It is also possible to work on the h ’ 0 limit (that is, the classical

¯

limit) of the quantum theory involving fermionic degrees of freedom

[4]. This requires the corresponding classical Lagrangian to have in

addition to the usual generalized coordinates and velocities, anti-

commuting variables and their time-derivatives. We must therefore

distinguish, at the quantum level, between those operators which are

even or odd under a permutation operator P

P ’1 AP = (’1)π(A) A (3.7)

where A is some operator and π(A) is de¬ned by

π(A) = 0 if A is even

= 1 if A is odd (3.8)

An even operator transforms even (odd) states into even (odd)

states while an odd operator transforms even (odd) states into odd

(even) states. In keeping with the properties of an ordinary commu-

tator, which as stated before are the same as those of the Poisson

brackets outlined in (3.5), we can think of a generalized commutator

(also called the generalized Dirac bracket) as being the one which is

obtained by taking into account the evenness or oddness of an opera-

tor. Thus setting π(A) = a, π(B) = b, and π(C) = c, the generalized

Dirac bracket A, B is de¬ned such that the following properties

hold

anti-symmetry: [A, B] = ’(’1)ab [B, A]

chain-rule: [A, B C] = [A, B]C + (’1)ab B[A, C]

© 2001 by Chapman & Hall/CRC

linearity: [A, B + C] = [A, B] + [A, C]

Jacobi identity: [A, [B, C]] + (’1)ab+ac [B[C, A]]

+(’1)ca+cb [C, [A, B]] = 0 (3.9a, b, c, d)

Clearly, these properties are the analogs of the corresponding ones

stated in (3.5). In the absence of any fermionic degrees of freedom it

is evident that (3.9) reduces to the usual properties of the commu-

tators.

The chain-rule allows us to recognize [A, B] as

[A, B] = AB ’ (’1)ab B A (3.10)

which implies that [A, B] plays the role of an anti-commutator when

A and B are odd but a commutator otherwise

[A, B] = AB + B A A and B odd

= AB ’ B A otherwise (3.11)

With the de¬nition (3.10) and the use of the linearity and chain-rule

properties, the Jacobi identity (3.9d) can be seen to hold.

To derive (3.10) it is instructive to evaluate [AB, C D], where C

and D are also operators. Applying (3.9b) in two di¬erent ways, we

get

[AB, C D] = [AB, C]D + (’1)π(AB)π(C) C[AB, D]

= [AB, C]D + (’1)(a+b)c C[AB, D] (3.12)

where we have used π(AB) = π(A) + π(B) = a + b and applied tee

chain-rule on C D. Next using (3.9a) and once again (3.9b) we arrive

at

[AB, C D] = (’1)bc [A, C]B D + A[B, C]D + (’1)ac+bc+bd

C[A, D]B + (’1)ac+bc C A[B, D]

(3.13a)

Applying now (3.9b) on AB we have

[AB, C D] = (’1)bc+bd [A, C]DB + A[B, C]D + (’1)ac+bc+bd

C[A, D]B + (’1)bc AC[B, D]

(3.13b)

© 2001 by Chapman & Hall/CRC

Since (3.13a) and (3.13b) are equivalent representations of [AB, C D]

we get on equating them

[A, C] B D ’ (’1)bd DB = AC ’ (’1)ac C A [B, D] (3.14)

(3.14) implies that the generalized bracket [X, Y ] involving two op-

erators X and Y can be identiti¬ed as

[X, Y ] = XY ’ (’1)π(X)π(Y ) Y X (3.15)

It is obvious that (3.15) is consistent with (3.11).

The generalized bracket (3.15) gives way to a formulation of the

quantized rule

lim [X, Y ]

= {X, Y } (3.16)

h

¯ ’ 0 i¯ h

where [X, Y ] has been de¬ned according to (3.15) and {X, Y } stands

for the corresponding classical Poisson bracket. Note that the clas-

sical system possesses not only commuting variables such as the q™s

and p™s but also additional anti-commuting degrees of freedom. So

the Poisson bracket in (3.16) is to be looked upon in a generalized

sense [5-12].

3.2 Some Algebraic Properties of the Gen-

eralized Poisson Bracket

Pseudomechanics or pseudoclassical mechanics as named by Casal-

buoni [5] is concerned with classical systems consisting of anti-

commuting as well as c-number variables in the form of coordinates

and momenta. Let θ± ™s be a set of anti-commuting or Grassmann

variables in addition to the coordinates qi ™s. Then the pseudoclassical

Lagrangian can be written as

™

L ≡ L q i , q i , θ± , θ ±

™ (3.17)

We assumt for simplicity that L is not explicity dependent upon the

time variable t. The corresponding Hamiltonian would be a func-

tion of even (bosonic) variables (qi , pi ) and odd (fermionic) variables

(θ± , π± ) where pi and π± are the corresponding canonical momenta

to the coordinates:

H ≡ H(qi , θ± , pi , π± ) (3.18)

© 2001 by Chapman & Hall/CRC

To develop a canonical formalism we need to impose upon the

coordinates and momenta the conditions (3.4), namely

{Qi , Qj } = 0, {Pi , Pj } = 0

{Qi , Pj } = δij (3.19)

but here Q and P denote collectively the coordinates (qi , θ± ) and the

momenta (pi , π± ).

