IL NUOVO CIMENT0 VOL. 35 A, N. 3 1 0t tobre 1976

Supersymmetries and the Pseudoclassical Relativistic Electron.

A. ]3ARDUCCI

Ist i tuto di Fisica Teorica dell' Universith - Fire,~ze

Ist i tuto Nazionale di Fisica Nucleare . Seziot~e di Firenze

t{. CASALBUONI al td L. LUSANNA

Isti tuto Nazionale di Fi.~ica Xucleare - Sezione di Firenze

(riccvuto il 10 Agosto 1976)

Summary ...... In this work we consider a version of the supcrsymmetry parametrized by antieommuting pseudovector and pseudoscalar variables instc~ad of spinorial ones. We construct a pseudoclassieal relativistic Lagrangian inwu'iant under the former supcrsymmetry. This Lagrangian is a degenerate one and we show that, after quantization, one of the constraints gives rise to the Dirac equ,~tion. Moreover, we introduce into our lm,grangian an interaction term with an external electromagnetic field, and we show that this is possible, in a consistent way, only if the anomalous magnetic moment is wmishing. It follows that this model represents a pseudoctassica.l description for the relativistic electron.

1. - Introduct ion .

I n a series of works (1-3) the classical mechan ics for a sy s t em descr ibed b y

c o m m u t i n g a n d a n t i e o m n m t i n g var iab les has been s tudied. I n pa r t i cu la r , a.

ce r t a in n u m b e r of models descr ib ing s p i n n i n g par t ic les was discussed.

(1) J . L . MARTIN: Proc. Roy. Soc., 251 A, 536 (1959); F. A. B~m.:ZlN aad M. S. ~/[A.RINOV : J E T P Lett., 21, 321 (1975). (2) R. CASALBUONI: 2¥U07)0 Cimento, 33 A, 115 (1976). (a) R. CASALBUONI: NUOCO Cimento, 33A, 389 (1976).

377

~ 7 ~ A. BARDUCCI, R. CASALBUONI ~ n d L. LUSANN&

In this p~per we w:mt to show how the idc~ of the supersp~ce (*,~) c~,n be

fruitfully extended to get ~ pseudoelassic~l version of the Dirac eqm~tion.

This is realized by extending the concept of supersymmetry to pseudovector

:~nd pseudosc~dur tmtieommuting vs~riablts.

After h,~ving studied in sect. 2 the ~lgebr:~.ic ~spects of Lhis new super-

symmetry , in sect. 3 we construct an invari~mt relativistic L,%~'angit~n. The

v~riables upper, ring in this L~grangian :~re the usual position four-vector und

five an t icommut ing v~ri~bles ~ a.nd ~, which beh~ve as ~ pseudovector ~nd

~s ~ pseudosca, l~r, respectively. The proposed L~gTangian gives ri;;e to v~rious constraints, one of which

is strongly reminiscent of the Dirge equ,~tion, with the ~nt icommuting v~ri~bles

playing the r61e of the y-m~trices. We ~n~lyse the nature of these eontr~,.ints ~md we introduce Dirac br,~ckets.

Fur thermore, we show tha t there exists ~ direction in the superspaee which

~llows for time-dependenL supersymmetry t ransformations under which the

theory is inv~ria.nt. We show ~lso the rela.tion of this symm(~try with the

v:didity of the Dirae constraint. I n sect. 4 we s tudy the equations of motion i~ the Ha miltonian form~flism,

~md we point out ~h,~t, in order to get the equivalence with the La.gr~ngian

formtdism, we need to introduce a further constraint . By qua,ntizing this theory, we find tha t the variables ~, ~md ~ go into the

matrices ~ y , and 7s, respectively; fm'thermore, the Dirge constr~tint becomes

condition over the st~tes, which is precisely the Dir%e eqm~tion. In sect. 5 we study the nonrelativistic limit and we show t h:~t we get the

Lagrangian for the spin studied in ref. (~,~). Moreover, ~v(, illustr~te how to

introduce, in ~ consistent w~y, the coupling wi~h an e]ectroma~'lwtic fiehl

a.nd we find tha t this is possible only for zero ~molna,lous m~gnetic mome~lt.

(*) Yu. A. GEL'FAND and E. P. LIKttTMAN: J E T P Lett., 13, 323 (1971); D. V. VOLKOV and V. P. AI~uLov: Phys. ]Jett., 46 B, 109 (1973); J. WESS and B. ZU~HNO, Nucl. Phys., 70B, 39 (1974); A. SALAlg and J. STRATHD]g:E, .~c l . Phys., 76 B, 477 (1974). See ~lso B. ZI~MIXO: Proc. o] the V I I International Con]erence on High-Energy Physics (London, 1974), Rutherford Laboratory, Chilton, Didcot (1974), pp. 1-254. A gre~t impact ill the development of the supersymmetries has been given by tim duM n,odels, see A. NEv]~u and 3. H. SCHWAI~Z: Nucl. Phys., 31 B, 86 (1971); P. RAMONI): Phys. Rev. D, 3, 2415 (1971). (5) See D. V. VOLKOV ~nd V. P. AKULOV (4), s e e tl~lso A. SALAM ~nd J. S'rI~h'rl:tI)l~E: Phys. Rev. D, 11, 1521 (1975). For more recent ~ttempts Mong this line, see L. N. CHA~(~, K. I. MACRAV" ~nd F. MANSOURI: Phys. Rev. D, 13, 235 0976); Y. M. CHO and P. G. 0. FR]~USD: Phys. Bey. D, 12, 1711 (1975); P. NATtl and R. AR~OWIT'r: Phys. Left., $6B, 171 (1975); B. ZuMIXO: Supersymmetry, CERN preprint TtI 2120, 1976.

