Lattice superspace and the Wheeler–DeWitt operator

Read-back of the definition module WheelerDeWittSuperspace. All definitions below live in one namespace. Throughout, indices $a,b,c,d,i,j$ range over $\{0,1,2\}$, and $X$ is an arbitrary type of "sites" (any universe). Unless stated otherwise, no finiteness, nonemptiness or decidability is assumed on $X$.

Global conventions (junk values). The definitions use total functions, so the following conventions apply silently everywhere:

• $\sqrt{t}$ is the real square root with $\sqrt{t}=0$ for every $t<0$.
• Real (and complex) division by $0$ and the inverse $0^{-1}$ both give $0$; in particular $(2\kappa)^{-1}=0$ when $\kappa=0$.
• For a real $3\times 3$ matrix $m$, $m^{-1}$ means $(\det m)^{-1}\,\mathrm{adj}(m)$. If $\det m = 0$ this is the zero matrix. No symmetry or positivity of $m$ is needed for $m^{-1}$ to be defined.
• The Fréchet derivative $Df(h)$ of a function at a point where it is not differentiable is the zero linear map, so every directional derivative built from it is $0$ there.

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1. Metric components Metric3. This is the type of all functions $\{0,1,2\}\times\{0,1,2\}\to\mathbb{R}$, i.e. arrays $(h_{ab})$ of nine real numbers, identified with $\mathbb{R}^9$. All nine entries are independent: there is no symmetry condition ($h_{ab}$ and $h_{ba}$ are distinct coordinates) and no positivity or nondegeneracy condition. As a normed space it carries the product structure (sup norm).

2. Configurations Config X. A configuration is any function $h : X \to$ Metric3, i.e. a family $h(x) = (h_{ab}(x))$ of nine arbitrary real numbers per site $x\in X$. If $X$ is empty there is exactly one configuration (the empty family).

3. metricAt. For a configuration $h$ and a site $x$, $\operatorname{metricAt}(h,x)$ is the $3\times3$ real matrix whose $(a,b)$ entry is $h_{ab}(x)$. Below I write $h(x)$ for this matrix.

4. IsPhysical. For a configuration $h$ on $X$, $\operatorname{IsPhysical}(h)$ is the proposition that for every site $x\in X$ the matrix $h(x)$ is positive definite in Mathlib's sense: it is symmetric ($h_{ab}(x)=h_{ba}(x)$ for all $a,b$, real Hermitian) and $\sum_{i,j} v_i\,h_{ij}(x)\,v_j > 0$ for every nonzero $v\in\mathbb{R}^3$. If $X$ is empty this holds vacuously. This predicate is only defined here; none of the other definitions assumes it.

5. volume. For any real $3\times3$ matrix $m$,

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

which equals $0$ whenever $\det m \le 0$ (including all non-symmetric or indefinite matrices with negative determinant).

6. deWitt. For any real $3\times3$ matrix $m$ and indices $a,b,c,d$,

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

The entries are used exactly as given (no symmetrisation of $m$). If $\operatorname{vol}(m)=0$ (i.e. $\det m\le 0$), then $G_{abcd}(m)=0$ for all indices.

7. invDeWitt. For any real $3\times3$ matrix $m$, writing $n = m^{-1}$ (the matrix inverse with the zero-matrix convention above),

$$G^{abcd}(m) = \frac{\operatorname{vol}(m)\,\bigl(n_{ac}n_{bd} + n_{ad}n_{bc} - 2\,n_{ab}n_{cd}\bigr)}{2}.$$

If $\det m \le 0$ then $\operatorname{vol}(m)=0$, so $G^{abcd}(m)=0$ (and if $\det m=0$ also $n=0$). Note the coefficient of the last term is $2$ here, versus $1$ in $G_{abcd}$.

