Wednesday, October 26, 2005

25 years of Polyakov action

In 2005, we also celebrate the 25th anniversary of the Polyakov action. Although many of us were still in kindergarden back in 1980, we just could not ignore this important contribution of the bloc of peace to string theory. The article about Fadějev-Popov ghosts in string theory seems appropriate for the week before Halloween.

Even more mysteriously, Saša Polyakov celebrates his 60th birthday, congratulations!

Polyakov - and several other people - introduced the BRST methods to string theory. BRST quantization is a powerful technology to deal with gauge invariance, especially non-Abelian gauge invariance. Some physics fans do not understand that BRST is mostly formalism - a calculational framework to deal with various unphysical states - but physics is only the very final product of the BRST quantization that could also be found by other means - for example, by light-cone and other gauge-fixed approaches.




Imagine that you have an Abelian gauge symmetry G. Because it is a gauge symmetry, the physical states must be invariant under all generators of the group called L_i. You may impose these dim(G) conditions as the single requirement that all the physical states are annihilated by
  • Q = c^i L_i

where "c^i" are arbitrary coefficients i.e. as the condition that "Q psi = 0". Also, all states of the form

  • psi = Q lambda

are pure gauge and therefore unphysical. It's because they are combinations of the variations "L_i psi", the differences between two infinitesimally similar states identified by the gauge symmetry. What I said also works if "c^i" are chosen to be new degrees of freedom, namely Fadějev-Popov ghosts. The advantage of making them anticommuting is that

  • Q^2 = 0

i.e. the operator Q is nilpotent, much like the exterior derivative operator "d" acting on the differential forms. The physical states can be found as those that are annihilated by "Q" but not of the form "Q lambda" - i.e. as cohomologies of "Q" which is a well-defined concept because of nilpotency.

At your Halloween party, you should not forget that there are good ghosts - the Fadějev-Popov ghosts - who allow us to fight against the bad ghosts - i.e. the states with a negative squared norm. A good ghost usually asks a bad ghosts whether he is BRST-closed but not BRST-exact, and if the bad ghost answers "No", he is simply screwed. This allows Fadějev, Popov, their ghosts, BRST, and Polyakov to keep the bad ghosts - the terrorists with a negative squared norm - away from the borders of the physical Hilbert space. Consequently, all of us are happy and we may enjoy our lives in the world where probabilities are guaranteed by the Hilbert space government to be positive. It's one of the exceptional examples in which a government bureaucracy is actually useful.

The real power of the BRST formalism of course only occurs if we consider non-Abelian gauge symmetries. In this case we must add the dual variables "b_i" (antighosts) associated with each "c^i" (with ghosts) and we must add a "b.c.c" term to "Q" to keep it nilpotent. It turns out that we must only add "1/2 c^i L_{i}^{ghost part}" because the ghosts, together with their own symmetry generator, prefer to be half-real, half-imaginary.

The ghosts "c^i" transform just like the parameters of the gauge transformations and "b_i" transform as the generators themselves. For conformal symmetry, "c^i" are fields of dimension "-1" while "b_i" have dimension "+2". There is a canonical action for the "bc"-ghosts, too.

For any theory with gauge (local) symmetry - which includes diffeomorphism and Weyl symmetries relevant for the string worldsheet - the contribution of 1-loop diagrams with "bc"-ghosts running in the loops cancels the contribution of unphysical modes, and it simplifies both the definition as well as calculations of loop amplitudes. Alternatively, in the path integral formalism, the one-loop determinant from the FP-ghosts cancels the unphysical degrees of freedom, and removes an unwanted Jacobian from a gauge-fixing.

In the case of string theory, the Polyakov (BRST) approach is very powerful because it allows us to rewrite the loop integrals over the moduli spaces of Riemann surfaces as very elegant integrals over the zero modes of the "b"-antighost. This simplifies the proofs of finiteness while we may keep the (Lorentz) covariance of the formulae manifest.

It was harder to keep the spacetime supersymmetry manifest, and Berkovits' pure spinor approach based on a new kind of ghosts is the first super-Poincaré quantum approach to the string worldsheet that we know.

My guess is that the BRST approach is more popular among the majority of theoretical physicists; it makes it even more impressive that Green and Schwarz preferred their light-cone gauge approach with manifest supersymmetry when they did their important discoveries peaked in 1984.

A generalization of the BRST approach, the Batalin-Vilkovisky approach, also involves ghosts for ghosts and ghosts for ghosts for ghosts, and so forth. Recall that ghosts - who themselves always violate the spin-statistics relation - also celebrate Halloween (together with antighosts) where they dress up as ghosts for ghosts; ghosts for ghosts (not to be confused with antighosts) are not quite like us, but they have the same spin-statistics relation as we do.

It is a relatively big question whether a new formalism analogous to the BRST approach will have to become important in the future when we figure out a more complete and unified way to formulate the theory of quantum gravity i.e. string/M-theory.

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