Operator ordering and supersymmetry

Nuclear Physics B296 (1988) 402-430 North-Holland, Amsterdam

OPERATOR ORDERING AND SUPERSYMMETRY v. DE ALFARO Dipartimento di Fisica Teorica, Universitd di Torino, INFN, Sezione di Torino, Italy S. FUBINI CERN, Geneva, Switzerland G. FURLAN Dipartimento di Fisica Teorica, Universitd di Trieste, INFN, Sezione di Trieste, International Centre for Theoretical Physics, Trieste, Italy M. RONCADELLI* CERN, Geneva, Switzerland Received 20 March 1987

We investigate supersymmetric quantum mechanics in curved space. In particular we discuss, in the framework of the canonical formulation, the role of supersymmetry (SUSY) and of invariance under general coordinate transformations (GCT) in solving the operator ordering ambiguity. It is shown that GCT invariance is instrumental to achieve such result. We also examine the connection with the Weyl ordering of operators and the relevance of our results for the path integral approach.

1. Introduction T h e q u a n t i z a t i o n of a p a r t i c l e in a curved space is a p r o b l e m that has b e e n with us for m a n y y e a r s [1]. I n this i n t r o d u c t o r y section we shall first offer a b r i e f survey o f t h e s u b j e c t in its d e v e l o p m e n t a n d then we shall illustrate the m a i n p o i n t s a n d n e w results o f o u r investigation. T h e classical l a g r a n g i a n for a system of N real c o o r d i n a t e s x ~ ( t ) on a m a n i f o l d w i t h m e t r i c g a , ( x ) reads, in a b s e n c e o f p o t e n t i a l L =

t,,

= 1...

N,

(1.1)

* On leave from Dipartimento di Fisica Nucleare e Teorica, and INFN, Universith di Pavia, Italy. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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and leads to the hamiltonian (1.2)

H = ½g~"(x)p~,p,.

At the quantum level there is a crucial problem: the non-vanishing commutator of x~ and p~ introduces an ordering ambiguity. We recall that a similar problem arises, in a milder form, for the quantization of a particle in an external electromagnetic field. At the classical level the lagrangian (1.1) exhibits invariance under general coordinate transformations (GCT) (1.3)

x '~' = F ~ ' ( x ) .

The fundamental requirement of form invariance of H under GCT has been recognized to be connected with the choice of the ordering by DeWitt in a classic paper [2]. He showed that the invariant hamiltonian is given by

(1.4)

H o = ½p~g~'~(x)p + ,

with "

In

= gl/4p~,g-1/4,

(1.5a)

p~- =p~, - ¼iOn, In g = g - X / 4 p , g l / 4 .

(1.5b)

He also proved that the most general invariant hamiltonian has the form H = H o + eR

(1.6)

for any constant c, R being the curvature. Anyway, the GCT invariant hamiltonian is uniquely defined for a fiat space (R = 0). As is well known, the ordering problem is present in this case also. If one starts from a Cartesian basis g(0) ~y = ~t~u

and performs a GCT, a metric is introduced on the (zero curvature) manifold OF ~' 9 F " g ~ " ( x ) = Ox" Ox a

(1.7)

In the path integral approach to quantization one may wonder where lies the trouble corresponding to the ordering ambiguity of the canonical approach since there are no operators and commutators. The point is that the action integral in the

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V. de A Ifaro et al. / Supersymmetry

functional context is, in general, not riemannian, but depends in an essential way on the detailed form in which the discrete sum is performed. This goes under the name of the Ito integral. Let us consider, for example, the expression -

X2)

(X

(1.8)

-

which appears in the expansion of the action in terms of small coordinate and time differences. The three choices X1 + X2

= xl'

~ = x2'

/~=

2

'

(1.9)

turn out to be not equivalent and lead to different results that are the exact counterpart of the ordering ambiguity for the canonical quantization. The connection is established by a result of Berezin [3] stating that the path integral obtained by the middle point choice defines the same quantum theory as the operator hamiltonian where the products of x and p are symmetrized; this is the so-called Weyl ordering (WO) prescription [4]. It is interesting to note that the expression (1.4) is not WO. For instance in the case of fiat space the difference H 0 - (Ho)wo is given by the compact expression [5] H0 - (Ho)wo

= aV=

! r x r~ o,,,

(1.10)

where F~x is the Christoffel symbol induced by the GCT as in (1.7). The problem outlined above is of particular physical interest. Indeed the quantum description of a relativistic spinless particle in a gravitational field exhibits similar features [6]. In turn this can be of relevance to the string case where, as emphasized by Maharana and Veneziano [7], the Fradkin-Polyakov treatment leads to analogous ordering problems [8]. A new dement in these considerations is represented by the introduction of supersymmetry which is expected to shed new light on the ordering problem. In fact it is known that for supersymmetric theories (in flat space) the ordering of the boson operators determines the corresponding form of the fermionic operators [9]. The matter has been examined by several authors [10]: in those works however the interplay between supersymmetry and general coordinate invariance has not been discussed in detail. In this paper we offer a systematic analysis of the ordering problem based on the combined use of supersymmetry (SUSY) and invariance under general coordinate transformations (GCT). Since the matter is rather subtle we prefer to approach the problem in the canonical operator formalism. This does not exclude that a functional treatment can be rather interesting.

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The main ingredient of our work is represented by the introduction, suggested by supersymmetry, of two sets of momenta p(X) and p~2) which can be considered as the natural generalization of the harmonic oscillator annihilation and creation operators, a and a ÷. They will be instrumental in proving the general coordinate invariance of the whole approach and in expressing the fundamental dynamical variables in a Weyl ordered form. In sect. 2 we discuss the supersymmetric formulation of the model, while sect. 3, still at the classical level, introduces the canonical formalism, the fundamental supersymmetry charges and the hamiltonian. Sect. 4 contains a discussion of quantization: we shall see that supersymmetry (contrary to a common belief) still allows an ambiguity in the ordering of the operators. This is eliminated if one adds, as we do in sect. 5, the requirement of general coordinate invariance, which thus plays the key role in selecting the unique form of the quantum hamiltonian. The main features related to the Weyl ordering of operators are dealt with in sect. 6: it is shown that, unlike the non-supersymmetric case, the unique, general coordinate invariant hamiltonian is Weyl ordered when the appropriate variables are used. In the concluding sect. 7 we point out the relevance of our results to the path integral approach. Some auxiliary calculations and examples are presented in the appendices A, B,C.

