Questions tagged [ghosts]

Ghosts are unphysical states that arise when quantizing gauge theories. Do not use this tag for 'ghosts' in the paranormal sense.

Filter by
Sorted by
Tagged with
1 vote
0 answers
20 views

Why does expressing the Faddeev-Popov determinant as this lead to such problems?

Background In the following, I am interested in the Schwinger function associated with the gluon propagator when one considers the Gribov no-pole condition in the partition function. Defining $\nabla^{...
user avatar
1 vote
0 answers
67 views

Definition of a determinant Peskin&Schroeder

In page 514 of Peskin&Schroeder we are given the definition of a determinant as $$ \det\left(\frac{1}{g}\partial_\mu D^\mu\right)=\int{\cal{D}cD\bar{c}\exp\left[i\int {d^4x\bar{c}(-\partial^\mu D_\...
user avatar
0 votes
1 answer
55 views

Compute the generating functional for the $bc$ theory

I need the generating functional for the $bc$ CFT, which has $$L=\frac{1}{2\pi}(b\bar{\partial}c + b\partial\bar{c}),$$ so I can compute the correlation function $$\langle b(z_1)c(z_2)\rangle =\frac{1}...
user avatar
2 votes
1 answer
108 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 ...
user avatar
  • 21
1 vote
1 answer
71 views

Conformal weights of space-time and ghosts

I'm studying the paper of Kaplunovsky https://arxiv.org/abs/hep-th/9205070 In particular in page 12 he says Hamiltonians $H$ and ̄$\bar{H}$ are totals of space-time, ghost and internal components; ...
user avatar
3 votes
3 answers
389 views

Scalar field theory with two scalars

Consider the following scalar field theory with a kinetic term as follows $$ \mathcal{L} = \frac{1}{2}\partial_{\mu}\phi_1\partial^{\mu}\phi_1-\frac{1}{2}\partial_{\mu}\phi_2\partial^{\mu}\phi_2 + V(\...
user avatar
  • 810
3 votes
0 answers
113 views

Wouldn't a simple scalar field fix the non-renormalizability of gravity?

It is well known that quadratic gravity is renormalizable. On the other hand it is possible to transform the partition function of Einstein-Hilbert + free minimally coupled complex scalar field into a ...
user avatar
1 vote
0 answers
40 views

Is there still a Gribov ambiguity when the Faddeev-Popov determinant is treated without ghosts?

In this document (Gribov Ambiguity by Thitipat Sainapha) the setup leading to the equation $3.77$ seems to strongly depend on the treatment of the Faddeev-Popov determinants with ghosts. Indeed the ...
user avatar
4 votes
1 answer
103 views

What is the derivative of $:bc:$?

For a $bc$ CFT (see e.g. Polchinski's string theory, section 2.5) given by \begin{equation} S=\frac 1{2\pi}\int dz^2 b \overline \partial c \end{equation} the energy-momentum tensor is: \begin{...
user avatar
  • 143
2 votes
0 answers
44 views

Open-closed amplitude in bosonic string theory

I want to know the scattering amplitude involving both open and closed string, more specifically, the amplitude between two gluons and one graviton in open closed set up. Is there a reference where ...
user avatar
1 vote
0 answers
51 views

How to read of the conformal dimension of $bc$ CFT to be $(2,-1)$ from the action $S_g$?

Quote Polchinski String Theory volume 1 page 89. $$S_f=\frac{1}{2\pi} \int d^2 z(b_{zz}\partial_{\bar z} c^z+b_{\bar z \bar z }\partial_zc^{\bar z})$$ Since the action... is weyl invariant, $b_{ab},c^...
user avatar
2 votes
0 answers
67 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 ...
user avatar
1 vote
0 answers
71 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 ...
user avatar
  • 153
3 votes
0 answers
63 views

General and geometric prescription of Picture-changing operator (PCO), Polchinski Vol.2, section 12.5

In section 12.5, Polchinski tried to give a general description of PCO from a super-riemann surface view. He gave the generalized amplitude, The measure on supermoduli space The expression $(5.4.19)$ ...
user avatar
1 vote
0 answers
39 views

Question about canonical quantization of the open string ghost system

In section 3.1.3 of Green, Schwarz and Witten book on superstrings, it is stated that the canonical anti commutation relations for the fermionic ghosts are $$ \{ b_{++}(\sigma, \tau), c^+(\sigma', \...
user avatar
4 votes
2 answers
294 views