8. superQuad. For real $3\times3$ matrices $m$ and $p$ (with $p$ arbitrary, not required symmetric),

$$\operatorname{superQuad}(m,p) = \sum_{a,b,c,d} G_{abcd}(m)\,p_{ab}\,p_{cd}.$$

9. superMap. For real $3\times3$ matrices $m,p$, the matrix with $(a,b)$ entry

$$\bigl(\operatorname{superMap}(m,p)\bigr)_{ab} = \sum_{c,d} G_{abcd}(m)\,p_{cd}.$$

10. invSuperMap. For real $3\times3$ matrices $m,k$, the matrix with $(a,b)$ entry

$$\bigl(\operatorname{invSuperMap}(m,k)\bigr)_{ab} = \sum_{c,d} G^{abcd}(m)\,k_{cd}.$$

11. velocityQuad. For real $3\times3$ matrices $m,k$, with $m^{-1}$ as above (zero if $\det m=0$),

$$\operatorname{velocityQuad}(m,k) = \operatorname{vol}(m)\Bigl(\operatorname{tr}\bigl(m^{-1}k\,m^{-1}k\bigr) - \bigl(\operatorname{tr}(m^{-1}k)\bigr)^2\Bigr),$$

using ordinary matrix products in the order written; $k$ is not required to be symmetric. It is $0$ whenever $\det m \le 0$.

12. coordDir. Assuming decidable equality on $X$, for a site $x$ and indices $a,b$, $\operatorname{coordDir}(x,a,b)$ is the configuration $e^{(x)}_{ab}$ given by

$$\bigl(e^{(x)}_{ab}\bigr)_{ij}(y) = \begin{cases}1 & \text{if } y=x,\ i=a,\ j=b,\\ 0 & \text{otherwise.}\end{cases}$$

It is the unit vector of the single coordinate $h_{ab}(x)$; in particular for $a\ne b$ it does not also move $h_{ba}(x)$.

13. partialD. Assume $X$ is finite with decidable equality (so Config X $\cong \mathbb{R}^{9|X|}$ is a finite-dimensional real normed space). For a function $\Psi :$ Config X $\to\mathbb{C}$ (no regularity assumed), a site $x$ and indices $a,b$, $\partial_{ab}(x)\Psi$ is the function on configurations

$$h \mapsto D\Psi(h)\bigl[e^{(x)}_{ab}\bigr],$$

where $D\Psi(h)$ is the real Fréchet derivative of $\Psi$ at $h$ (with $\mathbb{C}$ viewed as a real normed space). At any $h$ where $\Psi$ is not Fréchet differentiable, this value is $0$ (even if the one-variable partial derivative in that direction happens to exist).

14. partialR. Same as partialD, but for a real-valued $S :$ Config X $\to\mathbb{R}$: $\partial_{ab}(x)S(h) = DS(h)[e^{(x)}_{ab}]$, equal to $0$ wherever $S$ is not Fréchet differentiable.

15. kinetic. With $X$ finite and decidable equality, for $\Psi :$ Config X $\to\mathbb{C}$, a configuration $h$ and a site $x$,

$$\operatorname{kinetic}(\Psi,h,x) = \sum_{a,b,c,d} G_{abcd}\bigl(h(x)\bigr)\cdot \partial_{ab}(x)\bigl(\partial_{cd}(x)\Psi\bigr)(h)\ \in\mathbb{C},$$

where the real coefficient $G_{abcd}(h(x))$ is cast into $\mathbb{C}$. The inner derivative is in direction $(c,d)$ and the outer one in direction $(a,b)$; the outer derivative is the Fréchet derivative of the function $h'\mapsto \partial_{cd}(x)\Psi(h')$, which is $0$ wherever that function is not differentiable. The coefficient is evaluated at $h(x)$ and placed outside both derivatives. All coefficients vanish if $\det h(x)\le 0$.

