Wheeler–DeWitt kinetic operator on the conformal sector

kinetic_volume_sector. Setting. Let $X$ be any finite type of "sites" with decidable equality. It is nonempty, because a site $x_0 \in X$ is given. A configuration $h$ is a function assigning to each site $y \in X$ an arbitrary real $3\times 3$ array $h(y) = (h(y)_{ij})_{i,j\in\{0,1,2\}}$. All nine entries are independent coordinates, with no symmetry imposed, so the configuration space is $\mathbb{R}^{9|X|}$. Write $M := h(x_0)$ for the matrix at the distinguished site. For a real $3\times 3$ matrix $m$, the volume is

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

where the square root is Mathlib's real square root, which returns $0$ for negative arguments.

Data and hypotheses.

• $x_0 \in X$ is a site.
• $f : \mathbb{R} \to \mathbb{C}$ is a function that is $C^2$ on all of $\mathbb{R}$, as a map between real vector spaces.
• $h$ is a configuration.
• $M = h(x_0)$ is positive definite in Mathlib's sense: it is symmetric ($M^\top = M$) and $\xi^\top M \xi > 0$ for every nonzero $\xi \in \mathbb{R}^3$. Hence $\det M > 0$, and $v := \mathrm{vol}(M) = \sqrt{\det M} > 0$. Nothing is assumed about $h(y)$ for $y \ne x_0$.

The wavefunction. Define $\Psi$ on configuration space by

$$\Psi(h') = f\bigl(\sqrt{\det h'(x_0)}\bigr).$$

This uses the same truncated square root, so it equals $f(0)$ wherever $\det h'(x_0) \le 0$. $\Psi$ depends only on the nine entries at site $x_0$.

Derivatives. For a function $\Phi$ from configurations to $\mathbb{C}$, the partial derivative $\partial_{ab}\Phi(h)$ (with $a,b \in \{0,1,2\}$, taken at site $x_0$) is the Fréchet derivative of $\Phi$ at $h$ applied to the coordinate direction $E^{x_0}_{ab}$. That direction is the configuration with entry $1$ at site $x_0$, position $(a,b)$, and $0$ everywhere else. It is the ordinary partial derivative with respect to the single unsymmetrized entry $h(x_0)_{ab}$; the entries $h_{ab}$ and $h_{ba}$ are varied separately. By Mathlib convention, if $\Phi$ is not Fréchet differentiable at a point, its derivative there is taken to be $0$.

Kinetic term. For a real matrix $m$, the DeWitt coefficients are

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

The kinetic term of $\Psi$ at $(h, x_0)$ is the full sum over all $3^4 = 81$ index combinations:

$$K[\Psi](h,x_0) = \sum_{a,b,c,d=0}^{2} G_{abcd}(M)\; \partial_{ab}\bigl(\partial_{cd}\Psi\bigr)(h).$$

The inner derivative $\partial_{cd}\Psi$ is taken as a function on all of configuration space, and then $\partial_{ab}$ of that function is evaluated at $h$. The coefficient $G_{abcd}(M)$ is real and is regarded as a complex number.

Claim. Under the hypotheses above,

$$K[\Psi](h,x_0) = -\frac{3}{8}\Bigl(v\, f''(v) + 7\, f'(v)\Bigr), \qquad v = \sqrt{\det h(x_0)} \in \mathbb{R}_{>0},$$

with $v$ cast into $\mathbb{C}$.

• $f'(v) \in \mathbb{C}$ is the derivative of $f$ at $v$.
• $f''(v) \in \mathbb{C}$ is the second iterated derivative of $f$ at $v$, i.e. the derivative of $f'$ at $v$.
• Both are derivatives with respect to the real variable, and both exist genuinely because $f$ is $C^2$.

Edge cases and remarks.

• The hypotheses are satisfiable, so the statement is not vacuous. For example, take $h(x_0) = I$ and any $C^2$ function $f$.
• Positive definiteness is assumed only at the point $h$ where the identity is evaluated. It is not assumed on a neighbourhood. Nearby configurations may have nonsymmetric $h'(x_0)$, but $\det h'(x_0)$ stays positive near $h$, since the determinant is continuous.
• $f$ is required to be $C^2$ on all of $\mathbb{R}$, including at non-positive arguments. Only its 2-jet at $v > 0$ appears in the conclusion.
• The site set $X$ may consist of $x_0$ alone. Sites other than $x_0$ and their matrices play no role.
• The right-hand side involves only $v$, $f'(v)$ and $f''(v)$. It contains no other dependence on $M$.