2. The superfield formulation We denote by Q~ and Q2 the two SUSY generators which satisfy the algebra {Q1,Q2}=2H,

(Q1, Q1} = {Q2,Q2} = 0 . ,

(2.1)

Consider a set of real superfields

dp~(t ) = x"( t ) + i ~ ( t )O + itp~( t )O + F"( t )00,

(2.2)

to be viewed as coordinates of a N-dimensional manifold ~t' (1 ~<# ~
0_1 =

0 a0

iO

0 0t'

0.2= - ~ + i 0 - ~ ,

(2.3a)

(2.3b)

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the transformation law of q~' can be written as 8~ ~= -i(~1~) 1 + 420a)q~ ~,

(2.4)

where 41, 42 are infinitesimal, anticommuting parameters. Going to the component fields, one obtains the transformation rules

81x ~' = 41g,",

81F~ = i41t[,,,

81~" = 0,

81~ ~' = ~1(i2 ~' - F ' ) ,

(2.5a)

.&

82~k~ = 4 2 ( - i2~' - F~),

~2~ ~ = 0.

(2.5b)

Consider next the superfield A = ½g~,(q~)/)lq,~b2q~ ° - W(q~),

(2.6)

w h e r e / ) l and /)2 denote the covariant derivatives in superspace

61 =

0 0 O0 + iO-~,

0 _0 D: = ~-~ - i O - ~ .

(2.7a) (2.7b)

W(~) is a scalar function on 5K (its lowest component is the superpotential W(x) of the model). It is meaningful to reexpress A as

A = A + iSxO + iS20 + LO0,

(2.8)

with the new components given by: A = lg.,(x)qJ~'-

W(x),

(2.9a)

S x = ½g~,,(x)(V~'+ i2~')~ ~ - OxW(x)~X+ ½0xg~,,(x)~XqJ~'~" ,

(2.98)

S 2 = ½g,,(x)(F ~ - i 2 " ) ~ ' - axW(x)akx+ ~axg~,,(x)qJx~"~ ",

(2.%)

- ½iOxg~,,(x)(~/X~' _ ~,~/x)2,_ V(x, ~/, ~, F).

(2.9d)

V. de Alfaro et al. / Supersymmetry

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Finally the quantity V, which embodies all the dependence of the lagrangian on the auxiliary fields F, has the form V( x, ~/, ~,, F) = r~,,,,V~q," - ~g,,( x ) r"v" + a,,W( x ) V ~

_ OxO.W(x)qjx~. + ½axO°g..(x ) g,x~oq,.~.

(2.10)

and ip) is the Christoffel symbol. Since A is a real superfield (A + = A ) the rules of transformation for the components can be easily derived as in the case of q~'. From 8A = -i(~101

-[-

~202) A

it follows that 81,4 = ~1S2,

8181 = ~l(im - Z ) ,

8aS2 = 0,

81L = i~xS2,

82A = - ~ 2 S 1 ,

82S 1 ~-- 0,

82S2 = -~2(i~4 + L),

82L = i~2S 1.

(2.11a)

(2.11b)

The SUSY invariant action is finally

lsusv= f dt f dOdOA. Clearly L is the lagrangian dictated by the superfield formulation of a SUSY invariant theory. Yet, we will work in the component formalism. As it will be seen, the analysis will take a particularly simple form by making use of lagrangians which differ from L by a total time derivative• A glance at the transformation properties displayed in eqs. (2.11a, b) shows indeed the possibility of having two equivalent lagrangians whose transformation properties are particularly simple. By defining t~ = L

- i3 = ½g,.(x)S::: + ig..(x)~.,q."

+iYa,~,.Y:~% ~'+ iaxW(x)2 ~ - V(x, ~, ~, F)

(2.12a)

and L= = L

+ iA = lg,.(x):~.'::+ ig,.(x)O/.q."

-~r~,,,:'~,,l, ~- io~W(x):?'- V(x, ,I,,C~,r),

(2.12b)

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i t turns out that ~1L1 -- 0 ,

82L ~ ----2 i ~ 2 S 1

(2.13a)

61L2 = 2i~1S2,

~2L2 = O.

(2.13b)

The relation between the two lagrangians is L 2 - L 1 = 2iA. We notice from their definitions that L x and L 2 are not real but rather L i ~= L 2. In spite of this fact (of course not affecting the equations of motion) their role will result particularly useful as it will emerge from the developments of the next section discussing the canonical formulation of the model. The role of the auxiliary lagrangians L 1 and L 2 can be nicely understood in the superfield language by introducing the two superfields A 1 and A 2 A1, 2 = A + iOS 1 + iOS2 + OOL1, 2 • It is easily seen that A, A1, 2 are connected by a complex superfield transformation: A1, 2 = e :~i°~°/atA e + io~a/ot.

(2.14)

This explains the simple form of the transformation (2.13). Indeed Q1 is realized on A 1 as - 0 / 0 0 and Q2 is realized on A 2 as 0/O0. We proceed to the elimination of the auxiliary variables F v, as can be done by using the equations of motion. One has F ~ = g~%W+ rL¢•

°.

(2.15)

Accordingly the potential V(x, q~, 5, F ) takes on the form l;'( x, ~b, 5 ) = ½g'~O~WO.W + D~D~W~hp" + ½R,.o°q~q~"fP~°,

(2.16)

where

D.V.W= D,O.W=O,O.W-r/.o.W(x) and R~,~xp is the Riemann curvature tensor constructed with g~: (for all the definitions we follow the book by Weinberg [11]). The on-shell transformation rules read

81: . = 82 X#~---~ --~2~/~ ,

a"w+ rzoZp"+°),

(2.17a)

82~/~ ---~O,

(2.17b)

V. de Alfaro et al. / Supersymmetry

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We conclude the section by recalling the simplified formulation which is based on the use of the vielbeine fields e~(x). This will be particularly advantageous in the ensuing discussions because one can then employ fermionic variables which are invariant under reparametrizations. "Vielbeine" can be introduced via the definition* N a

"

a

g,.(x) = Y'.~e,(x)e.(x). 1

E~e~ae~b = ~ab

(2.18)

Correspondingly the desired fermionic quantities are

~a = e;~b",

Ca ~ e ; ~ I~ ,

(2.19)

and in these variables L reads

L = Ig~,~(x)Sc~Y¢" + l i ( ~ b ~ a - ~ C ~

) - i o ~ , a b Y c ~ k b - IT'(x ~, ~ , Ca), (2.20)

where the spin connection %, ab is defined as • , ab = --

e~xO~eb - ea,F~xeb .

