Lattice Hamiltonian constraints are second class

Read-back of constraints_not_first_class.

Setting. Let $X$ be any finite type of "sites" with decidable equality. No lower bound on its size is assumed, but the hypothesis $x \neq y$ below forces $|X| \ge 2$. A configuration $h$ assigns to every site $z \in X$ an arbitrary real $3\times 3$ array $h(z) = (h_{ab}(z))_{a,b \in \{0,1,2\}}$. All $9|X|$ entries are independent real coordinates, and symmetry is not built in. Configuration space $\mathcal C = (\mathbb R^{3\times 3})^X$ is a finite-dimensional real normed space. The phase space is $\mathcal C \times \mathcal C$, whose points are $(h,p)$. Here the momentum $p$ is likewise an arbitrary real $3\times 3$ array $p(z) = (p^{ab}(z))$ at each site, which need not be symmetric.

Definitions used (unfolded). For a real $3\times3$ matrix $m$:

• $\operatorname{vol}(m) = \sqrt{\det m}$, using the real square root. This returns $0$ when $\det m \le 0$.
• The supermetric is

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

The division is total, so this is $0$ when $\operatorname{vol}(m)=0$.

Fix real numbers $\kappa, \Lambda$ and a map $R : \mathcal C \times X \to \mathbb R$, written $R(h,z)$.

• The potential at site $z$ is

$$V_z(h) = -(2\kappa)^{-1}\,\sqrt{\det h(z)}\,\bigl(R(h,z) - 2\Lambda\bigr).$$

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

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

• For $S:\mathcal C\to\mathbb R$, the partial derivative $\partial_{ab(x)} S(h)$ is the Fréchet derivative of $S$ at $h$, applied to the coordinate direction $e_{x,ab}$. That direction has entry $1$ at site $x$, row $a$, column $b$, and $0$ everywhere else. The entries $(a,b)$ and $(b,a)$ are separate coordinates. If $S$ is not differentiable at $h$, this value is $0$ by convention.
• The Poisson bracket of $F,G : \mathcal C\times\mathcal C \to \mathbb R$ at $w=(h,p)$ is

$$\{F,G\}(w) = \sum_{z\in X}\sum_{a,b=0}^{2}\Bigl( DF(w)[(e_{z,ab},0)]\; DG(w)[(0,e_{z,ab})] - DF(w)[(0,e_{z,ab})]\; DG(w)[(e_{z,ab},0)] \Bigr).$$

Here $D$ is the Fréchet derivative on the product space, again defined as $0$ at points of non-differentiability. The sum runs over all 9 ordered index pairs, not only over symmetric combinations.

Hypotheses. The theorem quantifies over the following:

• $X$, as above.
• Reals $\kappa, \Lambda$ with $\kappa \neq 0$.
• A map $R$ such that for every site $z$ the function $h \mapsto R(h,z)$ is continuously differentiable ($C^1$) on all of $\mathcal C$, including non-physical configurations. Nothing else is assumed about $R$. It need not be local, and it need not be related to any curvature.
• A configuration $h$ that is physical: for every site $z$, the matrix $h(z)$ is symmetric and positive definite. In particular $\det h(z) > 0$ and $\sqrt{\det h(z)} > 0$ at every site.
• Two sites $x, y \in X$ with $x \neq y$.
• A dependence hypothesis: there exist indices $a,b \in \{0,1,2\}$ such that

$$\partial_{ab(x)} V_y(h) + \partial_{ba(x)} V_y(h) \neq 0.$$

This is the symmetrized derivative of the potential at $y$ with respect to the metric entry $(a,b)$ at the other site $x$. When $a=b$ it reads $2\,\partial_{aa(x)}V_y(h)\neq 0$.

What the dependence hypothesis amounts to. Since $x\neq y$, the factor $\sqrt{\det h'(y)}$ does not change along $e_{x,ab}$. At the physical $h$ it is also differentiable and positive, and $R(\cdot,y)$ is differentiable. So the hypothesis is equivalent to

$$\exists\, a,b:\quad \partial_{ab(x)} R(h,y) + \partial_{ba(x)} R(h,y) \neq 0.$$

Put differently, $R$ at site $y$ must depend on the metric at site $x$ through a symmetrized derivative. A dependence only through the antisymmetric part $h_{ab}(x)-h_{ba}(x)$ does not satisfy it.

The hypothesis can be satisfied. For example, $R(h',z) = h'_{00}(x)$ for all $z$ is $C^1$ and gives $\partial_{00(x)}V_y(h) = -(2\kappa)^{-1}\sqrt{\det h(y)} \neq 0$. So the hypotheses are jointly consistent whenever $|X|\ge 2$.

Conclusion. There exists a momentum configuration $p \in \mathcal C$ satisfying both of the following:

1. Every constraint vanishes: $C_z(h,p) = 0$ for every site $z \in X$, at the given physical configuration $h$.
2. The bracket of the two constraints is nonzero:

$$\{C_x, C_y\}(h,p) \neq 0.$$

The momentum $p$ ranges over arbitrary real $3\times3$ arrays per site. It is not required to be symmetric, and nothing ties it to $h$ beyond condition 1. The existence of such a $p$ is itself part of what the theorem asserts. The configuration $h$ is fixed as the one given in the hypotheses, not chosen.

Remarks on the bracket. At a physical $h$ with $R(\cdot,z)$ of class $C^1$, each $C_z$ is differentiable at $(h,p)$. So the junk-value convention for derivatives does not apply here. $C_z$ depends on momenta only through $p(z)$, and its kinetic part depends only on $h(z)$. Because $x\neq y$, the only terms of the bracket that can be nonzero are therefore

$$\{C_x,C_y\}(h,p) = \sum_{a,b}\Bigl(\partial_{h_{ab}(y)} V_x(h)\,\partial_{p^{ab}(y)}C_y(h,p) - \partial_{p^{ab}(x)}C_x(h,p)\,\partial_{h_{ab}(x)}V_y(h)\Bigr).$$

Not assumed. The statement does not involve $\hbar$, any symmetry requirement on $p$, any locality or neighbour structure on $X$, or any restriction on $\Lambda$.