SUPERSY~IM-lt, TRI]~S AND THe3 I~S]~UDOCLASSICAL RF, LATIVISTIC ]?,LECTI~ON 379

2. - S u p e r s y n n n e t r y .

The main idea abou t supersymnlet r ies (4) (':~n be tr.tced back to the notion of

associat ing to every point of the space- t ime the usual co-ordinates xu together with some other a .nt ieommuting co-ordinates (~,6). This p rob lem is formul ' t ted

b y requir ing t h a t a cer tain differenti~d form is invar innt under t ranslat ions

of the co-ordinate sys tem erected around the given point in the space-time.

I n the usual models, the differential form is

(2.1) o9~, -- dx l , - - tO* a~ d0 + i (10" al, 0,

where 0 and 0* are Weyl spinors. The form. 2.1 is invar ian t under the t rans-

fo rmat ion

I 0~ ~ 0~ -t- ~ , , (2.2) / x~ ~ x~, + i(~* a~ 0 - - 0* a~ ~) .

Then one considers <<fields ~> over the supeI'space pa.rametrized by ( x , 0,0~,),

and, by expanding these fields in powers of the an t i eommut ing w triables, one

obtains the usual killd of fiehls. However , in this scheme due to the t rans-

format ion (2.2) wc can relate together fermionic ~nd bosonie fields.

The generators of tr,~nsformation (2.2) satisfy ,~nticommutation relations,

so tha t , if ~hey are expressed as integrals of densities, it results t h a t they have to carry out half- integer spin, in order to satisfy the spin statist ics theorem.

i n rcf. (3,6) it was shown thv~t it: is possible to coJ~struet a. (, classical >) model involvinff Gr:~ssmam~ variables 0~ :rnd 0 z. This model is described by a La- grangian which is propor t iona l to the line clement associated with the differential

form (2.1). I t was found t h a t a.fter quant iza t ion the v~ri~bles 0~ and 0 z become, essentially, Fermi creation ~nd a.mfihilation oper 'rtors. I t follows tha t one can

const ruct coherent s tates of these Fermi opera lors, whose cio'(~nvalues will

be Grassmann variables (7). These states b(~eome superfi¢lds af ter 1he ((second quantizat ion )).

F r o m this poin t of view, it ;q)pears very hard to generalize the concept

of s u p c r s y m m e t r y to Grassm~ml wiri:~bles with integer spin.

However , in ref. (~,a), a, nonreh~tivistic model of the spin ½, described by

Grassma.un vt~riablcs as tra~)sforming like ~ vector under the ro ta t ional group,

was considered. The I)oint is tha t , af ter qlwntizat ion, these variables become

the genera tors of the real Clifford algebra C3, i.e. the Pauli matrices.

(6) The strict relation between this idea and relativity has been discussed in R. CASAI,- ]3co,I: t 'hys. Let$., 62 B, 49 (1976); see also ref. (3). (7) J. R. KLAUDER: Ann. o] _Phys., 11, 123 (1960).

3 8 0 A. BARDUCCI, R. CASALBUONI ~tnd L. LUSANNA.

Iqow, in the second-quantizat ion process, there is no more reference to the Fermi variables but for the spinorial indices.

We see tha t , by following this way, it is possible to t r y to generalize the supersymmetr ies to integer-spin variables thus avoiding the conflict with the

spin statistics theorem. The idea is to generalize the differential form (2.1) by choosing a different

realization of the four-vector added to dx, . The simplest way to do so is to introduce five real an t ieommut ing variables:

~ pseudovector ~, and a pseudoscalar ~ . Then we define the differential form

_ _ i 7 (2.3) .Q~, = dx~, + ifl ~ d ~ -]- - - d~# ~ mc mc '

where, following ref. (3), we have chosen ~, and ~a with the dimensions of the square root of an action. Fur thermore , fl and y are two numerical coefficients

which will be chosen later on. The form 2.3 is invar iant under the t ransformat ion

(2.4)

~. -+ ~ + ~ , ~ - ~ ~:~ + ~ ,

x ~ - + x ~ - ifl ~.~ _ ire ~ . ~ . m e m e

The commut'~tor of two infinitesimal t ransformat ions can be easily evalua ted:

(2.5a)

(2.5b)

(2.5c)

al (a . . x~) - a~(a,x,) = ~cc ( f l - 7)[~Y

(~1(a2~:#)- (~2((~15) ---- O,

We can introduce the generators of such t ransformat ions by put t ing

(2.6) ~x, = i [x~, s, G" + ,~,~ Gs]

~nd analogous relations for the other variables. By using the Jaeobi ident i ty , we get the following relations:

[X/~, [8~)V 'u + ~{51}G5, _(2)~_fe s~2,V~]] i (/~_ .~ (1)_,2) ~(2, o(,)1 ~u ',z -~- m c

(2.7) [~:.,,, t,~.~-'~' ~" ~- .~i'-) ~ , ~, ~,' + 4 ~' e~]] = o,

[~ , [~ ' ~ + ~:i '~5, ~ ' a~ + ~ ' a~]] = 0 ,b/~

SUPERSYMMF~TRI]~S A N D T H E P S E U D O C L A S S I C A L R E L A T I V I S T I C E L E C T R O N 381

In all these equations, eu ~nd s~ are ~nt icommuting parameters , thus it is con- sistent to require tha t they an t icommute with the generators G~ and G~.

