Canonical quantum gravity

From Wikipedia, the free encyclopedia

 

In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.

Canonical quantization

In the Hamiltonian formulation of ordinary classical mechanics the Poisson bracket is an important concept. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson-bracket relations, 

${\displaystyle \{q_{i},p_{j}\}=\delta _{ij}}$

where the Poisson bracket is given by

${\displaystyle \{f,g\}=\sum _{i=1}^{N}\left({\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right).}$

for arbitrary phase space functions ${\displaystyle f(q_{i},p_{j})}$ and ${\displaystyle g(q_{i},p_{j})}$. With the use of Poisson brackets, the Hamilton's equations can be rewritten as, 

${\displaystyle {\dot {q}}_{i}=\{q_{i},H\}}$,

${\displaystyle {\dot {p}}_{i}=\{p_{i},H\}}$.

These equations describe a ``flow" or orbit in phase space generated by the Hamiltonian ${\displaystyle H}$. Given any phase space function ${\displaystyle F(q,p)}$, we have 

${\displaystyle {d \over dt}F(q_{i},p_{i})=\{F,H\}.}$

In canonical quantization the phase space variables are promoted to quantum operators on a Hilbert space and the Poisson bracket between phase space variables is replaced by the canonical commutation relation: 

${\displaystyle [{\hat {q}},{\hat {p}}]=i\hbar .}$

In the so-called position representation this commutation relation is realized by the choice:  

${\displaystyle {\hat {q}}\psi (q)=q\psi (q)}$ and ${\displaystyle {\hat {p}}\psi (q)=-i\hbar {d \over dq}\psi (q)}$

The dynamics are described by Schrödinger equation: 

${\displaystyle i\hbar {\partial \over \partial t}\psi ={\hat {H}}\psi }$

where ${\displaystyle {\hat {H}}}$ is the operator formed from the Hamiltonian ${\displaystyle H(q,p)}$ with the replacement ${\displaystyle q\mapsto q}$ and 

${\displaystyle p\mapsto -i\hbar {d \over dq}}$.

Canonical quantization with constraints

Canonical classical general relativity is an example of a fully constrained theory. In constrained theories there are different kinds of phase space: the unrestricted (also called kinematic) phase space on which constraint functions are defined and the reduced phase space on which the constraints have already been solved. For canonical quantization in general terms, phase space is replaced by an appropriate Hilbert space and phase space variables are to be promoted to quantum operators. 

In Dirac's approach to quantization the unrestricted phase space is replaced by the so-called kinematic Hilbert space and the constraint functions replaced by constraint operators implemented on the kinematic Hilbert space; solutions are then searched for. These quantum constraint equations are the central equations of canonical quantum general relativity, at least in the Dirac approach which is the approach usually taken. 

In theories with constraints there is also the reduced phase space quantization where the constraints are solved at the classical level and the phase space variables of the reduced phase space are then promoted to quantum operators, however this approach was thought to be impossible in General relativity as it seemed to be equivalent to finding a general solution to the classical field equations. However, with the fairly recent development of a systematic approximation scheme for calculating observables of General relativity (for the first time) by Bianca Dittrich, based on ideas introduced by Carlo Rovelli, a viable scheme for a reduced phase space quantization of Gravity has been developed by Thomas Thiemann. However it is not fully equivalent to the Dirac quantization as the `clock-variables' must be taken to be classical in the reduced phase space quantization, as opposed to the case in the Dirac quantization. 

A common misunderstanding is that coordinate transformations are the gauge symmetries of general relativity, when actually the true gauge symmetries are diffeomorphisms as defined by a mathematician – which are much more radical. The first class constraints of general relativity are the spatial diffeomorphism constraint and the Hamiltonian constraint (also known as the Wheeler-De Witt equation) and imprint the spatial and temporal diffeomorphism invariance of the theory respectively. Imposing these constraints classically are basically admissibility conditions on the initial data, also they generate the `evolution' equations (really gauge transformations) via the Poisson bracket. Importantly the Poisson bracket algebra between the constraints fully determines the classical theory – this is something that must in some way be reproduced in the semi-classical limit of canonical quantum gravity for it to be a viable theory of quantum gravity. 

