View source for Wheeler–DeWitt equation

{{short description|Field equation, part of a theory that attempts to combine quantum mechanics and general relativity}}
{{technical|date=August 2011}}
{{Quantum field theory|cTopic=Equations}}
The '''Wheeler–DeWitt equation'''<ref name="DeWitt">{{Cite journal |last=DeWitt |first=Bryce S. |date=1967-08-25 |year=1967 |title=Quantum Theory of Gravity. I.&nbsp;The Canonical Theory |url=https://link.aps.org/doi/10.1103/PhysRev.160.1113 |journal=Physical Review |language=en |volume=160 |issue=5 |pages=1113–1148 |bibcode=1967PhRv..160.1113D |doi=10.1103/PhysRev.160.1113 |issn=0031-899X}}</ref> for [[theoretical physics]] and [[applied mathematics]], is a [[field equation]] attributed to [[John Archibald Wheeler]] and [[Bryce DeWitt]]. The equation attempts to mathematically combine the ideas of [[quantum mechanics]] and [[general relativity]], a step towards a theory of [[quantum gravity]].

In this approach, [[time in physics|time]] plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called "[[problem of time]]".<ref>{{Cite web |date=23 October 2013 |title=Quantum Experiment Shows How Time 'Emerges' from Entanglement |url=https://medium.com/the-physics-arxiv-blog/quantum-experiment-shows-how-time-emerges-from-entanglement-d5d3dc850933 |website=[[Medium (website)|Medium]] |publisher=The Physics arXiv Blog}}</ref> More specifically, the equation describes the quantum version of the [[Hamiltonian constraint]] using metric variables. Its commutation relations with the [[diffeomorphism constraint]]s generate the Bergman–Komar "group" (which ''is'' the [[diffeomorphism group]] [[on-shell]]).

== Motivation and background ==
In [[Canonical quantum gravity|canonical gravity]], spacetime is [[Foliation|foliated]] into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is <math>\gamma_{ij}</math> and given by
<math display="block">
 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.
</math>
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 <math>\gamma_{ij}</math> is the field, and we denote its conjugate momenta as <math>\pi^{ij}</math>. The Hamiltonian is a constraint (characteristic of most relativistic systems)
<math display="block">
 \mathcal{H} = \frac{1}{2\sqrt{\gamma}} G_{ijkl}\pi^{ij}\pi^{kl} - \sqrt{\gamma}\,{}^{(3)}\!R = 0,
</math>
where <math>\gamma = \det(\gamma_{ij})</math>, and <math>G_{ijkl} = (\gamma_{ik}\gamma_{jl} + \gamma_{il}\gamma_{jk} - \gamma_{ij}\gamma_{kl})</math> 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
<math display="block">
 \operatorname{tr}((g^{-1}dg)^2) - (\operatorname{tr}(g^{-1}dg))^2.
</math>

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
<math display="block">
 \hat{\mathcal{H}} = \frac{1}{2\sqrt{\gamma}} \hat{G}_{ijkl} \hat{\pi}^{ij} \hat{\pi}^{kl} - \sqrt{\gamma}\,{}^{(3)}\!\hat{R}.
</math>
Working in "position space", these operators are
<math display="block">\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}</math>

One can apply the operator to a general wave functional of the metric <math>\hat{\mathcal{H}} \Psi[\gamma] = 0,</math> where
<math display="block">
 \Psi[\gamma] = a + \int \psi(x)\gamma(x) \,dx^3 + \iint \psi(x, y)\gamma(x)\gamma(y) \,dx^3 \,dy^3 + \dots,
</math>
which would give a set of constraints amongst the coefficients <math>\psi(x, y, \dots)</math>. This means that the amplitudes for <math>N</math> 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 <math>\omega(g)</math> as an independent field, so that the wave function is <math>\Psi[\gamma, \omega]</math>.

== Mathematical formalism ==
The Wheeler–DeWitt equation<ref name=DeWitt/> is a [[Functional derivative|functional differential]] equation. It is ill-defined in the general case, but very important in [[theoretical physics]], especially in [[quantum gravity]]. It is a functional differential equation on the space of three-dimensional spatial metrics. The Wheeler–DeWitt equation has the form of an operator acting on a wave functional; the functional reduces to a function in cosmology. Contrary to the general case, the Wheeler–DeWitt equation is well defined in [[minisuperspace]]s like the configuration space of cosmological theories. An example of such a [[wave function]] is the [[Hartle–Hawking state]]. [[Bryce DeWitt]] first published this equation in 1967 under the name "Einstein–Schrödinger equation"; it was later renamed the "Wheeler–DeWitt equation".<ref>{{Cite book |last=Rovelli |first=Carlo |url=https://archive.org/details/arxiv-gr-qc0006061 |title=Notes for a brief history of quantum gravity |date=2001-01-23 |publication-place=Presented at the 9th Marcel Grossmann Meeting in Roma, July 2000 |language=English |arxiv=gr-qc/0006061}}</ref>

