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ON THE COVARIANT QUANTIZATION

arXiv:hep-th/9110059v1 21 Oct 1991QMW-91-18June 1991ON THE COVARIANT QUANTIZATIONOF THE 2nd. ILK SUPERPARTICLE.J.

L. V´azquez-Bello§Physics Department, Queen Mary and Westfield College,Mile End Road, London E1 4NS,UNITED KINGDOM.ABSTRACTThis paper is devoted to the quantization of the second-ilk superparticle usingthe Batalin-Vilkovisky method. We show the full structure of the master action.By imposing gauge conditions on the gauge fields rather than on coordinates wefind a gauge-fixed quantum action which is free.

The structure of the BRST chargeis exhibited, and the BRST cohomology yields the same physical spectrum as thelight-cone quantization of the usual superparticle.§ Work supported by CONACYT-MEXICO.

1. Introduction.Constraints of a dynamical system are classified as first and second class, ac-cording to their Poisson bracket relations [1,3].

However, the mixing of first andsecond class constraints, and the difficulty of their separation in a covariant wayhas proved to be a problem in the covariant quantization of superparticles andstrings [2,9].Covariant quantization of the original superparticle [4,5], together with its su-perstring generalizations, has proved problematic because of the mixing of first andsecond class constraints [7-10]. Progress has been made in the covariant quantiza-tion of the superparticle using the Batalin and Vilkovisky methods (BV) which canbe applied in the presence of second-class constraints [11,12], or using harmonicvariables for the separation of the fermionic constraints into first and second class[13,14].However, the latter approach suffers from some non-locality problems.Harmonic variables are additional bosonic variables added to the usual bosonicand fermionic coordinates.

Appropriate first class constraints are imposed on theharmonic variables to assure that they are purely gauge degrees of freedom [15].Following this idea of adding variables to separate constraints, an alternate possibil-ity is the introduction of fermionic coordinates to render all fermionic constraintsfirst class. Siegel has shown that by introducing a momentum conjugate to thefermionic variable θ and a gauge field for the fermionic world-sheet symmetry, onecan obtain systems with purely first class constraints [10,16].

However, althoughseveral alternatives have been proposed to address the quantization problem of thesuperparticle, which are based on modifications to the formulation given by Siegel[16], none of them has led so far to a satisfactory solution, essentially because asuitable gauge-fixing fermion has not been found or the BRST operator does notgive the correct cohomology [7,18-20,24]. To avoid some of the issues involved inthe covariant quantization of those formulations, further modifications have beenproposed.

In [21,22], two first-class formulations have been proposed, the first-ilkand the second-ilk superparticle. A similar model was proposed in [23].

Both of2

them allow covariant quantization. Another proposal was made in [17] where themodified action involves additional fermionic coordinates.

In each case, the BRSTcohomology gives the spectrum of N=1 super Yang-Mills.The treatment of the quantum second-ilk superparticle in [21] was incompleteand the purpose of this paper is to complete the analysis, using the methods ofBV. We explicitly show all the relevant steps in the calculation of the masteraction and the BRST charge, correcting the form of the master action given in[21].There, a BRST charge for the second-ilk superparticle was written downand one of our aims here is to compare this with the BRST charge that arises inthe BV approach.

Section 2 reviews the original superparticle action, the second-ilk superparticle action and their symmetries. In section 3, we analyse the ghoststructure which provides a representation of the BRST algebra to find the minimalset of fields which enter in the BV procedure.

In section 4, the master action of theBV method is obtained and is used in section 5 to determine the quantum actionand the BRST charge.2. Classical Actions and Symmetries.The original superparticle (which we shall refer to as SSP0) described by an ac-tion with mixed first and second class constraints was formulated in [4,5], and gen-eralized to superstrings by Green and Schwarz [6].

The evolution of the SSP0 super-particle is represented by a world-line in ten-dimensional superspace (xµ(τ), θ(τ))parameterized by τ, where µ = 0, ..., 9 and θ is an anti-commuting Majorana-Weylspinor. The SSP0 action is given bySSSP0 =Zdτpµ( ˙xµ −i¯θγµ ˙θ) −12ep2(2.1)where ˙θ = dθ/dτ, pµ is the momentum and e is the einbein on the world-line.

