Site-local potentials give Poisson-commuting constraints

Read-back of WheelerDeWittSuperspace.poisson_eq_zero_of_sitewise.

Setting. Let $X$ be any finite type of "sites" with decidable equality. It is not assumed nonempty as a type, but the statement takes points $x, y \in X$, so every instance that has content has $X \neq \emptyset$. A configuration is a map $h : X \to \mathbb{R}^{3\times 3}$, written $h_{ab}(z)$ for $z \in X$ and $a,b \in \{0,1,2\}$. All nine entries $h_{ab}(z)$ are independent real coordinates: symmetry is not built into the configuration space. So configuration space is $\mathbb{R}^{9|X|}$. Write $M_z(h)$ for the $3\times 3$ matrix $(h_{ab}(z))_{a,b}$. A phase-space point is a pair $w = (h, p)$ of two configurations: metric components $h_{ab}(z)$ and momentum components $p^{ab}(z)$. All $9$ momentum entries per site are independent as well, so phase space is $\mathbb{R}^{18|X|}$.

Binders and hypotheses. The theorem quantifies over:

• reals $\kappa$ and $\Lambda$, with the single hypothesis $\kappa \neq 0$ ($\Lambda$ is arbitrary);
• a function $R : (\text{configurations}) \times X \to \mathbb{R}$, $(h, z) \mapsto R(h, z)$ (a "potential", given as data), subject to:
  • (regularity) for every site $z$, the map $h \mapsto R(h,z)$ is $C^1$ on the whole of $\mathbb{R}^{9|X|}$. That includes non-symmetric and non-positive-definite "metrics", not only physical ones.
  • (locality) for every site $z$ and all configurations $h, h'$: if $h(z) = h'(z)$ as $3\times3$ arrays (all nine entries equal), then $R(h,z) = R(h',z)$. Equivalently, $R(h,z) = r_z(h(z))$ for some function $r_z : \mathbb{R}^{9} \to \mathbb{R}$, which by the regularity hypothesis is $C^1$ on $\mathbb{R}^9$.
• two sites $x, y \in X$. They are arbitrary and may coincide.
• a phase-space point $w = (h, p)$, with the hypothesis that $h$ is physical. For every site $z$, the matrix $M_z(h)$ is positive definite in Mathlib's sense: it is symmetric ($h_{ab}(z) = h_{ba}(z)$) and $v^{\mathsf T} M_z(h)\, v > 0$ for every nonzero $v \in \mathbb{R}^3$. There is no hypothesis on the momenta $p$. In particular $p^{ab}(z)$ need not be symmetric in $a,b$.

Definitions unfolded. For a $3\times3$ real matrix $m$, the volume is

$$\operatorname{vol}(m) = \sqrt{\det m},$$

where $\sqrt{\cdot}$ is the real square root, with the convention $\sqrt{t} = 0$ for $t < 0$. The DeWitt supermetric is

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

with the convention that division by $0$ gives $0$. So $G \equiv 0$ whenever $\det m \le 0$. For $m$ non-symmetric the formula is applied literally, entry by entry.

The potential at site $z$ is

$$V_z(h) = -(2\kappa)^{-1}\,\operatorname{vol}\big(M_z(h)\big)\,\big(R(h,z) - 2\Lambda\big).$$

The classical constraint at site $z$ is the function on phase space

$$C_z(h,p) = 2\kappa \sum_{a,b,c,d=0}^{2} G_{abcd}\big(M_z(h)\big)\, p^{ab}(z)\, p^{cd}(z) \;+\; V_z(h).$$

For $z \in X$ and $a,b$, let $e_{z,ab}$ be the configuration that is $1$ in the single entry $(z,a,b)$ and $0$ everywhere else. For a real function $F$ on phase space, write $\partial F/\partial h_{ab}(z)$ for the Fréchet derivative of $F$ at $w$ applied to the direction $(e_{z,ab}, 0)$, and $\partial F/\partial p^{ab}(z)$ for the one applied to $(0, e_{z,ab})$. Mathlib's convention is that the Fréchet derivative is the zero map at any point where $F$ is not differentiable. In that case all these partials are $0$. The Poisson bracket at $w$ is

$$\{F, G\}(w) = \sum_{z\in X}\ \sum_{a=0}^{2}\sum_{b=0}^{2}\left(\frac{\partial F}{\partial h_{ab}(z)}\frac{\partial G}{\partial p^{ab}(z)} - \frac{\partial F}{\partial p^{ab}(z)}\frac{\partial G}{\partial h_{ab}(z)}\right)\Bigg|_{w}.$$

The sum runs over all nine ordered pairs $(a,b)$, treated as independent coordinates, with no symmetrization or factor for $a \neq b$.

Conclusion. Under the hypotheses above,

$$\{C_x, C_y\}(w) = 0.$$

This is asserted at the given single phase-space point $w$, which is any point whose metric part is positive definite at every site and whose momentum part is arbitrary. Since $\kappa, \Lambda, R, x, y, w$ are universally quantified, it holds for all of them that satisfy the hypotheses.

Remarks on scope and edge cases.

• If $x = y$, the bracket is $\{C_x, C_x\}(w)$. Its summands cancel identically by antisymmetry, whatever the derivatives are.
• With $R$ local as above, both terms of $C_z$ depend on phase space only through the nine entries $h_{ab}(z)$ and the nine entries $p^{ab}(z)$ at the same site $z$.
• The derivatives are taken at $w$ in all $18|X|$ coordinate directions. That includes directions that make $M_z$ non-symmetric. Because $w$ is physical, $\det M_z(h) > 0$ at $w$ and hence on a neighbourhood of $w$, so near $w$ the volume and $G$ are given by the formulas without the $\sqrt{\text{negative}} = 0$ or $\div 0$ conventions. Away from physical points the conclusion says nothing.
• The hypotheses can be satisfied simultaneously, so the statement is not vacuous. For example, $R \equiv 0$ (constant, hence $C^1$ and local), $\kappa = 1$, any $\Lambda$, $X$ a one-point set, and $h$ the identity metric at every site.
• No quantum constant ($\hbar$) and no wavefunction appear in this statement.