Wheeler–DeWitt equation

Short description: Field equation, part of a theory that attempts to combine quantum mechanics and general relativity

The Wheeler–DeWitt equation 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 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

Defined formulations of string theory and M-theory are usually constructed with fixed asymptotic boundary conditions on the background spacetime. In such settings, the asymptotic structure determines a preferred time coordinate $t$, and therefore a corresponding definition of the Hamiltonian that generates time evolution. Because of this, string theory has generally not required the Wheeler–DeWitt equation as the central mechanism for describing dynamics.

It remains possible that a Wheeler–DeWitt-type framework could play a role in a more complete description of quantum gravity. However, in the development of quantum gravity research, other approaches, including string theory, have provided alternative frameworks, so the Wheeler–DeWitt equation has not been central to most work in string theory.

Motivation and background

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 Latin 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 ${\pi }^{ij}$. The Hamiltonian is a constraint (characteristic of most relativistic systems)

${\mathscr{H}}=\frac{1}{2\sqrt{\gamma }}{G}_{ijkl}{\pi }^{ij}{\pi }^{kl}-\sqrt{\gamma }{}^{(3)}R=0$

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

$\hat{{\mathscr{H}}}=\frac{1}{2\sqrt{\gamma }}{\hat{G}}_{ijkl}{\hat{\pi }}^{ij}{\hat{\pi }}^{kl}-\sqrt{\gamma }{}^{(3)}\hat{R}.$

Working in "position space", these operators are

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

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

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

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

which would give a set of constraints amongst the coefficients $\psi (x,y,...)$. 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 $\omega (g)$ as an independent field so that the wave function is $\mathrm{\Psi }[\gamma ,\omega ]$.

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

$\hat{H}(x)|\psi ⟩=0$

where $\hat{H}(x)$ is the Hamiltonian constraint in quantized general relativity and $|\psi ⟩$ 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 $|\psi ⟩$ may appear familiar, their interpretation in the Wheeler–DeWitt equation is substantially different from non-relativistic quantum mechanics. $|\psi ⟩$ is no longer a spatial wave function in the traditional sense of a complex-valued function defined on a 3-dimensional spacelike surface and normalized to unity. Instead, it is a functional of field configurations. This wave function contains all of the information about the geometry and matter content of the universe. $\hat{H}$ is still an operator acting on a Hilbert space of such functionals, but this Hilbert space differs from the nonrelativistic case, and the Hamiltonian no longer generates time evolution. Consequently, the Schrödinger equation $\hat{H}|\psi ⟩=i\hslash {\partial }_t|\psi ⟩$ does not apply. This property is known as timelessness. The reemergence of time can be described using semiclassical approximations, decoherence, and relational or “clock” degrees of freedom. Alternatively, an internal time parameter can be introduced via a matter field such as a scalar field.

Momentum constraint

We also need to augment the Hamiltonian constraint with momentum constraints

$\overrightarrow{{𝒫}}(x)|\psi ⟩=0$

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 coordinate label. What we interpret as time evolution of a physical system corresponds to a gauge transformation, analogous in spirit (though not identical in structure) to gauge transformations in QED such as $\psi \to e^{i\theta (\overrightarrow{r})}\psi$. In this context, the Hamiltonian does not generate physical time evolution but instead acts as a constraint: it restricts the space of kinematical states to those that are physically admissible, i.e., invariant under gauge transformations. For this reason, it is called the Hamiltonian constraint. Upon quantization, physical states are those annihilated by the Hamiltonian operator.

In generally covariant theories, the total Hamiltonian is a sum of constraints and therefore vanishes on physical states (on-shell), rather than being identically zero. This reflects the fact that time evolution is a gauge redundancy rather than a physical transformation.

See also

• ADM formalism
• Diffeomorphism constraint
• Euclidean quantum gravity
• Regge calculus
• Canonical quantum gravity
• Peres metric
• Loop quantum gravity

Index

Core theory Foundations Conceptual and interpretations Mathematical structure and systems Atomic and spectroscopy Wavefunctions and modes Quantum dynamics and evolution Measurement and information Quantum information and computing

Matter (by scale) Matter (by scale)

Applications and extensions Quantum optics and experiments Open quantum systems Quantum field theory Statistical mechanics and kinetic theory Condensed matter and solid-state physics Plasma and fusion physics Timeline Advanced and frontier topics

