Finite cochain Maxwell phase space

Fix nonnegative integers $n_0,n_1,n_2,n_3$ and ordered coordinate bases, and put $C^k:={\mathbb{R}}^{n_k}$ for $k=0,1,2,3$. The dual space $(C^k{)}^{\ast }$ is the vector space of linear functionals $C^k\to \mathbb{R}$, with the dual basis to the chosen coordinate basis. Let $D_0\in {\mathbb{R}}^{n_1\times n_0}$, $D_1\in {\mathbb{R}}^{n_2\times n_1}$, and $D_2\in {\mathbb{R}}^{n_3\times n_2}$ be linear maps satisfying $D_1{D}_0=0$ and $D_2{D}_1=0$. This data is a finite cochain complex. The transpose $D_k^{\mathsf{T}}$ denotes the representing matrix of the dual linear map in the dual bases. Let $M_1\in {\mathbb{R}}^{n_1\times n_1}$ and $M_2\in {\mathbb{R}}^{n_2\times n_2}$ be symmetric positive-definite matrices: $v^{\mathsf{T}}{M}_{k}v>0$ for each nonzero $v\in C^k$. They represent linear maps $M_1:C^1\to (C^1{)}^{\ast }$ and $M_2:C^2\to (C^2{)}^{\ast }$ in the chosen bases; in positive dimensions their inverses exist by the finite-dimensional positive-matrix result cited below, and the unique map on a zero-dimensional space is its own inverse.

The finite state space is $\mathcal{X}:=C^2\oplus (C^1{)}^{\ast }$, with state $(b,d)$, where $b$ is a column of $C^2$ coordinates and $d$ is a column in the dual basis of $(C^1{)}^{\ast }$. In this item, “phase space” means this finite-dimensional state space only; no symplectic form is asserted. On an open interval $J\subseteq \mathbb{R}$, a semidiscrete trajectory is a continuously differentiable map $t\mapsto (b(t),d(t))\in \mathcal{X}$ satisfying

$\dot{b}(t)=-D_1{M}_1^{-1}d(t),\dot{d}(t)=D_1^{\mathsf{T}}{M}_{2}b(t)\mathrm{.}$

Its two discrete constraints are $D_{2}b(t)=0$ in $C^3$ and $D_0^{\mathsf{T}}d(t)=0$ in $(C^0{)}^{\ast }$. The associated quadratic energy is

$\mathcal{E}(b,d):=\frac{1}{2}(b^{\mathsf{T}}{M}_{2}b+d^{\mathsf{T}}{M}_1^{-1}d)\mathrm{.}$

All objects here are finite-dimensional real linear-algebra data. The term “Maxwell” names this algebraic model and does not identify it with a continuum Maxwell or Einstein–Maxwell solution.

Closed Maxwell current and gauge boundary term

Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity

For a finite oriented cell complex, signed incidence matrices give one source of such maps: an entry is $+1$ or $-1$ according as the oriented lower cell occurs in the oriented boundary with agreeing or opposite orientation, and is $0$ when it does not occur. In a simplicial realization, the boundary of an ordered simplex is $\mathrm{\partial }[v_0,\dots ,v_k]={\sum }_{i=0}^k(-1{)}^i[v_0,\dots ,{\widehat{v}}_i,\dots ,v_k]$. Each codimension-two face obtained by deleting vertices $i<j$ occurs in ${\mathrm{\partial }}^2$ twice, with coefficients $(-1{)}^i(-1{)}^{j-1}$ and $(-1{)}^j(-1{)}^i$, which are opposites. Hence ${\mathrm{\partial }}^2=0$; taking transposes of the boundary matrices gives the two zero-composition identities for the cochain maps. This definition also admits any real matrices satisfying the displayed identities. Positive-definite mass maps are additional data and are not determined by incidence alone.

The cochain sequence is displayed below. The arrows are exactly $D_0,D_1,D_2$ in the ordered coordinate bases; the zero-composition identities are hypotheses of the definition and are not proved by the diagram.

C^0 \arrow[r,"D_0"] & C^1 \arrow[r,"D_1"] & C^2 \arrow[r,"D_2"] & C^3

The incidence identities support exact finite constraint calculations. They do not supply a family of meshes, commuting projections, approximation estimates, uniform energy or CFL bounds, or a continuum-limit theorem. The source’s cochain-sampling and Whitney-form results have those additional geometric and approximation scopes; none is claimed here.