The Wheeler--DeWitt Equation Canonical Quantum Gravity, Superspace, and the Problem of Time

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Abstract

The Wheeler--DeWitt equation is one of the central equations of canonical quantum gravity. It arises by expressing general relativity in Hamiltonian form, identifying the Hamiltonian and momentum constraints, and promoting the canonical variables to quantum operators. In its schematic form, H=0, the equation resembles a time-independent functional Schrödinger equation for the quantum state of spatial geometry and matter. Its absence of an external time parameter leads directly to the problem of time in quantum gravity. This article develops the equation from the ADM decomposition, introduces superspace and the DeWitt supermetric, discusses canonical quantization, factor-ordering and regularization problems, derives a minisuperspace model, and explains the semiclassical emergence of an approximate Schrödinger equation. The conceptual role and limitations of the Wheeler--DeWitt framework are also discussed.

Introduction

General relativity describes spacetime geometry through the metric g_ and Einstein's field equations: G_+ g_ = 8 Gc^4T_.

Ordinary quantum mechanics evolves a state according to the Schrödinger equation: i t = H.

Canonical quantum gravity attempts to quantize general relativity by treating the geometry of a spatial hypersurface as a dynamical canonical variable.

The central equation is schematically H=0.

This is the Wheeler--DeWitt equation.

Its most striking feature is the absence of an explicit external time parameter.

From Spacetime to Space Plus Time

To formulate general relativity canonically, spacetime is foliated into three-dimensional spatial hypersurfaces _t.

The spacetime metric can be written in ADM form: ds^2 = -N^2c^2dt^2 + h_ij (dx^i+N^icdt) (dx^j+N^jcdt).

Here: h_ij is the spatial metric, N is the lapse function, N^i is the shift vector.

The lapse specifies the proper-time separation between neighboring spatial slices, while the shift describes how spatial coordinates move from one slice to the next.

Extrinsic Curvature

The extrinsic curvature describes how a spatial hypersurface is embedded in spacetime.

Using units with c=1 temporarily, K_ij = 12N ( h_ij - D_iN_j - D_jN_i ).

Its trace is K=h^ijK_ij.

The Einstein--Hilbert action can then be rewritten, up to boundary terms, as S = 116 G dt\,d^3x\, Nh ( ^(3)R + K_ijK^ij - K^2 - 2 ).

This is the starting point of the ADM Hamiltonian formulation.

Canonical Momentum

The canonical momentum conjugate to the spatial metric is ^ij = L h_ij.

It is given by ^ij = h16 G ( K^ij-Kh^ij ).

Thus the canonical pair is (h_ij,^ij).

The spatial geometry itself becomes the configuration variable of the gravitational system.

The ADM Hamiltonian

The Hamiltonian of general relativity takes the constrained form H_ ADM = d^3x ( N H_ + N^i H_i ) + H_ boundary.

The lapse and shift act as Lagrange multipliers.

Variation with respect to them produces constraints: H_ &=0, H_i &=0.

These are the Hamiltonian and momentum constraints.

Hamiltonian Constraint

For pure gravity, one common convention gives H_ = 16 G h ( _ij^ij -12^2 ) - h16 G ( ^(3)R-2 ).

Here =h_ij^ij.

With matter, H_ = H_ grav + H_ matter.

The classical physical configurations satisfy H_=0.

Momentum Constraints

The momentum constraints can be written as H_i = -2D_j^j_i + H_i^ matter = 0.

They generate spatial diffeomorphisms.

At the quantum level, they require the wave functional to be invariant under spatial coordinate transformations.

Canonical Quantization

Canonical quantization promotes the classical variables to operators: h_ij(x) & h_ij(x), ^ij(x) & ^ij(x).

In the metric representation, h_ij(x) = h_ij(x), while ^ij(x) = -i h_ij(x).

The quantum state is therefore a functional: =, where represents possible matter fields.

The Wheeler--DeWitt Equation

The classical Hamiltonian constraint becomes a quantum constraint: H_=0 H_=0.

Thus, H_=0.

A schematic functional form is = 0, where convention-dependent density factors and operator-ordering choices may modify the precise expression.

The DeWitt Supermetric

The kinetic term of the gravitational Hamiltonian is governed by a metric on the space of spatial metrics.

A commonly used form is G_ijkl = 12 h ( h_ikh_jl + h_ilh_jk - h_ijh_kl ).

This object is known as the DeWitt supermetric.

It plays a role analogous to the metric in an ordinary configuration space, but now the configuration space is infinite-dimensional.

Superspace

The set of all spatial metrics is sometimes called Riem().

Metrics related by spatial diffeomorphisms describe the same intrinsic geometry.

The physical configuration space is therefore Superspace() = Riem() Diff().