I t follows tha t , up to the constants, we can determine from (2.7) the anti- commuta t ion relations satisfied by Gu and G~:

(2.8a) [G,, G~]+ = ag,~,

(2.8b) [Gs, G~]+ = b,

(2.8c) [Gu, Gs]+ = mcc ( f l - y) ~x~ ---- me '

where a and b ~re a rb i t ra ry constants. Fur thermore , Gz ~md G5 ~re a pseudo- vector ~nd ~ pseudoscalar, respectively, under Lorentz t ransformations, and they ~re t ransla t ional ly invariant , i.e. they eomnmte with P~.

A par t icular representa t ion of the algebra (2.8) c,~n be obtained in a space with fixed p2, by taking G, and G5 as line~r combinations of Dirac matrices:

(2.9)

These operators satisfy the algebra.

(2.1o)

= -

[~5, ~ ] + = - 2a~P~ + 262,

which is the same as the algebr~t (2.8) af ter the identification of the constants.

3. - L a g r a n g i a n .

In this section, we want to develop a pseudoclassical (2) model invar iant under the supersymmet ry t ransformations int roduced in the previous section.

To do this, we introduce, as usual, an invar iant parameter ~ to parametr ize the trajectories. This requires the Lagrangian to be homogeneous of the first

degree in the derivatives with respect to T. We see tha t the most general Laga'angian satisfying this requirement and

with the p roper ty to be quasi-invariunt under the t ransform~tion (2.4) will be

(3.1) V ( x . +

where the dot denotes the differentiation with respect to v, and the action

25 - I I Nuovo Cimento A .

3 8 2 A. BARDUCCI, R, CASALBUONI and L. LUSANNA

associated with this Lagrangian will be

~y

(3.2) S = j d ~ .5~.

In (3.1) a~ and as are parameters which will be determined later on. Under supersymmetries we have

d (3.3) ~.~f = izqe~5 + i~e ,~" = i ~ [ ~ e ~ + ~ t ~ " ] ,

from which the invariance of the equations of motion follows. The general expression for the generators of a canonical transformations

was given in ref. (~); in the present case, using the eqs. (2.4), we get ( 8 ~ = }.--$, , and so on)

(3.4) ~ = -- ~x,-P~ + ~,zt~ + ~5~5-- ic¢leS ~s-- ia~e,~" =

=--~5[~rs+iOtl~5--~P'~]--sl,[zt"+io~,~ ~' + ~ P ~ . ] ,

from which

(3.5) O~-~-- [xes + izq~5--mciJff"P'~] '

where Pu, ~tu and 7e5 are the canonical momenta. By using the standard Poisson brackets (~.8)

(3.7) { x . , P~} ----- - - g . , , {g., }.} ---- - - g ~ , {gs , }5} = - - i ,

we get

(3.s)

{Gs, Gs} ---- -- 2i31 ,

i {V~,, Os} -- ( f l - ~,)_v, ,

m ~

that is a Poisson-algebra realization for the generators of the graded Lie al- gebra (2.10).

In order to derive this result, we have used Poisson brackets; however, the Lagrangian (3.1) is a singular one and it gives rise to a certain number of constr.-Ants. By putting

SUPERSYMMETRIES AND TIIE PSEUDOCLASSICAL RELATIVISTIC ELECTRON ~3

for t h e canon ica l m o m e n t a we ge t

(3.10) lP. - -

(3.11)

(3.12) ~ 8 ~ 8V ~

~, = ~ = ~ + ~ ~ = ~ . , ~ + ~.~:.

F r o m (3.10) we ge t t he mass cond i t ion

(3.13) P ~ = m~c 2

F u r t h e r m o r e , eqs. (3.11) a n d (3.12) a re c o n s t r a i n t equa t ions , because t h e y

express ~r~ a n d n5 as func t ions of t he canon ica l va r iab les only .

W e will discuss these c o n s t r a i n t s fol lowing Di rac (s,8); thus , let us p u t

(3.14) Z = p2__ m s c 2 ,

(3.15) Zs = zrs-- icq~5 -- --ifl p . ~ , m c

(3.16) Z, = n . - - i~2~ + i y P , ~ 5 . m c

Now, b y us ing (3.7), we c o n s t r a i n t s

(3.17)

can ca lcu la te the Poisson b r a c k e t s a m o n g ti le

{Z, Z} = 0 ,

{Z, Zs} = 0 ,

{z, z.) = o ,

{z0, z0) = 2i~1,

{xs, z.} = i ~ - r p . , m c

{x . , x.} = 2i.~g.~.

W e see t h a t Z is a f irst-class cons t ra in t .

(s) P . A . M . DIRAC : Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University (New York, N.Y. , 1964).

3 ~ A. BARDUCCI, R. CASALBUONI a n d L. LUSANNA

To s tudy the charac te r of the other constraints , we have to s tudy their

mat r ix . In t roduc ing the nota t ion

(3.18) Za----- (Zs, Zu),

we have

(3.19) C~ ---- {Z~, g~} ----

2i(z, i flmc y

The de te rminan t of this m a t r i x is

det II C~ I[ : -- 4~,~2 q- - -

or, by using the cons t ra in t Z----O,

(~- y)~ p~, m 2 c 2

(3.20)

Now let us notice t h a t the cons t ra in t (3.15) recalls s t rongly the Dirac equa- t ion. We can hope to build up a pseudoclassical model for such an equat ion,

if we are able to choose the a rb i t r a ry p a r a m e t e r s t h a t appea r in (3.1) in such a way t h a t one of the const ra in ts Z~, or a l inear combinat ion of them, is a first- class one. This condit ion is necessary in order tha t , a f te r quant izat ion, this cons t ra in t becomes a condit ion on the states. Fu r the rmore , this cons t ra in t

mus t be expressed in t e rms of var iables whose algebra goes into the Dirae

a lgebra a f te r quant izat ion. In order to realize the first condition, it is sufficient t h a t the de t e rminan t

(3.20) vanishes ; thus we will impose on our p a r a m e t e r s the condit ion

(3.21) 4cqzq = (fl-- y)~.

