Space supersymmetries as hierarchies of ordinary hamiltonian symmetries

Physics Letters B 279 (1992) 336-340 North-Holland

PHYSICS LETTERS B

Space supersymmetries as hierarchies of ordinary hamiltonian symmetries Sergio De Filippo Dzpartimento di Fisca Teorica, Universita di Salerno, 1-84100 Salerno, Italy and IATN, Sezione di Napoli, 1-80125 Naples, Italy Received 14 October 1991

A recently proposed commutative setting for hamiltonian theories with fcrmions is applied to space-supersymmetric field theodes in one space dimension. It is shown that ordinary hamiltonian symmetry algebras for hierarchies of ordinary hamiltonian theories correspond in this setting to the original superalgebras. A space-supersymmetric version of the KdV equation is used to illustrate the general procedure.

A commutative setting for hamiltonian theories with fermion degrees of freedom was recently proposed [ 1,2 ]. In that context a superhamiltonian theor3' is seen to be equivalent to a whole hierarchy of ordinary hamiltonian systems. This is expected to be of use both for a deeper understanding of pseudoclassical mechanics of fermionic systems ~1 [5], in particular to characterize their integrability [ 2,6 ], and in connection with their quantization. As to the possible relevance of an ordinary hamiltonian context like this, in order to quantize systems with fermions, it should be stressed that the usual notion of a path integral for fermion degrees of freedom is merely algebraic [ 7 ], while a classical formulation in terms of commuting variables in principle avoids being forced to leave the realm of measure-theoretic integration. This would allow, for instance, that the euclidean path integral preserves its usual probabilistic interpretation [ 81. It was also suggested that, in the mentioned approach, the ordinary hamiltonian counterparts of supersymmetric theories have hamiltonian symmetry algebras, reflecting the original superalgebras [ 1,2 ]. In the present paper the relationship between space-

supersymmetry superalgebras of ( 1 q- 1 )-dimensional field theories and the corresponding hierarchies of symmetry algcbras is explicitly exhibited. As a specific example a supersymmetric extension of the KdV equation is used, owing to the relevance of this equation in the realm of string and superstring theories [6,9]. Field theories in 3 + 1 dimensions with super-Poincar6 supersymmetry will be considered elsewhere [ 10 ]. To be specific, consider a Z2 graded * algebra .~, i.e., such that G F = ( -- 1 )x(F)~(C')FG ,

( 1)

where F, G are pure elements of .~, i.e., of definite grading g ( F ) , g ( G ) =0, 1 eZ2. Furthermore assume that this algebra is endowed with a super-Poisson bracket (SPB), i.e., a graded bilinear antisymmetric c-map (1,, a ) e ~ × ~ { b ,

G}e~,

(2)

{G, V } = - ( - 1 )~(F)g(a){F, a } , g({F, G } ) = g ( F G ) = g ( F ) + g ( G )

(3) (mod 2 ) ,

(4)

satisfying the graded Jacobi identity ( - 1 )x~")*'("){r, {G, H}}

at A related approach consists in the algebraic formulation of dynamics [3 ], based on the algebraic characterization of tangent and cotangent bundles [4], and correspondingly of their lagrangian and hamiltonian vector fields. 336

+ ( - 1 )~(mx~G'){t/, {F, G}} + ( - 1 )g(ci)e~V){G, {H, F}} = 0 ,

(5)

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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and derivation property

{F, GH} = ( - 1 )g(v)g(~)G{F, 14} + {F, G}H,

(6)

where F, G, H in eqs. ( 3 ) - ( 6 ) are pure elements of

16 April 1992

by all component variables Ft¢ ~with [c] c S, in such a way that

{l~,G}t,q={Ftcl,Gs} s,

l',G~.~,[c]cS,

(12)

by which eq. ( 11 ) is equivalent to In the present paper ~ is an algebra including the observables of a space-supersymmetric ( 1 + 1 )-dimensional field theory. This amounts to assuming that the considered system is space translation invariant, i.e., such that {n,e}=0,

(7)

where H, P~,~, g(H) = g ( P ) = 0 respectively denote the time evolution and the space translation generators, and an odd element QE .~, g(Q) = 1, exists such that {H,Q}=0,

{P,Q}=0,

{a,a}~e.

