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The Wheeler–DeWitt equation[1] is an attempt to combine mathematically the ideas of quantum mechanics and general relativity, a step toward a theory of quantum gravity. In this approach, time plays no role in the equation, leading to the problem of time.[2] More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergmann-Komar "group" (which is the diffeomorphism group on-shell, but differs off-shell).

the Wheeler-DeWitt equation in one form or another is almost certainly necessary for a description of the most general geometries within the quantum mechanical framework.

Because of its connections with the low-energy effective field theory, it inherits all the problems of the naively quantized GR, and can't be used at multi-loop level etc., at least not according to the current knowledge.

And the equation hasn't played an important role in string theory so far because all properly enough defined and understood descriptions of string/M-theory deal with some fixed asymptotic conditions of the background, so at least at infinity, the "right" choice of the time coordinate "t" is determined in every description we use which also means that there is a preferred definition of the Hamiltonian/energy (with nonzero eigenvalues) and all the contrived tricks of the WdW equation to force the physical system to produce its own time dimensions dynamically is avoided.

But at the end, there should exist a WdW-like way to describe the bulk dynamics of any quantum theory of gravity. Most experts agree this is a potential that is still here and hasn't changed, but the decades after the equation was written down for the first time haven't brought us as clear results about the equation as some of the results building on completely different approaches.

Motivation and background[edit]

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 Roman 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^{kl}. 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.

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

\widehat{\mathcal{H}}=\frac{1}{2\sqrt{\gamma}}\widehat{G}_{ijkl}\widehat{\pi}^{ij}\widehat{\pi}^{kl}-\sqrt{\gamma}\,{}^{(3)}\!\widehat{R}.

Working in "position space", these operators are

 \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)}.

Derivation from path integral[edit]

The Wheeler–DeWitt equation can be derived from a path integral using the gravitational action in the Euclidean quantum gravity paradigm:[3]

Z = \int_{C}\mathrm{e}^{-I[g_{\mu\nu},\phi]}\mathcal{D}\bold{g}\, \mathcal{D}\phi

where one integrates over a class of Riemannian four-metrics and matter fields matching certain boundary conditions. Because the concept of a universal time coordinate seems unphysical, and at odds with the principles of general relativity, the action is evaluated around a 3-metric which we take as the boundary of the classes of four-metrics and on which a certain configuration of matter fields exists. This latter might for example be the current configuration of matter in our universe as we observe it today. Evaluating the action so that it only depends on the 3-metric and the matter fields is sufficient to remove the need for a time coordinate as it effectively fixes a point in the evolution of the universe.

We obtain the Hamiltonian constraint from

\frac{\delta I_{EH}}{\delta N}=0

where I_{EH} is the Einstein-Hilbert action, and N is the lapse function (i.e., the Lagrange multiplier for the Hamiltonian constraint). This is purely classical so far. We can recover the Wheeler–DeWitt equation from

\frac{\delta Z}{\delta N}=0=\int \left.\frac{\delta I[g_{\mu\nu},\phi]}{\delta N}\right|_{\Sigma} \exp\left(-I[g_{\mu\nu},\phi]\right)\,\mathcal{D}\bold{g}\, \mathcal{D}\phi

where \Sigma is the three-dimensional boundary. Observe that this expression vanishes, implying that the functional derivative also vanishes, giving us the Wheeler–DeWitt equation. A similar statement may be made for the diffeomorphism constraint (take functional derivative with respect to the shift functions instead).

Mathematical formalism[edit]

The Wheeler–DeWitt equation[1] is a 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 mini-superspaces 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".[4]

Simply speaking, the Wheeler–DeWitt equation says

\hat{H}(x) |\psi\rangle = 0

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 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. The reemergence of time requires the tools of decoherence and clock operators[citation needed] (or the use of a scalar field).

We also need to augment the Hamiltonian constraint with momentum constraints

\vec{\mathcal{P}}(x) \left| \psi \right\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 \rightarrow 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.

See also[edit]

References[edit]

  1. ^ a b DeWitt, B. S. (1967). "Quantum Theory of Gravity. I. The Canonical Theory". Phys. Rev. 160 (5): 1113–1148. Bibcode:1967PhRv..160.1113D. doi:10.1103/PhysRev.160.1113. 
  2. ^ https://medium.com/the-physics-arxiv-blog/d5d3dc850933
  3. ^ See J. B. Hartle and S. W. Hawking, "Wave function of the Universe." Phys. Rev. D 28 (1983) 2960–2975, eprint.
  4. ^ Go to Arxiv.org to read "Notes for a Brief History of Quantum Gravity" by Carlo Rovelli
  • Herbert W. Hamber and Ruth M. Williams (2011). "Discrete Wheeler-DeWitt Equation". Physical Review D 84: 104033. doi:10.1103/PhysRevD.84.104033.  Available at [1].
  • Herbert W. Hamber, Reiko Toriumi and Ruth M. Williams (2012). "Wheeler-DeWitt Equation in 2+1 Dimensions". Physical Review D 86: 084010. doi:10.1103/PhysRevD.86.084010.  Available at [2].

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Wheeler–DeWitt_equation — Please support Wikipedia.
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16 news items

 
New Scientist
Fri, 25 Oct 2013 12:44:20 -0700

One early attempt in the 1960s was the Wheeler-DeWitt equation, which managed to quantise general relativity – by leaving out time altogether. "It means that the universe should not evolve. But of course we see evolution," says Marco Genovese at the ...
 
Forbes
Tue, 05 Aug 2014 12:07:30 -0700

In 1967, while in his mid-50s, he helped devise the 'Wheeler-DeWitt equation,' an important mathematical attempt to combine general relativity with quantum mechanics. In his 60s, Wheeler co-wrote one of the most influential textbooks on general relativity.
 
Reason (blog)
Tue, 09 Sep 2014 09:44:25 -0700

Bill Nye the Science Guy ("He's not our Science Guy!" the Reason audience retorts) has waded into the Common Core debate. Per usual, he thinks those who disagree with him are—almost by definition—anti-science. After conceding one criticism of the ...
 
CNN (blog)
Fri, 28 Mar 2014 13:15:19 -0700

Touted as evidence for inflation (a faster-than-the-speed-of-light expansion of our universe), the new discovery of traces of gravity waves affirms scientific concepts in the fields of cosmology, general relativity, and particle physics. The new ...
 
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Thu, 08 May 2014 18:10:41 -0700

Revealing that they might in fact be part Time Lord, researchers at MIT have constructed an immense computer simulation of the 14 billion year history of the universe, from 12 million years after the Big Bang to the present, using both the stars and ...
 
New Scientist (subscription) (blog)
Mon, 14 Apr 2008 00:00:01 -0700

With Bryce DeWitt he dared to apply the strangeness of the quantum world to the universe as a whole, founding the field of quantum cosmology and writing down the infamous Wheeler-DeWitt equation - the "wavefunction of the universe" - or as DeWitt calls ...
 
Engadget
Sat, 26 Oct 2013 14:38:38 -0700

Early attempts to make them play nice (i.e., the Wheeler-DeWitt equation) solved some issues, but created another -- namely a static universe, where nothing every happens. Physicists then explored the idea that entanglement might explain things. The ...

PJ Media

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Thu, 31 Oct 2013 17:03:33 -0700

Time travel is a favorite trope of science fiction going back to at least A Connecticut Yankee in King Arthur's Court and The Time Machine. It took until the mid-40s for someone to come up with the grandfather paradox, which has been pretty well beaten ...
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