Abelian local jet BRST differential

Use the smooth off-shell coordinate model $\mathbb{M}={\mathbb{R}}^4$, with $x^0=c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}t$ in SI coordinates, metric $\eta =\mathrm{diag}(-1,1,1,1)$, increasing $x^0$ as the future direction, and standard orientation. The scalar time-ordering convention sets $c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}=1$, so $x^0$ is the length coordinate in that convention as well. Here off shell means that no field equation is imposed. Take smooth real components $a_{\mu }\in C^{\mathrm{\infty }}(\mathbb{M};\mathbb{R})$ and a smooth commuting matter field $\psi \in C^{\mathrm{\infty }}(\mathbb{M};{\mathbb{C}}^4)$. Use the normalized real potential $a=a_{\mu }d{x}^{\mu }$ and dimensionless coupling $g=q\sqrt{{\mu }_0{c}_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}\mathrm{/}\mathrm{\hslash }}$ of the charged-gauge-data convention; $a_{\mu }$ has units ${\mathrm{m}}^{-1}$, $g$ is dimensionless, and $\psi$ has units ${\mathrm{m}}^{-3\mathrm{/}2}$. The exact supplier convention is $a_{\mu }^{\mathrm{\prime }}=a_{\mu }+{\mathrm{\partial }}_{\mu }\alpha$, ${\psi }^{\mathrm{\prime }}=e^{ig\alpha }\psi$, for a smooth dimensionless real infinitesimal gauge parameter $\alpha$.

Let $\mathcal{P}$ be the free graded-commutative $\mathbb{C}$-algebra over $C^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$ generated by the finite jets of $a_{\mu }$, $\psi$, an odd ghost $c_{\mathrm{g}\mathrm{h}}$, an odd antighost $\bar{c}$, and an even auxiliary variable $B$. Every element of $\mathcal{P}$ is a polynomial in finitely many jet coordinates. The fields $a_{\mu }$ and $\psi$ have Grassmann parity $0$ and ghost number $0$; $c_{\mathrm{g}\mathrm{h}}$ has parity $1$ and ghost number $1$; $\bar{c}$ has parity $1$ and ghost number $-1$; and $B$ has parity $0$ and ghost number $0$. The ghost and nonminimal variables are formal algebraic generators; no state-space or positivity claim is made. The generators $c_{\mathrm{g}\mathrm{h}},\bar{c},B$ carry no physical unit assignment, and the differential below is dimensionless. The subscript distinguishes the ghost from the speed of light $c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}$.

Let $s$ vanish on the coefficient algebra, so $sf=0$ for every $f\in C^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$. Thus $s$ is $C^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$-linear.

Define the odd derivation $s:\mathcal{P}\to \mathcal{P}$ of ghost number $1$ by

Extend $s$ by the graded Leibniz rule $s(XY)=(sX)Y+(-1{)}^{\mathrm{\mid }X\mathrm{\mid }}X(sY)$ for homogeneous $X$, where $\mathrm{\mid }X\mathrm{\mid }$ is Grassmann parity, and by $s({\mathrm{\partial }}^{\beta }X)={\mathrm{\partial }}^{\beta }(sX)$ for every multi-index $\beta$. This is an off-shell local algebraic differential for the Abelian gauge transformations; it is not a BRST charge on a Hilbert space and it does not assert existence of a gauge-invariant quantum theory.

Charged gauge data and SI-natural conventions