Methods and tools Book:Quantum Collection/Methods and tools

Full contents

Foundations



1. Physics:Quantum basics
2. Physics:Quantum Postulates
3. Physics:Quantum Hilbert space
4. Physics:Quantum Observables and operators
5. Physics:Quantum mechanics
6. Physics:Quantum mechanics measurements
7. Physics:Quantum state
8. Physics:Quantum system
9. Physics:Quantum superposition
10. Physics:Quantum probability
11. Physics:Quantum Mathematical Foundations of Quantum Theory

Conceptual and interpretations



12. Physics:Quantum Interpretations of quantum mechanics
13. Physics:Quantum Wave–particle duality
14. Physics:Quantum Complementarity principle
15. Physics:Quantum Uncertainty principle
16. Physics:Quantum Measurement problem
17. Physics:Quantum Bell's theorem
18. Physics:Quantum Hidden variable theory
19. Physics:Quantum nonlocality
20. Physics:Quantum contextuality
21. Physics:Quantum Darwinism
22. Physics:Quantum A Spooky Action at a Distance
23. Physics:Quantum A Walk Through the Universe
24. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

Mathematical structure and systems



25. Physics:Quantum Density matrix
26. Physics:Quantum Exactly solvable quantum systems
27. Physics:Quantum Formulas Collection
28. Physics:Quantum A Matter Of Size
29. Physics:Quantum Symmetry in quantum mechanics
30. Physics:Quantum Angular momentum operator
31. Physics:Quantum Runge–Lenz vector
32. Physics:Quantum Approximation Methods
33. Physics:Quantum Matter Elements and Particles
34. Physics:Quantum Dirac equation
35. Physics:Quantum Klein–Gordon equation
36. Physics:Quantum pendulum
37. Physics:Quantum configuration space

Atomic and spectroscopy



Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

38. Physics:Quantum Atomic structure and spectroscopy
39. Physics:Quantum Hydrogen atom
40. Physics:Quantum number
41. Physics:Quantum Multi-electron atoms
42. Physics:Quantum Fine structure
43. Physics:Quantum Hyperfine structure
44. Physics:Quantum Isotopic shift
45. Physics:Quantum defect
46. Physics:Quantum Zeeman effect
47. Physics:Quantum Stark effect
48. Physics:Quantum Spectral lines and series
49. Physics:Quantum Selection rules
50. Physics:Quantum Fermi's golden rule
51. Physics:Quantum beats

Wavefunctions and modes



A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

52. Physics:Quantum Wavefunction
53. Physics:Quantum Superposition principle
54. Physics:Quantum Eigenstates and eigenvalues
55. Physics:Quantum Boundary conditions and quantization
56. Physics:Quantum Standing waves and modes
57. Physics:Quantum Normal modes and field quantization
58. Physics:Number of independent spatial modes in a spherical volume
59. Physics:Quantum Density of states
60. Physics:Quantum carpet

Quantum dynamics and evolution



61. Physics:Quantum Time evolution
62. Physics:Quantum Schrödinger equation
63. Physics:Quantum Time-dependent Schrödinger equation
64. Physics:Quantum Stationary states
65. Physics:Quantum Perturbation theory
66. Physics:Quantum Time-dependent perturbation theory
67. Physics:Quantum Adiabatic theorem
68. Physics:Quantum Scattering theory
69. Physics:Quantum S-matrix
70. Physics:Quantum tunnelling
71. Physics:Quantum speed limit
72. Physics:Quantum revival
73. Physics:Quantum reflection
74. Physics:Quantum oscillations
75. Physics:Quantum jump
76. Physics:Quantum boomerang effect
77. Physics:Quantum chaos

Measurement and information



78. Physics:Quantum Measurement theory
79. Physics:Quantum Measurement operators
80. Physics:Quantum Projective measurement
81. Physics:Quantum POVM
82. Physics:Quantum Weak measurement
83. Physics:Quantum Measurement collapse
84. Physics:Quantum Zeno effect
85. Physics:Quantum limit

Quantum information and computing



86. Physics:Quantum information theory
87. Physics:Quantum Qubit
88. Physics:Quantum Entanglement
89. Physics:Quantum Gates and circuits
90. Physics:Quantum Computing Algorithms in the NISQ Era
91. Physics:Quantum Noisy Qubits
92. Physics:Quantum random access code
93. Physics:Quantum pseudo-telepathy
94. Physics:Quantum network
95. Physics:Quantum money

Quantum optics and experiments



Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

96. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
97. Physics:Quantum optics beam splitter experiments
98. Physics:Quantum Ultra fast lasers
99. Physics:Quantum Experimental quantum physics
100. Physics:Quantum optics
101. Template:Quantum optics operators

Open quantum systems



102. Physics:Quantum Open systems
103. Physics:Quantum Master equation
104. Physics:Quantum Lindblad equation
105. Physics:Quantum Decoherence
106. Physics:Quantum dissipation
107. Physics:Quantum Markov semigroup
108. Physics:Quantum Markovian dynamics
109. Physics:Quantum Non-Markovian dynamics
110. Physics:Quantum Trajectories

Quantum field theory



Structural dependency map of quantum field theory.