To be t t e r unders tand this condition, let us s tudy the equat ions of mot ion

which follow f rom the Lagrangian (3.1); we have

_P~= 0 ,

We see tha t , when the condit ion (3.21) is satisfied, the equat ions for $~

and ~5 admi t a solution different f rom zero. Fu r the rmore , the equat ions of

8UPlgRSYMMIgTRIIgS AND THE P$lgUDOCLASSICAL RELATIVISTIC ]BLlgCTI~ON ~ 5

motion do not de te rmine all the variables, bu t a rb i t r a ry functions of t ime a.re present . I f we wan t only one of the Grassmann var iables to be an a rb i t r a ry

funct ion of t ime, then we mus t require a2 ~ 0 . Since our scale is a rb i t ra ry ,

we can choose ~z = ½. Now let us de te rmine the combinat ion of the constraints , which is a first-

class one. Buy pu t t ing

(3.23)

we get.

ZI) = aZ5 + b(P" Z),

{~D, )~5} : 2iaal-~ ib(/5--7)mc + ib(~mc ~)

f rom which

b /5- ~, _ 2~t ( 3 . 2 4 ) ~ ~ = - . , c - - ~ , o _ ?)me" (p

Z,

Up to this momell t , we do not have specified the values of the pa r ame te r s

fl and y. However , all the re levant Poisson brackets (3.8) and (3.17) depend on the difference / 5 - - y only. And tile same is t rue for the condition (3.20)

and for the equat ions of mot ion (3.22). This suggests the possibil i ty to per form

a eanonicM t rans fo rmat ion in such a w~y t h a t the original I~agrangian (3.1)

depends on /5--7 only. To this end, we notice that, we can rcwri te the differentia.1 form (2.3) in the

following w:~y :

(x~ i_yy ~ 5 ) + i f l -7 ~,d~. (3.25) ~Q~ = d -~ mc mc

By defining the new set of var iables

(3.26)

i)J

~ = ~ ,

the new Lagrangian will be

(3.27) ~(~, ~, ~) = - i ~ , ~ - i ~ . ~ . -

= ~f(x, , 6 , ~5).

3 8 6 A. BARDUCCI, R. CASALBUONI and L. LUSANNA

The co-ordinate t ransformation (3.26) is generated by the generating func- t ion p)

(3.28) ~v'[x~, P~, ~ , :~,, ~ , 7~] = -- Ix, + ~ ~ ] P~ + ~ + ~ : ~ ,

and the relation between old and new co-ordinates is given by

(3.29) ~ '

From these relations it follows tha t the expression of the new conjugate momenta is

(3.30) ~ : :r~ -[- mc

i7 : r 5 = : r5 - - - - P " ~ •

~nc

I t not necessary to emphasize tha t the Poisson brackets of the new canonical variables are the same as the old ones.

We see also tha t this canonical t ransformation can be interpreted as the following transformations of the parameters fl and 7:

(3.31) fl-->fl-- ~, ? - + 0 .

Wha t we have shown is tha t this resealing of the parameters is a canonical t ransformation and, consequently, we can use the La~rangian

(3.32) "Lf =-- izc~5~5-- i~2~'~'-- mc ~ ( ~' ~- i fl-- ~-~5) 2 m e

in place of the Lagrangian (3.1). 1V[oreover, the new expression for the generators of the supersymmetry and for the constraints can be obtained from eqs. (3.5), (3.6), (3.14), (3.15) and (3.16) by means of (3.31).

Final ly without loss of generality we can choose fl-- ~ = -- 1, from which, using (3.21) and a~= ½, we get ~-----½.

(9) We are using the same terminology as in H. C. CORBE~ and P. ST~HL~: Classical Mechanics, 2nd Edition (New York, N. Y., London and Sidney, 1965).

S U P E R S Y M M E T R I E S AND TI IE P S E U D O C L A S S I C A L R E L A T I V I S T I C E L E C T R O N ~ 7

As we saw, the constra ints (3.16)

i (3.33) i~u = ~u-- ~ ~ = 0

are second-class ones. Hence, we have to define Dirge brackets (s,3):

(3.34) (A, B)* = [A, B } - {A, Z.u}(C-1)/tv{Zv, B},

where C -~ is the inverse m~tr ix of

(3 .35) G~ = {Z~,, Z~} = igu~.

The only Dirge b racke t t h a t we nee(,[ is the following one:

(3.36) {~, ~,}* = igor.

lqow let us s tudy the covgrianee of the theory. The propert ies of t ransform-

at ion of our var iables under infinitesimal Lorentz t ransformat ions are

w h e r e (~uv ~ - - ¢o,,/t.

(3.37)

where

(3 .38)

(3.39)

x ~ - ~, --- ~x~ = ~o7, ~ x~,

I t follows t h a t the genera tor

L ~ = -- P . x~ ~ P~ x , ,

The Poisson algebr~ of these generators is

(3.40)

(3.4i)

of this t rans formgt ion is

{Lqz, Lu,} = gzuLq~-- g~uL~ ÷ gq, L~u-- gz, LQu,

{Sea, 8~,~} = gag S ~ -- g~u S ~ + g~ S ~ u - gz~ S ~ .

Under Dirac brackets , the second-clgss constr~dnts cgn be t aken to be strongly

~ A. BARDUCCI, R. CASALBUONI and L. L U S A N N A

zero~ thus we can re-express S., in the following way:

i (3.42) ~ = -- ~ [~ , ~,]_,

(3.44)

we see t ha t

where [,]_ is the commutator .

