored in the literature [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], but a systematic way to construct Dirac observables is lacking. Moreover, the explicit connection between this relational view and other approaches to the “problem of time” has remained unclear. In particular, a popular “solution” is the “semiclassical emergence of time”: time only exists when the wave function(al) of the gravitational field is in a semiclassical regime (see [24, 25, 39, 40] for a review and [41, 42, 43, 44, 45, 46, 47, 48] for phenomenological applications). For this reason, this emergent semiclassical time is often referred to as “WKB time” [49, 39, 40]. Recently [50], the author has argued that such a “semiclassical approach to the problem of time” coincides with a particular choice of gauge (i.e., time coordinate) and can be extended beyond the semiclassical level.

The purpose of this article is twofold: (1) to discuss a systematic method of construction of gauge-invariant operators in covariant quantum mechanics, i.e., operators which commute with the gauge generator and therefore have a physical interpretation which is independent of the time parametrisation (gauge) adopted; (2) to relate the heuristic notion of an “emergent semiclassical time” to the concrete and more fundamental framework in which the basic objects of the quantum theory are correlation functions of gauge-invariant operators.

The construction of operators corresponding to Dirac observables will be guided by an analogy to the Faddeev-Popov gauge-fixing method [51, 52] in conventional gauge theories (i.e., Yang-Mills theories). Our approach will be canonical (operator-based) and we will not make use of path integrals. Although the restriction to mechanical theories is for the sake of simplicity, the method here presented is directly applicable to all minisuperspace models of quantum cosmology and, hence, it is useful. The field-theoretical case (with the possible issues of regularisation and anomalies) will be left for future work.

The second objetive is a continuation of [50], in which it was extensively argued how the results of the usual “semiclassical approach to the problem of time” can be recovered from a complete quantum theory where the notion of “gauge fixing” was paramount. The work of [50] was, however, limited by the use of the indefinite Klein-Gordon inner product in the Hilbert space of physical states. In the present article, we adopt the positive-definite Rieffel induced inner product [53, 54, 55, 56, 57, 58, 59] (see also [60, 61, 62, 63]), and we show how the emergence of “WKB time” occurs in the simple example of a relativistic particle, which is sufficient to illustrate the connection between the “semiclassical time” and the exact relational dynamics at the fully quantum level.

The article is structured in the following way. In section 2, we review the classical theory and present the general formalism for the construction of Dirac observables in covariant quantum mechanics, comparing it with previous proposals. In the subsequent sections, we analyse concrete examples of this construction. In section 3, the relativistic particle is quantised and the corresponding Dirac observables are constructed. We show how they coincide with their non-relativistic counterparts in the appropriate limit, in which the “WKB time” also emerges. In particular, we construct the “time-of-arrival” Dirac observable (see [64, 37] and references therein) in the non-relativistic limit. In section 4, we analyse a cosmological model, the vacuum Bianchi I (Kasner) universe, and give an example of how the classical singularity may be avoided in the quantum theory. Finally, in section 5, we summarise our results and present our conclusions. We keep factors of and explicit.

## 2 The General Framework

### 2.1 Classical Theory

#### 2.1.1 Observables

In preparation to the quantum theory of Dirac observables, we review the fundamentals of generally covariant classical mechanics, which can be regarded as a toy model of general relativity in spacetime dimensions [65]. We refer to the one-dimensional background manifold as the worldline. Given two worldline vectors , we can define the (intrinsic) metric on the worldline as , where is a worldline scalar density called the einbein. Gauge transformations are worldline diffeomorphisms generated by a vector field [3, 4].

The dynamical variables are worldline tensors described in an arbitrary “frame” related to the choice of the worldline parameter . Under reparametrisations of the worldline, the components of tensors transform covariantly. In fact, there is no problem in defining physical quantities (observables) to be covariant rather than invariant under worldline reparametrisations (see the discussions in [66, 67, 68] as well as in [69, 70]). Thus, we can define observables to be worldline tensors. However, since one promotes the *initial values* of dynamical fields to operators in the quantum theory, we would like to be able to describe the initial values independently of the choice of parametrisation and, thus, in a gauge-invariant manner. In this way, we would like to construct Dirac observables, i.e., objects which commute with the phase-space constraints, to represent the *invariant extensions* of initial values of worldline tensors. These extensions will then be promoted to operators in the quantum theory. Let us see how this can be achieved systematically.

