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

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

Quantum Book II

Matter by scale

Quantum Book III

Methods and tools

Quantum Book IV

Data Analysis Techniques

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
25. Physics:Quantum measurement problem

Mathematical structure and systems



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

Atomic and spectroscopy



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

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

Wavefunctions and modes



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

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

Quantum dynamics and evolution



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

Measurement and information



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

Quantum information and computing



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

Quantum optics and experiments



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

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

Open quantum systems



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

Quantum field theory



Structural dependency map of quantum field theory.

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

Statistical mechanics and kinetic theory



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

Condensed matter and solid-state physics



142. Physics:Quantum Band structure
143. Physics:Quantum Fermi surfaces
144. Physics:Quantum Semiconductor physics
145. Physics:Quantum Phonons
146. Physics:Quantum Electron-phonon interaction
147. Physics:Quantum Superconductivity
148. Physics:Quantum Topological phases of matter
149. Physics:Quantum well
150. Physics:Quantum spin liquid
151. Physics:Quantum spin Hall effect
152. Physics:Quantum phase transition
153. Physics:Quantum critical point
154. 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.

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

Timeline



163. Physics:Quantum mechanics/Timeline
164. Physics:Quantum mechanics/Timeline/Pre-quantum era
165. Physics:Quantum mechanics/Timeline/Old quantum theory
166. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
167. Physics:Quantum mechanics/Timeline/Quantum field theory era
168. Physics:Quantum mechanics/Timeline/Quantum information era
169. Physics:Quantum mechanics/Timeline/Quantum technology era
170. Physics:Quantum mechanics/Timeline/Quiz

Advanced and frontier topics



171. Physics:Quantum topology
172. Physics:Quantum battery
173. Physics:Quantum Supersymmetry
174. Physics:Quantum Black hole thermodynamics
175. Physics:Quantum Holographic principle
176. Physics:Quantum gravity
177. Physics:Quantum De Sitter invariant special relativity
178. Physics:Quantum Doubly special relativity
179. Physics:Quantum arithmetic geometry
180. Physics:Quantum unsolved problems
181. Physics:Quantum Yang-Mills mass gap
182. Physics:Quantum gravity problem
183. Physics:Quantum black hole information paradox
184. Physics:Quantum dark matter problem
185. Physics:Quantum neutrino mass problem
186. Physics:Quantum matter-antimatter asymmetry problem

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.