Pedagogical introduction to BRST-1

CERN.TH-5195/88PEDAGOGICAL INTRODUCTION TO BRST*Antti J. NiemiCERN， CH-1211 Geneva 23, SwitzerlandAbstractWe review some recent results in the BRST quantization of constrained sys-tems. In particular, we explain how the Parisi-Sourlas extension of BRSTsupersymmetry emerges and how it implies formal equivalence with reducedphase space quantization. We construct the pertinent generators of Parisi-Sourlas superrotations for both first and second class constraint algebras, andas an explicit example we consider open bosonic strings.* To appear in the Proceedings of Common Trends in Particle and Condensed Matter Physics, Cargese, May 23 - June 4, 1988tPermanent Address: Research Institutefor Theoretical Physics, University of Helsinki, Finlandbitnet: ANIEMIFINUHCBRef.CERN.TH-5195/88 September 1988I. IntroductionRecently, progress has been made in understanding conceptual aspects of BRST invariance and its role in the quantization of constrained systems 1-6. In particular, it is now clear that BRST quantization is conceptually quite different from the conventional reduced phase space approach 7,8j. In the latter, unphysical phase space degrees of freedom are eliminated using the constraints i.e. fixing the gauge, and the physical degrees of freedom are then quantized. On the contrary, in BRST quantization unphysical degrees of freedom are not eliminated. Instead, the quantum theory is formulated in an extended phase space which is a Parisi-Sourlas superspace 9,10 with ghost degrees of freedom viewed as negative dimensional coordinates. The BRST operator is identified with the generator of a particular Parisi-Sourlas superrotation in this phase space, and the equivalence to the reduced phase space quantum theory is a consequence of the Parisi-Sourlas mechanism 2-6.In this article we shall review conceptual aspects of this new approach to the quantization of constrained systems, following the discussion in 3-6. In particular, we shall explain how the Parisi-Sourlas supersymmetry extends the BRST supersymmetry, and the role that it plays in the covariant phase space quantization of constrained systems. We explain why Parisi-Sourlas supersymmetry ensures unitarity and positivity, and in particular we discuss the formal equivalence between extended and reduced phase space quantum theories. We construct the Parisi-Sourlas Osp(l,l|2) extension of BRST for both first and second class constraint algebras, and as an explicit example we present this construction for free open bosonic strings.II. Reduced Phase Space ApproachThe basic idea in the reduced phase space quantization of constrained systems is very simple: We are interested in path integrals that can be conceptually viewed as infinite dimensional versions of ordinary D-2 dimensional Euclidean space integralsJ dD2xFx2) (2.1)where, with no loss of generality we take F to be a SO(D-2) rotation invariant function. In theories with constraints, the integration domain in the path integral version of (2.1) is specified only implicitly through constraints, as a subspace of some higher dimensional-2-space. In the simplest case we may view the constrained version of (2.1) to be an integral over the xi = X2 = 0 subspace of a D dimensional Euclidean space,J dDxS(xx)S(x2)F(x2) (2.2)with F now a SO(D) rotation invariant function. More generally, if we assume that tlie D-2 subspace is determined by two functions (x) = 2(x) = 0, instead of (2.2) we have integrals such as1(2.3)which reproduces (2.2) at least locally, when we solve the constraints i = a,4b: =cab k detCab / 0 (2.18.6)where Cab is an antisymmetric matrix, nondegenerate on the surface (2.18.a). Clearly, 4a can be viewed as a combination of a qapa with qa and pa the unphysical components of the phase space variables. Locally, the physical phase space T* is then determined byqa pa = 0 a = X,M (2.19)In these local coordinates the quantum theory is again defined by (2.12), (2.16) or equivalently by (2.13), but since 4a contains both qa and pa, i.e. a,4hexpi J piqx (2.20)-5-wliere we have assumed the reformulation (2.16). The Faddeev-Popov representation (2.15) can also be introduced, except that now only a single self-conjugate ghost rja appears.The previous discussion summarizes the reduced phase space method 7,8. In principle, it provides a solution to the problem of quantization for systems with irreducible first and second class constraint algebras. However, except for the simplest case with constant Ub, Fj*, Cab this quantization usually fails to be practical, and alternative formulations become necessary. Further difficulties become apparent in quantization of relativistic constraint algebras. For example in applications to Yang-Mills theory the variables xi,X2 in (2.2) correspond to the timelike and longitudinal components of the gauge field and their canonical conjugates, and if we eliminate these variables we also sacrifice manifest Lorentz invariance. This can be viewed as an analog of breaking the manifest SO(D) invariance of F and the measure in (2.2), (2.3). Similarly, in string theories x and x2 are the timelike and longitudinal components of the spacetime coordinates. Now elimination of these components also eliminates part of the Minkowski space, and while manifest co- variance can be restored in the pertinent version of (2.15) by introducing proper changes of variables, at the level o hamiltonian operator formulation covariance fails to be manifest.The difficulties with the reduced phase space approach become more serious when one tries