(8a,b,c)

According to the setting proposed in refs. [ 1,2 ], a generic element F of .~ is explicitly expanded as a formal power series in a given family fit, ~2, ..., ~, ... (possibly infinite) of Grassmann generators, (~h~k+ (k~h= 0), i.e., 1

F= ~ ~.. F(j,.j2,...j,) ~ ' (J2 "'" ~,~ ,

(9)

where here and henceforth summation is implied over repeated Grassmann indices, and the component variables Fu,,j~....a,) (which are completely antisymmetric in their Grassmann indices) are elements of an abelian algebra. Only even and odd monomials in Grassmann generators respectively appear in the sum i f F is an even or an odd element of .~; i f F is even, the component variable corresponding to k=0, the so-called body o f f [ 11 ], will be denoted by Fn. Consider now the evolution equation corresponding to a given hamiltonian H: d F = {F, H} dt,

(lO)

which can be rewritten componentwise, according to eq. ( 9 ) , as dFu~j2.....~k) = {F,/4) o, J2....a~) dt.

( 11 )

It can be proved that, once fixed a finite naturally ordered even subset, say S = - (1, 2 .... , 2n), of Grassmann indices, an ordinary Poisson bracket (PB){, }s can be introduced in the (abelian) algebra generated

dFl~j={Ft¢l,IIs}Sdt,

[c]mS,

(13)

where here and henceforth a letter in square brackets, like [c] in eqs. (12) and (13), denotes a naturally ordered Grassmann multi-index. The ordinary PB {, }s mentioned above is given by

{Figi,Gt~.l}s=o,

([g],[g']cS),

if [g] U [g'] ~ S,

(14a)

(Ftgl, Gig. i}s= ( - 1 )P'+m{F,G } l g l n t g , ] , if [ g ] U [ g ' ] = S ,

(14b)

where [g] n [g'] denotes the corresponding naturally ordered subset and P t, P2= 0,1 respectively the parity of the permutations S--, ( S \ [gl, [g] ) ,

(15a)

[g'] ~ ([g'] \ [g], [ g ' ] N [ g ] ) .

(15b)

Actually i0 ref. [ 1 ] this was stated and proved only for the particular case of an algebra .~- generated by two (possibly continuous) families of bosonic and fermionie variables, with F and G coinciding with two such variables, but the proof can be generalized [ 12 ]. If, in particular, the implied algebra . f is assumed to be formally generated by two continuous families (u(x))xc~, (~o(x))xo~ of pure elements

g(u(x))=O,

g((0(x))=l,

xe~,

(16)

respectively representing a bosonic and a fermionic field, the corresponding component variables obviously are ordinary fields, which for simplicity will be assumed to be real. (This choice, if the Grassmann generators are real as usual, amounts to considering complex boson and fermion ficlds, which could be made real by redefining component variables [ 11 ]. However, hermiticity requirements are of no concern here, since only pseudo-classical aspects are treated. ) Consider now the (infinitesimal) supersymmetry transformation

~F={F, ~I Q } = - { F , Q} ~rl,

(17) 337

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where r/is an anticommuting parameter, that, like any odd supernumber can be expanded as 1

r/--- ~ (2k+ 1)! r/u,j2.....j~,.,)(j,(,2... ¢j2,+,,

(18)

in terms of the ordinary parameters qu,,jz,...,i2k÷,) [ 11 ]. Componentwise eq. (17) can be written as 8Fie I ={F, 8q Q}tcl ={Ficl, (fir/Q)s} s = ~ (--1)"tdl{Ftcl,Qs\ldl}s ~5~/tol, [d]

(19)

where eq. (12) was used. Summation is performed on multi-indices which are naturally ordered subsets of S, and ~z[d] =0, 1 denotes the parity of the permutation S ~ ( [d], S \ [d] ). This shows that for a fixed S, say ( 1, 2, ..., 2n), the original supersymmetry generator Q gives rise to a whole family of 22"- l (ordinary ) symmetry generators Qt~ l for the (ordinary) hamiltonian flow generated by Hs according to eq. ( 13 ). As to the Lie-Poisson superalgebra defined by eqs. (7) and (8), its (ordinary) counterpart in the present setting is given by a hierarchy of algebras, each one corresponding to a fixed even subset of Grassmann indices, S, defined by

{Qldl, Qte,'l}s~Pte, lnI~'J, {Qtdj, ato.j }s=0,

{u(x),u(y)}=a'(x-yl, {~(x),~(y)}=-a(x-y), {u(x),~(y)}=o.

