Semiclassical limit: WKB phase obeys the Einstein–Hamilton–Jacobi equation

Theorem wdw_wkb (read-back).

Setting. Let $X$ be any finite type with decidable equality (a finite set of "sites"; it may a priori be empty, but the statement also takes a site $x \in X$, so it only has content when $X \neq \varnothing$). A configuration is a function $h : X \to \mathbb{R}^{3\times 3}$, i.e. at each site $y$ an arbitrary real $3\times 3$ array $h_{ab}(y)$, $a,b \in \{0,1,2\}$. All nine entries at each site are independent coordinates: configurations are not required to be symmetric, positive definite, or nondegenerate. The configuration space $\mathcal{C} = (\mathbb{R}^{3\times 3})^X \cong \mathbb{R}^{9|X|}$ is a finite-dimensional real vector space. For a site $y$ write $m = h(y)$ for the $3\times 3$ matrix with entries $m_{ab} = h_{ab}(y)$.

Data and hypotheses. The theorem is stated for

• real numbers $\kappa$, $\hbar$, $\Lambda$ with hypotheses $\kappa \neq 0$ and $\hbar \neq 0$ ($\Lambda$ is unrestricted);
• an arbitrary function $R : \mathcal{C} \times X \to \mathbb{R}$, $(h, y) \mapsto R(h, y)$ (no regularity, symmetry or locality assumed);
• a function $S : \mathcal{C} \to \mathbb{R}$ with the hypothesis that $S$ is $C^2$ (twice continuously Fréchet differentiable over $\mathbb{R}$) on all of $\mathcal{C}$;
• an arbitrary configuration $h \in \mathcal{C}$ and an arbitrary site $x \in X$ (no physicality condition on $h$).

Definitions used (unfolded).

• Volume density: for a real $3\times 3$ matrix $m$, $\sqrt{m} := \sqrt{\det m}$, where the real square root is taken to be $0$ for negative arguments. Hence $\sqrt{m} = 0$ whenever $\det m \le 0$.
• DeWitt supermetric:

$$G_{abcd}(m) = \frac{m_{ac}m_{bd} + m_{ad}m_{bc} - m_{ab}m_{cd}}{2\sqrt{\det m}},$$

with the convention that division by $0$ gives $0$. So $G_{abcd}(m) = 0$ for all indices whenever $\det m \le 0$. Since $m$ need not be symmetric, $G$ is formed literally from the given entries.

• Coordinate direction: $e_{x,ab} \in \mathcal{C}$ is the configuration that is $1$ in slot $(x,a,b)$ and $0$ everywhere else (only that single ordered entry; $e_{x,ab} \neq e_{x,ba}$ for $a \neq b$).
• Partial derivatives: for $F : \mathcal{C} \to \mathbb{R}$ (resp. $\mathbb{C}$), $\partial_{ab(x)} F(h) := DF(h)\,[e_{x,ab}]$, the real Fréchet derivative of $F$ at $h$ applied to $e_{x,ab}$ (for complex-valued $F$, $\mathbb{C}$ is treated as a real vector space). If $F$ is not Fréchet differentiable at $h$, this is defined to be $0$. Iterated: $\partial_{ab(x)}\partial_{cd(x)}F$ means first take $\partial_{cd(x)}F$ as a function on $\mathcal{C}$, then apply $\partial_{ab(x)}$ to it. Both derivatives are at the same site $x$.
• Kinetic term: for $\Psi : \mathcal{C} \to \mathbb{C}$,

$$K[\Psi](h,x) = \sum_{a,b,c,d=0}^{2} G_{abcd}(h(x))\; \partial_{ab(x)}\partial_{cd(x)}\Psi(h).$$

• Wheeler–DeWitt operator at site $x$:

$$\mathcal{H}[\Psi](h,x) = -2\kappa\hbar^2\, K[\Psi](h,x) \;-\; (2\kappa)^{-1}\sqrt{\det h(x)}\,\bigl(R(h,x) - 2\Lambda\bigr)\,\Psi(h).$$

• Einstein–Hamilton–Jacobi functional:

$$\mathrm{HJ}[S](h,x) = 2\kappa \sum_{a,b,c,d} G_{abcd}(h(x))\; \partial_{ab(x)}S(h)\; \partial_{cd(x)}S(h) \;-\; (2\kappa)^{-1}\sqrt{\det h(x)}\,\bigl(R(h,x) - 2\Lambda\bigr),$$

a real number.

Conclusion. Let $\Psi(h') = \exp\!\bigl(i\,S(h')/\hbar\bigr)$ for $h' \in \mathcal{C}$ (complex exponential, $S$ and $\hbar$ regarded as complex numbers). Then, for the given $h$ and $x$,

$$\mathcal{H}[\Psi](h,x) \;=\; e^{iS(h)/\hbar}\Bigl(\mathrm{HJ}[S](h,x) \;-\; i\,\hbar\,(2\kappa)\sum_{a,b,c,d=0}^{2} G_{abcd}(h(x))\;\partial_{ab(x)}\partial_{cd(x)}S(h)\Bigr),$$

an equality of complex numbers, where the real quantities $\mathrm{HJ}[S](h,x)$, $G_{abcd}$ and $\partial_{ab(x)}\partial_{cd(x)}S(h)$ are embedded in $\mathbb{C}$. The second-derivative term uses the same index order as the kinetic term ($\partial_{cd}$ applied first, then $\partial_{ab}$).

Edge cases and remarks.

• No positivity or symmetry is assumed on $h(x)$. If $\det h(x) \le 0$, then $\sqrt{\det h(x)} = 0$ and every $G_{abcd}(h(x)) = 0$, so both the potential and all supermetric terms vanish on both sides; the identity then reads $0 = 0$. If $\det h(x) > 0$ but $h(x)$ is not symmetric, the formulas are applied verbatim to the non-symmetric entries.
• Derivatives are with respect to each of the nine ordered entries separately; there is no symmetrization (no factor relating $\partial_{ab}$ and $\partial_{ba}$).
• The potential term on the left contains $\Psi(h) = e^{iS(h)/\hbar}$ and appears on the right inside $\mathrm{HJ}$; the function $R$ enters only through its value $R(h,x)$, so no regularity of $R$ is needed or assumed.
• The hypothesis $\hbar \neq 0$ excludes the case where $iS/\hbar$ would be evaluated with the convention $1/0 = 0$ (making $\Psi \equiv 1$); $\kappa \neq 0$ excludes $(2\kappa)^{-1} = 0$. Both hypotheses are satisfiable, as is $C^2$-smoothness of $S$ (e.g. $S$ constant or linear), so the statement is not vacuous. With $X$ nonempty, all binders can be instantiated.
• The $C^2$ hypothesis is global on $\mathcal{C}$; it guarantees the Fréchet derivatives involved exist, so the "derivative $=0$ when not differentiable" convention does not intervene for $S$ or $\Psi$.
• The identity is asserted pointwise for every single $h$ and $x$; it says nothing about $\mathcal{H}[\Psi]$ vanishing (no constraint equation is imposed) and makes no asymptotic claim in $\hbar$: it is an exact equality with an explicit term linear in $\hbar$.