111. Physics:Quantum field theory (QFT) basics
112. Physics:Quantum field theory (QFT) core
113. Physics:Quantum Fields and Particles
114. Physics:Quantum Second quantization
115. Physics:Quantum Harmonic Oscillator field modes
116. Physics:Quantum Creation and annihilation operators
117. Physics:Quantum vacuum fluctuations
118. Physics:Quantum Propagators in quantum field theory
119. Physics:Quantum Feynman diagrams
120. Physics:Quantum Path integral formulation
121. Physics:Quantum Renormalization in field theory
122. Physics:Quantum Renormalization group
123. Physics:Quantum Field Theory Gauge symmetry
124. Physics:Quantum Non-Abelian gauge theory
125. Physics:Quantum Electrodynamics (QED)
126. Physics:Quantum chromodynamics (QCD)
127. Physics:Quantum Electroweak theory
128. Physics:Quantum Standard Model
129. Physics:Quantum triviality

Statistical mechanics and kinetic theory



130. Physics:Quantum Statistical mechanics
131. Physics:Quantum Partition function
132. Physics:Quantum Distribution functions
133. Physics:Quantum Liouville equation
134. Physics:Quantum Kinetic theory
135. Physics:Quantum Boltzmann equation
136. Physics:Quantum BBGKY hierarchy
137. Physics:Quantum Relaxation and thermalization
138. Physics:Quantum Thermodynamics

Condensed matter and solid-state physics



139. Physics:Quantum Band structure
140. Physics:Quantum Fermi surfaces
141. Physics:Quantum Semiconductor physics
142. Physics:Quantum Phonons
143. Physics:Quantum Electron-phonon interaction
144. Physics:Quantum Superconductivity
145. Physics:Quantum Topological phases of matter
146. Physics:Quantum well
147. Physics:Quantum spin liquid
148. Physics:Quantum spin Hall effect
149. Physics:Quantum phase transition
150. Physics:Quantum critical point
151. Physics:Quantum dot

Plasma and fusion physics



Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

152. Physics:Quantum Fusion reactions and Lawson criterion
153. Physics:Quantum Plasma (fusion context)
154. Physics:Quantum Magnetic confinement fusion
155. Physics:Quantum Inertial confinement fusion
156. Physics:Quantum Plasma instabilities and turbulence
157. Physics:Quantum Tokamak core plasma
158. Physics:Quantum Tokamak edge physics and recycling asymmetries
159. Physics:Quantum Stellarator

Timeline



160. Physics:Quantum mechanics/Timeline
161. Physics:Quantum mechanics/Timeline/Pre-quantum era
162. Physics:Quantum mechanics/Timeline/Old quantum theory
163. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
164. Physics:Quantum mechanics/Timeline/Quantum field theory era
165. Physics:Quantum mechanics/Timeline/Quantum information era
166. Physics:Quantum mechanics/Timeline/Quantum technology era
167. Physics:Quantum mechanics/Timeline/Quiz

Advanced and frontier topics



168. Physics:Quantum topology
169. Physics:Quantum battery
170. Physics:Quantum Supersymmetry
171. Physics:Quantum Black hole thermodynamics
172. Physics:Quantum Holographic principle
173. Physics:Quantum gravity
174. Physics:Quantum De Sitter invariant special relativity
175. Physics:Quantum Doubly special relativity

References

• Herbert W. Hamber and Ruth M. Williams (2011). "Discrete Wheeler-DeWitt Equation". Physical Review D 84 (10): 104033. doi:10.1103/PhysRevD.84.104033. Bibcode: 2011PhRvD..84j4033H. 
• Herbert W. Hamber, Reiko Toriumi and Ruth M. Williams (2012). "Wheeler-DeWitt Equation in 2+1 Dimensions". Physical Review D 86 (8): 084010. doi:10.1103/PhysRevD.86.084010. Bibcode: 2012PhRvD..86h4010H.