I understand the general idea behind constructing the BRST symmetry: take a generic gauge transformation

$$\begin{equation} e^\omega, \end{equation}\tag{1}$$

where $\omega$ is Lie-algebra valued, and replace $\omega$ with some Grassmann number $\epsilon$ times the ghost field $c$:

$$\begin{equation} e^\omega\rightarrow e^{\epsilon c}. \end{equation}\tag{2}$$

Of course this doesn't tell you how the ghosts or Nakanishi-Lautrup fields ($B$) change. I've seen an additional term added, which is something like:

$$\begin{equation} \omega\rightarrow \epsilon c + \epsilon~\text{Tr}(c^2b), \end{equation}\tag{3}$$

but I can't quite work out how this term comes into play. I assume that: one, the extra term is fine because it goes as a BRST variation of the ghosts $\delta c\sim c^2$ and must be exact, and two, that the extra term somehow imposes conditions on the transformations for the antighost ($b$) and $B$ fields. I could try brute-forcing this calculation, but I can't really figure out how the Slavnov operator (or really gauge transformations in general) act on ghosts, antighosts or auxiliary fields. Specifically, gauge transformations act by conjugation (+ a derivative) on gauge bosons, since they are in the adjoint representation. Of course, so are $c$, $b$, and $B$, but acting naively on $c$ like $$c\rightarrow c'=g(c-Dg)g^{-1}\tag{4}$$ seems to only produce nonsense. Can someone help clarify this behavior?


1 Answer 1


TL;DR: The BRST transformation rules for the ghost fields follow from requiring that the BRST/Slavnov operator squares to zero (= is nilpotent in physics jargon).

More details: When studying the BRST formalism, one should be aware that there exists$^1$ at least 2 versions relevant to OP's question:

  1. The Hamiltonian Batalin–Fradkin–Vilkovisky (BFV) formalism. Assuming no second-class constraints, and that the first-class constraints $G_a(q,p)$ are irreducible,

    • the minimal field multiplet is the original position variables $q^i$ and the Faddeev-Popov (FP) ghost ${\cal C}^a$,

    • while the non-minimal field multiplet is the Lagrange multiplier $\lambda^a$ and the FP antighost $\bar{\cal C}_a$.

    On top of that there are the corresponding momenta, namely $p_i$, the FP ghost momenta $\bar{\cal P}_a$; the Nakanishi-Lautrup (NL) field $B_a$, and the FP antighost momenta ${\cal P}^a$, respectively.

    As usual, the above canonical pairs of operators can be realized via a suitable Schrödinger representation.

    Note that the bar/overline in $\bar{\cal C}_a$ and $\bar{\cal P}_a$ does not denote complex nor Hermitian conjugation. Rather, it signifies a negative ghost number.

    Also note that that the FP ghost ${\cal C}^a$ and antighost $\bar{\cal C}_a$ are not a canonical pair, i.e. they super-commute.

    The BRST charge operator $$\begin{align} Q~=~&Q_{\min} + Q_{\text{non-min}},\cr Q_{\min}~=~&G_a {\cal C}^a +\frac{1}{2}\bar{\cal P}_cf^c{}_{ab}{\cal C}^b{\cal C}^a +\ldots, \cr Q_{\text{non-min}}~=~&B_a{\cal P}^a, \end{align}$$ is (among other things) a Grassmann-odd operator of ghost number +1 that squares to zero $$ Q^2~=~0,$$ cf. Ref. 1. The above ellipses $\ldots$ refers to higher-order terms, where each term contains one more ghost ${\cal C}^a$ than ghost momenta $\bar{\cal P}_a$. Such term may be needed if the structure functions $f^c{}_{ab}(q,p)$ are not constant.

    See also e.g. this related Phys.SE post.

    The BRST transformations $$\delta~=~[Q,\cdot]$$ are given by the super-commutator. So e.g. the ghost transforms as $$\delta {\cal C}^c~=~[Q,{\cal C}^c]~=~\frac{1}{2}f^c{}_{ab}{\cal C}^b{\cal C}^a+\ldots,$$ cf. OP's title question.

  2. The same formalism where half of the variables (mostly momenta) have been integrated out. The remaining variables are the original position variables $q^i$, the FP ghost ${\cal C}^a$, the FP antighost $\bar{\cal C}_a$, and the NL auxiliary field $B_a$. This is the version usually found in textbooks.

    The field $b$ found in the string theory literature is either the antighost $\bar{\cal C}$ or the ghost momentum $\bar{\cal P}$, depending on context.


  1. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994; Chapter 9.


$^1$ There is also a Lagrangian BV formalism.

  • $\begingroup$ How does the ghost momentum relate to the antighosts? Is it just the momentum conjugate? Green-Schwarz-Witten outright calls them antighosts. $\endgroup$
    – y9QQ
    Feb 11 at 22:07
  • $\begingroup$ But what $\textit{is}$ the ghost momentum? And how should one impose a standard gauge transformation on ghosts? I would imagine it should have a similar rule to the gauge bosons, because they are in the adjoint representation, but I can't just add a covariant derivative, because that transforms as a vector. Is it reasonable to look for an expression like $c\rightarrow c+\delta c$ under a gauge transformation? $\endgroup$
    – y9QQ
    Feb 13 at 2:21
  • $\begingroup$ I'm led to belive that it's just the momentum conjugate. The explicit representation of the slavnov operator given in Weinberg Vol. II contains a term $-\frac{1}{2}{f^a}_{bc}c^bc^c\frac{\delta}{\delta c^a}.$ That derivative on the end looks awfully similar to the definition of the canonical momentum conjugate, and since it takes the form of a tangent vector, I imagine it corresponds to an antighost, so that ghosts (one forms) and antighosts are dual. Am I totally off-base here? $\endgroup$
    – y9QQ
    Feb 13 at 2:43
  • $\begingroup$ Page 39, eq. 15.8.9: $$ s=\omega^A\delta_A\phi^r\frac{\delta_L}{\delta\phi^r}-\frac{1}{2}\omega^B\omega^C{f^A}_{BC}\frac{\delta_L}{\delta\omega^A}-h^A\frac{\delta_L}{\delta\omega^{*A}}, $$ where the indices represent spacetime, Lorentz, color, etc. indices collectively. $\endgroup$
    – y9QQ
    Feb 13 at 4:47
  • $\begingroup$ I just realized I said "canonical momentum conjugate." I don't know what that means... I meant "momentum conjugate" $\endgroup$
    – y9QQ
    Feb 13 at 5:50

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