Canonical commutation relations and special relativity

As a way of studying the consequences of a possible breakdown of Lorentz invariance, the canonical commutation relations are enlarged to include time and energy as additional conjugate variables, and with this enlargement, they are shown to be invariant under the Lorentz transformation under the assumption that commutators of non-conjugate operators vanish. Although time does not commute with the energy operator, it does commute with the Hamiltonian. It is also shown that the invariance of the enlarged commutation relations holds for arbitrary non-singular linear transformations that describe a velocity boost to another uniformly moving frame, under the same assumptions about non-conjugate commutators. A covariant enlargement of the Poisson brackets, together with Dirac's interpretation, that would predict this invariance of the commutators is presented. There is a discussion of improper coordinate transformations, and a canonical time-reversal operator is introduced, and compared with the standard time-reversal operator T. It is concluded that even if the Lorentz transformation were to fail, and were to be replaced by another linear transformation, the enlarged canonical commutation relations could still hold