For simplicity, we assume the fundamental dynamical fields are worldline scalars. The gauge transformation of a scalar field reads

(1) |

In order for the dynamics to be reparametrisation invariant, the Lagrangian (where ) must be a worldline scalar density, such that it transforms as follows [65, 12]:

(2) |

This implies that the action

(3) |

is invariant if the infinitesimal diffeomorphism vanishes at the endpoints, i.e., . Otherwise, it is necessary to add boundary terms to the action to make it invariant [71]. In fact, given a worldline one-form , where transforms as in (2), then the quantity

(4) |

is an invariant (hence, observable) provided the integral converges and suitable boundary conditions are chosen for and . For example, one may restrict and to periodic boundary conditions . Similarly, one may let and if is integrable and . Objects of this form have been considered in [72, 73, 74, 75].

An important class of observables is given by the “evolving constants” [15, 12], as mentioned in the introduction. These objects encode the relational evolution of *on-shell* tensor fields, i.e., fields which are solutions to the equations of motion, and they yield invariant extensions of the initial values. They can be constructed by imposing a gauge condition, i.e., by defining a parametrisation of the worldline, in the following way. Let be an arbitrary initial parameter and define as a new time coordinate through the equation

(5) |

where is a worldline scalar which will be referred to as the gauge condition^{*}^{*}*Gauge conditions of the form given in (5) are sufficient for our purposes, although more general gauge conditions are possible [12].. The condition is admissible if

(6) |

which may be fulfilled only locally in the configuration-velocity space. In a region where (6) holds, one can solve it for to find the coordinate transformation

(7) |

If the gauge condition is admissible, defines a (field-dependent) diffeomorphism on the worldline, with which we can pullback tensor fields. The invariant extensions of initial values can then be obtained by writing the pullback in an arbitrary parametrisation^{†}^{†}†This corresponds to the statement that invariant extensions are obtained by writing gauge-fixed quantities in an arbitrary gauge.. To make this statement more precise, let us define the Dirac delta distribution

(8) | ||||

Then, given a scalar field , we can write (cf. (4))

(9) | ||||

provided (6) holds and the integral in (9) converges. Similar integral expressions have been considered in [72, 73, 74, 75]. For any *fixed* value of , eq. (9) defines an invariant extension (Dirac observable) of the initial value of in the sense that it is manifestly independent of the choice of . We will see in section 2.1.3 that this property implies that the quantity given in (9) Poisson-commutes with the phase-space constraint. In particular, the Dirac observable associated with the identity function is again the identity,

(10) | ||||

Eq. (10) is the “Faddeev-Popov resolution of the identity” for the gauge condition . The Dirac observable associated with the gauge condition itself is trivial

(11) | ||||

Similarly, given a one-form , we can define the Dirac observable

(12) |

As already noted, the integral expressions (9) and (12) are manifestly independent of the choice of and, thus, are gauge-invariant extensions for a fixed value of . However, they generally depend on the gauge condition given in (5). This is usually the case with invariant extensions [16, 18, 12, 17], i.e., they yield gauge-invariant but not gauge-independent objects. The physical interpretation of this procedure is particularly clear for the scalar Dirac observables (cf. (9)): they represent the value of the scalar field “when” the scalar has the value , i.e., they encode the (on-shell) relational evolution between the scalar fields.

#### 2.1.2 Hamiltonian and Gauge Generator

If the fundamental fields are worldline scalars, the Hamiltonian vanishes [65, 12]. To see this, we follow [65] and expand (2) and use (1) to obtain^{‡}^{‡}‡Summation over repeated indices is implied.

which yields

(13) |

where the momenta are defined in the usual way, . Eq. (13) also implies that the Lagrangian is singular [11]

(14) |

i.e., that one cannot invert to find the velocities as functions of coordinates and momenta. This entails that the momenta are not independent and are generally related by constraints ^{§}^{§}§Constraints of this type are called primary in the usual Rosenfeld-Dirac-Bergmann algorithm [5, 6, 13, 76].. For simplicity, we assume there is only one constraint, which amounts to imposing that the only gauge symmetry of the theory is given by the wordline diffeomorphisms. Thus, the constraint algebra is automatically first class and abelian.