In Dirac's approach it turns out that the first class quantum constraints imposed on a wavefunction also generate gauge transformations. Thus the two step process in the classical theory of solving the constraints ${\displaystyle C_{I}=0}$ (equivalent to solving the admissibility conditions for the initial data) and looking for the gauge orbits (solving the `evolution' equations) is replaced by a one step process in the quantum theory, namely looking for solutions ${\displaystyle \Psi }$ of the quantum equations ${\displaystyle {\hat {C}}_{I}\Psi =0}$. This is because it obviously solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because ${\displaystyle {\hat {C}}_{I}}$ is the quantum generator of gauge transformations. At the classical level, solving the admissibility conditions and evolution equations are equivalent to solving all of Einstein's field equations, this underlines the central role of the quantum constraint equations in Dirac's approach to canonical quantum gravity.

Canonical quantization, Diffeomorphism invariance and Manifest Finiteness

A diffeomorphism can be thought of as simultaneously `dragging' the metric (gravitational field) and matter fields over the bare manifold while staying in the same coordinate system, and so are more radical than invariance under a mere coordinate transformation. This symmetry arises from the subtle requirement that the laws of general relativity cannot depend on any a-priori given space-time geometry. 

This diffeomorphism invariance has an important implication: canonical quantum gravity will be manifestly finite as the ability to `drag' the metric function over the bare manifold means that small and large `distances' between abstractly defined coordinate points are gauge-equivalent! A more rigorous argument has been provided by Lee Smolin: 

“A background independent operator must always be finite. This is because the regulator scale and the background metric are always introduced together in the regularization procedure. This is necessary, because the scale that the regularization parameter refers to must be described in terms of a background metric or coordinate chart introduced in the construction of the regulated operator. Because of this the dependence of the regulated operator on the cuttoff, or regulator parameter, is related to its dependence on the background metric. When one takes the limit of the regulator parameter going to zero one isolates the non-vanishing terms. If these have any dependence on the regulator parameter (which would be the case if the term is blowing up) then it must also have dependence on the background metric. Conversely, if the terms that are nonvanishing in the limit the regulator is removed have no dependence on the background metric, it must be finite.” 

In fact, as mentioned below, Thomas Thiemann has explicitly demonstrated that loop quantum gravity (a well developed version of canonical quantum gravity) is manifestly finite even in the presence of all forms of matter! So there is no need for renormalization and the elimination of infinities. 

In perturbative quantum gravity (from which the non-renormalization arguments originate), as with any perturbative scheme, one makes the assumption that the unperturbed starting point is qualitatively the same as the true quantum state – so perturbative quantum gravity makes the physically unwarranted assumption that the true structure of quantum space-time can be approximated by a smooth classical (usually Minkowski) spacetime. Canonical quantum gravity on the other hand makes no such assumption and instead allows the theory itself tell you, in principle, what the true structure of quantum space-time is. A long-held expectation is that in a theory of quantum geometry such as canonical quantum gravity that geometric quantities such as area and volume become quantum observables and take non-zero discrete values, providing a natural regulator which eliminates infinities from the theory including those coming from matter contributions. This `quantization' of geometric observables is in fact realized in loop quantum gravity (LQG).

Canonical quantization in metric variables

The quantization is based on decomposing the metric tensor as follows,

${\displaystyle g_{\mu \nu }dx^{\mu }\,dx^{\nu }=(-\,N^{2}+\beta _{k}\beta ^{k})dt^{2}+2\beta _{k}\,dx^{k}\,dt+\gamma _{ij}\,dx^{i}\,dx^{j}}$

where the summation over repeated indices is implied, the index 0 denotes time ${\displaystyle \tau =x^{0}}$, Greek indices run over all values 0, . . ., ,3 and Latin indices run over spatial values 1, . . ., 3. The function ${\displaystyle N}$ is called the lapse function and the functions ${\displaystyle \beta _{k}}$ are called the shift functions. The spatial indices are raised and lowered using the spatial metric ${\displaystyle \gamma _{ij}}$ and its inverse ${\displaystyle \gamma ^{ij}}$: ${\displaystyle \gamma _{ij}\gamma ^{jk}=\delta _{i}{}^{k}}$ and ${\displaystyle \beta ^{i}=\gamma ^{ij}\beta _{j}}$, ${\displaystyle \gamma =\det \gamma _{ij}}$, where ${\displaystyle \delta }$ is the Kronecker delta. Under this decomposition the Einstein–Hilbert Lagrangian becomes, up to total derivatives,

