Questions tagged [ghosts]
Ghosts are unphysical states that arise when quantizing gauge theories. Do not use this tag for 'ghosts' in the paranormal sense.
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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^{...
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Definition of a determinant Peskin&Schroeder
In page 514 of Peskin&Schroeder we are given the definition of a determinant as
$$
\det\left(\frac{1}{g}\partial_\mu D^\mu\right)=\int{\cal{D}cD\bar{c}\exp\left[i\int {d^4x\bar{c}(-\partial^\mu D_\...
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Compute the generating functional for the $bc$ theory
I need the generating functional for the $bc$ CFT, which has $$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}...
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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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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; ...
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3
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Scalar field theory with two scalars
Consider the following scalar field theory with a kinetic term as follows
$$ \mathcal{L} = \frac{1}{2}\partial_{\mu}\phi_1\partial^{\mu}\phi_1-\frac{1}{2}\partial_{\mu}\phi_2\partial^{\mu}\phi_2 + V(\...
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Wouldn't a simple scalar field fix the non-renormalizability of gravity?
It is well known that quadratic gravity is renormalizable. On the other hand it is possible to transform the partition function of Einstein-Hilbert + free minimally coupled complex scalar field into a ...
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Is there still a Gribov ambiguity when the Faddeev-Popov determinant is treated without ghosts?
In this document (Gribov Ambiguity by Thitipat Sainapha) the setup leading to the equation $3.77$ seems to strongly depend on the treatment of the Faddeev-Popov determinants with ghosts. Indeed the ...
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What is the derivative of $:bc:$?
For a $bc$ CFT (see e.g. Polchinski's string theory, section 2.5) given by
\begin{equation}
S=\frac 1{2\pi}\int dz^2 b \overline \partial c
\end{equation}
the energy-momentum tensor is:
\begin{...
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Open-closed amplitude in bosonic string theory
I want to know the scattering amplitude involving both open and closed string, more specifically, the amplitude between two gluons and one graviton in open closed set up. Is there a reference where ...
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How to read of the conformal dimension of $bc$ CFT to be $(2,-1)$ from the action $S_g$?
Quote Polchinski String Theory volume 1 page 89.
$$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})$$ Since the action... is weyl invariant, $b_{ab},c^...
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Normal ordering constant value in String Theory and Old Covariant Quantization
Suppose you are approaching the quantization of the closed bosonic string for the first time (so we are in the so called Old Covariant Quantization (OCQ), and by now we know nothing about Lightcone ...
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Existence of ground states in $bc$ CFT
I am reading Polchinski's Vol. 1 on String Theory, and I have some basic doubts on how he introduces the $bc$ conformal field theory (see section 2.7, page 61).
He basically starts from the ...
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General and geometric prescription of Picture-changing operator (PCO), Polchinski Vol.2, section 12.5
In section 12.5, Polchinski tried to give a general description of PCO from a super-riemann surface view. He gave the generalized amplitude,
The measure on supermoduli space
The expression $(5.4.19)$ ...
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Question about canonical quantization of the open string ghost system
In section 3.1.3 of Green, Schwarz and Witten book on superstrings, it is stated that
the canonical anti commutation relations for the fermionic ghosts are
$$ \{ b_{++}(\sigma, \tau), c^+(\sigma', \...
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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-...
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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{...
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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= \...
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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 ...
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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 ...
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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 ...
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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) \...
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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?
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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 ...
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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_{...
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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}...
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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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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$, ...
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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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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?
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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 ...
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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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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} $$
which is ...
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Ghost Theory energy-momentum tensor CFT
Given the action
$$
\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=...
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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. ...
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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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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
$$ \begin{align}S_{A} =& i \int d\tau \Big(\frac{1}{2}A_{1}(\partial_{\tau}^2 - r^2)A_{1} + \...
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4th polarization in the massive vector boson field propagator
We can build propagators (in this case, for massive vector fields) with at least two approaches:
(1) from canonical quantization (or exact solutions, if you like)
$$\left[[(i\partial)^2-m^2]\delta_{...
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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 ...
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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, ...
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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 ...
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$bc$-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 ...
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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
$$T_F = -\...
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Longitudinal polarization of gluons in loop
I have a short question about the possible gluon polarization in loop diagrams. For external gluons, we only want the 2 transverse polarizations.
In Peskin-Schroeder it is explained that in Feynman-...
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Quantum field theory with only 3-point vertexes
Given an arbitrary quantum field theory, can I always write it in terms of another (different) quantum field theory containing only operators with 3 fields? (i.e. vertexes with 3 legs)
I guess that ...
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Ghosts use in the calculus of the scattering matrix
When you perform the calculus of $S$ matrix in, for instance, QCD maybe you need to add ghosts in external legs or internal loops. E.g.: $q\bar{q} \rightarrow gg$ needs 2 ghosts diagrams with ghosts ...
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Fermionic ghost path integral results in $\delta$ function?
This is related to a statement in pg 20 of hep-th/9408074 formula (2.39).
Suppose $$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where $\bar{\eta}$ to ...
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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 ...
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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 ...
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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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