(2.21)

Having in mind the same motivation which led us to introduce L 1 and L 2 starting from L expressed in terms of ~k~, C ~, one can likewise consider two new lagrangians L I and LII out of L expressed in terms of ~k~, ~a. These are

L t = L - iB = lg~(x)Yc~'Yc~ + i~a~ a - itolx,ab3C~Ca~b -- V ( X ~, ~a, ~ a ) ,

(2.22a)

Z + iB = Ig~,~(x)Yd':c" + i ~ a ~ a - i6o~,abfC~a~ b -- V ( x ~, ~a, C a ) ,

(2.22b)

Zli

---

where B - - i~k 1 a-a ~k •

(2.23)

3. The canonical approach

The analysis carded out in the previous section suggested that supersymmetry is most simply realized in terms of the lagrangians L 1 and L 2. Therefore, on our way to quantization, we shall find it rewarding to consider two independent canonical * For simplicitythe flat index "a" is referred to a compact case.

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V. de Alfaro et al. / Supersymmetry

formulations stemming from L 1 and L 2 beside the one based on the lagrangian L. This will allow us to derive for the fundamental SUSY charges Qt and Q2 two meaningful expressions, which have the remarkable advantage of retaining their simplicity in the quantized case. Let us therefore start by considering the canonical formalism arising from L 1. The bosonic and fermionic momenta are: p(1) -

OL 1 ~--

g.,(x)Yc" + iOn,W+"tFx,~,@ -~

"

(3.1)

and 0L1 O~/~,=tg~,,(x)q,

i~

(3.2)

respectively ( 0 / 0 ~ ~' acts as a "right" derivative). Thus in the present case the set of bosonic and fermionic coordinates and momenta is 121(x~', p~l); ~k~,i~,). Their transformation rules under supersymmetry are particularly simple in the ease 42 = 0, i.e.

~1x" =

~11~" ,

81~b~' = O,

81p~ 1) = 0 ,

81~ ~ = i41p~ 1) .

(3.3)

Let us turn to the canonical counterpart based on L 2. The bosonic and fermionic momenta are p(2) = °9L2

Oy¢----~ = g,,(x)Y:

-i~¢=-

aL 2

-

- iO~W + irx, ~,~"Lx'TY~',

= -ig~(x)q,~.

(3.4)

(3.5)

( 0 / 0 ~ ~ is a left-derivative). The set of canonical variables is thus ~'~2(X/~, p~2); ~/~, __ i ~ # ) With transformation properties which are remarkably simple in the case 41 = 0: 82x ~ = _ 4 2 y :

82~ ~'= 0,

,

S 2/"p. .(2) -_ 0,

82~k, = - i42p~2) .

(3.6)

The variables pi t) and p~2) are not real, rather P (1) m'~n(2)+r~, but they will nevertheless play a fundamental role in the subsequent discussion. More precisely, the usefulness of the canonical treatments based on the two sets of variables 91 and I~2 starts to become apparent when expressing the SUSY

E de Alfaro et al. / Supersymmetry

411

charges Q1 and Q2 in hamiltonian form. Consider Q1 first. The best choice for the lagrangian to be used in the Noether theorem is Lx, owing to its invariance under the transformations generated by Q~, 81L1 = 0. Taking then into account the form of L 1 and of the supersymmetry transformations (3.3), the Noether theorem yields Q1 =p~l)~,.

(3.7)

One proceeds in the same way in determining the form of Q2: by applying the Noether theorem to L 2, invariant under the transformations generated by Q2(82L2 = 0), we easily get

(3.8) In view of quantization, it is important to notice that the variables occurring in Q1 have vanishing Poisson brackets and the same occurs in the case of Q2. This is one of the main virtues of our canonical procedure. We proceed next to derive the canonical variables relevant to the description in terms of L1 and L H. The bosonic momentum is the same in both cases and reads pp_..~

~gL I

aYc~

=

OLII

= gp~,(X)~P__ "l£dl~,ab~-a~ b

0~c~

(3.9)

Notice that p~ is real (p~+ =p~,) to be contrasted with p~l) and p~2) which are complex objects. The fermionic momenta corresponding to L1 and LtI are 0LI = i ~ 0~ka OL n

-

= --iCa.

(3.10)

(3.11)

0q~" We thus introduce two new sets of bosonic and fermionic coordinates and momenta ~2x(x ~', p+,; ~", i~a ) and ~ii(x ~, PI~; ~a, __i~a), emerging from the canonical formulation based on L I and L n respectively. In the following there will be no need to distinguish any further between ~2t and I2ii and we shall refer to a single set ~2(x~, p~; ~ka, ~a). Let us now provide the explicit form of the hamiltonian. We have H = H o + I7,

(3.12)

H o = ½g+,~(x)~Yc p

(3.13)

where

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V. de Alfaro et al. / Supersymmetry

and D ,D,Wq,

,

+ ½R ,xo Nf x

p.

(3.14)

The hamiltonian can be finally expressed in terms of the three sets of variables (~21, 122, 12). It is easy, by using the appropriate definitions, to write 1) in terms of the (q,~', ~ , ) , ( ~ , g, ),(qJa, ~/") pairs. For what concerns H 0 we shall use the form Ho

=

½g~'(x)nlr,,

(3.15)

where % = g~,.(x)2" = p ~ l ) _ tFx,~,.~x~ _ iOt, W .

--is

=p~+i6O~,ab~a~ b

(3.16a) (3.16b) (3.16c)

according to the set of variables we want to select. As far as the supersymmetry charges Q1 and Q2 are concerned we shall use in all our discussions the remarkably simple expressions given in eqs. (3.8) and (3.9); we therefore skip the Noether theorem derivation based on L I or LII. Let us summarize our findings. Three sets of canonical variables have been selected: 121(x ~, p(X); q ~ i ~ ) , ~22(x~, p(2); ~ , _ iqJ~), 12(x ~, p~; ~b", ~o), each one providing a particularly simple and convenient expression for one of the generators Qt, Q2 and H respectively. The use of each set will be dictated by convenience, according to the particular aspect to be stressed. One immediately ascertains that the three sets have the same Poisson brackets and they are therefore canonically equivalent, connected by a complex canonical transformation. We report here the simple relations among the bosonic momenta p(t), p(2), p~ on which we shall focus some of the arguments for the quantum case. From the explicit expressions of L, L 1 and L 2 one finds p(2)_ p(1)= iOf, gxp ~pX~o _ 2iOn, W ,

(3.17a)

p(1)_p~ = ie~,a~,e~a~bb + iOn,W ,

(3.17b)

• J, --a b p(2) -- PI" = -- leaO~ e~b~b ~b -- iO~,W.

