Wheeler–DeWitt equation

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The Wheeler–DeWitt equation is a field equation. It is part of a theory that attempts to combine mathematically the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity. In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called 'problem of time'. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group" (which is the diffeomorphism group on-shell).

Quantum gravity

All defined and understood descriptions of string/M-theory deal with fixed asymptotic conditions on the background spacetime. At infinity, the "right" choice of the time coordinate "t" is determined (because the space-time is asymptotic to some fixed space-time) in every description, so there is a preferred definition of the Hamiltonian (with nonzero eigenvalues) to evolve states of the system forwards in time. This avoids all the need to dynamically generate a time dimension using the Wheeler–DeWitt equation. Thus, the equation has not played a role in string theory thus far.

There could exist a Wheeler–DeWitt-style manner to describe the bulk dynamics of quantum theory of gravity. Some experts believe that this equation still holds the potential for understanding quantum gravity; however, decades after the equation was published, completely different approaches, such as string theory, have brought physicists as clear results about quantum gravity.

Motivation and background

In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is ${\displaystyle \gamma _{ij}}$ and given by

${\displaystyle g_{\mu \nu }\,\mathrm {d} x^{\mu }\,\mathrm {d} x^{\nu }=(-\,N^{2}+\beta _{k}\beta ^{k})\,\mathrm {d} t^{2}+2\beta _{k}\,\mathrm {d} x^{k}\,\mathrm {d} t+\gamma _{ij}\,\mathrm {d} x^{i}\,\mathrm {d} x^{j}.}$

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 ${\displaystyle \gamma _{ij}}$ is the field, and we denote its conjugate momenta as ${\displaystyle \pi ^{kl}}$. The Hamiltonian is a constraint (characteristic of most relativistic systems)

${\displaystyle {\mathcal {H}}={\frac {1}{2{\sqrt {\gamma }}}}G_{ijkl}\pi ^{ij}\pi ^{kl}-{\sqrt {\gamma }}\,{}^{(3)}\!R=0}$

where ${\displaystyle \gamma =\det(\gamma _{ij})}$ and ${\displaystyle G_{ijkl}=(\gamma _{ik}\gamma _{jl}+\gamma _{il}\gamma _{jk}-\gamma _{ij}\gamma _{kl})}$ 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

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

Working in "position space", these operators are

${\displaystyle {\hat {\gamma }}_{ij}(t,x^{k})\to \gamma _{ij}(t,x^{k})}$

${\displaystyle {\hat {\pi }}^{ij}(t,x^{k})\to -i{\frac {\delta }{\delta \gamma _{ij}(t,x^{k})}}.}$

One can apply the operator to a general wave functional of the metric ${\displaystyle {\widehat {\mathcal {H}}}\Psi [\gamma ]=0}$ where:

${\displaystyle \Psi [\gamma ]=a+\int \psi (x)\gamma (x)dx^{3}+\int \int \psi (x,y)\gamma (x)\gamma (y)dx^{3}dy^{3}+...}$

which would give a set of constraints amongst the coefficients ${\displaystyle \psi (x,y,...)}$. This means the amplitudes for N gravitons at certain positions is related to the amplitudes for a different number of gravitons at different positions. Or, one could use the two-field formalism, treating ${\displaystyle \omega (g)}$ as an independent field so that the wave function is ${\displaystyle \Psi [\gamma ,\omega ]}$.

Derivation from path integral

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

${\displaystyle Z=\int _{C}\mathrm {e} ^{-I[g_{\mu \nu },\phi ]}{\mathcal {D}}\mathbf {g} \,{\mathcal {D}}\phi }$

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

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

where ${\displaystyle I_{EH}}$ is the Einstein–Hilbert action, and ${\displaystyle N}$ is the lapse function, i.e. the Lagrange multiplier for the Hamiltonian constraint. The demand for this variation of our gravitational action to vanish corresponds, in fact, to the background independence in general relativity. This is purely classical so far. We can recover the Wheeler–DeWitt equation from

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

where ${\displaystyle \Sigma }$ 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

The Wheeler–DeWitt equation 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 minisuperspaces 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".

Hamiltonian constraint

Simply speaking, the Wheeler–DeWitt equation says 

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

where ${\displaystyle {\hat {H}}(x)}$ is the Hamiltonian constraint in quantized general relativity and ${\displaystyle |\psi \rangle }$ 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 ${\displaystyle {\hat {H}}}$ and ${\displaystyle |\psi \rangle }$ may appear familiar, their interpretation in the Wheeler–DeWitt equation is substantially different from non-relativistic quantum mechanics. ${\displaystyle |\psi \rangle }$ 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. ${\displaystyle {\hat {H}}}$ 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 ${\displaystyle {\hat {H}}|\psi \rangle =i\hbar \partial /\partial t|\psi \rangle }$ no longer applies. This property is known as timelessness. The reemergence of time requires the tools of decoherence and clock operators (or the use of a scalar field).

Momentum constraint

We also need to augment the Hamiltonian constraint with momentum constraints

${\displaystyle {\vec {\mathcal {P}}}(x)\left|\psi \right\rangle =0}$

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 ${\displaystyle t}$ 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 ${\displaystyle \psi \rightarrow e^{i\theta ({\vec {r}})}\psi }$ where ${\displaystyle \theta ({\vec {r}})}$ 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.