TheSSP0 action is invariant under rigid space-time supersymmetry transformationstogether with world-line reparameterizations and a local fermionic symmetry, al-though there is no gauge field for this symmetry. In a covariant quantization it is3

necessary to find a covariant gauge choice for the fermionic symmetry. As there isno gauge field for the local fermionic symmetry, this can only be fixed by imposingconditions on (xµ, θ, e).

There have been numerous attempts to find a covariantquantization of the SSP0 superparticle given by (2.1), but there is no satisfactorycovariant gauge choice [8]. aIn this paper, we present the results of the covariant quantization of a fur-ther modification of the superparticle, the second-ilk superparticle of [21].

Thesecond-ilk superparticle has only first-class constraints.This new superparticleaction is formulated in a superspace with coordinates (xµ, θ0, . .

. , θ2n, .

. .

), whereθ0, . .

. , θ2n, .

. .

are anti-commuting spinors. The action isS0 = pµ ˙xµ −gp2 −iψ1/p(d0 −2/pθ0) + iXn=0+∞˙θ2n(d2n −/pθ2n)−Xn=0+∞λ2n+1(d2n + d2n+2 −2/pθ2n+2).

(2.2)and is invariant under global space-time supersymmetry and a number of localsymmetries. These symmetries are given byδθ0 = κ/p −iθ1,δxµ = 2ξpµ + iκγµ(d0 −2/pθ0) −/pγµθ0+Xn=0+∞θ2n+1γµ(θ2n −θ2n+2),δg = ˙ξ + 2iψ1/pκ,δλ2n+1 = ˙θ2n+1,δψ1 = ˙κ,δθ2n = −i(θ2n+1 + θ2n−1),δd2n = −2iθ2n+1/p.

(2.3)The gauge fields g, ψ1 and λ2n+1 are all lagrange multipliers imposing the infinite4

set of constraintsp2 = 0,/pd0 = 0,d2n + d2n+2 −2/pθ2n+2 = 0. (2.4)The momentum pµ is an auxiliary field whose algebraic equation of motion is givenbypµ = 12g(4iψ1pµθ0 + ˙xµ−iψ1γµd0 −iXn=0+∞˙θ2nγµθ2n +2Xn=0+∞λ2n+1γµθ2n+2).

(2.5)The remaining classical field equations are˙pµ = 0,/p ˙θ0 −p2ψ1 = 0,˙θ2n/p −iλ2n−1/p = 0.(2.6)3. The Ghost Structure of the Superparticle.For theories in which the classical gauge algebra closes off-shell, it is straight-forward to construct a BRST invariant action.

For theories in which the gaugealgebra only closes on-shell, however, the standard BRST approach do not workand it is convenient to use the BV method to quantize them [11,12]. Further-more, the quantum action constructed using BV or alternate procedures should beinvariant under BRST transformations which reflect the gauge invariance at theclassical level.

The first step towards the covariant quantization of models withopen gauge algebras, is to study the ghost structure in order to find the minimalset of fields that enter in the BV quantization procedure. The ghost structure isfound by demanding that the minimal set of fields provide a representation of theBRST algebra.Now consider the application of the BV method to determine the minimal setof fields of the second-ilk superparticle model given by (2.2).

The BRST trans-formation of any of the classical fields appearing in (2.2), is given by replacing5

the parameter of the gauge transformation with the corresponding ghost.Forthe action (2.2), we introduce ghosts (c, ˜θ1, . .

. , ˜θn, θ2n+1, ψn) corresponding to theclassical symmetries (2.3) with gauge parameters of opposite Grassmann parity tothe ghost set, (ξ, κ, .

. .