(3.17c)

4. Quantization In order to carry on the quantization procedure we shall replace the variables of the set f~(x ~', p~,; ~ka, ~a) by linear operators - denoted by the same symbols - acting in the Hilbert state space of the system.

V. de AIfaro et al. / Supersymmetry

413

In terms of the above variables the quantization of the model is defined by the non-vanishing canonical brackets

[x ~, p,,] = i8~,,

{

(4.1a)

(4.1b)

=Sab.

The bosonic operators are self-adjointwith respect to a suitably defined scalar product. W e will be working in the "coordinate" representationof the Schr6dinger picture, with the statesdescribed by wave functions ~(x, ~, t), where ~ collectively denotes the diagonal fermionic coordinates. Correspondingly, our choice for the scalar product is

(

~2)

= Z fdNx

(x,

t)ep2(x,

t)

(4.2)

and the absence of the ¢r~ factor in the integration should be noted. If use of the invariant measure 1/g dNx is preferred, one has to work with operators 0 related to ours by the similarity relation

~) = g - 1/4091/4.

(4.3)

The self-adjoint momentum p~, can be represented by -iO/ax~: this is possible because of the choice of the integration measure and because our fermionic states are built by application of the ~a operators, which commute with p~. These points are further elucidated in appendix B. Next we recall that SUSY suggested the two sets of basic canonical variables 121 and 122. In order to identify the quantum forms of the variables p~l) and pi E), we introduce the quantized version of eqs. (3.17). In such a procedure we face an ordering ambiguity. Since ¢4 and ~a do not anticommute after quantization, these operators can be ordered in many ways in the r.h.s, of eqs. (3.17) and, as we shall see, each choice is consistent with supersymmetry. Now we choose to quantize those equations as follows:

p~l) _ p = _ ½ie~ O~,e~b[ tp,~, ~bb] + iOn,W, p(2) _ p =

½ie~O~eva[ ~ba, ~b] _ iO~W,

(4.4a) (4.4b)

and as a consequence p(2)_ p~l)= ½iO~,gxo(x)[ ~x, ~p] _ 2iO~,W.

(4.4c)

These definitions imply p(1)+ =p(Z) (while M = P,)

(4.5)

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Supersymmetry

An important remark is that if we choose a different ordering, for instance by substituting in eqs. (4.4) _.

we can reobtain the form (4.4) by means of the substitution W ( x ) ~ l,V(x) = W ( x )

-

½1ne(x),

(4.6)

where e(x) =- det[e~(x)]. It is thus clear that any ordering is consistent with SUSY, since the substitution (4.6) does not affect the supersymmetry properties of the model. The difference however will show up when we shall consider invariance under general coordinate transformations, as will be done in the next section. Under GCT, In e transforms differently from the (scalar) potential W(x). As a consequence, we anticipate that only the combination of SUSY and GCT removes this ambiguity and leads to the unique choice given by eqs. (4.4). From eqs. (4.4) it follows that each of the two sets J2a and ~22 satisfies a set of canonical commutation relations whose non-vanishing elements are [x", p(1)] = iS~,,, ( ~b~, ~, } = 8~,

(4.7a)

and

[x',p}2)]=iS~,,, { ~ ' , ~b~} = 8~.

(4.7b)

This fact corresponds to the existence of the canonical Poisson brackets for the sets I21 and ~22. The quantum transformations among the different sets of variables are the quantum counterpart of complex canonical transformations for the classical case; they are similarity transformations that preserve commutation relations but not the scalar products in the Hilbert space. For the representation of p~l) and p~2) by differential operators in the Schroedinger picture consult appendix B, where it is shown that, in the basis in which fermionic states are built by application of covariant d~ or controvariant d~~ respectively, either p~t) or p~2) can be represented by the (non-hermitian) operator -ia~. The interest in the sets of variables ~x and ~2 at the quantum level becomes evident when we consider the expressions for the SUSY charges Q1 and Q2, given by the classical formulae (3.8) and (3.9). Thanks to the relations

[.p(1), tl~u] =0,

[p(2), ~/~,] =0

(4.8)

v. de .41faroet al. / Supersymmetry

415

no ambiguity arises in quantizing (3.8) and (3.9) and the quantum charges are simply given by Q1 = p~l)~k~',

(4.9a)

02=p~2)~ ~'.

(4.9b)

The formulae (4.9) are thus devoid of ordering ambiguities. As a consequence of the commutation relations we find Q? = Q~ = 0.

(4.10)

Finally, the quantum hamiltonian will be defined as the SUSY invariant expression H = ½(Q1, Q2}.

(4.11)

In order to proceed further and to compare with other works on the subject, it is convenient (as in sect. 3) to introduce the quantum operator given by: ¢r~= p~l)_ 21iF), [ ~x, ~b~] - iO~W,

(4.12a)

= p~:) - ½iF),[ 4~x, Y/] + iO,,w,

(4.12b)

--Pi, + ½io~¢,,b[~ a, tPb] "

(4.12C)

Starting from eqs. (4.9) it is then easy to prove that Q1 = ½{ ffU,~r~} +½i{~/~,O,W},

(4.13a)

Q== ½{ ~',¢r~} - ½ i { ~', O~,W}.

(4.13b)

A convenient way of expressing eqs. (4.13) is obtained by introducing

¢rf = gl/4¢t~g-1/4 = ~r~+ itguln gX/4, ~r~-= g-1/4~r~gl/4 = ~r~-- i O~ln gl/4 ,

(4.14)

0 1 - - * ' ( ~ 2 + i o : v ) = (~2 + i o . w ) : . ,

(4.15a)

Qa = ~t'(~r~+ - iO~W) = (~r~-- iO.W)~ ~ .