The Wheeler--DeWitt wave functional may be interpreted as a quantum amplitude on superspace: =.

Why There Is No External Time

In ordinary quantum mechanics, i t = H.

In canonical general relativity, however, the Hamiltonian is built from constraints.

For a closed universe, physical states satisfy H=0.

There is no external variable t appearing in the same way as in the ordinary Schrödinger equation.

This produces the famous problem of time.

The Problem of Time

General relativity treats time as part of the dynamical geometry.

Quantum mechanics normally assumes an external time parameter against which states evolve.

The conflict can be summarized as c Quantum Mechanics: external time General Relativity: dynamical time Quantum Gravity: what is time?

The Wheeler--DeWitt equation makes this conceptual tension explicit.

Relational Time

One possible approach is to define time relationally.

Suppose one dynamical degree of freedom behaves monotonically over a suitable regime.

Then other observables may be described relative to : O= O().

Instead of asking ``What happens at external time t?'' one asks ``What is the value of one observable when another has a given value?''

This is one route toward an emergent or internal notion of time.

Minisuperspace

The full Wheeler--DeWitt equation is an infinite-dimensional functional differential equation.

In quantum cosmology, symmetry can reduce the problem to a finite number of variables.

For a homogeneous and isotropic universe, ds^2 = -N^2(t)dt^2 + a^2(t)d_k^2, where a(t) is the scale factor.

The full functional is then approximated by (a,).

This reduced configuration space is called minisuperspace.

A Schematic Minisuperspace Equation

For a cosmological model containing a scale factor a and a homogeneous scalar field , the Wheeler--DeWitt equation often takes a form similar to (a,) = 0, where the exact coefficients depend on conventions, variables, lapse choice, factor ordering, spatial curvature, and matter content.

The function U(a,) acts as an effective potential on minisuperspace.

Quantum Cosmology

The Wheeler--DeWitt equation has been extensively studied as a possible framework for the quantum state of the universe.

One seeks solutions of H=0 subject to proposed boundary conditions.

Different proposals include: the Hartle--Hawking no-boundary proposal, tunneling boundary conditions, semiclassical expanding-universe conditions.

These proposals concern the interpretation and boundary conditions of cosmological wave functions and are not experimentally established as a unique description of the universe.

Semiclassical Approximation

A major question is how ordinary time-dependent quantum mechanics can emerge from a timeless Wheeler--DeWitt equation.

Consider a Born--Oppenheimer or WKB-type ansatz: = A[h] ( iM_P^2S_0[h] ) .

The gravitational degrees of freedom are treated as comparatively slow or semiclassical variables.

Expanding the Wheeler--DeWitt equation in inverse powers of the Planck scale produces, at leading order, a Hamilton--Jacobi equation for the classical geometry.

At the next order, one can obtain an approximate Schrödinger equation for matter fields: i t_ WKB = H_ matter.

Thus an approximate time variable may emerge from the semiclassical gravitational background.

Emergent Time

The semiclassical picture suggests the hierarchy c H=0 Semiclassical Geometry t_ emergent i_t= H_ matter.

In this interpretation, the familiar time of low-energy quantum mechanics need not be fundamental.

It can arise approximately from correlations between gravitational and matter degrees of freedom.

Factor-Ordering Ambiguity

Classically, h_ij^kl is simply a product.

Quantum mechanically, h_ij^kl ^kl h_ij.

Therefore different operator orderings can produce different Wheeler--DeWitt operators.

Schematically, G_ijkl ^ij^kl may become G_ijkl ^2 h_ij h_kl, or a covariant functional Laplacian with additional measure-dependent terms.

There is no universally accepted unique factor-ordering prescription.

Regularization Problems

The Wheeler--DeWitt equation contains products of functional derivatives at the same spatial point: ^2 h_ij(x) h_kl(x).

Such expressions are formally singular.

A mathematically complete quantum theory therefore requires: regularization, operator definition, renormalization or an equivalent ultraviolet prescription, a physical inner product.

These are among the major technical challenges of canonical quantum gravity.

Constraint Algebra

The classical Hamiltonian and momentum constraints satisfy the hypersurface-deformation algebra.

Schematically, \ H_i, H_j\ & H_k, \ H_i, H_\ & H_, \ H_, H_\ & h^ij H_j.

A consistent quantum theory should preserve the constraint structure without anomalies that destroy gauge consistency.

This is a highly nontrivial requirement.

Probability and the Inner Product

Ordinary quantum mechanics uses | with a positive-definite probability interpretation.

For the Wheeler--DeWitt equation, defining the correct physical inner product is more difficult.

The DeWitt supermetric has an indefinite structure, and the equation has features reminiscent of a Klein--Gordon-type equation on superspace rather than a standard Schrödinger equation.