${\displaystyle L=\int d^{3}x\,N\gamma ^{1/2}(K_{ij}K^{ij}-K^{2}+{}^{(3)}R)}$

where ${\displaystyle {}^{(3)}R}$ is the spatial scalar curvature computed with respect to the Riemannian metric ${\displaystyle \gamma _{ij}}$ and ${\displaystyle K_{ij}}$ is the extrinsic curvature,

${\displaystyle K_{ij}=-{\frac {1}{2}}({\mathcal {L}}_{n}\gamma )_{ij}={\frac {1}{2}}N^{-1}\left(\nabla _{j}\beta _{i}+\nabla _{i}\beta _{j}-{\frac {\partial \gamma _{ij}}{\partial t}}\right),}$

where ${\displaystyle {\mathcal {L}}}$ denotes Lie-differentiation, ${\displaystyle n}$ is the unit normal to surfaces of constant ${\displaystyle t}$ and ${\displaystyle \nabla _{i}}$ denotes covariant differentiation with respect to the metric ${\displaystyle \gamma _{ij}}$. Note that ${\displaystyle \gamma _{\mu \nu }=g_{\mu \nu }+n_{\mu }n_{\nu }}$. DeWitt writes that the Lagrangian "has the classic form 'kinetic energy minus potential energy,' with the extrinsic curvature playing the role of kinetic energy and the negative of the intrinsic curvature that of potential energy." While this form of the Lagrangian is manifestly invariant under redefinition of the spatial coordinates, it makes general covariance opaque. 

Since the lapse function and shift functions may be eliminated by a gauge transformation, they do not represent physical degrees of freedom. This is indicated in moving to the Hamiltonian formalism by the fact that their conjugate momenta, respectively ${\displaystyle \pi }$ and ${\displaystyle \pi ^{i}}$, vanish identically (on shell and off shell). These are called primary constraints by Dirac. A popular choice of gauge, called synchronous gauge, is ${\displaystyle N=1}$ and ${\displaystyle \beta _{i}=0}$, although they can, in principle, be chosen to be any function of the coordinates. In this case, the Hamiltonian takes the form

${\displaystyle H=\int d^{3}x{\mathcal {H}},}$

where

${\displaystyle {\mathcal {H}}={\frac {1}{2}}\gamma ^{-1/2}(\gamma _{ik}\gamma _{jl}+\gamma _{il}\gamma _{jk}-\gamma _{ij}\gamma _{kl})\pi ^{ij}\pi ^{kl}-\gamma ^{1/2}{}^{(3)}R}$

and ${\displaystyle \pi ^{ij}}$ is the momentum conjugate to ${\displaystyle \gamma _{ij}}$. Einstein's equations may be recovered by taking Poisson brackets with the Hamiltonian. Additional on-shell constraints, called secondary constraints by Dirac, arise from the consistency of the Poisson bracket algebra. These are ${\displaystyle {\mathcal {H}}=0}$ and ${\displaystyle \nabla _{j}\pi ^{ij}=0}$. This is the theory which is being quantized in approaches to canonical quantum gravity. 

It can be shown that six Einstein equations describing time evolution (really a gauge transformation) can be obtained by calculating the Poisson brackets of the three-metric and its conjugate momentum with a linear combination of the spatial diffeomorphism and Hamiltonian constraint. The vanishing of the constraints, giving the physical phase space, are the four other Einstein equations. That is, we have: 

Spatial diffeomorphisms constraints 

${\displaystyle C_{a}(x)=0}$

of which there are an infinite number – one for value of ${\displaystyle x}$, can be smeared by the so-called shift functions ${\displaystyle {\vec {N}}(x)}$ to give an equivalent set of smeared spatial diffeomorphism constraints, 

${\displaystyle C({\vec {N}})=\int d^{3}xC_{a}(x)N^{a}(x)}$.

