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15

This is a temperamental difference more than a physical one, but I feel like this question deserves an answer with a lot less formalism than what Urs is using. The physical point that you should never lose sight of is that gauge symmetries are not symmetries at all: they don't map one state to another one, but instead identify a priori different states as ...

10

The mystery here should disappear once one realizes that the BRST complex -- being a dg-algebra -- is the formal dual to a space , namely to the "homotopically reduced" phase space. For ordinary algebras this is more familiar: the algebra of functions $\mathcal{O}(X)$ on some space $X$ is the "formal dual" to $X$, in that maps $f : X \to Y$ correspond to ...

8

You seem to be talking about the "old covariant quantization" in which $L_n$ for positive $n$ and $(L_0-a)$ annihilate physical ket states $|\psi\rangle$, right? It's analogous to the Gupta-Bleuler quantization http://en.wikipedia.org/wiki/Gupta-Bleuler_quantization which was a standard procedure used already in electromagnetism. The idea is that the ...

5

A modern treatment of this subject can be found in Segreev's book on the Kahler geometry of loop spaces also available online. This line of research started with the seminal work of Bowick and Rajeev: The holomorphic geometry of the closed bosonic string theory and $Diff S^1/S^1$ (Spires) (and independently Kirillov and Yuriev (Please see the reference in ...

5

Your mechanism of "one ghost falling in" is a different way of talking about the "evaporation of FP ghosts by a black hole". So do black holes evaporate ghosts? This is not a well-defined question because FP ghosts are unphysical, too. They're just a mathematical method to deal with gauge symmetries, in this case the diffeomorphism symmetry. In this sense, ...

4

I) Let us first clarify the left and right derivatives. Left derivatives is explained between eq. (15.8.9) and (15.8.10) in Ref. 1. A left derivative means a derivative that acts from the left. E.g. if $F= \chi G$, where $G$ does not depend on $\chi$, then $\frac{\delta_LF}{\delta\chi}=G$. Similarly, a right derivative acts from the right. E.g. if $F= ... 3 "Something is conserved for an action" simply means that the action carries a zero overall value of "something" (for an additive quantity). In this case, the action has$N_{gh}=0$. It follows that the equations of motion derived from the action imply$dN_{gh}/dt=0$. Most typically, they imply$\partial_\mu J^\mu_{gh}=0$i.e. the local continuity law for a ... 3 I) The gauge-fixed pure Maxwell action is $$\tag{1} S[A,c,\bar{c}]~=~\int \! d^4x~ {\cal L}$$ with Lagrangian density$^1$$$\tag{2} {\cal L}~=~-\frac{1}{4}F_{\mu \nu}F^{\mu \nu} -\frac{\chi^2}{2\xi}-d_{\mu}\bar{c}~d^{\mu}c, \qquad \chi~:=~d_{\mu} A^{\mu}, \qquad \xi~>~0,$$ consisting of (i) the Maxwell term, (ii) the gauge-fixing term, and (iii) ... 3 answer for questions$1$,$2$, and$3a$1) Looking at$2.7.22$to$2.7.24$(and also$2.7.18abc$), one define the ghost number$N^g = \frac{-1}{2\pi i} \int_0^{2 \pi}dz :b(z)c(z):$, and that all the operators$c_n$increase the ghost number by one.$[N^g,c_m] = c_m$, So the field$c(z)$, made of operators$c_m$, increase the ghost number by one unit.So ... 3 As you already wrote, the$(3/2)\partial^2 c$term is needed for the current to be a one-form i.e.$(1,0)$tensor field; see also page 131 of Polchinski's String Theory, volume 1. This means that if you compute the OPE $$T(z) j^{BRST}(0)\sim \dots,$$ you want to get $$\dots \sim \frac{1}{z^2} j^{BRST}(0)+\frac{1}{z}\partial j^{BRST}(0),$$ see e.g. ... 2 The central charge counts the number of degrees of freedom only for matter fields living on a flat manifold (or supermanifold in the case of superstrings). An example where this counting argument fails for matter fields is the case of strings moving on a group manifold$G$whose central charge is given by the Gepner-Witten formula:$c = ...