The constraint defines a surface in phase space. We make use of Dirac’s weak equality sign to denote identities which hold only on the constraint surface [7]. Thus , since the canonical Hamiltonian is well defined only if . Hence, there is no loss of generality if we write , where is an arbitrary worldline scalar density which can be chosen to be the einbein:

(15) |

In this manner, the evolution in of a phase space function is given by

(16) |

where is the Poisson bracket

(17) |

Can gauge transformations be represented as canonical transformations in phase space? For worldline scalars with no explicit -dependence, we have

(18) |

Thus, the reparametrisations of such worldline scalars are *on-shell* canonical transformations generated by (called the gauge generator). For our present purposes, this is all that is needed^{¶}^{¶}¶Even if one allows to depend on the canonical variables , it is not possible to reach the gauge condition (5) by a canonical transformation, i.e., is not a canonical transformation. The reason for this is clarified in [70], where Pons et al. note that the map that produces the invariant extensions is not invertible, since it projects all the points in a gauge orbit to the same image where the gauge condition is satisfied. Hence, this map cannot be canonical. Using a formalism which is different from (but equivalent to) the one presented here, they show that the invariant extension can be seen as a limit of a one-parameter family of canonical transformations.. However, it is worth mentioning that it is possible to extend the phase space to include the einbein and its conjugate momentum as a canonical pair subject to the constraint . In this way, one can describe the gauge variations of worldline scalars and one-forms as on-shell canonical transformations generated by , where [69, 70, 4, 14].

#### 2.1.3 “Evolving Constants” are Invariant Extensions

We stated in section 2.1.1 that the quantity given by the integral expression in (9) represents an invariant extension of for each fixed value of because it is manifestly independent of the choice of the initial arbitrary parametrisation . As is well-known, this statement can be substantiated by proving that is a Dirac observable, i.e., it Poisson-commutes with the phase-space constraint and, therefore, with the gauge generator. To do this, we first note that the phase-space constraint generates evolution in “proper time”, defined as . Indeed, if is a phase-space function with no explicit time-dependence, we obtain (cf. (15))

(19) |

Thus, we can write (9) in terms of proper time,

(20) | ||||

where we assumed the scalars have no explicit time-dependence. From (19) and (20), we obtain

(21) |

This result holds if for fixed values of and the initial conditions .

Since is a Dirac observable for each fixed value of , once sees that there is a one-parameter family of invariant functions, each corresponding to one moment of the evolution. That is why such objects are often called “evolving constants” [15].

#### 2.1.4 Dynamics of Dirac Observables

The pullback of on-shell scalar functions under given in (7) is evidently dynamical (time-dependent) in general. Indeed, let us write for brevity. We can then write

(22) | ||||

where both the Hamiltonian (cf. (15)) and the Poisson bracket (cf. (17)) are taken with respect to the original set of fields and as opposed to the pulled-back fields. Only at the end of the calculation one sets (and ). Moreover, by setting in the above equation and using (11), we find

(23) |

where due to (6). In the context of minisuperspace quantum cosmology, eq. (23) yields the gauge-fixed lapse function. If we insert (23) on (22), we obtain

(24) |

which shows that the dynamics of observables is not “frozen” in general. In fact, eq. (24) yields the gauge-fixed (or “reduced”) equations of motion for the dynamical variables. The on-shell gauge-fixed evolution is generated by . Indeed,

Moreover, the right-hand side of (24) is a Dirac observable for each fixed value of . To see this, we use the right-hand side of (9) to write (cf. (12))

The equation

(25) |

was also obtained in [70] using a different method. Eq. (25) is of key importance for the quantum theory, since we expect that it can be promoted to a Heisenberg-picture equation of motion, both sides of which are well-defined operators (which commute with the constraint operator for each value of ). We will see in sections 2.2.2, 3.4 and 3.5 how this can be achieved.

### 2.2 Quantum Theory

#### 2.2.1 The Physical Hilbert Space

Following [53, 54, 55, 56, 57, 58, 59], we promote the classical phase-space constraint to a linear operator and assume that it is possible to choose the factor ordering such that is self-adjoint in an auxiliary Hilbert space of square-integrable functions equipped with an auxiliary inner product . In this way, has a complete orthonormal system of eigenstates

(26) | ||||

(27) |

where labels degeneracies. The symbol stands for a Kronecker or Dirac delta, depending on whether the spectrum of is discrete or continuous.

