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I have two related questions about some of the steps in the BRST section of Polchinski's String Theory book.

Question 1

Eq. \eqref{eq:4.2.8} of Polchinski's book reads $$ \begin{align} \epsilon \delta\langle f| i\rangle & =i\langle f| \delta_{\mathrm{B}}\left(b_a \delta F^a\right)|i\rangle \\ & =-\epsilon\langle f|\left\{Q_{\mathrm{B}}, b_a \delta F^a\right\}|i\rangle. \end{align}\tag{4.2.8}\label{eq:4.2.8} $$

I can show the first equality to hold true. But for the second equality to be true I would require that $$\delta_B(b_a \delta F^a)=i \epsilon \{Q_B,b_a \delta F^a\} \tag{1}\label{eq:1}$$ where $Q_B$ is the BRST conserved current.

My first question is: why does eq. \eqref{eq:1} here hold true?

Question 2

Polchinski says:

In order to move around in the space of gauge choices, the BRST charge must remain conserved. Thus it must commute with the change in the Hamiltonian.

That statement I am fine with, but he then writes down $$0=[Q_B,\{Q_B,b_a \delta F^a\}]. \tag{4.2.12}\label{eq:4.2.12}$$ I suppose that implies that the change in Hamiltonian is somehow related to the anti-commutator $\{Q_B,b_a \delta F^a\}$, but I don't see how.

My second question is: why does eq. \eqref{eq:4.2.12} hold true?

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Treb Neb
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1 Answers1

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  1. In the Hamiltonian formulation the infinitesimal BRST transformation $$-\delta_B(\cdot)~=~\epsilon \{Q_B,\cdot\}_{SPB}~=~\epsilon\frac{1}{i\hbar}[Q_B,\cdot]_{SC}$$ is a Hamiltonian vector field generated by the BRST charge $Q_B$. [Here $SPB$=super Poisson bracket and $SC$=super commutator.]

  2. Yes, in the Lagrangian action $$S_1+S_2+S_3\tag{4.2.3}$$ [the infinitesimal change of] the gauge-fixing term$^1$ $$S_2+S_3~=~\frac{1}{i\epsilon}\delta_B\Psi,\qquad \Psi~:=~b_AF^A, \tag{4.2.7}$$ [which is equal to the anticommutator in eq. (4.2.12)] is interpreted as [the infinitesimal change of] the gauge-fixing term of [minus] the Hamiltonian$^1$ $$H_{\Psi}~:=~H_B+\frac{1}{i}\{Q_B,\Psi\}_{SPB}$$ in the Hamiltonian action $S_H$. Eq. (4.2.12) states that the BRST charge $Q_B$ should $SPB/SC$ commute with [infinitesimal changes to] the Hamiltonian $H_{\Psi}$, so that it remains conserved.

References:

  1. J. Polchinski, String Theory Vol. 1, 1998; Section 4.2.

$^1$ Here DeWitt's condensed notation is implicitly used. It is also used that the form of the gauge-fixing term is preserved by the Legendre transformation, i.e. the process of introducing momentum variables.

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