Questions tagged [brst]

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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 ...
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1answer
66 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
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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
66 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 ...
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2answers
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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.
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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}...
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1answer
85 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 = -\...
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1answer
87 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 ...
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69 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}\...
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1answer
72 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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49 views

References to understand BRST Quantization

I'm looking for good, rigorous references that discuss BRST quantization in relation to how it leads to dealing with anomalies and ghost fields. I'm looking at high level references (i.e., assume ...
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1answer
78 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 ...
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1answer
282 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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48 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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129 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 ...
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1answer
117 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 ...
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1answer
57 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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48 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 ...
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110 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 \...
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1answer
142 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. ...
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322 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
450 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 (...
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1answer
131 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 ...
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1answer
101 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 ...
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178 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 ...
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225 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
77 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 ...
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1answer
64 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
630 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 ...
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1answer
182 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 ...
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1answer
268 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 ...
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1answer
207 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 ...
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2answers
704 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}(\...
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1answer
106 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
331 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 ...
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2answers
90 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 ...
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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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1answer
159 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 ...
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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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184 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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112 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 ...
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1answer
378 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 ...
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2answers
551 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 ...
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1answer
432 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}{}...
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1answer
165 views

Supertrace of holonomy of commutator

On page 47 of Surface operators in four-dimensional topological gauge theory and Langlands duality by Kapustin et al., the following expression is given \begin{equation} \delta\mathcal{N}=d(\omega_\...
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1answer
174 views

Problems with covariant action of the superstring

I was reading Kiritsis notes (http://arxiv.org/abs/hep-th/9709062), at page 105/106 (equation 10.1), where he has a covariant action of the superstring including the gravitino. I have problems showing ...
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1answer
302 views

Modified gauge fixing condition in Faddeev-Popov approach

Which gauge fixing conditions are allowed to choose for this approach? For example (following the steps of Peskin in 9.4) I can choose a "modified" lorenz gauge ( for a abelian theory): $$ (\...
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1answer
582 views

BRST quantization and norm

States with definite ghost number have zero norm (since ghost number is anti-hermitian and has real eigenvalues). E.G. when quantizing relativistic point particle, physical spectrum turns out to ...
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2answers
2k views

What is a ghost number?

I am currently studying CFT chapter of Becker,Becker,Schwarz and am trying to understand what the ghost number is in BRST Quantization. From what I gather BRST Quantization is used to add an extra ...