# How does the BRST transformation act on ghost fields?

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

$$$$e^\omega,$$\tag{1}$$

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

$$$$e^\omega\rightarrow e^{\epsilon c}.$$\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:

$$$$\omega\rightarrow \epsilon c + \epsilon~\text{Tr}(c^2b),$$\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?

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.

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.

References:

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

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

• How does the ghost momentum relate to the antighosts? Is it just the momentum conjugate? Green-Schwarz-Witten outright calls them antighosts.
– y9QQ
Feb 11, 2022 at 22:07
• 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?
– y9QQ
Feb 13, 2022 at 2:21
• 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?
– y9QQ
Feb 13, 2022 at 2:43
• 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.
– y9QQ
Feb 13, 2022 at 4:47
• I just realized I said "canonical momentum conjugate." I don't know what that means... I meant "momentum conjugate"
– y9QQ
Feb 13, 2022 at 5:50