In classical mechanics we encounter the notion of Poisson brackets in connection with transformations of the generalized coordinates and generalizaed momenta that leave the form of Hamilton’s equations of motion unchanged [1-3].Such transformations are called canonical and the main property of the Poisson bracket is its invariance with respect to the canonical transformations.In terms of the generalized coordinates q1, q2, . . . qn and generalized momenta p1, p2, . . . , pn the Poisson bracket in classical mechanics is defined by
{f, g}=n
j=1
∂f
∂qj
∂g
∂pj − ∂f
∂pj
∂g
∂qj
(3.1) for any pair of functions f ≡f(q1, q2, . . . qn;p1, p2, . . . pn;t) andg ≡ g(q1, q2, . . . qn;p1, p2, . . . pn;t).
Recall that whereas the Lagrangian in classical mechanics is known in terms of the generalized coordinates (qi), the generalized ve- locities ( ˙qi), and time (t), namelyL=L(q1, q2, . . . qn; ˙q1,q˙2, . . .q˙n, t), the corresponding Hamiltonian is given in terms of the generalized coordinates (qi), generalized momenta (pi), and time (t), namely
H = H(q1, q2, . . . qn;p1, p2, . . . pn;t) where pi = ∂∂Lq˙i, i = 1,2, . . . n.
The relationship between the Lagrangian and Hamiltonian is pro- vided by the Legendre transformation H =n
i=1
piq˙i−L and Hamil- ton’s canonical equations of motion are obtained by varying both sides of it
∂H
∂pi = ˙qi
∂H
∂qi = −p˙i
∂H
∂t = −∂L
∂t (3.2)
Relations (3.2) prescribe a set of 2nfirst-order differential equa- tions for the 2n variables (qi, pi).In contrast Lagrange’s equations involve n second-order differential equations for the n generalized coordinates
d dt
∂L
∂q˙i
= ∂L
∂qi i= 1,2, . . . n (3.3) For a given transformation (qi, pi)→(Qi, Pi) to be canonical we need to have
{Qi, Qj} = 0,{Pi, Pj}= 0,
{Qi, Pj} = δij (3.4)
These conditions are both necessary and sufficient.Often (3.4) is used as a definition for the canonically conjugate coordinates and momenta.Some obvious properties of the Poisson brackets are
anti-symmetry: {f, g}=−{g, f},{f, c}= 0 (3.5a) linearity: {f1+f2, g}={f1, g}+{f2, g} (3.5b) chain-rule: {f1f2, g}=f1{f2, g}+{f1, g}f2 (3.5c) Jacobi identity: {f,{g, h}}+{g,{h, f}}+{h,{f, g}}= 0 (3.5d) wherec is a constant and the functions involved are known in terms of generalized coordinates, momenta, and time.
The transition from classical to quantum mechanics is formu- lated in terms of the commutators from the classical Poisson bracket
relations.Indeed it can be readily checked that the commutator of two operators satisfies all the properties of the Poisson bracket summarized in (3.5). The underlying fundamental commutation re- lation in quantum mechanics being [x, p] = i¯h, the classical Poisson bracket may be viewed as an outcome of the following limit on the commutator
lim
¯ h→0
f, g
i¯h ={f, g} (3.6) wheref, g stands for the commutator of the two operators fand g.
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 operatorP
P−1AP = (−1)π(A)A (3.7) whereA is some operator andπ(A) is defined by
π(A) = 0 if A is even
= 1 ifA 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 settingπ(A) = a,π(B) = b, andπ(C) = c, the generalized Dirac bracket A, B is defined such that the following properties hold
anti-symmetry: [A, B] = −(−1)ab[B, A]
chain-rule: [A, BC] = [ A, B]C+ (−1)abB[ A, C]
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)abBA (3.10) which implies that [A, B] plays the role of an anti-commutator when Aand B are odd but a commutator otherwise
[A, B] = AB+BA A andB odd
= AB−BA otherwise (3.11) With the definition (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, CD], where C and D are also operators.Applying (3.9b) in two different ways, we get
[AB, CD] = [ AB, C]D + (−1)π(AB)π( C) C[ AB, D]
= [AB, C]D + (−1)(a+b)cC[ AB, D] (3.12) where we have used π(AB) = π(A) + π(B) = a+b and applied tee chain-rule onCD.Next using (3.9a) and once again (3.9b) we arrive at
[AB, CD] = (−1) bc[A, C] BD +A[B, C] D+ (−1)ac+bc+bd C[A, D] B+ (−1)ac+bcCA[B, D]
(3.13a) Applying now (3.9b) onAB we have
[AB, CD] = (−1) bc+bd[A, C] DB+A[B, C] D+ (−1)ac+bc+bd C[ A, D] B+ (−1)bcAC[ B, D]
(3.13b)
Since (3.13a) and (3.13b) are equivalent representations of [AB, CD]
we get on equating them
[A, C] BD −(−1)bdDB= AC−(−1)acCA[B, D] (3.14) (3.14) implies that the generalized bracket [X, Y] involving two op- eratorsX and Y can be identitified 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
¯ h→0
[X, Y]
i¯h ={X, Y} (3.16) where [X, Y] has been defined 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].