, ˜θn, θ2n+1, ψn). Then, considering the on-shell nilpotencycondition of the BRST transformations on all the classical and ghost fields, wewill construct the ghost spectrum of the superparticle (2.2) whose structure canbe represented by infinite towers of ghost fields.For the 10-dimensional superparticle defined by the action (2.2), the BRSTtransformations for the classical fields aresxµ = 2cpµ + i˜θ1γµ(d0 −2/pθ0) −/pγµθ0+Xn=0+∞θ2n+1γµ(θ2n −θ2n+2),sg = ˙c + 2iψ1/p˜θ1 + iψ2(d0 −2/pθ0),sθ0 = ˜θ1/p −iθ1,sλ2n+1 = ˙θ2n+1,sψ1 = −(˙˜θ1 −/pψ2),sθ2n = −i(θ2n+1 + θ2n−1),sd2n = −2iθ2n+1/p.

(3.1)The BRST transformations for the ghost fields are given by demanding the nilpo-tency of the generator s, up to terms that vanish when the classical equations ofmotion are satisfied, so that the BRST transformations becomes nilpotent on-shell.Thens˜θn = (−)n + 1/p˜θn+1,sψn+1 = (−)n + 1(˙˜θn+1 −/pψn+2),sc = i˜θ1/p˜θ1 −i˜θ2(d0 −2/pθ0),(3.2)where θ2n+1 and ˜θ1 are ghosts, while ˜θ2, . .

. , ˜θ2n+1 are ghosts-for-ghosts with Grass-mann parities and space-time chiralities that alternate with the level number.

How-ever, the construction of the quantum action requires the introduction of some6

new fields. For each n’th generation ghost field, one introduces an anti-ghost andNakanishi-Lautrup (NL) fields, plus ‘extra-ghosts’ together with the correspondingextra-NL fields, so that at the n’th generation the ghost is supplemented by BRSTdoublets.

This set of fields is always sufficient to construct a quantum action. Theminimal set of fields that enter in the BV quantization procedure is determinedby the classical gauge symmetries, together with the requirement that the BRSTtransformations of the classical fields and the ghosts should be nilpotent on-shell.This procedure also fixes much of the structure of the master action.

Then, theminimal set of fields for the first-class superparticle (2.2), based on the classicalsymmetries (2.3), consists of the classical and ghost fields introduced above,ΦAmin = {xµ, pµ, g, c, θ0, d0, ψ1, θ2n, d2n, λ2n+1, ˜θn, θ2n+1, ψn+1}. (3.3)A common feature of superparticle and superstring models is the infinite-reducibility of these systems.

The existence of an infinite number of ghost coordi-nates may seem to be a complication but they package together into an infinite-dimensional metaplectic representation of an orthosymplectic supergroup [8,26].4. BV QuantizationIn this section, the quantization of (2.2) will be discussed.

We begin by brieflyreviewing the BV procedure for constructing BRST transformations and the cor-responding quantum action, which works for arbitrary systems with open algebras[11]. The ‘minimal’ set of fields ΦA that enter in the BV method is determinedby the classical gauge symmetries, together with the requirement that the BRSTtransformations of the classical fields and the ghosts should be on-shell nilpotenti.e., s2 = 0 on any field, up to terms proportional to the equations of motion.

Foreach field ΦA a corresponding ‘anti-field’, ΦA⋆, of the opposite Grassmann parityis introduced. Then, the first step in determining the quantum action is to find7

the solution S(ΦA, ΦA⋆) to the master equation,∂rS∂ΦA∂lS∂ΦA⋆= 0,(4.1)subject to the boundary condition that the master action reduces to the classicalaction when the ‘anti-fields’ are set to zero, S(ΦA, ΦA⋆)|ΦA⋆=0 = S0(ΦA). Thesymbols r and l in (4.1) refer to right and left derivatives respectively, the orderbeing crucial due to the Grassmann nature of some of the fields.

Then, for anygauge fermion Ψ(ΦA), which is typically a sum of terms of the form (anti −field)×(gauge −condition), the corresponding quantum action is found by making thefollowing substitution for the anti-fields in S,ΦA⋆= ∂lΨ∂ΦA,(4.2)to give,SQ(ΦA) = S(ΦA, ΦA⋆)ΦA⋆=∂lΨ∂ΦA. (4.3)This quantum action is then invariant under the modified BRST transformationsgiven byˆsΦA =∂lS∂ΦA⋆ΦA⋆=∂lΨ∂ΦA(4.4)which are nilpotent up to terms which vanish when the equations of motion derivedfrom the quantum action SQ are satisfied.