The quantum analogue of the classical constraint surface is the linear subspace of states in the kernel of , which can be written as superpositions of . These states are invariant under the unitary flow of the constraint operator and their overlap reads

(28) |

The factor of is divergent if zero is in the continuous part of the spectrum of , which implies the auxiliary inner product cannot be used in this subspace. It is possible [53, 54, 55, 56, 57, 58, 59, 77] to define a regularised (induced) inner product on the kernel of in the following way:

(29) |

such that

(30) |

Now consider the superpositions

(31) |

where the sum over must be replaced by an integral if the degeneracies are labelled by continuous indices. Then, from (30), we obtain the (Rieffel induced) inner product for general invariant states

(32) |

The kernel of equipped with the inner product (32) is defined to be the physical Hilbert space of the theory.

#### 2.2.2 Matrix Elements of Quantum Dirac Observables

We are now in a position to propose a method of construction of operators which correspond to the classical Dirac observables. To begin with, according to (15), the Hamiltonian operator may be defined as , where factor ordering issues are avoided by choosing the arbitrary -parametrisation such that the einbein is not a function of the canonical variables and, thus, it is a c-number in the quantum theory. The simplest choice is (“proper time gauge”), which is the one we adopt. In this way, any classical observable of the form given in (4) can be promoted to the operator

(33) |

If the spectrum of is discrete, one may choose , whereas if the spectrum is continuous, we let and In any case, we find the matrix elements

(34) |

and the regularised matrix elements are [73]

(35) |

Now we would like to define an operator , such that^{∥}^{∥}∥We assume that the spectrum of is continuous in what follows, since this will be the case in the concrete examples analysed in later sections.

(36) |

is a symmetric quantisation of the classical scalar Dirac observable given in (9). In particular, we require that an operator version of the “Faddeev-Popov resolution of the identity” (10) holds, i.e.,

(37) |

which implies that the operator must satisfy the relation

(38) |

for all values of . In this point we differ from the work of Marolf in [73], in which the definition of the operator given in (36) was chosen in such a way that the invariant extension of the identity was not the identity operator, a result which we consider to be undesirable. Indeed, for the case of the relativistic particle (which we will analyse in section 3), the operator definition chosen in [73] yields . We believe that (38) should be the correct requirement. In fact, eq. (38) is equivalent to regularising the inner product by the “insertion of an operator gauge condition”, a procedure which was advocated in [12, 18].

How can we define ? Given a gauge condition operator which is self-adjoint in the auxiliary inner product, the (improper) projectors onto its eigenspaces are , where labels degeneracies of the eigenstates of . Since the classical gauge condition is only admissible if (6) holds, i.e., if , we consider the operator

(39) |

where the operators are included to project out the zero modes of . We can now define

(40) |

where we introduced the notation for brevity. The operator given in (40) is by construction an invariant for each fixed value of . Moreover, it is a symmetric quantisation of the classical expression for the invariant extension of , as can be easily verified. From (40), we obtain the symmetric resolution of the identity

(41) |

which leads to the sought-after definition

(42) |

Eq. (42) is the canonical (operator-based) analogue of the usual Faddeev-Popov procedure employed in path integrals [51, 52]. A similar canonical procedure was suggested in [18], although it was not specified which factor ordering was required and the need to include the operators was not recognised.

We are now in a position to define invariant extensions of operators other than the identity. Suppose is a scalar operator which commutes with the gauge condition and is self-adjoint with respect to the auxiliary inner product. Then and share a complete orthonormal system of eigenstates , where labels other possible degeneracies. We may write

(43) |

In analogy to (42), we *define*

(44) |

which amounts to defining the quantum Dirac observable (cf. (36)) via its spectral decomposition. We can also define invariant extensions of scalars which do not commute with the gauge condition in the following way. We first note that the (improper) projector onto the eigenspaces of can be written as

(45) | ||||

where if the spectrum of is discrete and if the spectrum of is continuous. If is a self-adjoint scalar operator which does not commute with , we can use (45) to generalise (44) to

(46) |

We take (36) together with (46) to be the general definition of the invariant extension of the scalar operator . Invariant extensions of scalar densities could be defined in a similar way but we shall have no need for them in what follows. In the next sections, we will apply the general formalism here presented to the concrete examples of the relativistic free particle and the vacuum Bianchi I model.