The structure of the Hilbert space of 2d CFT

In many textbooks, I found similar statements that in 2d CFT (which I hope I'm not misunderstanding), one can decompose the space of states into primaries and their Virasoro descendants, or into quas-...
user avatar
  • 497
2 votes
0 answers
61 views

Boundary conditions for the $bc$ system

In this question, I will be referring to chapter 2 of Polchinski String Theory vol. 1. In equation (2.7.29), he states that the boundary conditions for the $bc$ system of the open string are \begin{...
user avatar
1 vote
0 answers
59 views

Showing that a two-dimensional Euclidean CFT ghost action is hermitian

At the end of chapter 6 in Polchinski's String Theory book he says that the $c$ ghost is anti-hermitian. With that information, I tried to show that the action for the $bc$ system \begin{equation} S= \...
user avatar
  • 835
2 votes
0 answers
43 views

Accounting for ghosts in calculation of massless closed string emission

I've been trying to do problem 5.8 from String Theory in a Nutshell by Kiritsis, which is think, also related to problem 8.10 in Polchinski. My question is this: how do we account for ghosts in the ...
user avatar
3 votes
1 answer
132 views

Absence of negative-norm states in old covariant quantization of the bosonic string

This question is about the discussion about the absence of negative-norm states in the old covariant quantization of the bosonic string as presented e.g. in Becker Becker & Schwarz (BBS). Their ...
user avatar
  • 29.3k
3 votes
0 answers
90 views

Free field realisation for Vertex Operators

I wanted to know if for a generic CFT (in weak coupling limit) there exists free field realisation of the vertex operators. This seems like it should exist but I want to know a generic algorithm to ...
user avatar
  • 2,691
5 votes
1 answer
217 views

Bosonization of $\beta \gamma$ system

I am studying the bosonization of the $\beta \gamma$ ghost system, also called as the symplectic boson (See for example, section 2.3 of this paper. These have OPE, \begin{equation} \beta(z) \gamma(w) \...
user avatar
  • 211
2 votes
1 answer
66 views

Do Faddeev-Popov ghost contribute to vacuum polarisation?

I can imagine how one can draw a Feynman diagram for a boson self-energy with a ghost loop. My question is, shouldnt't the amplitude of that process be 0 as the ghosts are merely a mathematical tool?
user avatar
3 votes
4 answers
200 views

Various Definitions of 'Ghosts'

I have seen various different definitions of a ghost field in the literature. For example, one can find many examples where ghosts are simply defined as any field with a negative sign in the kinetic ...
user avatar
  • 2,227
1 vote
0 answers
66 views

Diagonalisation of a ghost Lagrangian

I have a ghost Lagrangian of the form $\mathcal{L}= \bar{c}M_{11}c + \bar{c}M_{12}b + \bar{b}M_{21}c + \bar{b}M_{22}b$ where $c,b$ are the ghosts and $\bar{c}, \bar{b}$ the anti ghost fields, $M_{...
user avatar
5 votes
1 answer
292 views

What's a minisuperspace in quantum cosmology?

I read this paper on Lorentzian Quantum Cosmology. But I couldn't understand the term minisuperspace which is used to define the path integral $$\int\mathcal{D}N \mathcal{D}\pi\mathcal{D}a\mathcal{D}...
user avatar
  • 2,815
0 votes
1 answer
69 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 ...
user avatar
  • 1,753
2 votes
1 answer
100 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 $s^2=0$, ...
user avatar
3 votes
1 answer
179 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 ...
user avatar
0 votes
0 answers
48 views

Yang-Mills massive ghosts

Is there any procedure to add a mass to Faddeev-Popov Lagrangian density of a pure Yang-Mills theory, other than just add it from nowhere?
user avatar
2 votes
1 answer
204 views

Physical States has ghost number 1 in String Theory

We know by no-ghost theorem (Polchinski I, section 4.4) that the physical Hilbert space have no longitudinal excitations ($X^0, X^1, b, c$). This is obvious by light-cone gauge quantization and in ...
user avatar
5 votes
2 answers
399 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 ...
user avatar
  • 165
0 votes
0 answers
79 views

