. 10
( 42 .)


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

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-
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

[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
[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]
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]

© 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)
¯ ’ 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)

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 }
[O, E]’ = i¯ {O, E}
[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


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( 42 .)