The gauge fermion must be chosen so asto remove the gauge degeneracy of the classical action and give invertible kineticterms. Using (4.4) one can define the generating functional W[J] as usual via thepath integralexp(iW[J]) =Z[dΦ] exp{i(SQ[Φ] + JiΦi)},(4.5)where W[J] must be regularised and normalised.

The functional integral will thenbe BRST invariant using the quantum action (4.3), provided that the measure is8

BRST invariant. If not, then one seeks local counterterms to cancel the variationof the measure, so that the functional integral is BRST invariant.

Within the BVformalism, this corresponds to seeking corrections to the master action of the formW = S + 0(¯h), such that W satisfies12(W, W) = −i¯h∆W + ¯haνcν + O(¯h2)(4.6)where aν are the anomalies and cν are the ghost fields. A remarkable result is thatif there is no local solution to the modified master equation (4.6), then the theoryis anomalous and the quantum theory is inconsistent.

A discussion of anomaliesin the BV formalism with an explicit regularization of the path integral is given in[25].Using the BV formalism, we find the solution Smin to the master equation (4.1)for the minimal set of fields (3.3), when expanded in powers of anti-fields, takesthe formSmin = S0 + S1 + S2= S0 +ZdτΦA ⋆(sΦA) + 12ZdτΦA ⋆ΦB ⋆EAB(Φ, Φ⋆),(4.7)where S0 is the classical action, (2.2). The term linear in anti-fields is given byS1 =Zdτnθ ⋆0 (/p˜θ1 −iθ1) −iXn=1+∞θ ⋆2n (θ2n+1 + θ2n−1)−Xn=0+∞(−)n˜θ ⋆n /p˜θn+1 + xµ ⋆Xn=0+∞θ2n+1γµ(θ2n −θ2n+2)+ xµ ⋆2cpµ + i˜θ1γµ(d0 −2/pθ0) −/pγµθ0−2iXn=0+∞d ⋆2n /pθ2n+1 + g ⋆[˙c + 2iψ1/p˜θ1 + iψ2(d0 −2/pθ0)]+Xn=0+∞λ ⋆2n+1 ˙θ2n+1 +Xn=1+∞(−)nψ ⋆n (˙˜θn −/pψn+1)+ ic ⋆[˜θ1/p˜θ1 −˜θ2(d0 −2/pθ0)]o,(4.8)9

while the term quadratic in anti-fields isS2 =Zdτhg ⋆θ ⋆0 ˜θ2 −g ⋆Xn=0+∞˜θ ⋆n ˜θn+2 −g ⋆Xn=0+∞ψ ⋆n ψn+2+ 4ig ⋆c ⋆˜θ1˜θ2 −ig ⋆xµ ⋆(˜θ1γµ˜θ1 + ˜θ2γµθ0)−xµ ⋆Xn=1+∞ψ ⋆n γµ˜θn+1 −c ⋆Xn=1+∞(−)nψ ⋆n ˜θn+2i. (4.9)This minimal action (4.7) corrects the one given in [21].

The xµ⋆and c⋆terms inS1 and the term g ⋆c⋆term in S2 differ from these of [21]. The full master actionis then given by adding to Smin the non-minimal terms Snon−min, where antighostfields ˆc⋆, ˆ˜θn⋆, ˆθ2n+1⋆together with the corresponding NL fields π, ˜πn, π2n+1 arerequired, so that at the n’th generation the ghost fields are supplemented by nBRST doublets.

The non-minimal term is thenSnon−min = ˆc ⋆π +Xn=1+∞ˆ˜θn ⋆˜πn +Xn=0+∞ˆθ2n+1 ⋆π2n+1. (4.10)There are terms in the master action which are quadratic in ghost anti-fields, whichcannot be found by solving the master equation (4.1) to first order in anti-fields,so that it is necessary to consider higher order terms in the anti-fields to find themaster action.5.