## 3 The Relativistic Particle

Let us now illustrate the general ideas presented in the previous section for the relativistic free particle, which is the archetypical example of a time-reparametrisation invariant system. We first present the construction of Dirac observables in the classical theory and their non-relativistic limit. We then quantise the theory and show that the notion of ‘‘WKB time’’ emerges in the non-relativistic limit of invariant transition amplitudes. This result, although expected, clarifies the relation between the ‘‘semiclassical approach to the problem of time’’ and the more complete quantum theory based on the induced inner product^{*}^{*}*The “semiclassical approach” was thoroughly analysed in [50]. The example of the relativistic particle here presented serves to elucidate how this approach is related to the fundamental reparametrisation invariance of the theory at the quantum level.. Moreover, we discuss the dynamics of quantum Dirac observables also in the non-relativistic limit to compare the formalism here presented with the results of [37, 38].

### 3.1 Classical Theory

#### 3.1.1 Observables

The action for a massive relativistic particle moving in the -dimensional Minkowski spacetime reads

(47) |

where are the coefficients of the Minkowski metric with signature and are the spacetime coordinates. The action is invariant under reparametrisations of which coincide with the identity at the endpoints. The Euler-Lagrange equations yield

(48) |

where are constants that satisfy the initial-value constraint

(49) |

The solutions of (48) are relational: we can determine the trajectories of one coordinate in terms of another. For example, we find

(50) |

by dividing the equation for by the equation for . We note that the relation (50) holds in any parametrisation (any gauge). In this way, the boundary values may be seen as an invariant extension of for a fixed value of , i.e.,

(51) |

is a Dirac observable. Indeed,

where we used (48). The observable given in (51) represents the value of “when” . This is true independently of the chosen parametrisation, i.e., it is a gauge-invariant statement. Similarly, we may construct an invariant extension of by writing it in terms of . We obtain

(52) | ||||

It is clear that the quantities given in (52) are invariants. The right-hand sides of (52) are well-defined provided . It is useful to note that the dynamics of (51) and (52) may be expressed in terms of Poisson brackets (as defined in (17) with the basic variables ) in the following way:

(53) | ||||

(54) |

and similarly for . In sections 3.4 and 3.5, we will find the quantum analogues of (53) and (54) as equations which determine the dynamics of quantum Dirac observables.

#### 3.1.2 On-Shell Action and the Hamilton-Jacobi Constraint

The constants may be eliminated in terms of the boundary values by using (49) and (50) evaluated at . The result is

(56) | |||

(57) |

Eq. (56) together with (48) implies that , which leads to . Using (56) and (57), we can now insert the relational solution (50) in the integrand of (47) to obtain the on-shell action:

which can be rewritten as

(58) |

This is the expected result from elementary relativity. One can readily verify that the on-shell action given in (58) is a solution to the Hamilton-Jacobi (HJ) constraint:

(59) |

similarly for the other endpoint. The unobservable label does not appear in these equations. Nevertheless, it is clear that the absence of “time” in (59) does not imply the absence of dynamics, but rather it signals that the dynamics is relational [33, 35]. It is worthwhile to note that the same disappearance of label time occurs in the quantum theory of gauge-invariant states. This does not imply that there is no quantum dynamics or that the dynamics can only be understood in a particular (semiclassical) regime (which is the view taken in the ‘‘semiclassical approach to the problem of time’’). As in the classical theory, the dynamics has to be understood in a relational way^{†}^{†}†In [50], it was argued that the notion of “gauge fixing” (i.e., the fixation of the time coordinate) is fundamental to the understanding of the dynamics in a time-reparametrisation invariant system. This means that the unobservable label can be chosen in order to describe the evolution of the dynamical fields in a relational way. This becomes clear when one considers the construction of invariant extensions through a gauge condition, as was done in sections 2.1.1 and 2.2.2 of the present article.. We will see how this occurs when computing matrix elements of quantum Dirac observables.