Questions tagged [brst]

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Cohomology of the Koszul-Tate complex for an irreducible symmetry vanishes in degree $-2$

There must be something really obvious that I am missing here but any help is appreciated. Suppose I have a theory with some action $S$ on some fields $\phi$ such that any function vanishing on-shell ...
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1answer
93 views

Batalin-Vilkovisky (BV) form of the Chern-Simons Action

As seen in Section 4 of Chapter 5 of Costello, K. "Renormalization and Effective Field Theory", or in section 5.2 $L_\infty$-Algebras of Classical Field Theories and the Batalin-Vilkovisky ...
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0answers
46 views

Resources on BRST and BV quantisation for local quantum field theories

This is a reference request, to ideally a textbook, monograph, set of lecture notes or lecture videos, on the topics of BRST quantisation and the Lagrangian BV formalism. My constraints are as follows:...
2
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1answer
50 views

Batalin-Vilkovisky quantization

Batalin-Vilkovisky (BV) quantization is way of quantizing a theory, which is apparently more powerful than BRST quantization. It has been used, for example, for string field theory, in the closed ...
5
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2answers
175 views

Questions about BRST formalism and BV formalism

This is from Pierre J. Clavier and Viet Dang Nguyen's paper Batalin-Vilkovisky formalism as a theory of integration for polyvectors. In section 2.3, it states: A symmetry is said to be open when it ...
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1answer
62 views

Eliminating residual gauge in BRST quantization of Yang-Mills theory

I would like to know if there is a procedure to completely fix a gauge, which I believe we must do in order to make sense of the path integral? In chapter 74 Sredniki introduces the Lagrangian $$ \...
2
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3answers
84 views

Gauge ghosts & unphysical states in gauge theory

I have a general question about a statement from Wikipedia about ghost states as occuring in gauge theory: "In the terminology of quantum field theory, a ghost, ghost field, or gauge ghost is an ...
2
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1answer
83 views

BRST symmetry, gauge invariance and longitudinal gauge bosons

While quantizing a non-Abelian gauge theory covariantly, we first demand that the BRST charge acting on the physical states of the Hilbert space must be zero. However such physical states still have ...
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0answers
19 views

During BRST quantization of non Abelian gauge fields, is it necessary for the quantized particles to be transverse? [duplicate]

While covariantly quantizing non-Abelian gauge theories, we first impose the condition that the action of the BRST charge on physical states must yield zero. Then we further demand that such states ...
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2answers
76 views

What happens to the symmetry after gauge fixing?

Given a theory with gauge symmetry. After gauge fixing where does the symmetry go? Does the gauge symmetry turn into a global symmetry? For example there is a way to quantize fields theory with BRST ...
2
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2answers
101 views

One question about BRST symmetry in reading Srednicki’s book: Why should the BRST charge $Q_B$ be nilpotent?

In p.453, Srednicki claims that since the BRST transformation of a BRST transformation is zero, $Q_B$, the BRST charge, must be nilpotent: $$Q_{B}^{2}=0.\tag{74.32}$$ I don't know why.
3
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1answer
107 views

BRST quantization: Explicit computation request

Following Green, Schwarz and Witten's book on Superstrings, the BRST charge is given by $$Q_B = c^i K_i-\frac{1}{2}f_{ij}^{~~~k}c^ic^jb_k\tag{3.2.4}$$ with $$[K_i, K_j] = f_{ij}^{~~~k}K_k,\tag{3.2.1}...
3
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1answer
93 views

Supercurrent of the $bc$-$\beta\gamma$ SCFT

In Polchinksi's Sec. 10.1, the $bc$-$\beta\gamma$ SCFT is introduced with action $$S_{BC} = \frac{1}{2\pi} \int d^2z (b \bar \partial c + \beta \bar \partial \gamma)$$ and supercurrent $$T_F = -\...
2
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1answer
95 views

Reference for proof of renormalizability

I have been trying to truly understand the renormalizability of quantum (i.e., without anomalies) gauge theories (after which I will focus on the case with spontaneous symmetry breaking). The problem ...
2
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0answers
85 views

BRST Charge in QED

The pure gauge QED Lagrangian density, including the ghost fields, is \begin{align*} \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2\xi}(\partial^\mu A_\mu)^2+\partial^\mu\overline{c}\...
2
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1answer
86 views

Questions about BRST symmetry [closed]

For a course about the standard model, I am writing a paper on BRST symmetry. For this I am mainly following the material developed in chapter 16.4 of Peskin and Schroeder. I am mostly done, however ...
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1answer
90 views

Dependence of BRST Quantization on the Choice of Gauge-Fixing Function

There is a point which confuses me about BRST procedure. One shows that, if we define physical states as the ones that are annihilated by BRST charge $Q$, the scattering amplitudes don't depend on ...
3
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1answer
316 views

Faddeev-Popov determinant and topology of the worldline

I am studying the path integral quantization of relativistic particles, using the BRST quantization method. I have to compute the integral \begin{equation} Z\sim \int Dx \det(\partial_\tau)e^{-\int_0^...
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53 views

Do longitudinal and scalar have anything to do with Faddeev-Popov ghosts?

