Scalar-field potential and Cauchy-data pairing

Let ${\mathbb{T}}^3:={\mathbb{R}}^3\mathrm{/}{\mathbb{Z}}^3$ with quotient map $q:{\mathbb{R}}^3\to {\mathbb{T}}^3$, and let $M:={\mathbb{R}}_t\times {\mathbb{T}}^3$. The one-forms $dx,dy,dz$ on ${\mathbb{T}}^3$ are the forms descended from the translation-invariant coordinate one-forms on ${\mathbb{R}}^3$; ${\mathrm{\partial }}_x,{\mathrm{\partial }}_y,{\mathrm{\partial }}_z$ denote their dual descended coordinate vector fields. For $f\in C^{\mathrm{\infty }}(M,\mathbb{R})$, spatial derivatives mean derivatives of its ${\mathbb{Z}}^3$-periodic lift to ${\mathbb{R}}_t\times {\mathbb{R}}^3$. Define $\mathrm{\nabla }f:=({\mathrm{\partial }}_{x}f,{\mathrm{\partial }}_{y}f,{\mathrm{\partial }}_{z}f),\mathrm{\mid }\mathrm{\nabla }f{\mathrm{\mid }}^2:=({\mathrm{\partial }}_{x}f{)}^2+({\mathrm{\partial }}_{y}f{)}^2+({\mathrm{\partial }}_{z}f{)}^2,\mathrm{\Delta }f:={\mathrm{\partial }}_x^{2}f+{\mathrm{\partial }}_y^{2}f+{\mathrm{\partial }}_z^{2}f\mathrm{.}$ Give $M$ the orientation $dt\wedge dx\wedge dy\wedge dz$ and flat metric $g=-d{t}^2+d{x}^2+d{y}^2+d{z}^2$, and write $\mathrm{v}\mathrm{o}\mathrm{l}=dt\wedge dx\wedge dy\wedge dz$. The constant-time cut ${\mathrm{\Sigma }}_t=\{t\}\times {\mathbb{T}}^3$ has orientation $dx\wedge dy\wedge dz$, equal to contraction of $\mathrm{v}\mathrm{o}\mathrm{l}$ with ${\mathrm{\partial }}_t$.

Distinguish this geometric cut from its representing chain. Let $i_t:{\mathbb{T}}^3\to M$ be the inclusion $i_t(p)=(t,p)$, with image ${\mathrm{\Sigma }}_t$. In positive affine coordinates on the standard simplex, put $T_3:=\{r\in {\mathbb{R}}^3:r_j\ge 0,r_1+r_2+r_3\le 1\}$ with orientation $d{r}_1\wedge d{r}_2\wedge d{r}_3$. For each permutation $\rho \in S_3$, put $v_0^{\rho }=0$ and $v_j^{\rho }:={\sum }_{k=1}^j{e}_{\rho (k)}$ for $1\le j\le 3$, where $e_1,e_2,e_3$ is the standard basis of ${\mathbb{R}}^3$. Define $a_{\rho }(r):={\sum }_{j=1}^3{r}_j{v}_j^{\rho },Q:={\sum }_{\rho \in S_3}\mathrm{sgn}(\rho )a_{\rho }\mathrm{.}$ Here $Q$ is a finite smooth singular three-chain in ${\mathbb{R}}^3$. For a smooth map $f$, its chain pushforward means composition of each simplex: $f_{\mathrm{\#}}({\sum }_j{c}_j{\sigma }_j):={\sum }_j{c}_j(f\circ {\sigma }_j)$. The representing chain in $M$ is $C_t:=(i_t\circ q{)}_{\mathrm{\#}}Q={\sum }_{\rho \in S_3}\mathrm{sgn}(\rho ){\tau }_{t,\rho },{\tau }_{t,\rho }:=i_t\circ q\circ a_{\rho }\mathrm{.}$ Every simplex has the stated domain orientation and coefficient $\mathrm{sgn}(\rho )$. The affine images are the six ordered-coordinate regions of the cube. Adjacent regions have identical ordered face maps and opposite signed face coefficients; on the exterior, opposite cube faces have matching ordered face maps after $q$, again with opposite coefficients. Thus $\mathrm{\partial }{C}_t=0$. By saying that $C_t$ represents the oriented cut ${\mathrm{\Sigma }}_t$, we mean precisely that for every smooth three-form $\alpha$ on $M$, ${\int }_{C_t}\alpha ={\int }_{[0,1{]}^3}(i_t\circ q{)}^{\ast }\alpha \mathrm{.}$ The right side integrates the coefficient relative to $dx\wedge dy\wedge dz$. This equality follows by summing the six simplex pullbacks with their orientation coefficients, as checked in step 3.1; it invokes only this explicit cube decomposition. Restrictions of fields and forms below are to the geometric cut ${\mathrm{\Sigma }}_t$, whereas boundary and chain-integration operations use $C_t$.

