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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 ...

11

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 ...

7

J.W. van Holten's "Aspects of BRST Quantization" arXiv:hep-th/0201124 might be what you're looking for...

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, ...

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

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 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 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: $$\delta c^{a} = \epsilon \{\Omega, c^a\} = i \epsilon f^{a}_{bc} c^b c^c$$ $$\delta \bar{c}^{a} = \epsilon \{\Omega, \bar{c}^a\} = - i \epsilon b^a$$ $$\delta b^{a} = \epsilon \{\Omega, b^a\} = ... 3 On the torus there are two real moduli \tau_1 and \tau_2, and two conformal Killing vectors corresponding to translations. This means that you need two insertions of b and two insertions of c in order to saturate the zero mode path integral. 3 I) The gauge-fixed pure Maxwell action is$$\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,$$consisting of (i) the Maxwell term, (ii) the gauge-fixing term, and (iii) ... 3 As you already wrote, the (3/2)\partial^2 c term is needed for the current to be a one-form i.e. (1,0) tensor field; see also page 131 of Polchinski's String Theory, volume 1. This means that if you compute the OPE$$ 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), $$see e.g. ... 3 "Something is conserved for an action" simply means that the action carries a zero overall value of "something" (for an additive quantity). In this case, the action has N_{gh}=0. It follows that the equations of motion derived from the action imply dN_{gh}/dt=0. Most typically, they imply \partial_\mu J^\mu_{gh}=0 i.e. the local continuity law for a ... 3 I) Since total divergence terms do not contribute to Euler-Lagrange (EL) equations, cf. e.g. this Phys.SE post, one could just integrate the Faddeev-Popov \bar{c}c term by part so that there are no more than first derivatives present and the standard form of the EL equations applies. II) Alternatively, in the presence of higher derivatives, the EL ... 2 The central charge counts the number of degrees of freedom only for matter fields living on a flat manifold (or supermanifold in the case of superstrings). An example where this counting argument fails for matter fields is the case of strings moving on a group manifold G whose central charge is given by the Gepner-Witten formula: c = ... 2 Bosonic path integrals :$$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]}$$are not mathematically well-defined, because of the presence of the imaginary unit in the exponential. ... 2 Kugo and Ojima's work was one of the major breakthroughs in understanding the role of BRST in the quantization of gauge theories. Historically BRST was discovered in the path integral formalism. The understanding of this theory as a cohomology theory started from the Kugo and Ojima's work. Now, the action is BRST invariant with and without the Gaussian ... 2 In your case, I think that \eta is playing the role of \epsilon in David Bar Moshe 's answer, that is : \eta is a real Lorentz-scalar Grassmann variable. So, you will have :$$\delta_\Omega \bar\psi= \overline{\delta_\Omega\psi} = \overline {i\eta\psi} = -i \eta \bar\psi = i \bar\psi \eta$$2 The expectation of the axial current divergence in a \theta shifted QCD vacuum is given by \partial_{\mu} \langle J^{\mu5}_{\mathrm{inv}} \rangle_{\theta} = 2m_q \langle \bar{q}i\gamma^5q \rangle_{\theta} + \langle \Xi \rangle_\theta, where the first term on the right hand side is the explicit breaking term due to the quark masses and the second ... 2 I did not furnish all the details because it would be too long, but I give some hints at the end of the answer. I have used the formulae :T^g: ~= ~:2(\partial c) b + c(\partial b): and :\frac{1}{2}cT^g: ~= ~:bc \partial c:, when there is an ambiguity in the calculus. We begin by :$$j_B = cT^m+:\frac{1}{2}:cT^g:+\frac{3}{2}\partial^2c=cT^m+:bc\partial ... 2 I got it.$\epsilon$anticommutes with$b_A$. 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 LSZ formula definitely applies to theories with massless gauge bosons, like QED and QCD. The S-matrix is given by the LSZ formula, which relates the former to correlation functions, which are in turn given by a path integral. The LSZ formula assumes asymptotic in- and out-states for particles of interest. In the path integral formalism, gauges can be ... 2 On one hand, by including the Lautrup-Nakanishi field$B^a$, we have an off-shell BRST formulation, i.e. we can prove the nilpotency of the BRST transformation without using the (Euler-Lagrange) equations of motion. On the other hand, for some applications, a simpler on-shell BRST formulation (where the Lautrup-Nakanishi field$B^ahas been integrated ... 2 You are always allowed to introduce a new integration variable as long as its not its already being summed over. This might be more clear in discrete form: \begin{align} \int d x \, f (x) & \rightarrow \Delta x\sum _i \,f ( x _i ) \\ & = \big( N \Delta y\sum _j g ( y _j ) \big) \Delta x \sum _i f ( x _i ) \\ \end{align} where N \Delta ...

2

You should think in terms of the norm of a state, and what happens here is that you have negative-norm states which you don't want, otherwise you can't construct a Hilbert space. Copying the formula in Wikipedia, $$\langle\vec{k}_a;\epsilon_\mu|\vec{k}_b;\epsilon_\nu\rangle=(-\eta_{\mu\nu}){1\over 2|\vec{k}_a|}\delta(\vec{k}_a-\vec{k}_b)$$ you see that ...

2

In a nutshell, the Grassmann-odd Faddeev-Popov ghosts fields appear from the exponentiation of the Faddeev-Popov determinant, i.e., when we write the determinant as a Gaussian integral over Grassmann-odd variables. The Faddeev-Popov determinant can roughly be viewed as a Jacobian factor in the path integral that appears because the path integral variables ...

2

Hints: The left-hand side $$-\frac{1}{2}g^2f^{abc}f^{cde}\left(A_{\mu}^{b}c^{d}c^{e}+{\rm cycl}(b,d,e)\right)$$ of eq. (16.47) can be relabelled as $$-\frac{1}{2}g^2\left( f^{abc}f^{cde}+{\rm cycl}(b,d,e)\right)A_{\mu}^{b}c^{d}c^{e}.$$ P&S assume that the structure constants $f^{abc}$ are totally antisymmetric, cf. text below eq. (15.79).

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

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. ...

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