Commutator of two lattice Wheeler–DeWitt constraints

Read-back of wdw_commutator. This theorem gives an exact formula, at one physical configuration, for the commutator of two lattice Wheeler–DeWitt operators at two distinct sites.

Setting and binders. Let $X$ be any finite type with decidable equality (the "sites"). A configuration is a map $h : X \to \mathbb{R}^{3\times 3}$. All nine entries $h_{ab}(x)$, $a,b\in\{0,1,2\}$, count as independent real coordinates, so symmetry is not built in. The configuration space $\mathcal{C}=\mathbb{R}^{X\times 3\times 3}$ is a finite-dimensional real normed space with the sup norm. All derivatives below are real Fréchet derivatives on $\mathcal{C}$. The statement quantifies over:

• real numbers $\kappa,\hbar,\Lambda$ with $\kappa\neq 0$ and $\hbar\neq 0$. $\Lambda$ is unrestricted;
• a real "potential datum" $R:\mathcal{C}\times X\to\mathbb{R}$, $(h,z)\mapsto R(h,z)$, which may depend on the whole configuration. The only assumption is that for every site $z$ the map $h\mapsto R(h,z)$ is $C^2$ on all of $\mathcal{C}$;
• a complex-valued function $\Psi:\mathcal{C}\to\mathbb{C}$ that is $C^4$ as a map between real vector spaces;
• a configuration $h$ that is physical: for every site $x$, the matrix $h(x)$ is symmetric positive definite (symmetric, and $v^\top h(x) v>0$ for all $v\neq 0$). In particular $\det h(x)>0$ at every site;
• two sites $x,y\in X$ with $x\neq y$.

If $X$ has fewer than two elements, no such $x,y$ exist and the statement is vacuous.

Definitions unfolded.

• Coordinate derivatives. $e^{(x)}_{ab}\in\mathcal{C}$ is the configuration that is $1$ in entry $(a,b)$ at site $x$ and $0$ everywhere else. For a function $F$ on $\mathcal{C}$ (real or complex valued), set

$$\partial^{(x)}_{ab}F(h) := DF(h)\big[e^{(x)}_{ab}\big],$$

the full Fréchet derivative at $h$ applied to that direction. By the Fréchet-derivative convention, $DF(h)$ is taken to be $0$ wherever $F$ is not differentiable. Iterated derivatives are nested: $\partial^{(x)}_{ab}\partial^{(x)}_{cd}F$ means first form $h'\mapsto\partial^{(x)}_{cd}F(h')$, then take $\partial^{(x)}_{ab}$ of that function.

• Volume density. $\sqrt{h(z)} := \sqrt{\det h(z)}$, with the real square root, which is $0$ for negative arguments.
• DeWitt supermetric at site $z$, for the matrix $m=h(z)$:

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

with the convention that division by $0$ gives $0$.

• Potential at site $z$:

$$V_z(h) := -\frac{1}{2\kappa}\,\sqrt{\det h(z)}\,\big(R(h,z)-2\Lambda\big).$$

• Lattice Wheeler–DeWitt operator at site $z$, acting on any $\Phi:\mathcal{C}\to\mathbb{C}$:

$$(\mathcal{H}_z\Phi)(h) := -2\kappa\hbar^2\sum_{a,b,c,d=0}^{2} G_{abcd}(h(z))\;\partial^{(z)}_{ab}\partial^{(z)}_{cd}\Phi(h)\;-\;\frac{1}{2\kappa}\sqrt{\det h(z)}\,\big(R(h,z)-2\Lambda\big)\,\Phi(h).$$

The supermetric stands to the left of both derivatives, and the second term is exactly $V_z(h)\,\Phi(h)$. Real quantities are embedded in $\mathbb{C}$.

Assertion. At the given physical $h$ and the given sites $x\neq y$:

$$\big(\mathcal{H}_x(\mathcal{H}_y\Psi)\big)(h)-\big(\mathcal{H}_y(\mathcal{H}_x\Psi)\big)(h) = -2\kappa\hbar^2\Big(T_{x,y}-T_{y,x}\Big),$$

where

$$T_{x,y}=\sum_{a,b,c,d=0}^{2}G_{abcd}(h(x))\Big(2\,\partial^{(x)}_{ab}V_y(h)\;\partial^{(x)}_{cd}\Psi(h)\;+\;\partial^{(x)}_{ab}\big(\partial^{(x)}_{cd}V_y\big)(h)\;\Psi(h)\Big)$$

and $T_{y,x}$ is the same expression with $x$ and $y$ swapped everywhere: the supermetric is $G(h(y))$, the derivatives are taken at site $y$, the potential is $V_x$, and $\Psi$ is the same function.

The function $\mathcal{H}_y\Psi$ is the whole map $h'\mapsto(\mathcal{H}_y\Psi)(h')$ on $\mathcal{C}$, and $\mathcal{H}_x$ is applied to it. On the left this needs Fréchet derivatives of $\mathcal{H}_y\Psi$ at points near $h$, not only at $h$. The identity is claimed at this single point $h$ only. In the first-order term the index pair $(a,b)$ goes on the derivative of $V_y$ and $(c,d)$ on the derivative of $\Psi$, and the sum is not symmetrized over swapping the pairs. The left side contains $\Lambda$ only through the operators, and the right side only through $V_x,V_y$.

Edge cases and points to note.

• $R(\cdot,z)$ is assumed $C^2$ for every site, globally on $\mathcal{C}$. No locality is assumed: $R(h,y)$ may depend on $h(x)$. This is why $\partial^{(x)}V_y$ can be nonzero.
• $\Psi$ is assumed $C^4$ on all of $\mathcal{C}$. Nothing restricts $\Psi$ or $R$ to physical configurations. Only the evaluation point $h$ is physical, and nearby non-symmetric configurations still lie in the domain of the derivatives.
• $\sqrt{\det}$ and $G_{abcd}$ are defined by junk conventions where $\det\le 0$ (square root $0$, division by $0$ gives $0$). Because $\det h(z)>0$ at the physical $h$, they are given by the displayed formulas on a neighbourhood of $h$.
• The non-vanishing hypotheses $\kappa\neq 0$ and $\hbar\neq 0$ exclude the junk value $(2\cdot 0)^{-1}=0$ in $V_z$. If $\hbar=0$, both sides would plainly be $0$, so that case is excluded but would not be problematic.
• The hypotheses are jointly satisfiable, for example with $X$ of two points, $h(x)=h(y)=I$, $R\equiv 0$ and $\Psi\equiv 0$, so the theorem is not vacuous whenever $|X|\ge 2$.