Wheeler–DeWitt equation

Motivation and background

In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is \gamma_{ij} and given by g_{\mu\nu},\mathrm{d}x^{\mu},\mathrm{d}x^\nu = (-N^2 + \beta_k\beta^k),\mathrm{d}t^2 + 2\beta_k,\mathrm{d}x^k,\mathrm{d}t + \gamma_{ij},\mathrm{d}x^i,\mathrm{d}x^j. In that equation the Latin indices run over the values 1, 2, 3, and the Greek indices run over the values 1, 2, 3, 4. The three-metric \gamma_{ij} is the field, and we denote its conjugate momenta as \pi^{ij}. The Hamiltonian is a constraint (characteristic of most relativistic systems) \mathcal{H} = \frac{1}{2\sqrt{\gamma}} G_{ijkl}\pi^{ij}\pi^{kl} - \sqrt{\gamma},{}^{(3)}!R = 0, where \gamma = \det(\gamma_{ij}), and G_{ijkl} = (\gamma_{ik}\gamma_{jl} + \gamma_{il}\gamma_{jk} - \gamma_{ij}\gamma_{kl}) is the Wheeler–DeWitt metric. In index-free notation, the Wheeler–DeWitt metric on the space of positive definite quadratic forms g in three dimensions is \operatorname{tr}((g^{-1}dg)^2) - (\operatorname{tr}(g^{-1}dg))^2.

Quantization "puts hats" on the momenta and field variables; that is, the functions of numbers in the classical case become operators that modify the state function in the quantum case. Thus we obtain the operator \hat{\mathcal{H}} = \frac{1}{2\sqrt{\gamma}} \hat{G}{ijkl} \hat{\pi}^{ij} \hat{\pi}^{kl} - \sqrt{\gamma},{}^{(3)}!\hat{R}. Working in "position space", these operators are \begin{align} \hat{\gamma}{ij}(t, x^k) &\to \gamma_{ij}(t, x^k), \ \hat{\pi}^{ij}(t,x^k) &\to -i \frac{\delta}{\delta \gamma_{ij}(t,x^k)}. \end{align}

One can apply the operator to a general wave functional of the metric \hat{\mathcal{H}} \Psi[\gamma] = 0, where \Psi[\gamma] = a + \int \psi(x)\gamma(x) ,dx^3 + \iint \psi(x, y)\gamma(x)\gamma(y) ,dx^3 ,dy^3 + \dots, which would give a set of constraints amongst the coefficients \psi(x, y, \dots). This means that the amplitudes for N gravitons at certain positions are related to the amplitudes for a different number of gravitons at different positions. Or, one could use the two-field formalism, treating \omega(g) as an independent field, so that the wave function is \Psi[\gamma, \omega].

Mathematical formalism

The Wheeler–DeWitt equation

Hamiltonian constraint

Simply speaking, the Wheeler–DeWitt equation says |border where \hat{H}(x) is the Hamiltonian constraint in quantized general relativity, and |\psi\rangle stands for the wave function of the universe. Unlike ordinary quantum field theory or quantum mechanics, the Hamiltonian is a first-class constraint on physical states. We also have an independent constraint for each point in space.

Although the symbols \hat{H} and |\psi\rangle may appear familiar, their interpretation in the Wheeler–DeWitt equation is substantially different from non-relativistic quantum mechanics. |\psi\rangle is no longer a spatial wave function in the traditional sense of a complex-valued function that is defined on a 3-dimensional space-like surface and normalized to unity. Instead it is a functional of field configurations on all of spacetime. This wave function contains all of the information about the geometry and matter content of the universe. \hat{H} is still an operator that acts on the Hilbert space of wave functions, but it is not the same Hilbert space as in the nonrelativistic case, and the Hamiltonian no longer determines the evolution of the system, so the Schrödinger equation \hat{H} |\psi\rangle = i \hbar \partial / \partial t |\psi\rangle no longer applies. This property is known as timelessness. Various attempts to incorporate time in a fully quantum framework have been made, starting with the "Page and Wootters mechanism" and other subsequent proposals. The reemergence of time was also proposed as arising from quantum correlations between an evolving system and a reference quantum clock system, the concept of system-time entanglement is introduced as a quantifier of the actual distinguishable evolution undergone by the system.

Momentum constraint

We also need to augment the Hamiltonian constraint with momentum constraints : \vec{\mathcal{P}}(x) |\psi\rangle = 0 associated with spatial diffeomorphism invariance.

In minisuperspace approximations, we only have one Hamiltonian constraint (instead of infinitely many of them).

In fact, the principle of general covariance in general relativity implies that global evolution per se does not exist; the time t is just a label we assign to one of the coordinate axes. Thus, what we think about as time evolution of any physical system is just a gauge transformation, similar to that of QED induced by U(1) local gauge transformation \psi \to e^{i\theta(\vec{r})} \psi, where \theta(\vec{r}) plays the role of local time. The role of a Hamiltonian is simply to restrict the space of the "kinematic" states of the Universe to that of "physical" states—the ones that follow gauge orbits. For this reason we call it a "Hamiltonian constraint". Upon quantization, physical states become wave functions that lie in the kernel of the Hamiltonian operator.

In general, the Hamiltonian vanishes for a theory with general covariance or time-scaling invariance.

References

References

1. DeWitt, Bryce S.. (1967-08-25). "Quantum Theory of Gravity. I. The Canonical Theory". Physical Review.
2. (23 October 2013). "Quantum Experiment Shows How Time 'Emerges' from Entanglement". The Physics arXiv Blog.
3. Rovelli, Carlo. (2001-01-23). "Notes for a brief history of quantum gravity".
4. Page, Don N.. (1983-06-15). "Evolution without evolution: Dynamics described by stationary observables". Physical Review D.
5. Rovelli, Carlo. (1990-10-15). "Quantum mechanics without time: A model". Physical Review D.
6. Boette, A.. (2016-06-27). "System-time entanglement in a discrete-time model". Physical Review A.
7. Boette, A.. (2018-09-12). "History states of systems and operators". Physical Review A.