Quantum Action and BRST-Charge.To define the quantum theory, it is necessary to ‘halve’ the extended configura-tion space (ΦA, ΦA⋆) by specifying a hypersurface which is defined by the condition(4.2), and the corresponding quantum action SQ is given by evaluating the masteraction S(ΦA, ΦA⋆) on (4.2) to give (4.3). However, the gauge fermion in (4.2) mustbe chosen so as to remove the gauge degeneracy of the classical action and giveinvertible kinetic terms, so that propagators are well defined.

The gauge fermiontypically includes a sum of terms consisting of anti-ghosts or extra-ghosts times agauge condition for each of the gauge fields.10

We now turn to discuss the choice of gauge. First, the classical gauge sym-metries (2.3) must be fixed.

We shall impose gauge conditions on the gauge fieldsg, ψn and λ2n+1 rather than on coordinates. The simplest gauge choice is g = 1,ψn = 0 and λ2n+1 = 0, which is implemented by the gauge fermion [21],Ψ(ΦA) =Zdτh(g −1)ˆc +Xn=1+∞ψnˆ˜θn +Xn=0+∞λ2n+1ˆθ2n+1i,(5.1)where ˆc, ˆ˜θn and ˆθ2n+1 are antighost fields.

However, these gauges can only beimposed locally, since each gauge field should be set equal to a constant modulusand these moduli should be integrated over. It will be convenient to consider aslightly more general class of gauges in which the gauge fields g, ψn and λ2n+1 areset equal to some fixed background fields ˜g, ˜ψn and ˜λ2n+1, so that g = ˜g, ψn = ˜ψnand λ2n+1 = ˜λ2n+1.Following the standard steps of the BV procedure, we derive a gauge-fixedquantum action which takes the free formSQ = pµ ˙xµ −p2 + (g −1)π + ˆc˙c + iXn=0+∞˙θ2nd2n+Xn=1+∞(−)nˆ˜θn˙˜θn +Xn=0+∞ˆθ2n+1 ˙θ2n+1+Xn=1+∞ψn˜πn +Xn=0+∞λ2n+1π2n+1,(5.2)after the following field redefinitionsd′2n = d2n −/pθ2n,π′ = π −p2,˜π′1 = ˜π1 −i/p(d0 −2/pθ0 + 2ˆc˜θ1),˜π′2 = ˜π2 + /pˆ˜θ1 + iˆc(d0 −2/pθ0),˜π′n = ˜πn + /pˆ˜θn−1 −(−)nˆcˆ˜θn−2,∀n ≥3,π′2n+1 = π2n+1 −d2n −d2n+2 + 2/pθ2n+2.

(5.3)We have dropped the primes for brevity. Further, the quantum action (5.2) canbe shown to be invariant under the modified BRST transformations (4.4), which11

satisfy ˆs2 = 0 when the field equations of motion derived from (5.2) are satisfied.As the quantum action defines a free theory, it is strightforward to quantize it byimposing canonical commutation relations on the operators corresponding to thevariables in (5.2). The modified BRST transformations for the classical fields arethenˆsxµ = 2cpµ + i˜θ1γµ(d0 −/pθ0) −/pγµθ0+ iˆc(˜θ1γµ˜θ1 + ˜θ2γµθ0)+Xn=0+∞θ2n+1γµ(θ2n −θ2n+2) −Xn=1+∞ˆ˜θnγµ˜θn+1,ˆsg = ˙c + 2iψ1/p˜θ1 + iψ2(d0 −/pθ0) −Xn=1+∞ˆ˜θnψn+2,ˆsλ2n+1 = ˙θ2n+1,ˆsθ0 = /p˜θ1 −iθ1 + ˆc˜θ2,ˆsθ2n = −i(θ2n+1 + θ2n−1),ˆsd0 = −i/pθ1 −p2˜θ1 −ˆc/p˜θ2,ˆsd2n = −i/p(θ2n+1 −θ2n+1),ˆsψ1 = −(˙˜θ1 −/pψ2 + ˆcψ3),(5.4)the modified BRST transformations for ghost and ghosts-for-ghosts fields areˆs˜θn = (−)n + 1/p˜θn+1 + (−)nˆc˜θn+2,ˆsψn+1 = (−)n + 1(˙˜θn+1 −/pψn+2) + (−)n + 1ˆcψn+3,ˆsc = i˜θ1/p˜θ1 −i˜θ2(d0 −/pθ0) + 4iˆc˜θ1˜θ2 −Xn=1+∞(−)nˆ˜θn˜θn+2,(5.5)and the modified BRST transformations for anti-ghosts and NL fields, plus ‘extra-ghosts’ together with the corresponding extra-NL fields are12

