= Gauge reduction of the Maxwell Hamiltonian
{title2=$H_T=\tfrac12\int[(\Pi^T)^2+(\nabla\times A^T)^2],d^3x$}
For a spatial potential $\mathbf A$ with $\mathbf E=-\dot{\mathbf A}-\boldsymbol\nabla A_0$, the <Maxwell Lagrangian> gives $\boldsymbol\Pi=-\mathbf E$ and $\Pi^0=0$. Its <Hamiltonian> is $\int[(\boldsymbol\Pi^2+\mathbf B^2)/2+A_0\boldsymbol\nabla\cdot\boldsymbol\Pi],d^3x$. Preserving the temporal momentum <first-class constraint> gives the <Gauss law constraint in gauge theory>. In free space, <Coulomb gauge> and decay at spatial infinity set $A_0=0$ and leave only transverse modes. The reduced <Dirac bracket> uses the projector $\delta_{ij}-k_i k_j/|\mathbf k|^2$, so quantization gives the <transverse equal-time commutator> and the positive Hamiltonian displayed above.
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