Generalized Dirac bracket and the role of the Poincaré symmetry in the program of canonical quantization of fields 1

Abstract

An elementary presentation of the methods for the canonical quantization of constraint systems with Fermi variables is given. The emphasis is on the subtleties of the construction of an appropriate classical bracket that could be consistently replaced by commutators or anti-commutators of operators, as required by canonical quantization procedure for bosonic and fermionic degrees of freedom respectively. I present a consequent canonical quantization of the Dirac field, in which the role of Poincaré invariance is made marginal. This simple example provides an introduction to the Poincaré-free quantization of spinor electrodynamics in the second part of the paper.

Generalized Dirac bracket and the role of the Poincaré symmetry in the program of canonical quantization of fields 2

“…In the w = 0 case the equations, when written in terms of A a , do not assume the usual form of Maxwell equations. This example reflects a general rule that it is the dynamics generated by H T , and not H E , which is equivalent to the Euler-Lagrange equations of the initial Lagrangian (see the discussion below the formula (II.20) of [14]) . However, if A 0 is transformed into Ã0 := A 0 − w then, in terms of the quantities Fab and Dψ that contain Ã0 in place of A 0 , the equations assume the standard form…”

“…and depends on arbitrary functions u 0 and w. The transformations corresponding to changes in these functions ought to be interpreted as gauge transformations, as explained in Sec.II.B of [14]. From (II.25) it is clear that the action of a general such transformation on Ḟ is…”

“…(II.6) where the upper sign applies whenever at least one of the variables F , G is even and the lower one corresponds to F and G odd (see formula (III.6) of [14] and the discussion therein). The consistency conditions for the time evolution of constraints are…”

“…It seems that there are three families of second class constraints, χ 1 , χ 2 , γ and one family of first class ones, γ 1 . However, as explained at the end of Sec.II.A of [14], the constraints are well separated if the number of first class constraints is possibly large, i.e. no first class constraints are hidden in the second class ones.…”

“…Now that we have all the gauge freedom explicitly described, we should proceed to quantize the theory. Since the only second class constraints are χ 1 and χ 2 , the generalized Dirac bracket is given simply by the formula (III.23) of [14], although the generalized Poisson bracket is now given by (II.6). If GDB's are promoted to the commutators of the operators in the quantum theory, the second class constraints can be consistently interpreted as strong operator equalities.…”