Questions tagged [brst]

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Deriving the Topological Descent Equations

I am trying to show that in a cohomological TQFT, given a physical operator $\phi^{(0)}$, one can construct a chain of non-local physical operators. In doing so, I need to show that a certain set of ...
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42 views

Why can we not observe Faddeev-Popov ghost fields? [duplicate]

I know that ghost fields de-couple from the gauge fields in abelian QED. But my question is how does decoupling prove that we cannot observe ghost fields?
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Faddeev-Popov for discrete gauge symmetry?

This excellent question here does not seem to have an acceptable answer. I have had precisely the same question recently. Namely, the way BRST quantization is usually presented relies on some kind of ...
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1answer
95 views

Wigner vs. BRST approach to Klein-Gordon

In Wigner's classification of particles Wigner, E. (1939). On Unitary Representations of the Inhomogeneous Lorentz Group. Annals of Mathematics, 40(1), 149–204. http://www.jstor.org/stable/1968551 the ...
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BRST invariant vertex operator

I'm trying to compute the commutator $\left[Q_{BRST}(z), V^{-1/2}_{v}(w)\right]$, where $V^{-1/2}_{v}(w)$ is the vertex operator corresponding to a massive fermion state. The vertex reads $$ V_{v}^{-\...
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How is cohomology theory used in quantum field theory?

Quantum field theory uses a large amount of mathematics and I was wondering about some applications of cohomology theory in QFT, I understand it has applications in string theory but I was wondering ...
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1answer
46 views

Amplitude of quark$+$antiquark $\rightarrow$ ghost$+$antighost in QCD

Since the BRST charge operator commutes with the Hamiltonian of QCD, a physical state such as $q+\bar q$ should not be allowed to evolve into an unphysical one like $\chi+\bar\chi$, where these two ...
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1answer
65 views

Nilpotency of BRST operator in gravity

I am going through the BRST quantisation in Perturbative quantum gravity and looked at the papers of Nishijima and Ojima. I am confused about the closure of the BRST operator; I.e $s^2=0$, ...
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108 views

Supersymmetry v.s. BRST symmetry: QFT examples

Questions: Can any expert contrast the differences and similarities of Supersymmetry (SUSY) v.s. BRST (global) symmetry? (Question 1) What are the RULES and CRITERIA that having one symmetry implies ...
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264 views

WHY BRST formulation works: Conditions imposed on QFT to find (how many) BRST parameters

question: WHY BRST formulation works? In more details: What are the conditions we need to impose on QFT to find the BRST (global) symmetry? Why can we demand the BRST parameter $\epsilon$ directly ...
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1answer
91 views

BRST as gauge symmetry or global symmetry or the generalization (e.g. in Peskin and Schroeder 16.4)

In Peskin and Schroeder (PS) Chap 16.4, such as after eq.16.45, in p.518, PS said: "local gauge transformation parameter $\alpha$ is proportional to the ghost field and the anti-commuting ...
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1answer
58 views

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
147 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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65 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
86 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 ...
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2answers
260 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
82 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 $$ \...
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105 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 ...
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1answer
113 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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2answers
100 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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145 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.
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135 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}...
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1answer
101 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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105 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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122 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
107 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
101 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
379 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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62 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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212 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
136 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
66 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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56 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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133 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
190 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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1answer
468 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
582 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
176 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
143 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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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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1answer
309 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
94 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
70 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
781 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
256 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
346 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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2answers
298 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
863 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
112 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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395 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 ...