===Hamiltonian constraint===

Simply speaking, the Wheeler–DeWitt equation says
{{Equation box 1
|indent = :
|equation = <math>\hat{H}(x) |\psi\rangle = 0,</math>
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where <math>\hat{H}(x)</math> is the [[Hamiltonian constraint]] in quantized [[general relativity]], and <math>|\psi\rangle</math> stands for the [[Universal wavefunction|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 <math>\hat{H}</math> and <math>|\psi\rangle</math> may appear familiar, their interpretation in the Wheeler–DeWitt equation is substantially different from non-relativistic quantum mechanics. <math>|\psi\rangle</math> 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 (mathematics)|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. <math>\hat{H}</math> 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]] <math>\hat{H} |\psi\rangle = i \hbar \partial / \partial t |\psi\rangle</math> 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.<ref>{{Cite journal |last=Page |first=Don N. |last2=Wootters |first2=William K. |date=1983-06-15 |title=Evolution without evolution: Dynamics described by stationary observables |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.27.2885 |journal=Physical Review D |language=en |volume=27 |issue=12 |pages=2885–2892 |doi=10.1103/PhysRevD.27.2885 |issn=0556-2821}}</ref><ref>{{Cite journal |last=Rovelli |first=Carlo |date=1990-10-15 |title=Quantum mechanics without time: A model |url=https://link.aps.org/doi/10.1103/PhysRevD.42.2638 |journal=Physical Review D |volume=42 |issue=8 |pages=2638–2646 |doi=10.1103/PhysRevD.42.2638 |pmid=10013133 }}</ref> 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.<ref>{{Cite journal |last=Boette |first=A. |last2=Rossignoli |first2=R. |last3=Gigena |first3=N. |last4=Cerezo |first4=M. |date=2016-06-27 |title=System-time entanglement in a discrete-time model |url=https://link.aps.org/doi/10.1103/PhysRevA.93.062127 |journal=Physical Review A |language=en |volume=93 |issue=6 |pages=062127 |arxiv=1512.07313 |doi=10.1103/PhysRevA.93.062127 |issn=2469-9926 |s2cid=119245348}}</ref><ref>{{Cite journal |last=Boette |first=A. |last2=Rossignoli |first2=R. |date=2018-09-12 |title=History states of systems and operators |url=https://link.aps.org/doi/10.1103/PhysRevA.98.032108 |journal=Physical Review A |language=en |volume=98 |issue=3 |pages=032108 |arxiv=1806.00956 |doi=10.1103/PhysRevA.98.032108 |issn=2469-9926 |s2cid=56101730}}</ref> 

===Momentum constraint===

We also need to augment the Hamiltonian constraint with [[momentum constraint]]s
: <math>\vec{\mathcal{P}}(x) |\psi\rangle = 0</math>
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 <math>t</math> 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 [[Quantum electrodynamics|QED]] induced by U(1) local gauge transformation <math>\psi \to e^{i\theta(\vec{r})} \psi,</math> where <math>\theta(\vec{r})</math> 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 (matrix)|kernel]] of the Hamiltonian operator.

In general, the [[Hamiltonian (quantum mechanics)|Hamiltonian]]{{clarify|date=August 2016}} vanishes for a theory with general covariance or time-scaling invariance.

==See also==
{{cols}}
*[[ADM formalism]]
*[[Diffeomorphism constraint]]
*[[Euclidean quantum gravity]]
*[[Regge calculus]]
*[[Canonical quantum gravity]]
*[[Peres metric]]
*[[Loop quantum gravity]]
{{colend}}

==References==

{{Reflist}}<!--added above categories/infobox footers by script-assisted edit-->

* {{Cite journal |last=Hamber |first=Herbert W. |last2=Williams |first2=Ruth M. |date=2011-11-18 |year=2011 |title=Discrete Wheeler-DeWitt equation |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.84.104033 |journal=Physical Review D |language=en |volume=84 |issue=10 |pages=104033 |arxiv=1109.2530 |bibcode=2011PhRvD..84j4033H |doi=10.1103/PhysRevD.84.104033 |issn=1550-7998 |s2cid=4812404}}
* {{Cite journal |last=Hamber |first=Herbert W. |last2=Toriumi |first2=Reiko |last3=Williams |first3=Ruth M. |date=2012-10-02 |year=2012 |title=Wheeler-DeWitt equation in 2 + 1 dimensions |url=https://link.aps.org/doi/10.1103/PhysRevD.86.084010 |journal=Physical Review D |language=en |volume=86 |issue=8 |pages=084010 |arxiv=1207.3759 |bibcode=2012PhRvD..86h4010H |doi=10.1103/PhysRevD.86.084010 |issn=1550-7998 |s2cid=119229306}}
{{quantum gravity}}

{{DEFAULTSORT:Wheeler-DeWitt equation}}
[[Category:Eponymous equations of physics]]
[[Category:Quantum gravity]]