Supersymmetric path integration

Volume 193, number 1 PHYSICS LETTERS B 9 July 1987

SUPERSYMMETRIC PATH INTEGRATION

Alice ROGERS Department of Mathematics, King's College, Strand, London WC2R 2LS, UK

Received 4 March 1987

The kernel of the evolution operator for the supersymmetric square root of the Schr6dinger equation of a supersymmetric quantum mechanical system is expressed as an integral in the space of paths in superspace. The measure is defined so that the supercharge rather than the hamiltonian is fundamental. It is shown that the odd and even parts of the paths in superspace are components of a single superpath. The Fourier mode expansion of the superpath is derived.

It is well known that in quantum mechanics the matrix elements of the evolution operator e x p ( - H t ) can be calculated as a Feynman path integral [ l ]. In the case of a particle in n dimensions, if the hamiltonian H has the form

H=½p 2 + V(x) , (1)

then the appropriate formula is

t

( a l e x p ( - H t ) l b ) = f D x e x p ( - f ( d x ( s ) / d ~ ) 2 - ~ - V(X(S)))dS, (2) 0

where Dx means integration is to be carried out in the space of all paths from a to b. (Imaginary time quantum mechanics will be considered throughout this paper.)

In supersymmetric quantum mechanics with a single hermitian super charge Q, the hamiltonian H is the square of Q. The object of this paper is to show that, if Q has the form

Q=( T'p)/x/2 + W(x, ~t) , (3)

( T being the Fermi operator), then the matrix elements of the super evolution operator e x p ( - H t - Q T ) [2 ], where ~ is an anticommuting "time" can be evaluated by integration in the space of paths x(t, z), T(t, z) in superspace according to the formula

(a, ~ l e x p ( - H t - Q z ) Ib, fl)

t + a ~

fOxO,exp( I 0

ds { [ Ox( s, a) / Os ] V'( s, a) + W( x( s, a), ~ ( s, a))} ). (4)

(Here a and fl are anticommuting parameters, and the integration variables s and a are respectively commuting and anticommuting.)

This result is useful for computation because the superpotential W is generally much isimpler than the poten- tial V. (An example is given at the end of the paper. This calculation also shows how the cancellations typical of supersymmetric theories are built into the method.) The result (4) is also interesting because it promotes

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Q to a fundamental role in the path integration of supersymmetric theories, which seems appropriate because in such theories Q is more fundamental than H. In other situations, such as supersymmetric field theory or string theory, the analogue of the quantity J. ~ + W(x, ~), which occurs in the exponent in the integrand of (4), may prove significant. This quantity is not the square root of the lagrangian, but relates to the supercharge Q rather as the lagrangian relates to the hamiltonian H.

It is also shown in this paper that x(t, z) and ~(t , z) may be expressed in terms of the superpath X(t, z)=b(t)+izfl(t)/x/~, where b(t) is an ordinary bosonic path or brownian motion and flU) is the fermionic equivalent [ 3 ].

Before establishing (4), a rather more rigorous formulation is required. Supersymmetric quantum mechanics in n dimensions, n being an even number, will be considered [4]. The bosonic operators x ~ and pi and the fermionic operator ~ ( i= 1 ..... n) obey canonical commutation relations

[pi, xq=it~'J, {~', ~s)}=2~o , (5)

all other commutators being zero. In the Schr6dinger representation of these commutation relations [5,6 ], x', ff and ~u~ are represented as operators on the space of complex functions of n commuting real variables x ~ and n anticommuting real variables 0 i by

xi=x i, pi=iO/Oxi, ~i=Oi+O/O0 i. (6)

As usual, the evolution of the system is determined by the SchriSdinger equation

dfldt= -Hf . (7)

In purely bosonic theories, in order to avoid infinite dimensional normalisation factors, the ill-defined measure Dx and the free terms in the action f~(dx/ds)2dsare combined into a single measure d/~. Thus (2) is written

l

(alexp,-Ht, lb>= f dlz[a,b,t] exp( - f +V(x, dt) , (8, 0

where, if 0 < tl < ... < tN < t,

f dlt[a, b, t]f(x(fi), x(t2) ..... x(tu))

~f f d"xl ...d"XNP(a, xl, tl )P(xj , x2, t 2 - t l )...

×P(xN_ 1, XN, tu-- tN--I)P(xN, b, t - tu)f(xl ..... XN) ,

with

(9)

P(x, y, s) gf(2zcs) -,/2 exp[ - (x-y)a/2s] . (10)

The validity of (8) follows essentially from the fact that the function P(x, y, s) is the kernel of e x p ( - H o t ) , H0 being the free hamiltonian p2/2. (A rigorous proof of this result, which goes back to work by Feynman and Kac, may be found in the book of Simon [ 7].) Recently the author has developed an analogous theory for purely fermionic systems [ 3 ].

