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3.4 A digression: Observables and the problem of time

When one attempts to interpret the quantum theories coming from the Chern-Simons formalism or covariant canonical quantization, one finds an immediate and rather profound difficulty. The gauge-invariant observables - the traces of the holonomies - are automatically nonlocal and time-independent, and one obtains a “frozen time formalism,” or what Kuchaƙ has called “quantum gravity without time”  [173] . In one sense, this is a good thing: One knows from general arguments that the diffeomorphism-invariant observables in any quantum theory of gravity must have these features  [257] . On the other hand, it is not at all easy to see how to extract local geometry and dynamics from such a picture: If our only observables are nonlocal and time-independent, how can we recover a classical limit with local excitations that evolve in time?

Quantum gravity in 2+1 dimensions offers a possible answer to this dilemma. Note first that the problem is already present classically. A geometric structure determines a spacetime, and must contain within it all of the dynamics of that spacetime. On the other hand, the basic data that fix the geometric structure - the transition functions, or, often, the holonomies - have no obvious dynamics. In principle, the classical answer is simple:

  1. Use, say, the holonomies to determine a spacetime geometry.
  2. Select a favorite time-slicing.
  3. Read off the spatial metric and its time derivatives from the spacetime metric of Step  1 in this slicing.

This procedure can be understood as a concrete realization of the isomorphism described in Section  3.3 between the phase space and the space of classical solutions, with the Cauchy surface C fixed by the choice of time-slicing.

For the simple case of the torus universe, these steps can be transcribed almost directly to the quantum theories. Equations (42View Equation, 43View Equation, 44View Equation) become definitions of operators,

( - it/l + - it/l)( - it/l + -it/l)-1 ^tt = ^r1 e + ^r1 e ^r2 e + ^r2 e , ^p = - --il---(^r+eit/l + ^r-e- it/l)2, t 2sin 2lt 2 2 (62) 2 H^ = l- sin 2t(^r-^r+ - ^r+ ^r-), t 4 l 1 2 1 2
where the operator ordering has been chosen to respect the modular transformations (50View Equation). The parameter t is now merely a label for a one-parameter family of diffeomorphism-invariant observables. These observables obtain their physical significance from the classical limit: ^tt, for example, is the operator whose expectation value gives the mean value of the modulus on a time slice of constant mean curvature T = - 2cot 2t l l . Such observables are examples of what Rovelli has called “evolving constants of motion”  [236237] .

From this point of view, we should think of Chern-Simons/covariant canonical quantization as a sort of Heisenberg picture, with time-independent states and “time”-dependent operators. To obtain the corresponding Schrödinger picture, we proceed as in ordinary quantum mechanics: We diagonalize ^tt, obtaining a transition matrix + - + - K(t, t;t| r2 ,r2 ) = <t,t;t|r2 ,r2 > that allows us to transform between representations  [6888Jump To The Next Citation Point] . The resulting “time”-dependent wave functions obey a Schrödinger equation of the form (52View Equation, 53View Equation), but with the Laplacian in H^ replaced by the weight 1/2 Maass Laplacian D1/2 of Equation (54View Equation). In  [118], it has been shown that these wave functions are peaked around the correct classical trajectories. (Different operator orderings in Equation (62View Equation) give different weight Laplacians  [70] .)

As a useful byproduct, this analysis allows us to solve the problem of the poorly-behaved action of the modular group discussed at the end of Section  3.2   [8889] . If we start with a reduced phase space wave function ~ y(t, t;t) and use the transition matrix K to determine a Chern-Simons wave function + - y(r 2 ,r2 ), we find, indeed, that + - y(r 2 ,r2 ) is not modular invariant. Instead, though, the entire Hilbert space of Chern-Simons wave functions splits into “fundamental regions,” orthogonal subspaces that transform into each other under the action of the modular group. Any one of these fundamental regions is equivalent to any other, and each is equivalent to the Hilbert space arising from reduced phase space quantization. Moreover, matrix elements of any modular invariant function vanish unless they are taken between states in the same fundamental region. Modular invariance thus takes a slightly unexpected form, but can still be imposed by restricting the theory to a single fundamental region of the Hilbert space.

We can also begin to address the problem raised at the end of Section  3.1, the limited and slicing-dependent range of questions one can ask in reduced phase space quantization. The operators (62View Equation) introduced here on the covariant canonical Hilbert space were obtained from a particular classical time-slicing, and answer questions about spatial geometry in that slicing. In principle, however, we can choose any other slicing, with a new time coordinate t, and determine the corresponding operators ^t t, ^p t, and ^H t . The operator ordering of such operators will, of course, be ambiguous, though one might hope that the action of the modular group might again restrict the choices. But such an ambiguity need not be seen as a problem with the theory; rather, it is merely a statement that many different quantum operators can have the same classical limit, and that ultimately experiment must decide which operator we are really observing.

There is, to be sure, a danger that the “Schrödinger pictures” coming from different time-slicings may not be consistent. Suppose, for example, that we choose two slicings that agree on an initial and a final slice S1 and S2, but disagree in between. If we start with an initial wave function on S1, we must check that the Hamiltonians coming from the different slicings evolve us to the same final wave function on S2 . For field theories, even in flat spacetime, this will not always happen  [258] . For (2+1)-dimensional gravity, on the other hand, there is evidence that one can always find operator orderings of the Hamiltonians that ensure consistent evolution  [95] . If this ultimately turns out not to be the case, however, it may simply mean that we should treat the covariant canonical picture as fundamental, and discard the Schrödinger pictures of time-dependent states.

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