The constraints (3.33) are invar iant under supersymmet ry t ransformations, thus there is no change in the algebra for Gu and Gs.

By looking at the expressions ~or G, and G5 and using (3.33):

(3.43) 05 = - m + g ~ ,

(3.45) ( Thus this combination of generators is weakly zero. Bu t this means t h a t the theory must be invar iant under r -dependent translat ions along a part ic- ular direction in the (~ , ~5)-space. The general t ransformat ion is

G = e, G" + e5 G~ .

By using (3.45) we have

1 e~P.) V . ,

and, for t ransformat ions such tha t e , = (1/mc)esP,, G is weakly zero. I t fol- lows tha t the theory must be invar iant under the r -dependent supersymmetry t ransformat ion

(3.46)

1

~5 -~ ~5 + ~ ( ~ ) ,

This invariance can be checked explicit ly on the Lagrangian (3.32) or, more simply, on the equations of mot ion (3.22). In fact these equations depend only on tlle combinat ion ~ , - - (1/mc)-Pu~5, which is invar ian t under (3.46).

8 U P E R S Y M M E T R I E S A N D T H E P S E U D O C L A S S I C A L R E L A T I V I S T I C E L E C T R O N 3 ~

4. - Equations o f mot ion and quantization.

As we saw in the previous section, in this theory there are two first-class constraints . I t follows t h a t the more general Hami l ton ian will be (s)

(4.1) = ql(P 2 - m2c ~) + q~(P.~-- imc~5-- lmc~a) .

We recall again t ha t our final choice of the p a r a m e t e r s was y = O, fl = - - 1,

0~ 1 = 6¢ 2 = ½ .

I n the present case, ~2 mus t be an odd Gra.ssmann var iable ; thus the

mos t general covarial~t choice will be

(4.2) ~2 = ~1~5 + ~2~

with ~ and ~2 constants . Fur the rmore , to have conne(.tion with the s e a h r rcl~tivistic particle, we

will pu t (lo)

1 (4.3) O~ -- 2me "

Thus we get

2mc (P~-- m~c2) + 21zr5 P ' ~ - - mc~s + ~2~5(P'~-- imc~5) .

The equat ions of mot ion will be given by

(4.5) A ={A,W}*,

where the Dirac b racke t is defined in (3.34). For our dynamica l var iables X~, ~ , ~5, ~5, P , , i t follows t h a t

(4.6)

1 2 ° I t~e

(1) /~, = 0 .

(lo) See, for instance, R. CASALBUONJ, J. God, iS and G. LONGm: Nuovo Cimento, 24 A, 249 (1974).

3 9 0 A. BARDUCCI, R. CASALBUONI a n d L. LUSANNA

F r o m these equations we easily get

(4.7)

where

4 2 (lzl (41n~ + 4s~) + ~os()lln~ + 4s~) = 0 ,

(4.8) (o s = - - 2 i ~ 1 ) l s P s - m s es( 1 2 1 - - i 2 s ) ~ .

Let us notice t h a t 41 and 4s m u s t be, respect ively, real and pure i m m a g i n a r y

quant i t ies , as 5/f has to be a real quan t i t y (11). We can observe here a r a the r curious phenomenon ; af ter quant iza t ion, f rom

eq. (3.36) and f rom the quant iza t ion rules establ ished in ref. (a), i t follows t h a t [ ~ , ~ ] + = - g~. This means t h a t ~o will become an an t i -He rmi t i an op-

e ra tor (we will have ~-+75~+~). Consequently, the hermi t ic i ty proper t ies of H will be ve ry complicated.

However , one can easily realize t h a t H will be He rmi t i an with respect to the

indef ini te metr ic ~o~ i f 41 and +~s are chosen pure i m m a g i n a r y and real~ respec- t ively. This means t h a t a f te r quant iza t ion we mus t redefine the Hamiltonian~

b y changing the real i ty propert ies of 41 and 4~.. This is not very s t range, be- cause it is wha t we usual ly do in the theories with an indefinite metr ic (12).

We also notice t ha t this indefinite metr ic will not give rise to t roubles if

~5 will go into an Hermi t i an opera tor a f ter quant izat ion, because, in this case, the constra.int ZD : 0 will re la te the an t i - I t e rmi t i an opera tor ~o to H e r m i t i a n

operators . Analogously, by choosing a gauge ~ ~ xo, we will have to choose Qs ~ 4~o,

and 4 will be pure immag ina ry in the pseudoelassical case and real in the

q u a n t u m case. Wi th this in mind let us come back to the equations of mot ion. When

p s : m sc s, we have

co 2 = - m " e S ( ½ 41 + i 4 s ) s ,

and co will be pure i m m a g i n a r y in the pseudoclassical case and real in the

q u a n t u m one. By pu t t i ng

(4.9) o~ = -- imc(½ 41 q- i42),

(U) A real quantity in this context means real with respect to the standard involution defined in the Grassmann algebra; see, for instance, F. A. BE~EZIN: The Method of Second Quantization (New York, N. Y. and London 1966). (zs) K. L. NAGY: State Vector Spaces with Indefinite Metric in Quantum Field Theory (Groningen, 1966).

SUPERSYM~fETRIES AND THE PSEUDOCLASSICAL RELATIVISTIC ELECTRON ~ 1

the solution of (4.7) is

(4.1o)

where

21z~(r) + 2 ~ ( r ) ---- cl exp [ia)r] ÷ e~ exp [-- ia)r] ,

(4.11) C 1 -~- C 2 = 21:7/'5(0 ) -~- 22~5(0) .