(4.15b)

and correspondingly

E de Alfaro et aL / Supersymmetry

416

Performing the fundamental anticommutator (4.11) and using eqs. (4.15) we finally obtain for H the expression H = H 0 + V,

(4.16)

where n o = :1, , -- g pv ( x ) % +

(4.17a)

v = ~g,,,(x ) ~ g : D , w + ~n,,D,w[ ~,,',,~'1+ ~R,,,,,[~,,',~'][~, ~,1. (4.17b) Computing

I~=g-1/4Hgl/4

(4.18)

we find complete agreement in form and in ordering with the hamiltonian given in ref. [12]. The main point to be discussed remains the ordering property of eqs. (4.17). This will be done in sect. 6.

5. General coordinate transformations In the previous discussion we have stressed the relevance of GCT in order to solve all ordering ambiguities and we have anticipated that the choice (4.4) is actually imposed by GCT requirements. In this section we shall examine the quantum formulation of GCT and finally show that the hamiltonian H(x ~, p,; ~a, ~a) is indeed invariant. Our proof will make essential use of the two sets of basic variables 121 and 122 introduced earlier. An infinitesimal GCT is written as x " -~ x ' " = x " + F ( x )

(5.1)

or

ax, = x', - x,'= /,'(x ) , where f~,(x) is infinitesimal. Under (5.1) ~° and ~ka are scalar quantities A I ~ a ~-~ A I ~ a = 0 .

(5.2)

So, the generator G of unitarily implemented GCT does not contain fermionic variables and therefore has the same form as in the non-supersyrnmetric case and is uniquely given by [1]

G=~(p,,/,'}.

(5.3)

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As a consequence we shall write for the total variation AA of a dynamical variable A

AA=i[G,A]+SA,

(5.4)

where 8.4 denotes the variation in the functional form of A due to the dependence ~W. on et~a , ea. The variation A~ of the fundamental variables ~-= (x ~, p~.. ~a. ~a) of the set I2 are indeed given by

= i[o,

(5.5)

In order to proceed further and to evaluate the complete variation of a dynamical variable A ( x ~', p~,, q:a, ~,,) we have to take into account that in general A is a functional of the quantities e~, e~ and W also. The form of these quantities is changed by a GCT according to the laws

8e~ = e~"(x) - e~(x) = - e ~ ( x ) a ~ , f X ( x )

- Oxe~,(x)fX(x),

8e~ = eX~(x)axf~'(x) - a x e ~ ( x ) f X ( x ) , 8( e~O~,e,,h) = - e ~ , O x e , b O j X ( x )

- e;exbO~,O,fX(x) -- Ox( e~O~,e~b)fx(x),

8e = 8(det e~(x)) = -ea~,f~'(x) - OxefX(x),

8 W = - OxWfX(x).

(5.6a) (5.6b) (5.6c) (5.6d) (5.6e)

We are now in the position to show that An(x',pp.,lpa,~t/a) ~ 0 , which will be achieved by proving that AQ1 = 0,

AQ2 --- 0.

In order to obtain this result we shall make constant use of the variables of the sets I2x and 122, which, as already stressed, allow a simple and unambiguous expression of the generators Q1 and Q2The proof will be made easier by the introduction of the two auxiliary operators

G,=½{Q,,/~}=½{:"(x),pJ')}-½ia,,f"(x)[~",~,], (5.7) G==½{Q,.f~k~.}=½{f~'(x).p~=)}-½iO.f~'(x)[~'.~.].

(5.8)

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where we have used the basic definitions of Q1 and Q2 Q1 -- P(1)iP ~ ,

Q2 =p(2)~.

By using the fact that Q~ and Q2 are nilpotent, we have that [Q1, G1] = 0,

(5.9a)

[Q2, G2] = 0.

(5.9b)

The next step is to use the quantum definitions of This leads us finally to the fundamental separation

p~(~),p(2) given in eqs. (4.3). (5.10)

G1, 2 = G + X1,2,

where X 1 and X 2 are

X 1 = - ½i[~ a, ~b](f~e~O~e,b+ O~f'eUae,b) + if~O~W,

(5.11a)

[a,a ~ " 2 ~ !2 i t'~"

(5.11b)

~b] (f~'e~O~,e~,~+ O~f~e~e~a) -- if~O~,W.

The importance of X x (and Xz) lies in the fact that the commutators of XI(X2) with the variables of the set ~21(#z) coincide with the variation in form of those variables considered as functions of the variables of the set £, namely i = 1,2.

~ i = i[ gi, ~i],

(5.12)

Indeed a direct computation (based on the use of eqs. (5.6)) shows that

i[ X~, q~'] = q:(e'aaJ" -f"a,e~) = ~"Se~ --- 8ff ~' ,

i[ X 1, ~,]=

(5.13a)

-~(e:~,f,+ f~:~,,,) (5.13b)

= ~a~e,a ~ ~ l z ,

i[ )(1, p~')] =

-iO~,( fXOxW ) + {i[ ~/', ~] Ot,O~,fx + ½i[ ff", ~b]( e~oOxe,bO~fx + fXaxe;a,e,b) --Sp (') . (5.13c)

As a consequence we find that the total variation of any variable ~ of f2~ can be written as = i[G,

- i[Gx,

+

~1].

=

+

(5.14)

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By the same chain of arguments one can easily obtain the any variable ~2 of the set I22 verifies a~ 2 = i[G2, ~2].

(5.15)

A glance at all these results shows that there is a correspondence between each operator G, G 1, G 2 and the three sets of variables /2, 121,/~2. Indeed, each of the G, G 1, G 2 generates a total transformation A when commuted with the quantifies ~, ~1, ~2 respectively. Eqs. (5.14) and (5.15) represent the key result in our proof of the GCT invariance of the hamiltonian. By applying it to the supersymmetric generators Q1 and Q2 we have AQ1 = i[G1, a l ] ,

(5.16)

aQ2 = i[62, Q2].

(5.17)

Using now eqs. (5.9) we reach the conclusion that AQ1 = AQ2 = 0. The supersymmetric representation for H, eq. (4.11), leads finally to the announced result A H - - 0, the invariance of the hamiltonian under GCT. It is important to remark that all our arguments are based on the transformation properties of e~ and W given in eqs. (5.6). Had we chosen a different ordering in the definitions of p~l) and p~2) all the equations would have remained the same apart from the replacement of W ( x ) by l~(x): l,~'(x) = W ( x ) - ½1ne(x).

