5. Quantum Mechanics

Since the classical dynamics was formulated in terms of the well established Lagrangian formalism, it seems natural to quantize the system by interpreting the phase space variables R and P_R as the basic observables, or quantum operators acting on a Hilbert space ${\cal H}$. Then the correspondence principle

  

$$\rightarrow \hat{P} _R = - i \hbar \frac{1}{R} \frac{\partial}{\partial R} R$$ P_R (59)

$$\rightarrow \hat{R} = R$$ R (60)

leads to the canonical commutation relation: $\left[ \hat{R}, \hat{P} _R \right] = i \hbar$. The Hamiltonian constraint of the classical theory is then imposed on the quantum states

$$\hat{H}( P_R, R ) \vert \Psi \rangle = 0 \quad ,$$ (61)

thus selecting a linear physical subspace of ${\cal H}$.

Equation (61) is nothing but the Wheeler-DeWitt equation for the quantum theory of ``shell-dynamics".

Unfortunately, this straightforward procedure leads to a non local form of the Hamiltonian (55) which, as a quantum operator, is plagued by ordering ambiguities. The point of fact, then, is that any manipulation of that Hamiltonian has a degree of arbitrariness that goes with it. For instance, a suggestion was made in ref.[4,5], whereby, after a canonical transformation, eq.(61) is transformed into a finite difference equation (the representation used there differs, however, from eqs.(59-60).

Alternatively, extrapolating the result of Section IV, namely, that the effective dynamics of a shell is simulated by the motion of a classical particle in a given potential, one may interpret the relation (61), as the stationary equation for a quantum particle of rest mass m. As a consistency check on this interpretation, let us consider the limit, as $G_N \rightarrow 0$, of the Hamiltonian $\hat{H}$ corresponding to the case, mentioned earlier, of a shell of dust separating an interior Minkowski spacetime from an exterior Schwarzschild one. In this case, we have A_in=1, A_out= 1 - 2 G_N M / R, $\kappa R^2 = m = const$, $\sigma _{in} = \sigma _{out} = +1$, and the corresponding Hamiltonian operator is given by

$$\hat{H} = m - \frac{2R}{G_N} \left[1 -\frac{ G_N M}{R} - \... ...\frac{G_N \hat{P}_R}{R} \right) \right] ^{\frac{1}{2}} \quad$$ (62)

with $\hat{P} _R$ given by equation(59). Expanding in power of G_N, with a suitable choice of ordering, one obtains

$$\hat{H} \sim m - \frac{R}{G_N} \left[ \frac{G_N^2 M^2}{R^2... ... = m - \left[ M^2 - \hat{P}_R^2 \right] ^{\frac{1}{2}} \quad .$$ (63)

Therefore, in the limit $G_N \rightarrow 0$, the Wheeler-DeWitt equation reduces to

$$m \Psi = \sqrt{M^2 - \hat{P}_R^2} \Psi \quad .$$ (64)

Squaring the above relation, yields the Klein-Gordon equation for an S-wave particle of rest mass m and energy (ADM energy) M, which seems to be a natural result for a spherical shell of dust, once the gravitational interaction has been switched off.

Coming back to the Wheeler-De Witt equation (61), although its full solution is presently out of reach, a special class of solutions exists in the literature representing a sufficiently significant sample of the shell quantum dynamics. These are the WKB solutions of the form

$$\Psi = e ^{\frac{i}{\hbar} S _{eff}} \quad .$$ (65)

However, rather then reviewing the explicit construction of these solutions (see ref.[15]), in the next section we will discuss the application of the WKB formalism to some quantum tunneling processes which are forbidden according to the classical dynamics discussed in Section IV.