These generate spatial diffeomorphisms along orbits defined by the shift function ${\displaystyle N^{a}(x)}$.

Hamiltonian constraints 

${\displaystyle H(x)=0}$

of which there are an infinite number, can be smeared by the so-called lapse functions ${\displaystyle N(x)}$ to give an equivalent set of smeared Hamiltonian constraints, 

${\displaystyle H(N)=\int d^{3}xH(x)N(x)}$.

as mentioned above, the Poission bracket structure between the (smeared) constraints is important because they fully determine the classical theory, and must be reproduced in the semi-classical limit of any theory of quantum gravity.

The Wheeler-De-Witt equation

The Wheeler-De-Witt equation (sometimes called the Hamiltonian constraint, sometimes the Einstein-Schrödinger equation) is rather central as it encodes the dynamics at the quantum level. It is analogous to Schrödinger's equation, except as the time coordinate, ${\displaystyle t}$, is unphysical, a physical wavefunction can't depend on ${\displaystyle t}$ and hence `Schrödinger's equation' reduces to a constraint: 

${\displaystyle {\hat {H}}\Psi =0.}$

Using metric variables lead to seemingly un-summountable mathematical difficulties when trying to promote the classical expression to a well-defined quantum operator, and as such decades went by without making progress via this approach. This problem was circumvented and the formulation of a well-defined Wheeler-De-Witt equation was first accomplished with the introduction of Ashtekar-Barbero variables and the loop representation, this well defined operator formulated by Thomas Thiemann. 

Before this development the Wheeler-De-Witt equation had only been formulated in symmetry-reduced models, such as quantum cosmology.

Canonical quantization in Ashtekar-Barbero variables and LQG

Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to quantum operators because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekars new variables. Ashtekar variables describe canonical general relativity in terms of a new pair canonical variables closer to that of gauge theories. In doing so it introduced an additional constraint, on top of the spatial diffeomorphism and Hamiltonian constraint, the Gauss gauge constraint.

The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation, in the context of Yang-Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of Gauss gauge invariant states. The use of this representation arose naturally from the Ashtekar-Barbero representation as it provides an exact non-perturbative description and also because the spatial diffeomorphism constraint is easily dealt with within this representation. 

Within the loop representation Thiemann has provided a well defined canonical theory in the presence of all forms of matter and explicitly demonstrated it to be manifestly finite! So there is no need for renormalization. However, as LQG approach is well suited to describe physics at the Planck scale, there are difficulties in making contact with familiar low energy physics and establishing it has the correct semi-classical limit.

The problem of time

All canonical theories of general relativity have to deal with the problem of time. In quantum gravity, the problem of time is a conceptual conflict between general relativity and quantum mechanics. In canonical general relativity, time is just another coordinate as a result of general covariance. In quantum field theories, especially in the Hamiltonian formulation, the formulation is split between three dimensions of space, and one dimension of time. Roughly speaking, the problem of time is that there is none in general relativity. This is because in general relativity the Hamiltonian is a constraint that must vanish. However, in any canonical theory, the Hamiltonian generates time translations. Therefore, we arrive at the conclusion that "nothing moves" ("there is no time") in general relativity. Since "there is no time", the usual interpretation of quantum mechanics measurements at given moments of time breaks down. This problem of time is the broad banner for all interpretational problems of the formalism.

The problem of quantum cosmology

The problem of quantum cosmology is that the physical states that solve the constraints of canonical quantum gravity represent quantum states of the entire universe and as such exclude an outside observer, however an outside observer is a crucial element in most interpretations of quantum mechanics.

Field equation

From Wikipedia, the free encyclopedia

 

In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equation are mathematical functions which correspond directly to the field, as functions of time and space. Since the field equation is a partial differential equation, there are families of solutions which represent a variety of physical possibilities. Usually, there is not just a single equation, but a set of coupled equations which must be solved simultaneously. Field equations are not ordinary differential equations since a field depends on space and time, which requires at least two variables.

Whereas the "wave equation", the "diffusion equation", and the "continuity equation" all have standard forms (and various special cases or generalizations), there is no single, special equation referred to as "the field equation".