Understanding the gauge condition in light-cone gauge

When going over Cambridge String Theory Notes (page 38) I came across the following gauge conditions: $$X^+ (t, \sigma) = x^+(t)\hspace{5mm}, \hspace{5mm} P_−(t, \sigma) = p_−(t) \tag{1} $$ which is ...
user avatar
3 votes
1 answer
250 views

Ghost Theory energy-momentum tensor CFT

Given the action $$ \mathcal{S_g}=\frac{1}{2\pi}\int{d^2x\left(b\bar{\partial}c+\bar{b}{\partial}\bar{c}\right)} $$ where $b$ and $c$ are ghosts. How can I calculate the energy-momentum tensor$$ T=...
user avatar
4 votes
0 answers
111 views

Picture Number Operator in String Theory

My question concerns the ghost picture charge/picture number operator in the RNS formalism of Superstring theory. In particular I refer to page 403 of "Basic Concepts of String Theory" by R. ...
user avatar
  • 165
2 votes
3 answers
174 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 ...
user avatar
2 votes
1 answer
141 views

Finding ghost terms for super Yang-Mills theory in background field gauge

I want to find the ghost terms (2.16) for the action in this paper. The gauge field action is given by $$ \begin{align}S_{A} =& i \int d\tau \Big(\frac{1}{2}A_{1}(\partial_{\tau}^2 - r^2)A_{1} + \...
user avatar
  • 902
0 votes
0 answers
109 views

4th polarization in the massive vector boson field propagator

We can build propagators (in this case, for massive vector fields) with at least two approaches: (1) from canonical quantization (or exact solutions, if you like) $$\left[[(i\partial)^2-m^2]\delta_{...
user avatar
  • 63
2 votes
1 answer
92 views

Ghost exorcisms of fields?

In Mack's paper "D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes", he makes the following ...
user avatar
0 votes
0 answers
59 views

Critical dimension from the symmetries of the string action

(Related: This post and this post.) In this thesis it is said (on page 13) that just by assuming that we have some general action with the same symmetries as the Polyakov action (Poincare invariance, ...
user avatar
  • 747
2 votes
0 answers
34 views

Time reversal symmetry of the Faddeev-Popov determinant

I am studying the Faddeev-Popov procedure to quantize a non-Abelian gauge theory, and I got confused by the status of the time reversal symmetry. People have different definitions of the time reversal ...
user avatar
2 votes
1 answer
182 views

$bc$-system energy momentum tensor

I have a (maybe silly) question regarding the expression of the energy momentum tensor of the $bc$-system in equations $(2.5.11a)$ and $(2.5.11b)$ in Polchinski's String Theory, page 50. I know that ...
user avatar
  • 165
3 votes
1 answer
116 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 = -\...
user avatar
  • 1,834
2 votes
1 answer
165 views

Longitudinal polarization of gluons in loop

I have a short question about the possible gluon polarization in loop diagrams. For external gluons, we only want the 2 transverse polarizations. In Peskin-Schroeder it is explained that in Feynman-...
user avatar
4 votes
1 answer
111 views

Quantum field theory with only 3-point vertexes

Given an arbitrary quantum field theory, can I always write it in terms of another (different) quantum field theory containing only operators with 3 fields? (i.e. vertexes with 3 legs) I guess that ...
user avatar
  • 1,774
4 votes
0 answers
194 views

Ghosts use in the calculus of the scattering matrix

When you perform the calculus of $S$ matrix in, for instance, QCD maybe you need to add ghosts in external legs or internal loops. E.g.: $q\bar{q} \rightarrow gg$ needs 2 ghosts diagrams with ghosts ...
user avatar
  • 1,174
2 votes
1 answer
176 views

Fermionic ghost path integral results in $\delta$ function?

This is related to a statement in pg 20 of hep-th/9408074 formula (2.39). Suppose $$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where $\bar{\eta}$ to ...
user avatar
  • 391
3 votes
1 answer
188 views

Gauge fixing while preserving supersymmetry

In supersymmetric gauge theories, the vector potential is a part of a vector supermultiplet which is represented by a real superfield $V$. Expanded out in components, the Lagrangian for such a field ...
user avatar
  • 8,014
4 votes
0 answers
78 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 ...
user avatar
1 vote
0 answers
286 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 ...
user avatar