Power identity: $P = \sum_x \sum_a v(x)\cdot(W - W^{\mathsf T})\, v(x+e_a)$

Setting. The statement takes natural numbers $q, L, d$ and assumes $L \neq 0$. There are no other hypotheses on $q$ or $d$. A site is any function $x : \{0,\dots,q-1\} \to \mathbb{Z}/L\mathbb{Z}$, meaning a point of the discrete torus $(\mathbb{Z}/L\mathbb{Z})^q$. There are $L^q$ sites. The coordinates $x_a$ are elements of $\{0,\dots,L-1\}$, and addition on them is taken modulo $L$.

For a site $x$, a direction $a \in \{0,\dots,q-1\}$ and a step $s \in \mathbb{Z}/L\mathbb{Z}$, the shift $\sigma_a^s(x)$ is the site that agrees with $x$ in every coordinate except $a$. Its $a$-th coordinate is replaced by $x_a + s \bmod L$:

$$\sigma_a^s(x)_b = \begin{cases} x_a + s \bmod L, & b = a,\\ x_b, & b \neq a.\end{cases}$$

The two shifts that appear are $s = 1$ and $s = -1$:

• $1$ means $1 \bmod L$.
• $-1$ means the additive inverse of $1$ in $\mathbb{Z}/L\mathbb{Z}$, which is $L-1$ when $L \ge 2$.

So $\sigma_a^{1}(x)$ is the neighbour one step forward in direction $a$, with wrap-around. $\sigma_a^{-1}(x)$ is the neighbour one step backward, also with wrap-around. There are two degenerate cases:

• $L = 1$: $1 = -1 = 0$, so both shifts are the identity: $\sigma_a^{\pm1}(x) = x$.
• $L = 2$: the forward and backward neighbours coincide.

Data. Both of the following are completely arbitrary. No symmetry, sign, boundedness or other condition is imposed on them.

• $W$ assigns a real $d \times d$ matrix $W_{x,a}$ to each site $x$ and each direction $a$.
• $v$ assigns a vector $v_x \in \mathbb{R}^d$ to each site $x$.

The dot product is $u \cdot w = \sum_{i=0}^{d-1} u_i w_i$. The product $M w$ is the ordinary matrix–vector product.

The defined quantity ("power"). The quantity is

$$P(W,v) \;=\; \sum_{x \in (\mathbb{Z}/L)^q} \; \sum_{a=0}^{q-1} \; v_x \cdot \Big( W_{x,a}\, v_{\sigma_a^{1}(x)} \;-\; W_{\sigma_a^{-1}(x),\,a}\, v_{\sigma_a^{-1}(x)} \Big).$$

The second term uses the matrix attached to the backward neighbour $\sigma_a^{-1}(x)$ in direction $a$ (not the matrix at $x$). That matrix acts on the vector at the backward neighbour.

Assertion. For every $q, L, d$ with $L \ge 1$, and for every $W$ and $v$ as above:

$$P(W,v) \;=\; \sum_{x \in (\mathbb{Z}/L)^q} \; \sum_{a=0}^{q-1} \; v_x \cdot \Big( \big(W_{x,a} - W_{x,a}^{\mathsf T}\big)\, v_{\sigma_a^{1}(x)} \Big),$$

where $W_{x,a}^{\mathsf T}$ is the transpose of $W_{x,a}$.

Degenerate cases included by the quantifiers:

• $q = 0$: there is exactly one site (the empty tuple) and no directions. The inner sum over directions is empty, so both sides equal $0$.
• $d = 0$: all vectors are empty and every dot product is $0$, so both sides are $0$.
• $L = 1$: there is a single site $x$ and every shift is the identity. The left side becomes $\sum_a v_x \cdot (W_{x,a} v_x - W_{x,a} v_x) = 0$. The right side becomes $\sum_a v_x \cdot ((W_{x,a} - W_{x,a}^{\mathsf T}) v_x)$.
• $L = 0$: excluded by the assumption $L \neq 0$.