In his this book, Hatfield calls ghosts the negative states appearing in the covariant (Gupta-Bleuler) quantization prescription of the electromagnetic field (page 89). When discussing Yang-Mills ...
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150 views

One-loop correction to triple gluon vertex in QCD

I'm trying to calculate a one loop correction to the 3 gluon vertex, which is given by a circular correction, and another that's due to the 4 gluon vertex. However I'm unsure how the ghost fields ...
2
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1answer
128 views

Equivalence between ghosts?

Ok. I'm trying to get the terminology right about the term ghost in physics. Is there any equivalence between these terms? Faddeev-Popov ghosts Paul-Villars ghosts Landau ghost The vanishing ...
3
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1answer
58 views

Question on BRST closed vectors which are also co-closed

I'm studying the BRST quantization formalism from this reference. I have a question though, about page 44. The author introduces a co-BRST operator on the extended Hilbert space (which also include ...
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0answers
50 views

How is BRST symmetry related to local integrals of motion?

I'm hoping someone can confirm or check my reasoning below: In this wiki, they describe caos in a classical system as the spontaneous symmetry breaking of a BRST. In this stackexchange, they clarify ...
2
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0answers
126 views

BRST quantization and physical states (Polchinski)

I'm reading section 4.2 of Polchinski's string theory vol. 1. In the page 131, there are sentences "In fact, only states $\mid k,\downarrow \rangle$ satisfying the additional condition $ b\mid \...
3
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1answer
163 views

What is the Grassmann parameter $\epsilon$ in the BRST transformation?

Whenever I learn about anything involving fermions and the path integral, I get confused about Grassmann numbers. I'm currently following Weigand's notes, specifically the section on BRST symmetry. ...
6
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1answer
382 views

BRST cohomology and Gupta-Bleuler$.$

Let $Q$ be the BRST operator. Define physical state as those in $\mathrm{ker}\,Q$ (modulo its image): $$ Q|\psi_\mathrm{physical}\rangle\equiv 0\tag1 $$ It is often claimed1 that this condition ...
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1answer
495 views

What is the physical meaning of nilpotent operator?

I would like to know that what does the nilpotent physically represents? For example, in BRST quantization of point particle, BRST charge is nilpotent means square of this operator gives zero (...
2
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1answer
148 views

What is the major difference between Dirac and BRST quantization of point particle?

I have derived the action for the bosonic point particle and now I want to quantize it but there are two formalism: one is Dirac and the other one is BRST. I want to know what is the major difference ...
2
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1answer
108 views

Faddeev-Popov Ghosts in the canonical formalism

In the Lorenz gauge in electrodynamics, the timelike and longitudinal components can be eliminated by prescribing the Gupta-Bleuler condition $\partial^{\mu}A_{\mu}|\Psi)$ on physical states. This ...
4
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2answers
190 views

TQFT- Adding a $Q$-exact term which is equal to the action itself

It is known that Witten-type topological quantum field theories (TQFT) are invariant when $Q$-exact terms are added to the classical action, where $Q$ is the BRST charge. But for these theories, the ...
5
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1answer
254 views

Can we choose other than Gaussian integral for Faddeev-Popov gauge fixing?

for $U(1)$ field $A_\mu$ and its longitudinal gauge component $\partial_\mu \alpha(x)$, Faddeev-Popov gauge fixing written in Peskin (eq.9.56) is: $$ N(\xi)\int \mathcal{D}\omega\hspace{0.1cm}\text{...
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1answer
84 views

Is the space on which the BRST $Q$ operator acts a Hilbert space?

When I looked at the BRST symmetry in Yang-Mills-Theories I was puzzled by the statement: Suppose we go back to canonical quantisation with a Hilbert space $\mathcal{H}$. The BRST symmetry leads to ...
2
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1answer
66 views

Orthogonality between the BRST-Closed Subspace and the non-BRST-Closed One

Consider the BRST quantisation of free Maxwell theory, in one of the averaged Lorenz gauges, $$S = \int - \frac{1}{4} F^2 + \frac{1}{2} (\partial\cdot A)^2 + i \bar{C} \partial^2 C.$$ Calling the ...
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2answers
689 views

Geometric Interpretation of BRST Symmetry

BRST quantization (and BRST symmetry in general), at least in this point in my understanding of them, seem rather arbitrary and slightly miraculous. However, the cohomological nature of the BRST ...
5
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1answer
212 views

How does Faddeev-Popov work for higher-spin fields? (or does it?)