In supersymmetric theories, by introducing an anticommuting time z in addition to ordinary (imaginary) time t, one may construct a square root of the Schrrdinger equation [2]. This is the equation

D,.~f= Qf, (11)

where

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D,,, gf 0/0r + z OlOt, (12)

and f is a function of x ~, 0 r ( i= 1, ..., n), t and z. The solution to this equation with initial condition

f (x, O, t, r) =g(x, O) when t=0, z = 0 , (13)

is given by

f = exp( - H t - Q z ) g . (14)

Just as the purely bosonic path integral is based on the kernel of exp(-Hot) the supersymmetric path integral will be based on the kernel of exp( -Hot -Qoz) , where Qo is the free supercharge

ao = (l/q/~) ~V.p. (15)

It is useful to note that

Ho = O ~ . (16)

The kernel of an operator K on the space of functions of n commuting and n anticommuting variables is defined to be the function [K](a, b, a, r ) of 2n commuting variables a i, b i and 2n anticommuting variables oJ, fir such that

Kf(a, a) = J dnbdnfl [K](a, b, a, fl)f(b, fl) . (17)

Here the integration over the anticommuting variables pr is the usual Berezin integral [ 8 ]. It is well known that the trace of K can be obtained from the formula

traceK= f dnad"a [K](a, a, a, - a ) . (18)

The operator e x p ( - H o t - Q o r ) has kernel

[exp( -Hot -Qoz)] (a , b, a, fl)= j d"~c P(a, b, a, fl, x, t, z) , (19)

where the variables x ~, ..., x" are anticommuting and

e(a, b, a, r, ~c, t, z) gf(2nt) -'/2 exp{- [ b - a - ( i / v /2)r ( f l+ix)]2 /2 t - ix . (a- f l ) } . (20)

This suggests defining a measure d/z[a, b, or, r , x, t, r] on the space of paths (x(t, r), ~(t , z)) by setting

f dlt[a, b, a, fl, x, t, z] f (x( t l , '~'1), ~(f i , zl), ...,x(tN, ru), 7t(tN, ZN))

~eef f dnNx dnNO d"Np {P(a, x~, a, 0~, x, tt, z l ) P(Xl, x2, 0 l, 02, Pl, t2 -- tl + r2 r~, r2 -- z j ) ...

×P(xN_I, XN, ON_I, ON, PN--I, tu-tN_~ + zuru_~, rN--ZN_I)P(XN, b, Ou, 13, pu, t--tN +ZrN, Z--'I~N)

x f ( ' ~ l , O, -['ipl . . . . , Xu, ON"JI-ipN) } , (21)

if 0<f i <.. .<tu. A useful result, which follows directly from this definition, is that

{exp( - -Hot l -Qoz l ) f (x , ~)...exp[--Ho(tN--tu_l +Zu'Cu_t)--Qo(Zu--1:N-l)]fu(x, ~)

×exp[ - -Ho( t - - tu+ZZu) -Qo(z -zu ) ]} (a , b, a, fl)

= f d " x du[a, b, a ,p , x, t, r lA(X(tl , T~), 7'(t,, Tt))...fu(x(tu, Tu), ~'(tu, ru ) ) , (22)

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if 0< tl <... < tN. (Here it is assumed that

f~(x, ~ ) = Zf~.(x)~Pu, (23) #

where the f,# are ordinary functions, g is a multi-index #~.../Zk with 1 ~<#1 <...</tk~<n, and ~u= 7,a...~u. This avoids ambiguities in operator ordering.)

In terms of this measure on supe~aths, the key result (4) may be expressed as t+a~"

[exp(_Ht_Qr)](a,b,a,,)=IdOxd#[a,b,a,,,x,t, Tlexp(_fda y dsW(x(s,a),7'(s,o))). (24) 0

This result is not easily proved by direct assault, but may be proved by an extension of a method developed by Guerra, Rosen and Simon for proving the bosonic Feynman-Kac formula [9]. Three formulae from ele- mentary (1, 1 )-dimensional supercalculus are required: first, the fundamental theorem of supercalculus,

t + t r ' r

Ida f dsD,,j(s,e)=f(t,,)-f(O,O), (25) 0

is required. Secondly, there is the formula for differentiation with respect to the limits of integration,

t + o - r

Dr,, (J" da f dsf(s,a))=f(t,,), (26, 0

and thirdly there is the formula for change of variable

t + o r t - u + o ( r - r l ) + t l ~

u + rx r t 0

Combining (25) and (26) one may deduce that t + G z t + ~ z t + o z

exp( -~d t r f d s g ( s , o ) ) = l - ~ d r / I du[g(u, rl)exP(-fde ~ dsg(s, tr))]. (28, 0 0 u + o q

