Questions tagged [brst]
The brst tag has no usage guidance.
120
questions
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Vanishing of path integral over internal d.o.f. of test particle in $SU(N)$ gauge theory
In Ch-2 (Yang-Mills theory) of David Tong’s notes on gauge theory. Tong writes an action $$S_w=\int d\tau \hspace{2pt}i w^{\dagger} \frac{dw}{d\tau}+\lambda(w^{\dagger}w-k)+w^{\dagger}A(x^{\mu})w\tag{...
2
votes
1
answer
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How does the BRST transformation act on ghost fields?
I understand the general idea behind constructing the BRST symmetry: take a generic gauge transformation
$$\begin{equation}
e^\omega,
\end{equation}\tag{1}$$
where $\omega$ is Lie-algebra valued, and ...
5
votes
1
answer
164
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Intuition for Hilbert space of a quantized gauge theory
In the standard explanation, the physical Hilbert space of a quantized gauge theory (such as QCD) is given by the cohomology of the BRST charge acting on some larger, unphysical Hilbert space.
More ...
2
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1
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73
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Does the Slavnov-Taylor identity still hold for scalar Yang-Mills?
I want to renormalize the minimally-coupled scalar Yang-Mills theory:
$$\mathcal{L}_{YM\phi}=(D_\mu\phi)^\dagger(D^\mu\phi)-\frac{1}{4}F_{\mu\nu}^a{F^{\mu\nu}}^a-\frac{1}{2\xi}(\partial_\mu {A^\mu}^a)^...
2
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0
answers
79
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What does cohomology of $Q_B$ mean in BRST quantization in Polchinski?
While proving no-ghost theorem ($4.4$ Polchinski) the term cohomology of $Q_B$ is used quite a lot of time. From what I understand this has to be a set since "cohomology of $Q_B$" is ...
1
vote
0
answers
14
views
Idea/Intuition behind using commutator of light cone number operator and BRST generator
In ch-$4$ of Polchinksi while proving no-ghost theorem we are introduced to number operator of light cone oscillators
$$N^{lc}=\sum_m{\frac{1}{m}:\alpha^+_{-m}\alpha^-_m:}\tag{4.4.6}$$
$$\alpha^{\pm}...
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0
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23
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How does commutator of $Q_B$ with change in $H$ results in moving around in gauge space
In ch-$4$ Polchinksi states following:
In order to move around in space of gauge choices, the BRST charge must remain conserved. Thus it must commute with change in the Hamiltonian.
Commutation with ...
2
votes
1
answer
45
views
Counting sign definiteness in BRST cohomology of string
In Polchinksi ch-$4$ following manipulation is done:
$$|\psi_1\rangle=(e\cdot\alpha_{-1}+\beta b_{-1}+\gamma c_{-1})|0,\textbf{k}\rangle .\tag{4.3.25}$$
$$\langle\psi_1|\psi_1\rangle=\Big(e^*\cdot e+(\...
2
votes
1
answer
67
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BRST variation of $\delta_{\alpha}F^A$ in $S_3$ in BRST section of Polchinski
The Faddeev-Popov action reads
$$S_3=b_Ac^{\alpha}\delta_{\alpha}F^A(\phi).\tag{4.2.5}$$
I want to find the BRST variation of the gauge variation of $F^A$ in $S_3$ i.e. $$b_Ac^{\alpha}\color{red}{\...
2
votes
0
answers
66
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Normal ordering constant value in String Theory and Old Covariant Quantization
Suppose you are approaching the quantization of the closed bosonic string for the first time (so we are in the so called Old Covariant Quantization (OCQ), and by now we know nothing about Lightcone ...
1
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0
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70
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Existence of ground states in $bc$ CFT
I am reading Polchinski's Vol. 1 on String Theory, and I have some basic doubts on how he introduces the $bc$ conformal field theory (see section 2.7, page 61).
He basically starts from the ...
2
votes
0
answers
59
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Current state of the Gribov-Zwanziger formalism and the softly broken BRST symmetry
I hope to get a little bit more clarification about the topics. My confusion arises from the fact that some authors (Lavrov et al) state that a gauge theory with softly broken BRST symmetry is ...
3
votes
1
answer
140
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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 ...
1
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0
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53
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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 ...
2
votes
1
answer
111
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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 ...
3
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111
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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}^{-\...
5
votes
3
answers
345
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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 ...
0
votes
1
answer
68
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 ...
2
votes
1
answer
100
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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 $s^2=0$, ...
3
votes
1
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185
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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 ...
5
votes
3
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330
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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 ...
3
votes
1
answer
175
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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 ...
3
votes
1
answer
64
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 ...
2
votes
1
answer
196
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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 ...
5
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0
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125
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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
votes
1
answer
187
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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
votes
2
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398
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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 ...
1
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1
answer
137
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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
$$
\...
2
votes
3
answers
173
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Gauge ghosts & unphysical states in gauge theory
I have a general question about a statement from Wikipedia about ghost states as occuring in gauge theory:
"In the terminology of quantum field theory, a ghost, ghost field, or gauge ghost is an ...
3
votes
1
answer
164
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BRST symmetry, gauge invariance and longitudinal gauge bosons
While quantizing a non-Abelian gauge theory covariantly, we first demand that the BRST charge acting on the physical states of the Hilbert space must be zero. However such physical states still have ...
2
votes
2
answers
198
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What happens to gauge symmetry after gauge fixing?
Given a theory with gauge symmetry. After gauge fixing where does gauge symmetry go?
Does the gauge symmetry turn into a global symmetry?
For example there is a way to quantize fields theory with BRST ...
2
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2
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299
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One question about BRST symmetry in reading Srednicki’s book: Why should the BRST charge $Q_B$ be nilpotent?
In p.453, Srednicki claims that since the BRST transformation of a BRST transformation is zero, $Q_B$, the BRST charge, must be nilpotent:
$$Q_{B}^{2}=0.\tag{74.32}$$
I don't know why.
3
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BRST quantization: Explicit computation request
Following Green, Schwarz and Witten's book on Superstrings, the BRST charge is given by $$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}...
3
votes
1
answer
116
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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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1
answer
137
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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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0
answers
184
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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
votes
1
answer
143
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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 ...
1
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1
answer
119
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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
votes
1
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480
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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^...
4
votes
0
answers
78
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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 ...
1
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0
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284
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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
votes
1
answer
145
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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
votes
1
answer
83
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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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vote
0
answers
65
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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
votes
0
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150
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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 \...
4
votes
1
answer
234
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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. ...
7
votes
1
answer
633
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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 ...
2
votes
1
answer
716
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 (...
3
votes
1
answer
199
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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 ...
5
votes
1
answer
240
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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 $\partial^{\mu}A_{\mu}|\Psi\rangle=0$ on physical states. ...