Furthermore, differentiating (4.10) and using again eqs. (4.6), we get

2i2x2~ (4.12) cl-- c~. -- - -

(D imc [1 -- i2~) 22~d0)) e .~(o) + ~ (~ 2, G ~ o ( 0 ) - •

If we impose on the solutions the first-class constraints Z and %o, we can show tha t

(4.13) cl = 0,

1 (4.14) c~ = ~i~- 2 ~ [2~JrdO) + 2i21~(0) + 221)~2P.~(0)].

By using solution (4.10) ~nd the previous result for c~ and c~, it is easy to integrate the remaining equations, and the result is

(4.15)

(4.16)

(4.17)

(4.18)

&(T) =

1 : ~1 -~- 2i2 ' i [ 2 i 2 1 ~ 5 ( 0 ) ( e x p [ - i6o~]- 1) -~ ~5(0)(2i22 e x p [ - - i o ) T ] -~- ~1)] ,

re(T) =

1 -- 21 -}- 2i22 [22~d0)(exp [ - - i w r ] - 1) + ~(0)(2~ exp [--iwr] + 2i2~)],

2i P~ (21~(0) q- 22~(0))(exp [--io~r]-- 1) G(~) = ~(0) + 21~ 2i2~ m--e

P~ i x~(~) = x.(0) + - - ~ - - ( ~ ( 0 ) + ~&(0) )~ (0 ) (exp [ - i ~ ] - ~).

mc e)

The last te rm in the expression for xu(r) is tile pseudoclassical analogous of the zitterbewegung (13). However, as we previously noticed, the quantum- mechanical zitterbewegung consists of very fast oscillation, contrary to the classical one, which is given by a real exponential.

In the gauge fixed by 21 and 2~, the solutions of the equations of motion for the Grassmann variables are given in terms of six constants ~u(0), Sd0) and ~d0), which, however, %re not independent, due to the constraint Xo = 0.

(la) p. A. ~¢[. DIRAC: The Pri'nwiples o] Quantum Mecha~ics, 3rd Edition (Oxford, 1947).

392 A. BARDUCCI, g . CASALBUONI a n d r~. L U S A N N A

H o w e v e r , we saw in sect . 3 t h a t we h a v e o n l y fou r i n d e p e n d e n t L a g r a n g i a n

e q u a t i o n s of mo t ion . This m e a n s t h a t we h a v e t h e f r e e d o m to fix a f u r t h e r

r e l a t i o n a m o n g t h e s e c o n s t a n t s .

I t is i m m e d i a t e l y seen t h a t th i s r e l a t i o n can be chosen of t h e k i n d

~5(0) =/z~5(0), w h e r e # is a n a r b i t r a r y c o n s t a n t . I n o r d e r to d e t e r m i n e t h i s

c o n s t a n t , we c a n use t h e i n v a r i a n e e of t he t h e o r y u n d e r s u p e r s y m m e t r i e s to

fix t h e (( or ig in ~> of ou r f r a m e for $~(0) a n d us(0).

tt =--i/2, because , in th i s case, we h a v e

A c o n v e n i e n t choice is

( 4 . 1 9 ) ~5 + ~ 5 , s~G5 + s~G~' = - - i s s - - m'-~

a n d we see t h a t , u n d e r th i s choice , t h e n e w c o n s t r a i n t is i n v a r i a n t u n d e r t h e

z - d e p e n d e n t s u p e r s y m m e t r y t r a n s f o r m a t i o n i n t r o d u c e d in t h e p r e v i o u s s ec t ion

(see eq. (3.46)).

F u r t h e r m o r e , t h i s c o n s t r a i n t is a c o n s t a n t of m o t i o n b e c a u s e f r o m (4.6)

we h a v e

i 1 1 (4.20) ~---~(~5 +~,5)=i(i22--~ 21)(P.,-- imc~5--7~mc,5)~O.

Our n e w set of c o n s t r a i n t s is

(4.21)

w i t h t h e D i r a c b r a c k e t s

g = / ) 2 m 2 c 2 ,

1 ZD = P" ~-- imc~5-- ~ mc~5 ,

i

{Z, X~}* = 0,

(~.~) {z°, z'~}* = 0,

t z~, z'~}* i

W e see t h a t our choice for Z~ is such t h a t Z a n d ZD r e m a i n f i r s t -c lass con-

s t r a i n t s , whi le Z: is a second-c lass one (1,). I t fo l lows t h a t we h a v e to i n t r o -

(1,) We notice that , for a different choice of /t, Zv is a second-class constraint. But i t is possible to show tha t in any case there exist four variables ~, with an algebra such that , after quantization, i t goes into the algebra of the y~ matrices. In this case, the wave function will satisfy the constraint (p2 m2c 2)[~> = 0, and, furthermore, ]~> will be a basis for the algebra of the 7g matrices. This is sufficient to show tha t I~> satisfies the Dirac equation. The choice /~ = - - i/2 gives rise to thc same results, but in a more t ransparent way.

SUPERSYMMETRIES AND THE PSEUDOCLASSICAL RELATIVISTIC ELECTRON 3 9 3

duce new Dirac brackets , which will be given by (1~)

(4.23) {A, ~}** = {A, B}*-- i{A, X'~}* {Z~, B}*.

The only new bracke t is

(4.24) {~, ~}** = -- i .

We see also t h a t the new const ra in t destroys all the s u p e r s y m m c t r y t rans-

format ions bu t the z-dependent one defined in (3.46).

To summar ize the si tuation, we see t h a t now our sys tem is described by

five Grassmann var iables ~ , 25 sat isfying the Dirac algebra

(4.25) {~,, ~}** = 0,

{~0, ~}** = - i .

Fur the rmore , these var iables mus t satisfy the first-class constra int

(4.26) Z~ = P ' ~ - - mc~5 ~ O .