16. wdw. With $X$ finite and decidable equality, for real parameters $\kappa$ (kappa), $\hbar$ (hbar), $\Lambda$ (Lam), an arbitrary function $R :$ Config X $\times X\to\mathbb{R}$ (any "potential" depending on the whole configuration and on the site; no relation to curvature is imposed), $\Psi :$ Config X $\to\mathbb{C}$, a configuration $h$ and a site $x$,

$$\operatorname{wdw}(\Psi,h,x) = -2\,\kappa\,\hbar^2\,\operatorname{kinetic}(\Psi,h,x) \;-\; \Bigl[(2\kappa)^{-1}\operatorname{vol}\bigl(h(x)\bigr)\bigl(R(h,x)-2\Lambda\bigr)\Bigr]\,\Psi(h)\ \in\mathbb{C},$$

where the bracket is computed in $\mathbb{R}$ and cast to $\mathbb{C}$. No sign or nonzero assumptions on $\kappa,\hbar,\Lambda$: if $\kappa=0$ both terms vanish (since $(2\cdot 0)^{-1}=0$) and $\operatorname{wdw}=0$; if $\hbar=0$ only the potential term remains.

17. hamiltonJacobi. With $X$ finite and decidable equality, for real $\kappa,\Lambda$, arbitrary $R$ as above, $S :$ Config X $\to\mathbb{R}$, a configuration $h$ and a site $x$,

$$\operatorname{HJ}(S,h,x) = 2\kappa\sum_{a,b,c,d} G_{abcd}\bigl(h(x)\bigr)\,\partial_{ab}(x)S(h)\,\partial_{cd}(x)S(h) \;-\; (2\kappa)^{-1}\operatorname{vol}\bigl(h(x)\bigr)\bigl(R(h,x)-2\Lambda\bigr)\ \in\mathbb{R}.$$

Here there is no $\hbar$ factor; the first-derivative partials are those of item 14 (zero where $S$ is not differentiable). If $\kappa=0$ the value is $0$.

18. potentialAt. For any $X$ (no finiteness needed), real $\kappa,\Lambda$, arbitrary $R$, a configuration $h$ and a site $z$,

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

which is $0$ if $\kappa=0$ or $\det h(z)\le 0$.

19. Phase space Phase X. The type of pairs $w=(h,p)$ of two configurations on $X$: $h$ ("metric") and $p$ ("momenta"), each assigning nine arbitrary real numbers to every site, with no symmetry condition. As a normed space it is the product of two copies of Config X.

20. classicalConstraint. For any $X$, real $\kappa,\Lambda$, arbitrary $R$, a site $z$ and a phase-space point $w=(h,p)$,

$$C_z(w) = 2\kappa\sum_{a,b,c,d} G_{abcd}\bigl(h(z)\bigr)\,p_{ab}(z)\,p_{cd}(z) \;+\; V_z(h),$$

with $V_z$ from item 18 (so the potential enters with the minus sign built into $V_z$). It is $0$ if $\kappa=0$; its kinetic part is $0$ if $\det h(z)\le0$.

21. poisson. With $X$ finite and decidable equality, for arbitrary functions $F,G :$ Phase X $\to\mathbb{R}$ (no regularity assumed) and a point $w$,

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

where $D$ is the Fréchet derivative on the product space, $(e,0)$ moves only the metric coordinate $h_{ab}(z)$ and $(0,e)$ only the momentum coordinate $p_{ab}(z)$. All 9 index pairs $(a,b)$ are summed separately (the $(a,b)$ and $(b,a)$ coordinates are independent). If $F$ (resp. $G$) is not differentiable at $w$, its derivative there is $0$ and the corresponding products vanish. If $X$ is empty, $\{F,G\}=0$ identically.

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Edge cases summarised. (i) Nothing outside IsPhysical restricts metrics: every definition accepts non-symmetric, singular or indefinite arrays, where $\operatorname{vol}=0$ forces $G_{abcd}=G^{abcd}=\operatorname{velocityQuad}=0$, and the kinetic and potential terms vanish. (ii) All nine entries $h_{ab}$ (and $p_{ab}$) are separate coordinates, and partial derivatives are taken along single entries. (iii) Derivative-based quantities silently return $0$ at non-differentiable points. (iv) $\kappa=0$ kills every term carrying $(2\kappa)^{-1}$ or the factor $2\kappa$. (v) For empty $X$, there is one configuration, IsPhysical holds vacuously, site-indexed quantities have no site to be evaluated at, and the Poisson bracket is identically $0$.