(22a) (22b) (22c)

The supergenerator of the space translations P[u, ~o] and a supersymmetry generator Q[u, ~0], namely a functional fulfilling eqs. (8b), (8c) are then easily seen to be given by Plu, ~01= f dx ½(u2-~o~0'l,

(23/

Q[u, ~] = f dx u~o,

(24)

with, to be specific, {Q, Q}= - 2 P .

(25)

A particular hamiltonian fulfilling eqs. (7), (8a) corresponds to one of the several considered fermionic extensions [ 1,2,6,13 ] of the KdV equation: (26)

(20a)

whose hamiltonian, with respect to the PB defined by eq. (22a1, is

(20b)

. ~ [ u ] = f dx (u3+ ~u '2) .

if [ d ] U [ d ' ] = S ,

if [ d ] U [ d ' ] ~ S ,

nations of respectively odd and even order derivatives of the 8-function, which, to take only the lowest possible derivatives, amounts, up to arbitrary multiplicative constants, to putting

d u = (6uu'-u") dt,

{Htcl, HI,.. i }s = {Ptcl, PI~'I }s= {Ht~l, p[¢71 }s

={nt~l,Qtd]}s={etcl,Qldl}s=o,

16 April 1992

where[c], [c'] denote even and [d], [d'] odd naturally ordered subsets of S. (Nilpotency of these Lie algebras follows from the abelian character of the space-time algebra generated by H and P in the present particular case of one space dimension. ) To be specific consider now an explicit realization of the considered superalgebra in terms of the fields u and ~0in eq. ( 16 ). The simplest class of translationinvariant PBs is defined by

(27)

Observing that eq. (26) is invariant with respect to the scaling transformation

x--,c~x,

u-,a-2u,

t--,a3t,

(28a)

the field u can be assigned the same dimensions as x -2. If now the (infinitesimal) transformation generated by Q in eq. (24) is considered, i.e., 8u(x) = 8~/~0'(x) = -~0'(x) 8r/,

(29a)

{u(x), u ( y ) } = f ( x - y ) ,

8~0(x) = 8q u(x) = u(x) &l,

(29b)

{~0(x), ~0(y)} = g ( x - y ) ,

the fermionic field is consistently given the dimensions of x - 3/2 (and parenthetically q ~ x i/2 ). Then, if invariance under transformation (28a) (which, generalized by t - , a t +2~t, keN, is a characteristic feature of the whole KdV hierarchy [ 14 ] ), properly extended by

{u(x),~o(y)}=O,

(21)

wherefand g respectively are odd [ f ( - x ) = - f ( x ) ] and even [ g ( - x ) = g ( x ) ] (ordinary) distributions. If locality is required, f a n d g are finite linear combi338

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~0~a -3/2~o,

(28b)

is preserved, the most general fermionic extension of the KdV hamiltonian is obtained by adding terms proportional to J d x (u~0') and j'dx (~0"). Supersymmetry requirement, eq. (8a), then leads to the superhamiltonian

H[u,(o] = f dx

(u3+ ~u'2-2u~0~0'+ ½~0~0"),

(30)

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The PB, defined in general by eq. (14), is then given by

{UB, uB}S={u~,,2), u~,,2)}~={U, V} s = {~0(l), ~0(l)}s= {~ot2), tp(2)} s = 0 ,

(34a)

{uB(x), u¢~.2)(y)}S=6'(x-y) ,

(34b)

{~0~,)(x), ~o~2)(y)}S=,~(x-y),

(34c)

d u = {u, H} d t = (6uu'-u"-2~o~o") dt,

(31a)

where omitted indices or arguments are meant to be arbitrary. Relative to this PB, the generator of the subsystem of system (32)given by eqs. (32a), (32b), (32c), with i= l , j = 2 , is

d~o=(GH}dt=(-~o"+2u'~o+4u~o')dt,

(31b)

