Covariant phase space and boundary ambiguities — Examples

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

Let $M={\mathbb{R}}^4$ with coordinates $(X^1,X^2,X^3,X^4)=(t,x,y,z)$, orientation $dt\wedge dx\wedge dy\wedge dz$, and flat metric $-d{t}^2+d{x}^2+d{y}^2+d{z}^2$. For one-forms use $\ast dt=-dx\wedge dy\wedge dz,\ast dx=-dt\wedge dy\wedge dz,\ast dy=dt\wedge dx\wedge dz,\ast dz=-dt\wedge dx\wedge dy\mathrm{.}$ Here the massless wave equation means $(-{\mathrm{\partial }}_t^2+{\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2+{\mathrm{\partial }}_z^2)u=0$. Let $u_1:=t^2+x^2$ and $u_2:=t$, and define the smooth field family ${\psi }_{a,b}:=a{u}_1+b{u}_2$ on $P={\mathbb{R}}^2$. Both $u_1,u_2$ solve the massless wave equation, so $\psi =0$ at $(a,b)=(0,0)$ is a solution and ${\delta }_1\psi :={\mathrm{\partial }}_a{\psi }_{a,b}=u_1$, ${\delta }_2\psi :={\mathrm{\partial }}_b{\psi }_{a,b}=u_2$ are commuting variations solving that same equation. Use the massless scalar Lagrangian $\mathcal{L}=-\frac{1}{2}d\psi \wedge \ast d\psi$, with selected potential $\theta (\psi ;\delta \psi )=-\delta \psi \ast d\psi$ and current $\omega ({\delta }_1,{\delta }_2)={\delta }_2\psi \ast d{\delta }_1\psi -{\delta }_1\psi \ast d{\delta }_2\psi$. For $0<t_1<t_2$ and $L>0$, put $a_0=(t_1,0,0,0)$ and $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,{\mathrm{\ell }}_3,{\mathrm{\ell }}_4)=(t_2-t_1,L,1,1)$. Let $T_4:=\{r\in {\mathbb{R}}^4:r_j\ge 0,{\sum }_{j=1}^4{r}_j\le 1\}$ with orientation $d{r}_1\wedge d{r}_2\wedge d{r}_3\wedge d{r}_4$. For each permutation $\rho \in S_4$, define vertices $v_0^{\rho }:=a_0,v_j^{\rho }:=a_0+{\sum }_{k=1}^j{\mathrm{\ell }}_{\rho (k)}{e}_{\rho (k)}(1\le j\le 4),$ where $e_1,e_2,e_3,e_4$ is the standard basis in the ordered coordinates $(t,x,y,z)$. Define the 24 affine simplex maps and their oriented sum by ${\sigma }_{\rho }(r):=v_0^{\rho }+{\sum }_{j=1}^4{r}_j(v_j^{\rho }-v_0^{\rho }),C:={\sum }_{\rho \in S_4}\mathrm{sgn}(\rho ){\sigma }_{\rho }\mathrm{.}$ Each map has domain orientation $d{r}_1\wedge d{r}_2\wedge d{r}_3\wedge d{r}_4$; the coefficient $\mathrm{sgn}(\rho )$ orients it relative to $dt\wedge dx\wedge dy\wedge dz$.

For $t\in \{t_1,t_2\}$, let $b_t=(t,0,0,0)$ and $(m_1,m_2,m_3)=(L,1,1)$. With $T_3:=\{r\in {\mathbb{R}}^3:r_j\ge 0,{\sum }_{j=1}^3{r}_j\le 1\}$ oriented by $d{r}_1\wedge d{r}_2\wedge d{r}_3$, define, for $\rho \in S_3$, $w_0^{t,\rho }:=b_t,w_j^{t,\rho }:=b_t+{\sum }_{k=1}^j{m}_{\rho (k)}{\widehat{e}}_{\rho (k)}(1\le j\le 3),$ where ${\widehat{e}}_1=e_2$, ${\widehat{e}}_2=e_3$, and ${\widehat{e}}_3=e_4$, and put ${\sigma }_{t,\rho }(r):=w_0^{t,\rho }+{\sum }_{j=1}^3{r}_j(w_j^{t,\rho }-w_0^{t,\rho }),{\mathrm{\Sigma }}_t:={\sum }_{\rho \in S_3}\mathrm{sgn}(\rho ){\sigma }_{t,\rho }\mathrm{.}$ Each ${\mathrm{\Sigma }}_t$ is oriented by $dx\wedge dy\wedge dz$. Define the side chain by $B:=\mathrm{\partial }C-{\mathrm{\Sigma }}_{t_2}+{\mathrm{\Sigma }}_{t_1}$, with the signed simplex-face multiplicities inherited from the boundary of $C$. The coordinate-order boundary calculation shows that $B$ is supported on the six side faces of the box, each with outward-normal-first orientation from $dt\wedge dx\wedge dy\wedge dz$; their oriented volume forms are $+dt\wedge dy\wedge dz$ at $x=0$, $-dt\wedge dy\wedge dz$ at $x=L$, $-dt\wedge dx\wedge dz$ at $y=0$, $+dt\wedge dx\wedge dz$ at $y=1$, $+dt\wedge dx\wedge dy$ at $z=0$, and $-dt\wedge dx\wedge dy$ at $z=1$. Then the bulk current is closed on the box, and by definition of $B$, $\mathrm{\partial }C={\mathrm{\Sigma }}_{t_2}-{\mathrm{\Sigma }}_{t_1}+B$, but ${\mathrm{\Omega }}_{{\mathrm{\Sigma }}_{t_2}}-{\mathrm{\Omega }}_{{\mathrm{\Sigma }}_{t_1}}=-L(t_2^2-t_1^2)\mathrm{\ne }0.$ The oriented side flux is $L(t_2^2-t_1^2)$, so the finite-region balance is ${\mathrm{\Omega }}_{{\mathrm{\Sigma }}_{t_2}}-{\mathrm{\Omega }}_{{\mathrm{\Sigma }}_{t_1}}=-{\int }_B\omega$. The box is compact; the example uses no asymptotic region or falloff premise.

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §§2.2–2.4

The following is the $t$–$x$ projection of the four-dimensional box; the $y,z$ directions are suppressed. It shows the two constant-time cuts and the faces $x=0$ and $x=L$. Boundary orientations are induced from $dt\wedge dx\wedge dy\wedge dz$ by the outward-normal-first convention.