(5.18)

Bff'(x ) = - f ~3~1~(x ) - ½0~f ~ .

(5.19)

Now (see eqs. (5.6))

The extra term would show up in our calculation and yield AQ1 =/=0, AQ2 ÷ 0. We thus realize that the choice of ordering made in eqs. (4.4) is the only one which guarantees general reparametrization invariance. An alternative way of proving invariance under GCT is suggested in the second part of appendix B, where it is shown that za~r+ = - O j X ( x )

• ~r~-,

A0rS = -~rf0,fX(x).

As a consequence both H 0 and V as given by eqs. (4.17) are separately invariant under GCT. This is completely analogous to what happens in the purely bosonic case, eq. (1.6). As in that case, pure invariance under GCT doesn't fix uniquely the

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appropriate hamiltonian. A unique determination is only given by combination of GCT and SUSY invariance requirements.

6. The ordering property Let us refer explicitly to the form of the hamiltonian given in eq. (4.17): it is important to notice that the discussion leaves open the choice of the operator basis (12, 121,2) to be used to express H. Indeed we shall see that the ordering properties will not be the same in the three basic sets. We first consider the set 12(x~', p~; ~k~, ~ ) . One can identify two sources of deviation from Weyl ordering (WO)*. (I) The purely bosonic term can be written as

lg-1/4p~gl/2gt'~p,g-1/4=X(p~,ppg"~)wo+~(R+g~"F~XFXo),

(6.1)

thus leading to a deviation

avi= {( R + g'°r~xr)o).

(6.2)

(II) The four-fermion term reads

-f~(g 6Ov.,abO)v,cd--Rabcd)([l~a, ~b], [ ~t/c,tpdl } .

(6.3)

Using the ordering identity (6.4) we obtain a deviation A V n = I ( - R + g'ua'tdp.,abtd,v, ab ) .

(6.5)

Hence the overall deviation is

AV=

AV I +

AVII ~g (CxF~o + tO.,ab~,,ab)-

(6.6)

We thus see that although the terms in R cancel, a non-vanishing contribution is left: the conclusion is that the hamiltonian expressed in the variables 12 is not Weyl ordered. Let us now make use of the variables 121(x~, p~l); ~b~,,i~g). Again the following deviations from Weyl ordering are found. (I) The purely bosonic term 1(7-- I / 4 rj (1)o.p,vn.(1)o.-- 1/4

* General properties of WO and some details of the computations are provided in appendix A.

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can be treated in the same way as before and leads to the deviation AV,'" =

R + g'or:

r:o ) .

(6.7)

(II) For the four-fermion piece

-~(R~oo8- g.'r~,..ro,.8){ [~x, ~.], [fro, ~8]) we find, via eq. (6.4), the deviation (6.8) Combining the results (6.7), (6.8) we achieve a complete cancellation i.e. AV~ ~) + AV~1) -- a V ( " = O.

Hence the hamiltonian written in terms of the variables ~t (or 02) is Weyl ordered. This result may look somewhat surprising, even if the fact that things are different for different sets of variables is not unexpected, owing to the presence of a non-linear transformation between ~ and ~1- In order to clarify why ~21 and ~-~2are privileged, we offer an alternative derivation of the same result, based on the simple behavior of the ~1 and ~2 variables under SUSY transformations. Our strategy will be to trace the ordering properties of H from those of the simpler operators Qx and Q2. Let us thus express Q2 = p ( 2 ) ~ in terms of the variables ~1- It is easy to check that Q2 is Weyl ordered:

+ i( ~o~xg"Xg°"a.g.o JT )WO •

(6.9)

Next we compute H through the fundamental anticommutator of Q1 and Q2- In such calculation we are indeed helped by the great simplicity of QI in the variables ~2x. This simplicity becomes apparent if one adopts a useful device which simplifies the calculation and explains immediately why WO is preserved. The hamiltonian H can be most conveniently rewritten by introducing an anticommuting parameter e as follows:

Q2( e) = eeQl/2Q2e-eQ1/2 = Qz + l e ( Q1, Q2 } .

(6.10)

H = fdeQ2(e ) .

(6.11)

Then

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By inserting into eq. (6.10) the expression for Q2 provided in (6.9), one easily ascertains that Q2(e) can be expressed in terms of the linear combinations Xt~( e) = eeQ,/2xthe-~Q1/2= X p. + e~b ~ ,

(6.12a)

~p,(e) = eeQl/2~l~e-eQl/2=

(6.12b)

~, + ep (1) ,

(6.12c) (6.12d)

=

It is important to notice that each of the expressions (6.12a,d) linearly combines only commuting operators; these combinations therefore obey the same commutations relations as the original variables. This entails that Q2(e) is WO when expressed through the variables (6.12), since the original Q2 is WO in the basic variables $21. Finally from eqs. (6.10), (6.11) we obtain H=

~_(n(1)n(1)~,,u] __ 1.i[ ~,hOl"o p~l) } [ ~ a , 2\/"/~ /"P O ] W O 4.'Ld~ ~l~P '

~tt]

(6.13) It is instructive to verify again the absence of WO by inserting the variables $2 in eq. (6.13). One sees that the only deviation from WO comes from the new four-fermion term introduced by the non-linear transformation between p(X) and p~. This is

_ ~(e~O~e~beOcO,e#dg ~, + gXVeo~eOdF;ve~Oxe~b){[

I//a, ~b], [l~c, l~d]}

and leads via eq. (6.4) to the deviation

AV=

__ ~(eag/abebO~e/~ag 1 a fl ~t~ +

fl o a g XVeobeaF~reoaxe~b)

.