ˆsˆc = π + p2,ˆsˆ˜θ1 = ˜π1 + i/p(d0 −/pθ0 + 2ˆc˜θ1),ˆsˆ˜θ2 = ˜π2 −/pˆ˜θ1 −iˆc(d0 −/pθ0),ˆsˆ˜θn = ˜πn −/pˆ˜θn−1 + (−)nˆcˆ˜θn−2,ˆsˆθ2n+1 = π2n+1 + d2n + /pθ2n + d2n+2 −/pθ2n+2,ˆsπ = 0,ˆsπ2n+1 = 0,ˆs˜π1 = −2iπ/p˜θ1,ˆs˜π2 = /p˜π1 −i(d0 −/pθ0)π,ˆs˜πn = /p˜πn−1 + (−)nˆc˜πn−2 + ˆ˜θn−2π. (5.6)Taking into account the change of variables (5.3), the action (5.2) is invariant underthe BRST transformations generated by the BRST chargeQBRST = cp2 −Xn=0+∞θ2n+1(d2n + /pθ2n) −Xn=0+∞θ2n+1(d2n+2 −/pθ2n+2)+ i˜θ1/p(d0 −/pθ0) −iˆc˜θ2(d0 −/pθ0) + iˆc˜θ1/p˜θ1−Xn=1+∞ˆ˜θn/p˜θn+1 −ˆcXn=1+∞(−)nˆ˜θn˜θn+2.

(5.7)After the following change of variablesd2n →t2n,ˆ˜θn →˜tn,(5.8)and using the following definitionsdn = −tn + /pθn,qn = −tn −/pθn,(5.9)the BRST charge takes a simple formQBRST = Q0 + cp2 −ˆcf,(5.10)13

whereQ0 = −Xn=0+∞[˜tn/p˜θn+1 −θ2n+1(q2n + d2n+2)](5.11)andf =Xn=0+∞(−)n˜tn˜θn+2 −i˜θ1/p˜θ1(5.12)and we have defined ˜t0 = id0. Further, the BRST charge (5.10) is both conservedand nilpotent, so that the physical spectrum of the first-class superparticle corre-sponds to the cohomology classes of the BRST charge QBRST .

Our BRST operator(5.10) has exactly the same structure as that of [21], which was computed usingdifferent methods. The BRST cohomology is derived in [21] and yields the samephysical spectrum as the light-cone quantization of the usual superparticle.To summarise, we have used the methods of Batalin and Vilkovisky to quantizea second-ilk superparticle, (2.2), which is free of second class constraints.

The BVquantization of this model was also considered in [21] using a different approach.∗By solving the BV master equation (4.1) and using the gauge fermion (5.1) wefound a quantum action which, after some field redefinitions, led to the free quan-tum action (5.2), which is invariant under the BRST transformations generated bythe BRST charge (5.10). It is straightforward to perform an operator quantizationof (2.2), as in [21], and study the BRST cohomology to find the physical spectrum,as the quantum action defines a free theory.

This gives the physical spectrum ofthe ten-dimensional super Yang-Mills theory. It should be of interest to study thestructure of second-ilk superparticles and possible generalizations to superstrings,since at the classical level the evolution of the superparticle is represented by aninfinite set of classical fields which at first generation level in the BRST transfor-mations involve an infinite set of ghosts which at higher level generations requireghosts-for-ghosts, so that new infinite towers of ghosts-for-ghosts are involved.∗Although our expressions for the BRST operator agree, (4.7)-(4.9) correct errors in theexpression for the master action given in [21].14

Acknowledgements: The author wishes to express his gratitude to C. M. Hullfor encouragement, helpful discussions and reading of the manuscript. I wish tothank to M. B.

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