This cons t ra in t gi~es rise also to the mass condit ion (Z-constraint), because

we have

(4.27) {Z-, Z~}** ~-- i( P ~ ' - m2c2) •

This sys tem can be immedia te ly quantized by following the general prescrip- t ion given in ref. (2,3) ; in par t icular , we get the following an t i eommuta t ion rela-

tions :

(4.28) [~ , ~]+ = - - ]/g~, [~ , ~5]+ --:- 0, [$5, ~5]+ = ]~-

We see t h a t this algebra can be realized with the positions

V (4.29)

~ = g~'~.

(1~) We recall that the construction of the Dirac brackets can be made, step by step, due to their iterative property; sec A. J. HANSON and T. REGO]~: A n n o] Phys . , 87, 498 (1974).

3 9 4 A. BARDUCCI~ 1~. CASALBUONI a n d T.. LUSANNA

57ow the first-class const ra in ts become conditions on the s ta tes

(4.30) (p2_ m~c~)[~, ) = O,

(4.31) (P -- mc) J~v} ---- O.

Fu r the rmore , we can fix our q u a n t u m Hami l ton i an b y pu t t ing

1 ~ = i2o,

and choosing

1 ~= +~.

With these positions we have

1 Y f - - 2me

2 - - (PZ-- m ~c~ + ~ s P ' ~

and, by t ak ing into account (4.29),

1 (4.32) ~ ---- - - 2me (P~-- m2e~) -k -P'F.

We notice tha t , with our choice of the cons tan ts ~ and 23, H is H e r m i t i a n

with respect to the metr ic 70.

5. - The nonrelativistic limit and the electromagnetic interaction.

We s ta r t again with the Lagrang ian (3.1)

This expression can be rewr i t ten in the following way:

(5.2) ~e = - ~ 5 ~ 5 - ~ 5 ~ ~ + i ~ .

I f we wan t to t ake the nonrelat ivis t ie l imit, i t is convenient to pu t

S U P E R S Y M M E T R I E S A N D T I I E P S E U D O C L A S S I C A L R E L A T I V I S T I C E L E C T R O N 395

= Xo = ct, and we get, for the nonrelativist ic Lagrangian (LfNa = cL~°),

(5.3) 1 ,

z e ~ = - ~ 2 + ~ ~. ~ + 5 ~ - ~ , ~ 5 ~ - ~ ~o ~0 + ~ o ~ + o ,

where, now, the dot denotes the derivat ive with respect to the t ime t. This Lagrangian is near ly the one studied in ref. (~,3), apar t the mass t e rm

and the te rms in ~ and ~o~ However , we can write

•

(5.4) ~ e ~ = - ~ + ~ . ~ + ~ m v ~ - ~ ( ~ o - ~ ) ( ~ o - ~ ) + ~ ( ~ o ~ ) + o ,

and we see tha t ~ R depends only on the difference to -- ~5. Now, if we consider the non relativistic limit of the Dirae constra int (4.26), we have

(5.5) P ' ~ - - m c ~ 5 - ~ m c ( ~ o - - ~5),

and we see tha t , when this constra int is satisfied, our effective Lagrangian is

(5.6) ~e.R = ~g.~ + ~ m v 2 + o ,

where we have neglected a to ta l t ime derivative. But the Lagrangian (5.6) is just the G~ Lagrangian considered in reI. (1,3). Thus we see tha t the Dirac con- s traint is precisely what we need in order t ha t the Lagrangian (3.1) reduces to the Lagrangian Ga in the nonrelat ivist ic limit.

I t is interest ing to s tudy the gauge Xo ---- T; to this end, we can impose the fur ther constra int

(5.7)

The Dirac brackets are

(5.8)

~ O ~ - - - X o - T .

(Z, Zo}** = 2Po,

{Z-, Z0}** = to,

{Xo, Zo}** = 0.

Thus the mat r ix of the constraints Zo, Z, Z- is given by

(5.9) C =

I 0 -- 2/)o - - : o ]

2Po 0

to 0 i Z j

3 9 6 A. BARDUCCItp It. CASALBUOlqI a n d L. LUSAlqN/k

I t is e a s y to rea l i ze t h a t t h e i nve r se m a t r i x ex i s t s o n l y if Z # 0 ; in f a c t we

h a v e

(5.10) C -~ =

m m 1

0 0 2Po

1 @o 0

2/)0 2 P o z

i~o i

2Po g Z _ 0

T h a t m e a n s t h a t we c a n choose a f i r s t -c lass c o m b i n a t i o n of ou r con-

t r a i n t s .

A c o n v e n i e n t c o m b i n a t i o n is ~ogv; in f ac t , we h a v e

(5.11) [ {a xo, Xo}** = ~ = o ,

{a x~,, ~o xo}** = o,

{axo, x}**= o.

I t fol lows t h a t we c a n def ine n e w D i r a c b r a c k e t s r e l a t i v e to t h e second-

class c o n s t r a i n t s Xo a n d Z, a n d t h a t our n e w H a m i l t o n i a n wil l be (lo)

(5.12) • ¢ ~ = M o ( t " . ~ - mc~5) + Po.

B y choos ing ~ = 2/h as in t h e c o v a r i a n t gauge , we h a v e , a f t e r q t t an t i -

z a t i o n (:7),

(5.13) ~:/f---- y o P ' y + m c y o ,

t h a t is, u s i n g t h e s t a n d a r d n o t a t i o n s ,

(5.14) ~ f = P ' a + m c f l .

W e see t h a t in th i s g a u g e t h e t t a m i l t o n i a n is a H e r m i t i a n o p e r a t o r .