2

Kugo and Ojima's work was one of the major breakthroughs in understanding the role of BRST in the quantization of gauge theories. Historically BRST was discovered in the path integral formalism. The understanding of this theory as a cohomology theory started from the Kugo and Ojima's work. Now, the action is BRST invariant with and without the Gaussian ...

2

The reason for the discrepancy is that the BRST operator is not self adjoint $\Omega^{\dagger} \ne \Omega$, This can easily be seen from its action on the ghosts: $$\delta c^{a} = \epsilon \{\Omega, c^a\} = i \epsilon f^{a}_{bc} c^b c^c$$ $$\delta \bar{c}^{a} = \epsilon \{\Omega, \bar{c}^a\} = - i \epsilon b^a$$ $$\delta b^{a} = \epsilon \{\Omega, b^a\} = ... 2 Bosonic path integrals :$$Z = \int D\phi ~e^{-i \large \int ~ dx [\frac{1}{2}\phi (\square+m^2)\phi]}$$or Femionic path integrals (like Fadeev-Popov ghosts) :$$Z = \int D\eta D \tilde \eta ~e^{-i \large \int ~ dx [\tilde \eta^a \square \eta^a]}$$are not mathematically well-defined, because of the presence of the imaginary unit in the exponential. ... 2 I think 't Hooft and Kugo are solving different problems. 't Hooft addresses the issue that the anomaly involves a topological term. As a result, in perturbation theory there is no theta dependence and the anomaly equation by itself does not solve the U(1) problem. He shows that topological objects (semi-classically, instantons) generate theta ... 2 The expectation of the axial current divergence in a \theta shifted QCD vacuum is given by \partial_{\mu} \langle J^{\mu5}_{\mathrm{inv}} \rangle_{\theta} = 2m_q \langle \bar{q}i\gamma^5q \rangle_{\theta} + \langle \Xi \rangle_\theta, where the first term on the right hand side is the explicit breaking term due to the quark masses and the second ... 2 I did not furnish all the details because it would be too long, but I give some hints at the end of the answer. I have used the formulae :T^g: ~= ~:2(\partial c) b + c(\partial b): and :\frac{1}{2}cT^g: ~= ~:bc \partial c:, when there is an ambiguity in the calculus. We begin by :$$j_B = cT^m+:\frac{1}{2}:cT^g:+\frac{3}{2}\partial^2c=cT^m+:bc\partial ...

1

To test whether $|0\rangle$ is physical, you would apply $\partial^\mu A_\mu^+$ to it. So $|0\rangle \in V$. Note that 7.48 does not mean that $\partial_\mu A^\mu = 0$ as an operator identity in this space. It means that all it's matrix elements in this space are zero. You have noticed that the state you created, $\partial_\mu A^\mu |0\rangle$, lives not ...

1

In your case, I think that $\eta$ is playing the role of $\epsilon$ in David Bar Moshe 's answer, that is : $\eta$ is a real Lorentz-scalar Grassmann variable. So, you will have : $$\delta_\Omega \bar\psi= \overline{\delta_\Omega\psi} = \overline {i\eta\psi} = -i \eta \bar\psi = i \bar\psi \eta$$

1

There are no actual states with zero energy, $P_0=0$ (I don't even need to talk about $P_i$), unless they correspond to modes of massless states and only massless scalars – moduli – should be considered real. Note that relativity implies that $P_0=0$ states would have to have zero mass and zero spatial momenta in both compact and noncompact dimensions, too. ...

1

I) Let us reformulate OP's question(v1) as How can hermiticity$^1$ be maintained for the gauge algebra $$\tag{1} [\hat{G}_a , \hat{G}_b ] ~=~ i\hbar~\hat{G}_c ~\hat{f}^{c}{}_{ab}$$ of first-class operator constraints $\hat{G}_a$, if the structure operators$^2$ $$\tag{2} \hat{f}^{c}{}_{ab}~=~f^{c}{}_{ab}(\hat{q}^i,\hat{p}_j)$$ depend on the ...

1

I'd say BRST ghosts have more elements of reality compared to longitudinal gauge bosons which are pure gauge. Look at the inner product structure. Physical states belong to the BRST cohomology of BRST-closed modulo BRST-exact. pure gauge longitudinal gauge bosons are BRST-exact and have zero inner product with any other BRST-closed state. BRST ghosts have ...

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