Consequently, the interpretation of probability in quantum cosmology is a subtle problem.

Connection to the Planck Scale

The Wheeler--DeWitt equation is intended to describe regimes where quantum effects of geometry become important.

The characteristic length is _P = Gc^3, and the characteristic energy is E_P = c^5G.

Near the Planck regime, the classical approximation g_=smooth classical geometry may cease to be adequate.

The Wheeler--DeWitt framework replaces a single classical geometry with a quantum state over possible geometries.

Connection to the Graviton

The graviton arises by perturbatively quantizing small metric fluctuations: g_ = _ + h_.

The Wheeler--DeWitt approach is conceptually different.

Instead of quantizing only small perturbations around a fixed background, it attempts to quantize the spatial geometry itself: h_ij h_ij.

In a suitable semiclassical weak-field limit, graviton physics should emerge from the more general quantum-geometrical description.

Background Independence

One attraction of canonical quantum gravity is that it does not require a fixed spacetime background in the same way as ordinary perturbative quantum field theory.

The quantum state is defined over geometries themselves: .

This reflects a central principle inherited from general relativity: geometry is dynamical.

However, implementing background independence consistently at the quantum level is technically difficult.

Conceptual Structure

The derivation can be summarized as c Einstein--Hilbert Action 3+1\ ADM Decomposition (h_ij,^ij) H_=0, H_i=0 ^ij-i/ h_ij H_=0 Wheeler--DeWitt Equation.

Wheeler--DeWitt Versus Schrödinger Equation

p0.30p0.28p0.30 Property & Schrödinger Theory & Wheeler--DeWitt Framework State & (q,t) & Configuration & Particle or field & Spatial geometry and matter Time & External parameter & No fundamental external time Equation & i_t= H & H=0 Configuration space & Finite or field space & Superspace Gravity & Usually external & Quantized geometry

Relation to Other Quantum-Gravity Programs

The Wheeler--DeWitt equation belongs primarily to the canonical approach to quantum gravity.

Its ideas are related to several other frameworks.

p0.28p0.60 Framework & Relation Canonical Quantum Gravity & Direct quantization of gravitational constraints Loop Quantum Gravity & Reformulates canonical variables using connections and holonomies Quantum Cosmology & Uses symmetry-reduced Wheeler--DeWitt-type equations Path-Integral Gravity & Uses sums over geometries rather than canonical evolution String Theory & Provides a different ultraviolet framework containing quantum gravity Holography & Describes gravitational systems using nongravitational quantum theories in appropriate settings

Major Open Problems

Important unresolved questions include: What is the correct operator ordering? How should the functional equation be regularized? What is the correct physical inner product? How should observables be defined in a diffeomorphism-invariant theory? How does classical spacetime emerge? How does ordinary quantum time emerge? How should singularities be treated? How is the Wheeler--DeWitt framework related to a complete ultraviolet theory of quantum gravity? Can distinctive predictions be experimentally tested?

What the Equation Does and Does Not Establish

The Wheeler--DeWitt equation is a historically and conceptually important candidate equation for canonical quantum gravity.

It demonstrates how the Hamiltonian constraint of general relativity leads, under canonical quantization, to a timeless quantum constraint.

However: it is not an experimentally confirmed final equation of quantum gravity, its precise operator form is not unique without additional choices, its mathematical definition requires regularization, its probability interpretation is subtle, its full solution space is not known, the correct emergence of classical spacetime remains an active conceptual problem.

Conclusion

The Wheeler--DeWitt equation represents one of the most direct attempts to combine the canonical structure of general relativity with quantum mechanics.

Its symbolic form, H=0, contains a profound conceptual message.

In general relativity, time is part of the geometry. Once geometry itself is quantized, the external time parameter of ordinary quantum mechanics disappears from the fundamental constraint equation.

The familiar Schrödinger evolution may then arise only in a semiclassical or relational limit: Timeless Quantum Geometry Semiclassical Spacetime Emergent Time Ordinary Quantum Evolution.

The Wheeler--DeWitt equation therefore connects several of the deepest questions in theoretical physics: What is quantum spacetime, and what is time itself?

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Wheeler1968 J. A. Wheeler, ``Superspace and the Nature of Quantum Geometrodynamics,'' in Battelle Rencontres: 1967 Lectures in Mathematics and Physics, edited by C. DeWitt and J. A. Wheeler (1968).

ADM1962 R. Arnowitt, S. Deser, and C. W. Misner, ``The Dynamics of General Relativity,'' in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, 1962).

Kiefer2012 C. Kiefer, Quantum Gravity, 3rd ed., Oxford University Press (2012).

HartleHawking1983 J. B. Hartle and S. W. Hawking, ``Wave Function of the Universe,'' Physical Review D, 28, 2960--2975 (1983).