Questions tagged [brst]
The brst tag has no usage guidance.
128
questions
2
votes
1
answer
45
views
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 $c$ ...
2
votes
1
answer
49
views
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 ...
2
votes
1
answer
50
views
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 $$Q^2\phi=0\tag{16....
3
votes
1
answer
117
views
Constructing a field theory action for the point particle in curved space
The point particle action in the Hamiltonian formalism is
$$
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}
$$
where I explicitly displayed the ...
2
votes
0
answers
63
views
Constructing the BRST operator for the superstring
In page 133 of Polchinski's String Theory textbook (ref. [1]), it is stated that given a set of constraints $\{G_I\}$ satisfying the algebra
$$[G_I,G_J] = i {g^k}_{IJ}G_K \,, \tag{4.3.12}$$
the BRST ...
3
votes
2
answers
377
views
BRST Symmetry and Single Particle States
I am studying about BRST symmetry from the book of P&S (Peskin's and Schroeder's "An Introduction to QFT", Chapter 16.4). The authors construct a nilpotent charge operator and then they ...
2
votes
1
answer
95
views
Antifields in BV formalism - do they also have gauge transformation laws?
I am studying Weinberg Vol 2 and the BV formalism of the gauge theory.
There, the antifields are introduced somewhat out of thin air. I am a little bit confused about their properties.
For example, ...
2
votes
1
answer
76
views
Vanishing of path integral over internal d.o.f. of test particle in $SU(N)$ gauge theory
In Ch-2 (Yang-Mills theory) of David Tong’s notes on gauge theory. Tong writes an action $$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{...
3
votes
1
answer
223
views
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
192
views
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
votes
1
answer
88
views
Does the Slavnov-Taylor identity still hold for scalar Yang-Mills?
I want to renormalize the minimally-coupled scalar Yang-Mills theory:
$$\mathcal{L}_{YM\phi}=(D_\mu\phi)^\dagger(D^\mu\phi)-\frac{1}{4}F_{\mu\nu}^a{F^{\mu\nu}}^a-\frac{1}{2\xi}(\partial_\mu {A^\mu}^a)^...
2
votes
0
answers
95
views
What does cohomology of $Q_B$ mean in BRST quantization in Polchinski?
While proving no-ghost theorem ($4.4$ Polchinski) the term cohomology of $Q_B$ is used quite a lot of time. From what I understand this has to be a set since "cohomology of $Q_B$" is ...
1
vote
0
answers
19
views
Idea/Intuition behind using commutator of light cone number operator and BRST generator
In ch-$4$ of Polchinksi while proving no-ghost theorem we are introduced to number operator of light cone oscillators
$$N^{lc}=\sum_m{\frac{1}{m}:\alpha^+_{-m}\alpha^-_m:}\tag{4.4.6}$$
$$\alpha^{\pm}...
1
vote
0
answers
41
views
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
53
views
Counting sign definiteness in BRST cohomology of string
In Polchinksi ch-$4$ following manipulation is done:
$$|\psi_1\rangle=(e\cdot\alpha_{-1}+\beta b_{-1}+\gamma c_{-1})|0,\textbf{k}\rangle .\tag{4.3.25}$$
$$\langle\psi_1|\psi_1\rangle=\Big(e^*\cdot e+(\...
2
votes
1
answer
73
views
BRST variation of $\delta_{\alpha}F^A$ in $S_3$ in BRST section of Polchinski
The Faddeev-Popov action reads
$$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
88
views
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
vote
0
answers
78
views
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
66
views
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
153
views
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
0
answers
72
views
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
124
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 ...
3
votes
0
answers
125
views
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
484
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 ...
0
votes
1
answer
80
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
133
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$, ...
3
votes
1
answer
277
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 ...
5
votes
3
answers
395
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 ...
4
votes
1
answer
264
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 ...
3
votes
1
answer
69
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 ...
3
votes
1
answer
222
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 ...
5
votes
0
answers
152
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
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 ...
5
votes
2
answers
506
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 ...
1
vote
1
answer
176
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
$$
\...
3
votes
3
answers
231
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 ...
4
votes
1
answer
220
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 ...
2
votes
2
answers
248
views
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
votes
2
answers
386
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.
3
votes
1
answer
186
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
$$[K_i, K_j] = f_{ij}^{~~~k}K_k,\tag{3.2.1}...
3
votes
1
answer
146
views
Supercurrent of the $bc$-$\beta\gamma$ SCFT
In Polchinksi's Sec. 10.1, the $bc$-$\beta\gamma$ SCFT is introduced with action
$$S_{BC} = \frac{1}{2\pi} \int d^2z (b \bar \partial c + \beta \bar \partial \gamma)$$
and supercurrent
$$T_F = -\...
2
votes
1
answer
174
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 ...
2
votes
0
answers
232
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}\...
2
votes
1
answer
196
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 ...
1
vote
1
answer
153
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 ...
3
votes
1
answer
557
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^...
4
votes
0
answers
83
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 ...
1
vote
0
answers
329
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 ...
2
votes
1
answer
161
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 ...
3
votes
1
answer
100
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 ...