Now let Bt., be the operator with kernel

[Bt.r](a, b, ot, fl)= f dnx~ dg[a, b, a,,a, x, t, T]exp(- f do t + O T

0

(29)

Taking (28) with g(s, a) = W(x(s, tr), ~(s, tr)), changing variable according to (27), and integrating with measure d,xd#[a, b, or, r , x, t, T] gives

(30)

[B,, ,](a, b, a, fl)= [exp( -Hot-Qoz)](a, b, a, fl) t+ r /¢

-j'd j" du texpt- (Hou+Qo )] b. 0

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A further change of variables gives

B,.~(a, b, a, f l )= [ exp ( -Ho t -Qoz ) ] (a, b, or,/~)

0 / ¢ #

-ida J ds [exPI-Ho(t-s-a,)] +a0(r-a) WBs,.](a, b, a,p). (31) l - - a'¢

Direct differentiation, recalling that Q= Qo+ w, now gives

Dt.tBt.r = QBt.r • (32)

Clearly Bo.0 is the identity operator, and so one may conclude that

Bt , r = e x p ( - H t - Q z ) , (33)

which establishes (24). In order to actually use this formula, it is useful to expand the superpaths x(t, r) and ~v(t, z) in terms of

independent gaussian modes. To expand the superpaths, one evaluates the covariance fdl~[a, b, a, fl, x, T, X]od(s, cr)xJ(t, r) (and so on), and exploits the fact that gaussian processes are characterised by their covariances. These covariances additionally show how x(s, a) and ~V(s, or) may be expressed in terms of the superpath X(s, a). In fact it is only useful to consider X=0, and sufficient to set a, b, a, fl and x to zero, and T to 1, regaining the general case by translations and rescaling. The results are then

covariance (x'(s, tr), xJ(t, ~ ) ) = c ~ a s ( l - t ) - trz/2 ,

covariance (x~(s, tr), ~vJ( t, T))=c~a(is~-ia)/x/~,

covariance (~i (s , or), xJ( t, z ) ) = d / J ( - i z / x / ~ + i x / ~ ( 1 - t ) a ) ,

covariance (~ ' (s , a), ~vJ(t, r ) ) = c~a(- 1+ 2crz) , (34)

where 0 < s < t< 1. This suggests that one sets

x~(s, a) =bi( s) + iafl'(s)/x/~ , ~Vi(s, a) =fl~(s)-x/~itrci(s) , (35)

where b(s) is a conventional bosonic path (or, more technically, the n-dimensional brownian bridge), fl(s) is a fermionic path (that is, an n-dimensional Grassmann brownian bridge [3]) and c(s) is db(s)/ds. (Strictly such a c(s) does not exist, but c(s)ds can be handled by Ito integrals). Thus (using an unconventional expansion of b(s), so that b(s) and c(s) are both expressed in terms of the same set of orthonormal functions), one has

b~(s) = ~ x/~(cosB, sPt~+sinBrsQ~'+Sir)/B~+il2x//3, r = l

c'(s)= ~ v/-2(-sinB~sP~r +cosB,sQ'O , r = l

~Si(s) = ~. x/~(sin CrsXir +cos Crs yir)/cr , (36) r = l

where B~=2rrc, C,= (2r+ 1 )rr, p~r, Q,r, SJ~ are independent three-dimensional even gaussian random variables with distribution

exp[(p,r)2 + (Q~r)2 +2(S,02 _S~p~,] , (37)

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Volume 193, number 1 PHYSICS LETTERS B ... 9 July 1987

and .V r, y~r are two-dimensional odd gaussian random variables with distributions

exp( - i Y ~r, X ~r ) . (38)

Eq. (35) also shows that one may express both x(s, a) and ~(s, a) in terms o f a superpath

S(s, a) ~fb(s) +itrp(s)/v/-2, (39)

with

x(s, a)=X(s, a ) , ~(s, t )= -v/~iDs, ,X(s, a) . (40)

This result remains true for more general values of the parameters a, b, a, fl, x if b(s) is a pinned brownian motion between a and b over time t and f ( s ) is a fermionic brownian motion with parameters a, p and x [ 3 ].

As an example of the use of the superpath integration formula (24), the kernel [ e x p ( - H t ) ] ( a , a, or, - a ) will be calculated for the supersymmetric harmonic oscillator in two dimensions which has

Q ~ ' Q o "~-(.I)( P ' X 2 - - ~'/2X I ) / W ~ . (41)

The formula (24) gives

[exp(-Ht)](a,a, ot, -ct)=[d~x[dlt[a,a,a, - a , x , t , 0] d d

t

× e x p ( - I d a f ds (l/v/-2)co( ~'(s , a)x2(s, a ) - 712(s, a)x ' (s, a ) ) ) . (42) 0