to quantize Lorentz invariant reducible constraint algebras such as antisymmetric tensor fields that naturally appear in supergravity and superstring theories, or higher rank constrained systems such as (super)membranes. Similarly, the quantization of second class constraint algebras such as Green-Schwarz superstring or Brink-Schwarz superparticle has not been very succesful in the reduced phase space formalism. In order to quantize these and other complicated systems that are presently becoming popular, it is necessary to introduce manifestly covariant techniques. In the following we shall describe one such technique, the Parisi-Sourlas reformulation of the BUST quantization of constrained systems 3-6.III. Parisi-Sourlas SupersymmetryIn this section we shall outline a conceptual reformulation of (2.1)-(2.3) that preserves the SO(D) invariance of the measure and the function F in (2.2), (2.3), and reproduces (2.1) without explicitly restricting the integrals to D-2 dimensional subspaces. In appli-6-cations to covariant constrained systems this implies that symmetries such as manifest Lorentz invariance can be preserved. Such unrestricted representations of (2.1) in higher dimensional spaces are possible in the Parisi-Sourlas formalism 9,10: Instead of reducing the integral as in (2.2), (2.3) we shall directly reproduce (2.1) using analytic continuation in dimensions, i.e. something likej dD2F(x2) im21 dDxddyF(x2 + y2) (3.1)We then realize the negative dimensional coordinates in (3.1) by anticommuting variables, and in this way we find a representation of the D-2 dimensional integral as an integral over a D+2 dimensional superspace. In particular, the D dimensional rotation invariance of the measure and F in (2.2), (2.3) does not have to be compromised, in fact it will be extended into a D+2 dimensional Parisi-Sourlas superrotation invariance. In phase space applications these Parisi-Sourlas superrotations then generalize the conventional BRST supersymmetry.In order to explain how the negative dimensional coordinates in (3.1) can be realized by anticommuting variables, consider an extended D+2 dimensional superspace S which in addition to the D Euclidean (or Minkowskian) coordinates includes two anticommuting coordinates 9 and B such that $2 = d2 = $ - $ = 0. We denote coordinates on E byya = (yyy9) - (3.2)and generalize D dimensional rotations into D+2 dimensional superrotations by replacing x2 byy2 ga0yay0 = + (3.3)where the nonvanishing components of the metric tensor ga0 are9 = # y (3.9)are nilpotent. In a Minkowski space with coordinates we find similarly that light-cone generators such as R9 are nilpotent: In the following section we shall argue that in phase space applications the BRST operator can be identified with the nilpotent generator R9 of an Osp(l， l|2) algebra.We define integration over anticommuting variables by(3.10)and consider superspace integrals of functions F that depend on their variables in Osp(D|2) invariant combinationsj dPxdUB F(x2 +99) = i J dD2xF(x2) - F(oo) = j dD2xF(x2) (3.11)where we have restricted to functions F with F(oo) 0. From this we conclude that in Osp(D|2) invariant integrations the 0 and 0 variables cancel two bosonic variables.-8Consequently B and 0 can be viewed as coordinates in a minus two dimensional space in the sense of analytic continuation as in dimensional regularization. Indeed,OOd2xdDyF(x2+y2) = da j d2xdDye-a、 x2+y2、 Pa)二 ir j d2xded0F(x2+ 0)o(3.12)The l.h.s. of (3.11) is then the desired representation of (2.1). In particular, since Xi and xi are now cancelled against 6 and B instead of being eliminated explicitly as in (2.2),(2.3) , in the representation (3.11) of (2.1) the D dimensional rotation invariance in (2.2),(2.3)(corresponding to e.g. manifest Lorentz invariance in phase space versions) remains unbroken, and is in fact extended into a D+2 dimensional superrotation invariance.In applications to constrained systems, the integrands in the path integral versions of (3.11) in general do not have the simple 0sp(D|2) form of (3.11). However, it turns out that in the unphysical sector, these integrands can always be related to integrands of the form (3.11) by a simple change of variables. In order to explain this change of variables, we consider a generalization of (3.5) with now a nontrivial function of the coordinatesya, . .(3.13.a)e e + =4- (3.13.6)o e + o,R0= 6 (3.13.c)where N is an integer introduced for later convenience, and 屯 (y) is an arbitrary function of the superspace coordinates ya generalizing to a coordinate dependent quantity. Even though the squared length (3.3) remains invariant under (3.13) the measure in (3.11) fails to be invariant. Instead we have a super JacobianI dDxdUe Fy2) j dDxd$d9l - , + 0)F(y2) (3.14)We repeat (3.14) N times in a basis (e.g. light-cone basis) with R9 is nilpotent. In the limit N oo the Jacobian exponentiates,(l- e-_ (3.15)and we getJ dDxdUBF(y2) - J dDxSdO eR， l(， Fy2) (3.16.a)-9-In particular, since the l.h.s. of (3.16.a) is independent of 审 ，the r.h.s. must also be independent of which constitutes a proof3 of the Fradkin-Vilkovisky theorem 1 which states the gauge i.e.审 independence of the path integral in the phase space version of (3.16.a).Notice that with another nilpotent operator such that i?, , /2 0 and $ another arbitrary function of ya, (3.16.a) generalizes intoI dDxd$de F(y2) = I dDxdede + $,RF(3/2) (3.16.6)and so on.In the following section we shall find that in the phase space versions the integrands always have the simple Osp structure of (3.16) in the unp