Alternative symplectic structures in second–class constrained systems

Abstract

The classical description of second-class constrained systems using alternative pairs, comprising a symplectic structure and a primary Hamiltonian other than the canonical ones, is investigated. It is exhibited that the dynamical equations do not fix a unique pair and instead there may exist infinitely many of them whose difference is not exhausted by canonical transformations. The consequences on the symplectic structure of the reduced phase space are addressed, namely the effect of different pairs on the construction of Dirac bracket is explored. Since constrained systems can also be studied via the Faddeev–Jackiw symplectic analysis, the implementation of alternative pairs on this formalism is considered as well. It is shown that Dirac’s algorithm outcomes are independent of the type of alternative pairs considered here irrespectively to whether there are also first-class constraints in the system. Finally, some comments on the implications of our results in the path integral quantisation scheme are pointed out.