H(l,2) = J" dx [3u~u(l,2) +U~U]).2)

whose Hamilton cquations are

and were obtaincd in ref. [ 13 ] by means of the superspace approach. The explicit transcription of eqs. (31) in terms of component fields [ 1,2] is given by the system

dub = (6uBu~--u~') dt,

(32a)

d~o(,) = ( - tp~'~)+ 2uh~p(~) + 4uR ¢ff~o) dt,

(32b)

--2UB(~(I )(fl~2)--q~(2)~0~l)) +~0(1)~'¢'~) ] ,

while the generators of transformations (33a), (33b), (33c) corresponding to parameters r/(~), r/(2) respectively are Q(2) = f d x (u13~(2)) ,

-Q(~)= j dx (--UB~0tt)) . -" +2(ao)~ao)) t! dt, - 2v-(o(a~) !

(32c)

which is invariant with respect to the componentwise expression of the supersymmetry transformation in eq. (29):

(35)

(36)

Finally the generator of space translations is given by P(t.2) = f dx (uau(~,2) -~(i)~0~2)) ,

(37)

while the bodies of H and P respectively are

PB = ~ cL~ ~un.l 2

SuB(x) = 0 ,

(33a)

HB[UB]=,~[UB],

8~o(i)(x)=uB(x) 8q(,) ,

(33b)

8uti.~)(x) =~P'u)(x) St/(o -~0~o(x) 8qt~),

(33c)

It is straightforward to verify that functionals given by eqs. (35)-(38) fulfil eq. (20), giving in particular

8~o~;j,~)(x) = u~,.~)(x)

8r/<,~-

{Q~I), Q(2)}s=2pB,

u~,.,~(x) 8rk, )

+ u(i..i) (x) 8rlt~) + u~ (x) 8q(,,;,~) ,

(33d)

(38)

(39)

thus generating (one of) the simplest element(s) of the mentioned hierarchy of Lie-Poisson algebras.

~U(i,j,h,k ) (X) =~O~(j,h.k)(X) 6q(i) --~O)i,h,k ) (X) ~qO) Financial support from INFN and MURST (Italy) is acknowledged. +~o},) (x) 8qo.,~,~) -~oi~)(x) 8rh,j,~) "~'(PO) (X)

~(i,h,k)--~0(i) (X) 8q(j.h,k),

(33e)

Consider now for simplicity the simplest possibility of two fixed Grassmann indices, say S= (1, 2).

References [ 1 ] S. De Filippo, A bosonization procedure for hamiltonian theories with fermions, J. Phys. A, to appear. [ 2 ] S. De Filippo, Phys. Lett. B 265 ( 1991 ) 127.

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[3] S. De Filippo, G. Landi, G. Marmo and G. Vilasi, Phys. Lett. B 220 (1989) 576, and references therein. [4] S. De Filippo, G. Landi, G. Marmo and G. Vilasi, Ann. Inst. H. Poincar6 50 (1989) 205, and references therein. [5] R. Casalbuoni, Nuovo Cimento 33 A (1976) 389; F.A. Berezin and M.S. Marinov, Ann. Phys. 104 (1977) 336. [6] B.A. Kupershmidt, Phys. Left. A 102 (1984) 213; Yu.l. Manin and A.O. Radul, Commun. Math. Phys. 98 (1985) 65; M. Chaichian and P.P. Kulish, Phys. Lett. B 183 (1987) 169; P. Mathieu, Phys. Lett. B 203 (1988) 287. [7]F.A. Berezin, The method of second quantization (Academic Press, New York, 1966); L.D. Faddeev, in: Methods in field theory, eds. R. Balian and J. Zinn-Justin (North-Holland, Amsterdam, 1976).

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[8] K. Symanzik, J. Math. Phys. 7 (1966) 510; J. Glimm and A. Jaffe, Quantum physics - a functional integral point of view (Springer, Berlin, 1981), and references therein. [9] J.L. Gervais, Phys. Lett. B 160 (1985) 277, and references therein. [ 10] S. De Filippo and L. Mercaldo, work in progress. [ 11 ] B. DeWitt, Supermanifolds (Cambridge U.P., Cambridge, 1984). [ 12 ] S. De Filippo, in preparation. [ 13 ] P. Mathieu, J. Math. Phys. 29 (1988) 2499. [14] S. De Filippo, Phys. Lett. B 131 (1983) 65, and references therein.