(6.14)

Using eq. (2.21) one can verify the identity between the expressions (6.14) and (6.6) for AV in the variables $2. 7. Conclusions

We have seen that the combined use of SUSY and GCT invariance fixes the ordering in the hamiltonian in a totally unique way. Our treatment makes clear that both requirements are needed: SUSY alone leaves still some ambiguity. We stress that our two sets of variables $21 and $22 play a very privileged role throughout the whole discussion. Actually, these variables are canonical and appear

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in the two SUSY charges in a remarkably simple way. As a consequence, they are instrumental in proving the GCT invariance of the whole approach. On the other hand, we have shown that our reparametrization-invariant hamiltonian is Weft-ordered when expressed in terms of those variables. It is rather striking that the role of the sets I21 and ~22 is unique in this respect: when transforming to the familiar variables 12- which are suggested by hermiticity (and are not simple with respect to SUSY) - Weyl ordering in the hamiltonian is lost. It is amusing to speculate on the implications of our result in the context of the path-integral approach. The equivalence between Weyl-ordering and middle-point prescription is well-known in the bosonic case [3]. Although we have not been able to find an analogous statement in the literature for the fermionic case, we are convinced that the equivalence can also be extended to this case. As a consequence, it seems reasonable to assess that in the SUSY case the path-integral method can be straightforwardly applied to the GCT invariant hamiltonian with the middle-point prescription for bosons and fermions, provided either the set ~'~1 or ~"~2 is used as integration variables. The reason is due to the striking boson-fermion cancellation given by eqs. (6.7), (6.8), which implies that no further effective potential has to be added to the hamiltonian. This situation should be contrasted with what happens by expressing our reparametrization-invariant hamiltonian in terms of the variables $2, where - owing to eq. (6.6) - the modified hamiltonian H+AV=H'

should be used in the path integral. As Gervais and Jevicki have shown [5], an analogous situation occurs in the non-SUSY case. A further use of the variables p(1), p(2) is related to the simple form of the supersymmetric wave functions for the ground states of the model corresponding to the empty (tk0) and filled (q'F) sectors of the fermion Fock space. Indeed the beautiful result given in ref. [12] q~o - e - Wgl/4,

qJF -- eWg-1/4

can be obtained with amusing simplicity by requiring that those states are annihilated by p(1) and p(2) respectively p (2) lo) = o,

F> = o.

In this sense p(2) and p(1) can be considered as the supersymmetric generalization of the celebrated annihilation and creation operators for the harmonic oscillator [131. We believe that the method proposed here is particularly suitable to extend the present discussion to supersymrnetric theories in more than one dimension. The two-dimensional case has become particularly relevant, due to the renewed interest

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V. de Alfaroet al. / Supersymmetry

in string modds. It has been shown by Maharana and Veneziano [8] that the existence of Ward identities in the Fradkin-Polyakov approach to the string depends on the possibility of performing general canonical transformations in the path integral. We hope to come back to this interesting question in the near future.

Appendix A We wish to list some properties of Weyl ordering which are useful for our investigation; for the general definitions the reader can refer to the works quoted in ref. [4]. We shall consider pairs of bosonic conjugate operators qi and pj, and of fermionic conjugate operators ~ and ~#. For sake of definiteness ~b,, will be identified as the coordinate and g,# as the momentum. When dealing with expressions containing a single bosonic or fermionic momentum, Weyl ordered products coincide with anticommutators or commutators:

(Pif(q))wo = ½(Pif(q) + f(q)Pi),

(A.1)

( # ' ~ a ) w o = ½( ~b~a - ~a~b~),

(A.2)

=

t

tp

+

~,~p,,~b~).

(A.3)

If there is one bosonic and one fermionic momentum, we simply have

(p,~.f(q)~b~)wo=¼(pJ(q)+f(q)pi)(~/.,o-tp~,~).

(A.4)

We turn now to expressions containing either two bosonic or two fermionic momenta:

( PiPJf(q))wo = ¼( P~PJf + PJPJ + PJfPi + fPgPJ)

(A.5)

= ¼(6#tpatk~q% + qj#q, Cvtpa + tpsCvtp.qJa + qJ q%~aqJa). (A.6) Two fundamental identities, valid in the bosonic and fermionic cases respectively, will also be needed. The first one is

g-1/4pigl/2gOpjg-1/4= ( piPjgiJ)wo +/iV, /iV= ¼[ Oi Ojgij _ 4g-1/% ( gijgX/2Oig-1/4) ]

(A.7)

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and after a somewhat cumbersome calculation one obtains*

R + gXor:xr;o).

(A.8)

For fermionic operators we shall rely on the identity

which can be easily verified.

Appendix B A.1. Our aim is to discuss the realization of the momentum operators p~, p(1)p~2), % on the states in the Hilbert space. Let us start from the states Ix,0)

(B.1)

)(~lx,O) = x~'lx,O),

(B.2)

~balx,0) = 0.

(B.3)

defined by

( X denotes the coordinate operators). The action of p~ on these states is defined by p~lx,0) -- i3~,[x,0).

(B.4)

We now introduce the "vierbein" states with a definite number of vierbein fermion levels occupied:

Ix,{ak}}=~ Since p~, and

~a commute,

. . . ~ l x , O),

(O<~k<~N).

(B.5)

we have as a consequence of (B,4) that

p~lx, { ak }) = iO~lx, { ak })-

(B.6)

Thus on the wave functions

@{a,}(x) -- ({ a k }, x[~)

(B.7)

the operator p~ is represented by - i0F,, a hermitian operator since we choose in the * The calculation is essentially the same as in ref. [5]. The curvature term R is absent there, since the case of flat space was investigated.

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normalization integral the non-invariant measure

(8.8)

x5 = f dNx¢(x)x(x)-

We wish now to determine another set of states on which the (non-hermitian) operator ff~l) acts as ia~, where fro) recalling eq. (4.4a) of the text, is given by ff(1)=p(1) +

ia¢ln g l / 4

icgW_- g-1/4p(1)gl/4

ia~,W

= p, + ie~ atte,a~btv.

(8.9)

Using (B.3) and (B.4) it follows then that fr°)lx , 0 5 = i a~l x, 0).

(B.10)

Now let us define the "covariant" states Ix, (Ark }) = ~.1 "'" ~¢k Ix'O)"

(B.11)

On these states, since [p = ia~lx, {#k}>"

(B.12)

(Obviously, owing to our definition (B.8) of the norm, ff~) is not hermitian.) A similar treatment can be given for fig2) =p~2) _ ia~ln gl/4 + ia~W= g~/4p(2)g-1/4 + ia~W (B.13)

= Pt~ -- i e ~ t ~ez'a~b~a"

If we introduce the "contravariant" states Ix, ( r k } ) = ~'~ ... ~Tklx,0),

(B.14)

ff~(2)l, ( pk } ) = ia~lx, ( vk } ).