The mode expansion for x, ~ in this case may be found by using the fact that (with an obvious notation) (x(s, or), ~(s, tr))[a, a, a, - a , x, t, 0] has the same distribution as

(x/~tx(s/t, alv/tt)+a+itr(ix-ot)/x/~, P(s/t, alx/~t)+ik-a)[O, 0, 0, 0, 0, 1, 0 ] ,

apart from a normalisation factor t-'/2exp( - 2 ix .a) . (This result may be found by comparing the distributions defined by (21 ) for the two different sets of parameters.) This leads to

[exp( -Ht)](a, a, a, - a )

=fd 'x (2n t ) - ' I ~=, f i ( d~P~ d'Q~ d"S~ d"Y'exp[(P~)2+(Q~)2+ 2(S~)2-S"P~-iY"Xq

X exp(to~Ix/-2 ) ( .~, 2( Y'" y2. _X,.X2.)IC. + (21C~),,2[ y , . (a2 _ ik2) _ y2.(a ~ - i x ' )1

\ lr 2r 2r lr 2 1 I 2 ) - i ( P Q - P Q )/(B~) ] + 2 ( i x - a ) ( i x - a z ) - 2 i x . a

=Id'x(to/2nsinhoot)[cosh(tot/2)+i(-a ~ + i x ~ ) ( - a ~ + i x 2 ) ( - a 2 + ix 2 )sinh(cot/2)] exp( - 2ix.a).

(43)

In supersymmetric quantum mechanics, divergence cancellation due to supersymmetry means that the kernel of the Witten index [ 10] [Trace(~ 5 exp( -Ht))](a, a) should not diverge as t tend to zero. (Here Trace denotes the Clifford algebra trace.) In this case,

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[Trace(y5 exp( -Ht))](a, a) = Jd"x d~ a{ (o9/2n sinh ~ot) exp( - 2 i x . a )

× [ i ( a ~ + i x l ) ( a 2 +ix2)cosh(ogt/2) + i ( i x I - o ? ) ( i x 2 -a2)sinh(ogt/2)]}

= 2o9/n, (44)

which is clearly f inite as t tends to zero. In this paper a new approach to pa th integrat ion in supersymmetr ic theories has been developed, based on

the supercharge ra ther than the hamil tonian . Only the s implest supersymmetr ic system - supersymmetr ic quantum mechanics with a single he rmi t i an supercharge - has been considered; the extension to more general quantum mechanical systems, with more supercharges ( and more corresponding an t i commut ing t ime parameters ) should be straightforward. Fur the r extensions, to field and string theories, should show there is a considerable advantage in using supercharges ra ther than hami l tonians , because the supercharges are both more s imple and more fundamenta l .

Other extensions o f this work current ly being considered by the author are an analyt ical ly r igorous devel- opment o f the approach, and its appl ica t ion in curved superspace.

Stochastic different ial equat ions with an t i commut ing t ime var iables as well as commut ing t ime have been considered by Haba [11 ], but the stochastic processes in that paper are different from the process in t roduced

here.

The au thor is grateful to Professor I.M. Singer for useful discussions in which he suggested the invest igat ion o f superpath methods for the s tudy o f the super evolut ion operator . The author is also grateful to the Science and Engineering Research Counci l o f Grea t Bri tain for f inancial suppor t under advanced research fellowship no B/AF/687.

References

[ 1 ] R.P. Feynman and A. Hibbs, Quantum mechanics and path integrals (McGraw Hill, New York, 1965). [2] D. Friedan and P. Windey, Nucl. Phys. B 235 (1984) 395. [3] A. Rogers, Fermionic path integration and Grassmann brownian motion, King's College preprint, to be published in Commun.

Math. Phys. [4] F.A. Berezin and M.S. Marinov, Pis'ma Zh. Eksp. Teor. Fiz. 21 (1975) 678 [JETP Let. 21 (1975) 320];

L. Brink, S. Deser, B. Zumino, P. Di Vecchia and P. Howe, Phys. Lett. B 64 (1976) 435; P. Di Vecchia, L. Brink and P. Howe, Nucl. Phys. B 118 (1976) 77.

[ 5 ] B.S. DeWitt, Supermanifolds (Cambridge U.P., Cambridge, 1984 ). [6] LD. Fadeev and A.A. Slavnov, Gauge fields, introduction to quantum theory (Benjamin/Cummings, London, 1980). [7] B. Simon, Functional integration and quantum mechanics (Academic Press, New York, 1979). [ 8 ] F.A. Berezin, The method of second quantization (Academic Press, New York, 1966). [9] F. Guerra, L. Rosen and B. Simon, Ann. Math. 101 (1975) 111.

[ 10] E. Witten, Nucl. Phys. B 202 0982) 253. [ 11 ] Z. Haba, Supersymmetric brownian motion, Universit~it Bielefeld preprint 181 (1986).

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Supersymmetric path integration

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