To deal with the odd variables it is necessary to identify properly

the processes of left and right di¬erentiation. At the pure classical

level where we deal with even variables only (like coordinates and

momenta), such a distinction is not relevant. However, in a pseudo-

classical system in which the dynamical variable X is a function of

Q and P , its di¬erential needs to be speci¬ed as [12]

δX(Q, P ) = X,Q dQ + dP ‚P X (3.20)

where a right-derivative is taken with respect to the coordinates Q

and a left-derivative with respect to the momenta P . By accounting

for the permutations correctly we can write

‚Q X = (’1)π(Q)[π(Q)+π(X)] X,Q (3.21)

A consequence of (3.21) is that

‚θ O = O,θ , ‚π O = O,π

‚θ E = ’E,θ , ‚π E = ’E,π

‚q O = O,q , ‚p O = O,p

‚q E = E,q , ‚p E = E,p (3.22)

where O and E denote odd and even variable respectively.

It is clear from (3.20) that the canonical momenta are to be

de¬ned as P = L,Q implying that since the Lagrangian is an even

™

function of the underlying variables we should have

‚L ‚L

pi = , π± = ’ (3.23)

™

‚ qi

™ ‚ θ±

with {π ± , θβ } = 0, ± =β.

© 2001 by Chapman & Hall/CRC

To derive the generalized Hamilton™s equation of motion we set

up a Legendre transformateon from the classical analogy

™

H= pi q i +

™ π± θ ± ’ L (3.24)

±

i

Varying H(qi , pi , θ± , π± ) and keeping in mind the rules (3.20) and

(3.21), the equations of motion emerge as

‚H ‚H

qi =

™ , pi = ’

™

‚pi ‚qi

‚H ‚H

™

θ± = , π± =

™ (3.25)

‚π± ‚θ±

Noting that the equation of motion of a dynamical variable X

is given in terms of the Poisson bracket as dX = ‚X + {X, H} and

dt ‚t

{X, H} is de¬ned according to

{X, H} = X,Q ‚P H ’ H,Q ‚P X (3.26)

[where we have followed (3.1) but made a distinction between the

left and right derivatives], it is trivial to check using (3.22) that

™

{θ, H} = θ and {π, H} = π. ™

More generally, the generalized Poisson bracket for various cases

of even and odd variables may be summarized as follows

‚E1 ‚E ‚E2 ‚E1 ‚E1 ‚E2 ‚E2 ‚E1

{E1 , E2 } = ’ +’ +

‚q ‚p ‚q ‚p ‚θ ‚π ‚θ ‚π

‚E ‚O ‚O ‚E ‚E ‚O ‚O ‚E

{E, O} = ’ ’ +

‚q ‚p ‚q ‚p ‚θ ‚π ‚θ ‚π

‚O ‚E ‚E ‚O ‚O ‚E ‚E ‚O

{O, E} = ’ + +

‚q ‚p ‚q ‚p ‚θ ‚π ‚θ ‚π

‚O1 ‚O2 ‚O2 ‚O1 ‚O1 ‚O2 ‚O2 ‚O1

{O1 , O2 } = + + +

‚q ‚p ‚q ‚p ‚θ ‚π ‚θ ‚π

(3.27a, b, c, d)

An interesting feature with the structure of (3.27) is that the canon-

ical relations (3.19) between the coordinates and momenta are auto-

matically preserved. This enables us to derive a classical version of

the supersymmetric Lagrangian in a straightforward manner.

© 2001 by Chapman & Hall/CRC

Finally, the classical h ’ 0 limit of the quantization rule (3.16)

¯

may be written down with respect to the even and odd operators by

making use of (3.8) and (3.10).

[E1 , E2 ]’ = i¯ {E1 , E2 }

h

[O, E]’ = i¯ {O, E}

h

[O1 , O2 ]+ = i¯ {O1 , O2 }

h (3.28)

where the right hand side denotes the generalized Poisson bracket

with respect to both commuting and anti-commuting sets of vari-

ables and where ’ and + in the left hand side corresponds to the

commutator and anti-commutator, respectively. It may be remarked

that from (3.27b) and (3.27c) we have {O, E} = ’{E, O}. The re-

spective expressi,n for the Poisson bracket in (3.28) are those given

by (3.27a), (3.27c) and (3.27d). We notice that only the odd-odd

operators are quantized with respect to the anti-commutator while

the remaining ones are quantized with respect to the commutator.

3.3 A Classical Supersymmetric Model

We now seek the classical supersymmetric Hamiltonian in the form

HScl = {Q, Q+ } (3.29)

with {Q, Q} = {Q+ , Q+ } = 0. Utilizing the Hamiltonian™s equation

of motion in Poisson bracket notation we have

™

Q = {Q, HScl }

= {Q, {Q, Q+ }}

= ’{Q, {Q, Q+ }}

™

= ’Q (3.30)

™ ™

where Jacobi identity has been used. So Q = 0 and similarly Q+ = 0.

These give at once

{Q, HScl } = Q+ , HScl = 0 (3.31)

implying that the conservation of Q and Q+ is in-built in (3.29).

© 2001 by Chapman & Hall/CRC

We can also write down explicit representations for Q and Q+

by setting

1 1