(B.15)

we obtain ([i~(2), ~k'] = 0)

Now let us devote our attention to the representation of the operator (see eq. (4.12) in the text)

% -_ g . , x.r =pt,+il i t ¢¢ , a b t[,T,a V , t~b]

•

(8.16)

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As a consequence of the previous definitions we check that ~r~ acts as a spinor covariant derivative on the vierbein-types states: ~r~,lx, { a k ) ) = ig~,lx, ( a k } ) + iZbt%ablX, {bt} ) = iDfbl x, {bt) ) . (B.17) Analogously, ~r~ = g-1/4%g1/4 acts as a covariant derivative on the covariant states:

g - 1 / ~ g l / 4 1 x, ( ~ k ) ) = iO~lx, (/~k)) --Y'~'F~zklX, ~" ( / ~ } ) =iD~lx, (l~k))

(B.18)

Finally %+--gila%g-i~4 acts as the covariant derivative on the contravariant states. A.2. At this point it is instructive to apply the formalism of sect. 5 to the study of the transformation properties of p~, p tl), p(2) and ~rz under GCT. We have

Ap~=i[G, p~] = _ 1 ( p~, O~f~}.

(B.19)

One should notice however that p~, like all the variables of the set ~2, is not invariant under local transformations in the vielbeine indexes a, b. On the contrary the sets $21, 122 are invariant under that group, which means that we have not to worry about such transformations when dealing with QI, Q2 (and H). It turns out that the variables p(1) and p(2) obey more complicated rules of transformation i.e.

A p(~l) = -p(xl)O, f X( x ) + i~kffx g~,a,f X( x ) ,

(B.20)

Ap(2) = -p~2)gjX(x) + i~x~,gzOxf'(x).

(B.21)

It is amusing to notice that ~r,, corresponding to the covariant derivative on "vielbein" fermion states (eq. (B.17)), has the simple transformation law

a..=-:{ 1

7r

0./:(x)),

(B.22)

while it is invariant under local (a, b) rotations. The vectors rt~+ and ~r~- also have the simple transformation law

&¢-= -a.fX (x) ~r ; ,

(B.23)

Artz- = - ~ r x a j X ( x ) .

(B.24)

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From these formulae we see that Ho, as given in (4.17a), is separately invariant under reparametrizations. The same occurs for V. This means that any combination Ho+ cg

is invariant under GCT. The value c = 1 is finally fixed by SUSY.

Appendix C A simple illustration for the fermion case of the connection between Weyl ordering and the middle-point rule in the path integral approach, can be offered by considering a single fermion mode. (For the formalism adopted in this appendix we refer to the book by Lee [4].) Let us consider the standard annihilation and creation operators a, a+, with the anticommutation relations a,a

=1,

{a, aJ={a+,a+}=O,

(C.1)

and the occupation number N = a+a.

(C.2)

The states can be classified in terms of the eigenstates of N: NI0 ) - - 0 ,

N i l ) - - I1).

(c.3)

In order to define the coordinate representation for the fermion states, we introduce the anticommuting variable z such that a = z,

a + = d/dz.

(C.4)

The wave functions are (zlO) = z ,

(z[1)-- 1,

(OIz) = - 1,

(llz) =z,

(c.5)

and obey the completeness relations

f (alz)(zll3) dz = Sa~, Y~,~(z l [ a ) ( a l z 2 ) = z2 - zl = 8 ( zx - z 2 ) .

(C.6)

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We also introduce the "Fourier transform"

f ( k ) = f exp(kz)F(z) dz in terms of the anticommuting momentum k. We have f exp[ k ( z I - z2) ] d k = 3(z 1 - z2).

(C.7)

Let us now consider the Weyl ordered hamiltonian

H= ½(a+a- aa+),

(c.8)

for which it is easy to show that

(zl[H[z2)=½(Zl+Z2)=fexp[k(za-z2)](-k)½(za+z2)dk.

(C.9)

The for the short time action one finds

(zlle-matlz2)=fexp[k(zl-z2)+½iatk(zl +z2)] d k .

(C.10)

If we now notice that z=~k,

k=~

(C.11)

are the "classical" analogues of a and a ÷, eq. (C.10) can be rewritten in the form

where Hx = ~k7( 1 + if2)

(C.13)

We thus realize that H x, corresponding to the Weyl ordered expression (C.8), is indeed computed with the middle point prescription. This elementary example shows explicitly the connection between Weyl ordering and the middle point recipe and hints at its general validity. References [1] B.S. DeWitt, Phys. Rev. 85 (1952) 653; M. Omote and H. Sato, Progr. Theor. Phys. 47 (1972) 1367; L.S. Schulman,Techniquesand applicationsof path integration(Wiley,1981), ch. 24 [2] B.S. DeWitt, Rev. Mod. Phys. 29 (1957) 377

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[3] F.A. Berezin, Theor. Math. Phys. 6 (1971) 194 [4] H. Weyl, Group theory and quantum mechanics (Dover, 1931); F.A. Berezin, Usp. Fiz. Nauk 132 (1980) 497; T.D. Lee, Particle physics and introduction to field theory (Harwood, 1981) [5] J.L. Gervais and A. Jevicki, Nucl. Phys. Bll0 (1976) 93 [6] L. Brink, P. Di Vecchia and P. Howe, Nucl. Phys. Bl18 (1977) 76; A. Neveu and P. West, Phys. Left. B182 (1986) 343 [7] J. Maharana and G. Veneziano, Phys. Lett. B169 (1986) 177; Nucl. Phys. B283 (1987) 126 [8] A.M. Polyakov, Phys. Lett. 103B (1981) 207; E.S. Fradkin and A.A. Tseythn, Nucl. Phys. B261 (1985) 1 [9] E. Witten, Nucl. Phys. B188 (1981) 513 [10] A. Nakamura and F. Palumbo, Phys. Lett. B147 (1984) 96; A.C. Davis, A.J. Macfarlane, P.C. Popat and J.W. van Holten, J. Phys. 17A (1984) 2945; A.J. Macfarlane and P.C. Popat, J. Phys. 17A (1984) 2955; A.V. Smilga, Nucl. Phys. B292 (1987) 363 [11] S. Weinberg, Gravitation and cosmology (Wiley, 1972) [12] M. Claudson and M.B. Halpern, Nucl. Phys. B250 (1985) 689 [13] V. De Alfaro, S. Fubini and G. Fudan, ISAS preprint, Trieste 35/85/EP (1985), in E.S. Fradkin Festschrift (A. Hilger, Bristol), to be published