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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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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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 $\{c^\alpha,b_\beta\}=\delta^{\alpha}_{\beta}$):
$$
\Omega=c^{\...
- 469
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Calculating a Gaussian-like path integrals with Grassmann variables and real variables
I want to compute the following path integral
$$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \prod_{i=1}^{n}d\overline{\theta}_id\theta \: \exp{\left(-\overline{\theta}_i \partial_j w_i(x)\theta_j -\...
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Ghost detection at the level of equations of motion
My question is about how to detect ghostly degrees of freedom at the level of equations of motion. It is not clear for me how does this work. Let me explain with an example:
Consider the following ...
- 547
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Faddeev-Popov ghost in the Standard-Model
When we quantize $SU(N)$ gauge theories using the path integral formalism, we must introduce Faddeev-Popov ghosts and will appears as scalar fermions coupled to our gauge bosons in the Lagrangian of ...
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Notation of ghost fields $b$, $\tilde{b}$, $c$, and $\tilde{c}$ in Polchinski
I am terrifically confused by the notation in Polchinski's string theory book from chapter 3 to chapter 4. The ghost action of the bosonic string in conformal gauge is (3.3.24)
$$S = \frac{1}{2 \pi} \...
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Constraint in BRST quantization of point particle
On page 130 of Joe Polchinski's String Theory volume 1 book, the Constraint or the missing equation of motion for point particle after gauge fixing is $H = 0$, and the BRST operator is the ghost $c$ ...
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Typo of P&S' QFT eq.(16.40)
First, begin with P&S's QFT eq.(16.39):
$$ \frac{1}{2}\left[\left(i \mathcal{M}^{\mu \nu} \epsilon_\mu^{-*} \epsilon_\nu^{+*}\right)\left(i \mathcal{M}^{\prime \rho \sigma} \epsilon_\rho^{+} \...
- 1,287
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Noether current for ``local" conformal transformation?
If the fields $b$ and $c$ have conformal weight $\lambda$ and $1-\lambda$ and action is:
$$S = \frac{1}{2\pi} \int d^2z \, b \bar \partial c,$$
under conformal transformations $z \rightarrow z+\...
- 423
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Motivation of Grassmann fields in the Faddeev-Popov method for free Gluon fields
The Faddeev-Popov approach to make the generating functional corresponding to free gluon fields well defined, introduces two independent Grassmann fields. Since these are scalar, their quanta can be ...
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Ghost modes in Lattice-Boltzmann Method (LBM)
I've read a lot of documents on Lattice-Boltzmann Method (LBM) and "ghost modes" are commonly referred to. It is explained that they are unphysical modes that can pollute the numerical ...
- 111
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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 ...
- 945
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Weyl Anomaly for Old Covariant Quantization in String Theory?
In the context of quantization in string theory, the modern approach is the path integral/modern covariant quantization approach. As known from QFT, we fix our gauge and represent the arising Fadeev-...
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Green Schwarz Witten (7.1.23) - Ghost decoupling
In the Superstring Theory book by Green Schwarz Witten, when trying to show ghosts decouple in tree level string scattering amplitudes, I find equation (7.1.23)
$$[(L_m-L_0-m+1),V_N\Delta]=0 \tag{7.1....
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BRST Symmetry and Single Particle States
I am studying about BRST symmetry from the book of P&S (Peskin's and Schroeder's "An Introduction to QFT", Chapter 16.4). The authors construct a nilpotent charge operator and then they ...
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Applying the Optical Theorem in Non-Abelian Gauge Theories
I am reading P&S (Peskin's and Schroeder's book on QFT), Chapter 16.3 entitled Ghosts and Unitarity. The authors employ the optical theorem to calculate the imaginary part of a $f\bar{f}\...
- 3,174
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Computing the functional integral over the gauge and ghost fields
In Peskin&Schroeder page 517 the authors mention that the functional integration of the gauge fields and ghost fields yields the following determinants
$$
(\det[-\partial^2])^{-d/2}\cdot(\det[-\...
- 1,947
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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^{...
- 2,268
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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
$$
\det\left(\frac{1}{g}\partial_\mu D^\mu\right)=\int{\cal{D}cD\bar{c}\exp\left[i\int {d^4x\bar{c}(-\partial^\mu D_\...
- 1,947
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Compute the generating functional for the $bc$ theory
I need the generating functional for the $bc$ CFT, which has $$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
$$ \mathcal{L} = \frac{1}{2}\partial_{\mu}\phi_1\partial^{\mu}\phi_1-\frac{1}{2}\partial_{\mu}\phi_2\partial^{\mu}\phi_2 + V(\...
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Wouldn't a simple scalar field fix the non-renormalizability of gravity?
It is well known that quadratic gravity is renormalizable. On the other hand it is possible to transform the partition function of Einstein-Hilbert + free minimally coupled complex scalar field into a ...
- 2,268
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Is there still a Gribov ambiguity when the Faddeev-Popov determinant is treated without ghosts?
In this document (Gribov Ambiguity by Thitipat Sainapha) the setup leading to the equation $3.77$ seems to strongly depend on the treatment of the Faddeev-Popov determinants with ghosts. Indeed the ...
- 2,268
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What is the derivative of $:bc:$?
For a $bc$ CFT (see e.g. Polchinski's string theory, section 2.5) given by
\begin{equation}
S=\frac 1{2\pi}\int dz^2 b \overline \partial c
\end{equation}
the energy-momentum tensor is:
\begin{...
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Open-closed amplitude in bosonic string theory
I want to know the scattering amplitude involving both open and closed string, more specifically, the amplitude between two gluons and one graviton in open closed set up. Is there a reference where ...
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How to read of the conformal dimension of $bc$ CFT to be $(2,-1)$ from the action $S_g$?
Quote Polchinski String Theory volume 1 page 89.
$$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})$$ Since the action... is weyl invariant, $b_{ab},c^...
- 3,034
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111
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Normal ordering constant value in String Theory and Old Covariant Quantization
Suppose you are approaching the quantization of the closed bosonic string for the first time (so we are in the so called Old Covariant Quantization (OCQ), and by now we know nothing about Lightcone ...
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Existence of ground states in $bc$ CFT
I am reading Polchinski's Vol. 1 on String Theory, and I have some basic doubts on how he introduces the $bc$ conformal field theory (see section 2.7, page 61).
He basically starts from the ...
- 153
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General and geometric prescription of Picture-changing operator (PCO), Polchinski Vol.2, section 12.5
In section 12.5, Polchinski tried to give a general description of PCO from a super-riemann surface view. He gave the generalized amplitude,
The measure on supermoduli space
The expression $(5.4.19)$ ...
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Question about canonical quantization of the open string ghost system
In section 3.1.3 of Green, Schwarz and Witten book on superstrings, it is stated that
the canonical anti commutation relations for the fermionic ghosts are
$$ \{ 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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93
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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 ...
- 2,316
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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 ...
- 1,761
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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 ...
- 4,108
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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?
- 2,268
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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 ...
- 165
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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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