Pedagogical introduction to BRST

PHYSICS REPORTS (Review Section of Physics Letters) 184, Nos. 2-4 (1989) 147-165. North-Holland, Amsterdam 147PEDAGOGICAL INTRODUCTION TO BRSTAntti J. NIEMICERN, CH-1211 Geneva 23, Switzerland*Abstract:We review some recent results in the BRST quantization of constrained systems. In particular, we explain how the Parisi-Sourlas extension of BRST supersymmetry emerges and how it implies formal equivalence with reduced phase space quantization. We construct the pertinent generators of Parisi-Sourlas superrotations for both first and second class constraint algebras, and as an explicit example we consider open bosonic strings.1. 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, 8. 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 refs. 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 sj)ace 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.2. 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 integrals* Permanent address: Research Institute for Theoretical Physics, University of Helsinki, Finland.0370-1573/89/$6.65 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)148 Common trends in statistical physics and field theoryJ dD2xFx2), (2.1)where, with no loss of generality we take F to be an 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 space. In the simplest case we may view the constrained version of (2.1) to be an integral over the 心=x 2 = 0 subspace of a D-dimensional Euclidean space, J(2.2)with F now an SO(D) rotation invariant function. More generally, if we assume that the Z) - 2 subspace is determined by two functions (x) = = 0, instead of (2.2) we have integrals such as*)|d5()8(2)det|F(x2), (2.3)which reproduces (2.2) at least locally, when we solve the constraints = b = Cab, dctCab #0 , (2.18a, b)where Cab is an antisymmetric matrix, nondegenerate on the surface (2.18a). Clearly, (f)a can be viewed as a combination of a contains both qa and pa, i.e. fl)Vdet丨，d exp(i J qlpj , (2.20)where 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 7a 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 Ucab, Vba, Cab this quantization usually fails to be practical, and alternative formulations become necessary. Further difficulties become apparent in the quantization of relativistic constraint algebras. For example in applications to the Yang-Mills theory the variables x2 in (2.2) correspond to the timelike and longitudinal componentsA.J. Niemi， Pedagogical introduction to BRST 151of 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 Fand the measure in (2.2)，(2.3). Similarly, in string theories xx 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 covariance can be restored in the pertinent version of (2.15) by introducing proper changes of variables, at the level of 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 self-dual tensor fields, or higher rank constrained systems such as (super)membranes. Similarly, the quantization of second class constraint algebras such as the Green-Schwarz superstring or the Brink-Schwarz superparticle has not been very successful 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 BRST quantization of constrained systems 3-6.3. 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 applications 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 likeI dD2x F(x2) - im J dDx ddy F(x2 + 2). (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 Z) 2-dimensional integral as an integral over a Z) + 2-dimensional superspace. In particular, the D-dimensional rotation invariance of the measure and of fin (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 X which in addition to the D Euclidean (or Minkowskian) coordinates includes two anticommuting coordinates 0 and 0 such that 02 = d2 = 00 06 = 0. We denote coordinates on 2 byya = (y, /, ye) = (x,e,d) (3.2)and generalize D-dimensional rotations into D + 2-dimensional superrotations by replacing x by152 Common trends in statistical physics and field theory(3.3)y2 = gaPyayp = x2 + ee,where the nonvanishing components of the metric tensor ga(i are=%v， See = = 2 - (3-4)The invariance of (3.3) determines the group of orthosymplectic superrotations, or Osp(D|2). Its bosonic parts are rotations SO(D) that leave x invariant and Sp(2) that leave 80 invariant. But in addition there are also graded transformations that mix with 6 and 0, for example/ / + 6- d - , dd (3.5a, b, c)with an (infinitesimal) anticommuting D-vector. By considering the complete invariance of (3.3)，the graded Lie algebra of Osp(Z)|2) can be derived 10,尺 a“， RyS = gPyRa8 - (-)y8gSRay - (-ygayRd + (广 ( ， (3.6)where the grading (-)a/3 equals minus one if both a and /3 are anticommuting indices, plus one otherwise.Since (3.6) is a grading of ordinary rotations, its representations can be constructed by grading representations of ordinary rotations. For example, if we grade the differential representation of ordinary rotations we find the differential representation of (3.6),R = (-YY-ya , (3.7)whereda= ga = d!dx Side, side). (3.8)Notice in particular that generators such as(3.9)are nilpotent. In a Minkowski space with coordinates x we find similarly that light-cone generators such as Re are nilpotent. In the following section we shall argue that in phase space applications the BRST operator can be identified with the nilpotent generator Rd of an Osp(l, 1|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(Z)|2) invariant combinationsA.J. Niemi, Pedagogical introduction to BRST 153f dDxd0de F(x2 + 00) = I d“2x F(x2) - F() = | dD2x Fx2), (3.11)where we have restricted ourselves to functions F with F(oo) = 0. From this we conclude that in Osp(D|2) invariant integrations the 6 and 6 variables cancel out two bosonic variables. Consequently 6 and 6 can be viewed as coordinates in a minus-two-dimensional space in the sense of analytic continuation as in dimensional regularization. Indeed,00lim J d2x dDy F(x2 + 2) = Jim j da j d2x dDy ea(x2+y F(a)o= 7r J d2x dd d0 F(x2 + 06). (3.12)The lhs of (3.11) is then the desired representation of (2.1). In particular, since xx and x2 are now cancelled against 8 and 6 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 Osp(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 eM now a nontrivial function of the coordinates ya,xvxv + (1/N)xv, R， = (l/NyPrT0 , (3.13a)+ /N)l = ff + (l/N) , 00 + (l/N)d, Rd = et (3.13b, c)where N is an integer introduced for later convenience, and (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 dDxd0ddF(y2)- f dDx d6 de 1- 士 % Re + o) F(y2). (3-14)We repeat (3.14) N times in a basis (e.g. light-cone basis) with Re nilpotent. In the limit the Jacobian exponentiates,(3.15)and we getf dDxde d6F(y2) = j dDxd3d0el， ,rBteF(y2).(3.16a)154 Common trends in statistical physics and field theoryIn particular, since the lhs of (3.16a) is independent of 少，the rhs must also be independent of 少，which constitutes a proof*) of the FradkinVilkovisky theorem 1 which states the gauge i.e.屮 independence of the path integral in the phase space version of (3.16a). _Notice that with Rv another nilpotent operator such that RP, , = 0 and b = a,ijb = -81. (4.4)These variables then complete the Parisi-Sourlas phase space. For each a its unphysical subspace is an eight-dimensional graded phase space, with an equal number of bosonic and fermionic variables. Half of these variables are coordinates and half are momenta, hence the unphysical coordinate and momentum subspaces each are four dimensional. This implies that for each a the relevant ortho- symplectic supergroup must be either Osp(2|2) or Osp(l, l|2). We shall now establish that it is Osp(l,l|2).For each a we introduce a four-dimensional Parisi-Sourlas superspace with light-cone position variablesqaa = (xa, xa, 0% da) = (F-7T a) (4.5)and momentum variablesdefPca = (Pa， K， aJ = (Ga,Aa,a,ija). (4-6)156 Common trends in statistical physics and field theoryThe Poisson brackets of these variables ar