For one-forms, define the Hodge map by $\ast dt=-dx\wedge dy\wedge dz,\ast dx=-dt\wedge dy\wedge dz,\ast dy=dt\wedge dx\wedge dz,\ast dz=-dt\wedge dx\wedge dy,$ and extend it linearly with smooth coefficients. Let $\psi \in C^{\mathrm{\infty }}(M,\mathbb{R})$, let $\mu \ge 0$ be a fixed real parameter, and take $\mathcal{L}=-\frac{1}{2}d\psi \wedge \ast d\psi -\frac{1}{2}{\mu }^2{\psi }^2\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{.}$ All objects here are mathematical; no physical units or empirical interpretation are assigned. Let $u_1,u_2\in C^{\mathrm{\infty }}(M,\mathbb{R})$ satisfy $(-{\mathrm{\partial }}_t^2+\mathrm{\Delta }-{\mu }^2)u_i=0(i=1,2),$ and define the smooth two-parameter field family ${\psi }_{a,b}:=a{u}_1+b{u}_2$ on $P={\mathbb{R}}^2$. At $(a,b)=(0,0)$ the base solution is $\psi =0$; the commuting parameter variations are ${\delta }_1\psi :={\mathrm{\partial }}_a{\psi }_{a,b}=u_1$ and ${\delta }_2\psi :={\mathrm{\partial }}_b{\psi }_{a,b}=u_2$. The chosen potential and current are $\theta (\psi ;\delta \psi )=-\delta \psi \ast d\psi ,\omega ({\delta }_1,{\delta }_2)={\delta }_2\psi \ast d{\delta }_1\psi -{\delta }_1\psi \ast d{\delta }_2\psi \mathrm{.}$ On the oriented finite smooth singular cycle $C_t$ defined above, the integrated form is ${\mathrm{\Omega }}_{C_t}({\delta }_1,{\delta }_2)={\int }_{[0,1{]}^3}({\delta }_1\psi {\mathrm{\partial }}_t{\delta }_2\psi -{\delta }_2\psi {\mathrm{\partial }}_t{\delta }_1\psi )dxdydz\mathrm{.}$ The integral on the right is over the fundamental cube for the descended three-form. The initial-data pair used here is $(\psi {\mathrm{\mid }}_{{\mathrm{\Sigma }}_t},{\mathrm{\partial }}_t\psi {\mathrm{\mid }}_{{\mathrm{\Sigma }}_t})$. Write $\mathcal{L}={\mathcal{L}}_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}}\mathrm{v}\mathrm{o}\mathrm{l}$; then ${\mathcal{L}}_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}}=\frac{1}{2}({\mathrm{\partial }}_t\psi {)}^2-\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }\psi {\mathrm{\mid }}^2-\frac{1}{2}{\mu }^2{\psi }^2\mathrm{.}$ Define its canonical momentum density by $\pi :=\frac{\mathrm{\partial }{\mathcal{L}}_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}}}{\mathrm{\partial }({\mathrm{\partial }}_t\psi )}={\mathrm{\partial }}_t\psi \mathrm{.}$ Thus the displayed form is the canonical pairing of the variations of $(\psi {\mathrm{\mid }}_{{\mathrm{\Sigma }}_t},\pi {\mathrm{\mid }}_{{\mathrm{\Sigma }}_t})$.

Covariant phase space with boundaries, §3.2

Advanced Lectures on General Relativity, §1.3.1

Some Properties of Noether Charge, §3