The topic broadly splits into equations of classical field theory and quantum field theory. Classical field equations describe many physical properties like temperature of a substance, velocity of a fluid, stresses in an elastic material, electric and magnetic fields from a current, etc. They also describe the fundamental forces of nature, like electromagnetism and gravity. In quantum field theory, particles or systems of "particles" like electrons and photons are associated with fields, allowing for infinite degrees of freedom (unlike finite degrees of freedom in particle mechanics) and variable particle numbers which can be created or annihilated.

Generalities

Origin

Usually, field equations are postulated (like the Einstein field equations and the Schrödinger equation, which underlies all quantum field equations) or obtained from the results of experiments (like Maxwell's equations). The extent of their validity is their extent to correctly predict and agree with experimental results. 

From a theoretical viewpoint, field equations can be formulated in the frameworks of Lagrangian field theory, Hamiltonian field theory, and field theoretic formulations of the principle of stationary action. Given a suitable Lagrangian or Hamiltonian density, a function of the fields in a given system, as well as their derivatives, the principle of stationary action will obtain the field equation.

Symmetry

In both classical and quantum theories, field equations will satisfy the symmetry of the background physical theory. Most of the time Galilean symmetry is enough, for speeds (of propagating fields) much less than light. When particles and fields propagate at speeds close to light, Lorentz symmetry is one of the most common settings because the equation and its solutions are then consistent with special relativity. 

Another symmetry arises from gauge freedom, which is intrinsic to the field equations. Fields which correspond to interactions may be gauge fields, which means they can be derived from a potential, and certain values of potentials correspond to the same value of the field.

Classification

Field equations can be classified in many ways: classical or quantum, nonrelativistsic or relativistic, according to the spin or mass of the field, and the number of components the field has and how they change under coordinate transformations (e.g. scalar fields, vector fields, tensor fields, spinor fields, twistor fields etc.). They can also inherit the classification of differential equations, as linear or nonlinear, the order of the highest derivative, or even as fractional differential equations. Gauge fields may be classified as in group theory, as abelian or nonabelian.

Waves

Field equations underlie wave equations, because periodically changing fields generate waves. Wave equations can be thought of as field equations, in the sense they can often be derived from field equations. Alternatively, given suitable Lagrangian or Hamiltonian densities and using the principle of stationary action, the wave equations can be obtained also.

For example, Maxwell's equations can be used to derive inhomogeneous electromagnetic wave equations, and from the Einstein field equations one can derive equations for gravitational waves.

Supplementary equations to field equations

Not every partial differential equation (PDE) in physics is automatically called a "field equation", even if fields are involved. They are extra equations to provide additional constraints for a given physical system. 

"Continuity equations" and "diffusion equations" describe transport phenomena, even though they may involve fields which influence the transport processes.

If a "constitutive equation" takes the form of a PDE and involves fields, it is not usually called a field equation because it does not govern the dynamical behaviour of the fields. They relate one field to another, in a given material. Constitutive equations are used along with field equations when the effects of matter need to be taken into account.

Classical field equation

Classical field equations arise in continuum mechanics (including elastodynamics and fluid mechanics), heat transfer, electromagnetism, and gravitation. 

Fundamental classical field equations include

• Newton's Law of Universal Gravitation for nonrelativistic gravitation.
• Einstein field equations for relativistic gravitation
• Maxwell's equations for electromagnetism.

Important equations derived from fundamental laws include:

• Navier–Stokes equations for fluid flow.

As part of real-life mathematical modelling processes, classical field equations are accompanied by other equations of motion, equations of state, constitutive equations, and continuity equations.

Quantum field equation

In quantum field theory, particles are described by quantum fields which satisfy the Schrödinger equation. They are also creation and annihilation operators which satisfy commutation relations and are subject to the spin–statistics theorem. 

Particular cases of relativistic quantum field equations include

• the Klein–Gordon equation for spin-0 particles
• the Dirac equation for spin-1/2 particles
• the Bargmann–Wigner equations for particles of any spin

In quantum field equations, it is common to use momentum components of the particle instead of position coordinates of the particle's location, the fields are in momentum space and Fourier transforms relate them to the position representation.

Wheeler–DeWitt equation

From Wikipedia, the free encyclopedia

 

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.