Take for example a spin $2$ field $h_{\mu\nu}$ and some gauge-invariant Lagrangian. Does the Faddeev-Popov trick work here? by work I mean: does it lead to a consistent and unitary theory? is the ...
2
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1answer
311 views

BRST Quantization of the Bosonic String: Nilpotence of BRST transformation (Polchinski)

Currently I am studying string theory and I encountered a bunch of interrelated problems in the context of BRST quantization which I can't solve for myself although I tried hard for some days. My ...
2
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1answer
230 views

BRST Quantization of the Point Particle: Sign of Structure Functions (Polchinski)

Currently I am studying string theory and I encountered a bunch of interrelated problems in the context of BRST quantization which I can't solve for myself although I tried hard for some days. The ...
6
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2answers
741 views

Why Faddeev-Popov ghost cannot exist in external line?

I was studying the path integral quantization of non-abelian gauge field. After the path integral quantization, the action becomes $$\mathcal{L}=-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}-\frac{1}{2\zeta}(\...
2
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1answer
107 views

How is the state $|a_0 a_i\rangle$ physical?

For a state $|\psi\rangle$ to be physical we require that: $$\langle\psi|a_0^\dagger a_0|\psi\rangle = \langle\psi|a_i^\dagger a_i|\psi\rangle$$ It is always said that physical state must contain ...
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1answer
352 views

Negative norm states

If we have negative norm states such that $[a,a^\dagger]=-1$ how do we treat the normalization of two particle states ? Suppose: $|aa\rangle = N a^\dagger a^\dagger |0\rangle$, after some work we ...
2
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2answers
92 views

Diagonalizing Faddeev-Popov Lagrangian $U(1)$

How can we diagonalize the U(1) Faddeev-Popov Lagrangian in a consistent manner. I can't seem to find any papers on this but I can't believe that they don't exist. Any pointers would be greatly ...
2
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0answers
165 views

Weaker Gupta-Bleuler conditon $<\psi|(\partial_\mu A^\mu)^2|\psi> = 0$

Some context In the Gupta Bleuler quantization procedure for gauge fields we introduce the gauge fixing term: $$S_{GB} = \int dV-\frac{1}{2}(\partial_\mu A^\mu)^2$$ to the Lagrangian. After ...
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2answers
208 views

Why does it make sense to talk about the first-quantized BRST formulation of a relativistic point particle?

My question is about the BRST quantization of a point particle in Polchinski, Vol.1, Section 4.2. The BRST quantization starts from the effective action for the gauge-fixed path-integral. But for the ...
4
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2answers
2k views

How do Faddeev-Popov (FP) ghosts help path integrals?

How does the inclusion of Faddeev-Popov ghosts in a path integral help to fix the problem of over counting due to gauge symmetries? So, after exponentiating the determinant for the inclusion of ...
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0answers
197 views

Anticommutator BRST charge and $c$-ghost mode

My goal is to compute the anticommutator $\{Q_B, c_m\}$ where $Q_B$ is the BRST charge in string theory and $c_m$ is the $m$th mode of the $c$ ghost field $$ c(z) = \sum_m \frac{c_m}{z^{m-1}} $$ (...
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0answers
124 views

Covariant quantization of an interacting relativistic particle

A method of covariant quantization for a free relativistic particle appears in the first part of some introductory string theory texts (Tong, Zwiebach,...). None of them (as far as I hae seen) give an ...
3
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1answer
406 views

Why do Faddeev-Popov ghost anti-commute?

I'm trying to understand why the Faddeev-Popov ghost that appear in the quantization of non-abelian gauge theories are anti-commuting fields. I've seen a number of books (chapters), lecture notes and ...
3
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2answers
604 views

Where is the BRST symmetry?

When quantizing YM we start from the gauge fixed path integral (to remove redundancy of integrating over Gauge symmetric configurations) $$\int \mathcal{D}A \delta(G(A)) \text{det} \Delta_{FP}e^{i\int ...
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1answer
1k views

What is the relationship between BRST symmetry and gauge symmetry?

As far as i know the BRST symmetry is an infinitesimal (and expanded) version of gauge symmetry. Recently I read the following: "when QFT was reformulated in fiber bundle language for application to ...
3
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1answer
472 views

BRST quantization (Green, Schwarz, Witten)

In Green, Schwarz, Witten Volume 1, section 3.2, BRST quantization is introduced in a general way. A Lie algebra $G$ is defined with elements $$ [K_i,K_j] = f_{ij}{}^k K_k \tag{3.2.1}$$ where $f_{ij}{}...