惠勒－德威特方程

在理论物理中，惠勒-德维特方程（英语：Wheeler-DeWitt equation，简称惠-德方程）是一个描述宇宙波函数 ψ 必须满足量子引力理论的方程。 其中一个波函数的例子是哈妥-霍金态。

简单说，惠-德方程的数学形式为： H|ψ ⟩ =0

其中 H 是量子化广义相对论中的全部哈密顿约束。

广义来说，在一个时间尺度不变性的理论中，哈密顿算符会是零。

虽然符号上， H 与 |ψ ⟩ 和传统非相对论性量子力学所用符号相同，然而诠释上，惠勒-德维特方程则与非相对论性量子力学中的方程大相径庭。 |ψ ⟩ 不再是传统上空间波函数的观点(即一复数值的函数，定义于3维类空表面，且归一化。相对地，它是个定义于时空整体的场结构的泛函。此项波函数包含了所有关于宇宙几何以及物质内涵的所有信息。 H 依然是作用在希尔伯特空间中各个波函数上的一项算符，但是这个希尔伯特空间已与非相对论性量子力学中的希尔伯特空间不同，而且哈密顿算符不再决定系统的演化（所以薛定谔方程—— H|ψ ⟩ =iħ ∂ /∂ t|ψ ⟩ ——不再适用）。

此方程源自于ADM形式。

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).

Motivation and background

In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is γ ᵢⱼ and given by （公式略）

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 γ ᵢⱼ is the field, and its conjugate momenta are denoted as π ^(ij). The Hamiltonian is a constraint (characteristic of most relativistic systems) （公式略）

where γ =det(γ ᵢⱼ), and G_(ijkl)=(γ ᵢₖγ _(jl)+γ _(il)γ ⱼₖ-γ ᵢⱼγ _(kl)) is the Wheeler–DeWitt metric. In index-free notation, the Wheeler–DeWitt metric on the space of positive definite quadratic forms g in three dimensions is tr ((g⁻¹dg)²)-(tr (g⁻¹dg))².

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 the operator （公式略） is obtained.

Working in "position space", these operators are （公式略）

One can apply the operator to a general wave functional of the metric （公式略） where （公式略）

which would give a set of constraints amongst the coefficients ψ (x,y,…). This means that the amplitudes for N gravitons at certain positions are related to the amplitudes for a different number of gravitons at different positions. Or, one could use the two-field formalism, treating ω (g) as an independent field, so that the wave function is Ψ [γ,ω ].

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

where H(x) is the Hamiltonian constraint in quantized general relativity, and |ψ ⟩ 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 H and |ψ ⟩ may appear familiar, their interpretation in the Wheeler–DeWitt equation is substantially different from non-relativistic quantum mechanics. |ψ ⟩ 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. 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 the evolution of the system, so the Schrödinger equation H|ψ ⟩ =iħ ∂ /∂ t|ψ ⟩ no longer applies. This property is known as timelessness. Various attempts to incorporate time in a fully quantum framework have been made, starting with the "Page and Wootters mechanism" and other subsequent proposals.