(is) We recall tha t according to Dirac (13) the Hamil tonian is a linear combination of the first-class constraints only. We notice tha t the presence of the second term is necessary, because, in order to generate the r ight equations of motion, we must have "JY = Po, when the constraints are satisfied (15). (17) We observe that , going from the classical case to the quantum one, we have some ambiguities which are the analogous of the order ambiguities in the usual case. The actual ambiguities are in terms like ~ ; however, the quantum Hamil tonian is fixed by the same criterion as the classical one, i.e. when the constraints are satisfied, we must have 5~ = Po. This corresponds to quantizing the expression (5.12) formally.

SUPERSYMMETRIES AND THE PSEUDOCLASSICAL RELATIVISTIC ELECTRON ~ 7

Now we want to introduce in our scheme the interaction with an electro-

magnet ic field. First. of all, let us notice tha t in n theory with first-class contraints

we cannot introduce an interuction in an urbi t rary way. In f~ct, we h,~ve to

preserve the eh,~raeter of the constraints. Otherwise, there is no way to have

a smooth limit when we turn off the interaction. This is due to the fact that ,

if the inter:~ction modifies the first-class constraints into second-class ones,

then the new Dirac brackets we need to introduce will be singular for zero

coupling constant . I t follows tha t any consistent interaction must not change ]irst-

class constraints into second-class ones.

In our c~se, we will proceed in the following way: we w~nt to reproduce

the Dirac equation for a charged particle interact ing with an external electro-

magnetic field; this means tha t the constraint Z- will become

\ (5.15) Z- = [ P - - _e A | m c ~ 5 .

\ e ]~

Furthermore, we suppose tha t the interaction will not change the second-

class constraints

i

(5.16) , i

in order not to modify the basic algebra.

In the free ca.se, the Dira.c brucket of Xv with itself gives the other con-

straint X- Thus we will require tha t {XD, X~}** is a constraint of our theory;

we have

[( (5.17) {ga, ZD}** = i P - - A m2c ~

where F,~ is the electromagnetic tensor. Now the crucial point is to verify

tha t {ZD, Z} * * = 0. In general, we have

} **

£'~a ~ •

As we know, a is proportiom~l to the tot:~l magnetic moment of the particle

and we see tha t the interaction is consistent only if there is no anomalous

26 - I I N u o v o G t m * ~ o A .

~ A. B A R D U C C I , R . C A S A L B U O N I a n d ~.. L U S A N N A

magnetic moment , tha t is if ~ ----- ie/c. Thus our first-class constraints will be

(5.19) Z~ = ~.-- mc~5,

(5.20) )~ = P - - A m~c ~

The LagTangian which gives rise to the constraints (5.19), (5.20) and to

the first of (5.16) is

2

(5.21) ~ : - - = ~ - - : ~ . ~ -- m~c ~ - 2.-- c ~-~"~ ---e~'A'c

which can be rewri t ten as

(5.22) . ~ = - - ~ ~ - -

. 2

The last t e rm in this expression is the usual minimal interaction te rm;

the second te rm in the square p~renthcsis is the interaction of the field with the spin (ls.~9); the third t e rm is a sort of renormalization of the mass and it

is proportionul to E . B . Let us verify tha t the Lagrangiun (5.21) gives rise to the right constraints ;

we have

V - ie ~ :~.-- (i/mc) ~.~5 e (5.23) P . ---- m 2 c ~ -- - F . ~ s ~

c V ( ~ . - - (i/mc)~.~5) 2 ÷ c A " ,

i (5.24) ~. = ~ ~.,

(5.25) = 5 = ~ 5 - ~ c c P eA ~".

From (5.23) we immediately get eq. (5.20). Fur thermore, by imposing again

the relation ~5 -}- (i/2) ~5 = 0, f rom (5.25) it follows eq. (5.19). We notice tha t ,

also in this case, ~5-~ (i/2)~5 is a constant of motion.

(is) j. I. FRENKEL: Zeits. Phys., 37, 247 (1926); see also A. O. ]~ARUT: Electrodynamics and Classical Theory o/ Fields and Particles (New York, N.Y., 1964). (19) Recall from eq. (3.42) that - - i ~ . ~ are the generators of the internal Lorentz group.

SUPERSYMMETRIES AND THE PSEUDOCLASSICAL RELATIVISTIC ELECTRON 399

A f t e r h ~ v i n g c o m p l e t e d th i s work , we r e c e i v e d a p r e p r i n t b y F . A. BEREZIN

g n d M . S . M A R ~ o v , e n t i t l e d (~ P~r t i e l e s Sp in D y n u m i e s as t h e G r u s s m ~ n n

Vgri,~nt of Clussical Mechan ic s ~, p r e p r i n t Moscow 1976, in wh ich r e su l t s v e r y

s im i l a r to ours were r euehed , a . l though t h e i r s t ~ r t i n g p o i n t is qu i t e d i f ferent .

R I A S S U N T 0

In questo lavoro si considera una versione della supersimmetria la eui parametrizza- zione 6 effettuata in termini di variabi l i ant ieommutant i pseudovettorial i e pseudosea- lari inveee the spinoriali. Si eostruisee una Lagrangiana relativist ica pseudoelassica invariante rispetto all~ preeedente supersimmetria. Questa Lagrangiana risulta dege- nere e si mostra ehe, dopo quantizzazione, uno dei vineoli d~ luogo alia equazione di Dirae. Nella nostra Lagrangiana si introduce inoltre un termine di interazione con un c~mpo elettromagnc~ieo esterno, e si mostra the ei6 ~ possibilc in maniera consi- stente solo se il momento m~gnetieo anomalo ~ hullo. Segue che questo modello rap- presenta una descrizione pseudoelassica per l 'elettrone relativistico.

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