Questions tagged [brst]

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Mass dimension of ghost Lagrangian in BRST quantization

It seems from the BRST transformation rules that the ghost fields should be dimensionless: For eg. in the Abelian case in 4D: $$A_{\mu} \to A_{\mu} + d_{\mu}c.$$ Then the ghost Lagrangian density $\...
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A question on BRST current of the Bosonic string, why we choose $c(z)$ as generator?

I'm new to the forum, I will try to make my asking as clear as possible. I'm currently writing a 40-minutes talk on the BRST quantization of the Bosonic string, mostly following Polchinski's Book. The ...
Joshua's user avatar
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What is the full algebra of BRST-invariant observables for general relativity?

The Hamiltonian formulation of general relativity - either in the ADM formalism or in Ashtekar variables - is straightforwardly a gauge theory. While the BRST formalism has primarily been developed to ...
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Introduce Ghost Field to eliminate unphysical degrees of freedom in case of Photon Field

In wikipedia's article about ghost fields is stated the following which requires a bit more clarification: An example of the need of ghost fields is the photon, which is usually described by a four ...
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Constraints Generating Gauge Transformations and BRST

Given a gauge-invariant point particle action with first class primary constraints $\phi_a$ of the form ([1], eq. (2.36)) $$S = \int d \tau[p_I \dot{q}^I - u^a \phi_a]\tag{1}$$ we know immediately, ...
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Why are physical states not eigenstates of BRST charge?

In many texts in quantum field theory or string theory, it is stated that the BRST charge $Q$ must annihilate physical states because the states are required to be BRST invariant. Since $Q$ generates ...
Chang Hexiang's user avatar
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The BRST variation of the gauge fixing condition

Following Polchinski volume I, p 126 onwards, The BRST variation of fields $\phi^{i}$ is given by $$\delta_{B} \phi_{i} = - i \epsilon c^{\alpha} {\delta}_{\alpha}\phi_{i} \; .\tag{4.2.6a}$$ My ...
unifymchn_MCR's user avatar
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BRST structure functions in gravity?

In the classical Hamiltonian BRST formalism, there arise structure functions $\Omega^{\beta_1...\beta_n}_{\alpha_1...\alpha_{n+1}}$ ($n\geq0$) --- see https://inspirehep.net/literature/221897 for ...
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An explicit form for the co-BRST operator?

Take a theory with 1st class constraints $M_{\alpha}$. We gave ghosts $c^\alpha$ and their conjugates $b_\alpha$ for every constraint. The BRST operator $\Omega$ has ghost number $+1$ and has an ...
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Does the following limit exist in the BRST formalism?

Consider the BRST operator $\Omega$ (which has ghost number $+1$) and the gauge fermion operator $\rho$ which has ghost number $-1$. Given an exact state $|\Phi\rangle$ (i.e. $|\Phi\rangle=\Omega|\Psi\...
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Square of BRST operator

The BRST operator $\Omega$ can be expanded in powers of the ghost fields $c^{\alpha}$ and their conjugates $b_{\alpha}$ (which satisfy $\{c^\alpha,b_\beta\}=\delta^{\alpha}_{\beta}$): $$ \Omega=c^{\...
dennis's user avatar
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Renormalisation of Yang-Mills Breaks Gauge Invariance?

Consider the Lagrangian (renormalised + counterterm) of QED: $$\mathcal{L} = -\frac{1}{4} F_{\mu \nu}F^{\mu \nu} - \frac{1}{2 \xi}(\partial_{\mu} A^{\mu})^2 + \bar{\psi}(i \displaystyle{\not} D - m)\...
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Confused about the relation between BRST invariant states and 'group averaging'

https://arxiv.org/abs/hep-th/0111270 by O.Y. Shvedov tries to relate the 'group averaging procedure' (see https://ncatlab.org/nlab/show/group+averaging and references therein) to the BRST formalism so ...
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Difference between Gauge invariance and BRST invariance

Which is the difference between gauge invariance and BRST invariance? Is it the same symmetry? Is the BRST the extention of the gauge symmetry even on the ghost fields?
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Non-linear symmetry and symmetry at quantum level

Can anyone explain me what does the statement mean: "the BRST symmetry is a non-linear symmetry, so the BRST is also a symmetry at the quantum level"? What does "at the quantum level&...
nabla_quadro's user avatar
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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 $c$ ...
Roy's user avatar
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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 ...
Simplyorange's user avatar
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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....
Simplyorange's user avatar
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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, ...
Keith's user avatar
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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 ...
y9QQ's user avatar
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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 ...
nodumbquestions's user avatar
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1 answer
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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)^...
Mauro Giliberti's user avatar
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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 ...
aitfel's user avatar
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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 ...
Alessio Fontanarossa's user avatar
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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 ...
Slz2718's user avatar
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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 ...
J. H's user avatar
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1 answer
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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 ...
CoffeeCrow's user avatar
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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 ...
SvenForkbeard's user avatar
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1 answer
157 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}^{-\...
Nathanael Noir's user avatar
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3 answers
791 views

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 ...
Unmotivated L-function's user avatar
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1 answer
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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 ...
Anthony Guillen's user avatar
2 votes
1 answer
181 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 $s^2=0$, ...
Pratik Chatterjee's user avatar
4 votes
1 answer
414 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 ...
ann marie cœur's user avatar
6 votes
3 answers
531 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 ...
ann marie cœur's user avatar
4 votes
1 answer
435 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 ...
ann marie cœur's user avatar
3 votes
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 ...
Ivan Burbano's user avatar
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4 votes
1 answer
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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 ...
Ivan Burbano's user avatar
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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
421 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 ...
samario28's user avatar
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5 votes
2 answers
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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 ...
Andrews's user avatar
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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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