Questions tagged [brst]

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Constraint in BRST quantization of point particle

On page 130 of Joe Polchinski's String Theory volume 1 book, the Constraint or the missing equation of motion for point particle after gauge fixing is $H = 0$, and the BRST operator is the ghost $c$ ...
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Peskin and Schroeder's discussion of the BRST operator

On page 519 of Peskin and Schroeder, the authors have the following discussion on the nilpotent BRST operator $Q$ that commutes with the Hamiltonian $H$. Many eigenstates of $H$ must be annihilated ...
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Nilpotency of the BRST operator

I'm styding chapter 16 of Peskin and Schroeder, in section 16.4 on the BRST symmetry, Peskin and Schroeder first checks (on page 518) that if $Q$ is the BRST symmetry operator, then $$Q^2\phi=0\tag{16....
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Constructing a field theory action for the point particle in curved space

The point particle action in the Hamiltonian formalism is $$ S = \int d\tau \Big( -p_\mu \dot{x}^\mu - \frac{e}{2}(g^{\mu\nu} p_\mu p_\nu - m^2) \Big) \ ,\tag{1} $$ where I explicitly displayed the ...
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Constructing the BRST operator for the superstring

In page 133 of Polchinski's String Theory textbook (ref. [1]), it is stated that given a set of constraints $\{G_I\}$ satisfying the algebra $$[G_I,G_J] = i {g^k}_{IJ}G_K \,, \tag{4.3.12}$$ the BRST ...
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BRST Symmetry and Single Particle States

I am studying about BRST symmetry from the book of P&S (Peskin's and Schroeder's "An Introduction to QFT", Chapter 16.4). The authors construct a nilpotent charge operator and then they ...
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Antifields in BV formalism - do they also have gauge transformation laws?

I am studying Weinberg Vol 2 and the BV formalism of the gauge theory. There, the antifields are introduced somewhat out of thin air. I am a little bit confused about their properties. For example, ...
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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{...
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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 ...
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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 ...
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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)^...
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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 ...
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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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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 ...
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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+(\...
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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}{\...
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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 ...
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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 ...
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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 ...
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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 vote
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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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1 answer
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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 ...
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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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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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1 answer
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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
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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 ...
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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 ...
4 votes
1 answer
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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 ...
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1 answer
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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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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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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
259 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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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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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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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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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
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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 ...
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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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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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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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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
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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 ...
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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 ...
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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^...
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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 vote
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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 ...
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1 answer
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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 ...
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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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