Noether Identities in Phase-space and Dirac's Conjecture

Based on the equations of motion for a Hamiltonian system with primary first-class and second-class constraints, the authors discuss the origin of Dirac's conjecture at first. Then it is supposed that the constraint multipliers are the functions of time and canonical variables, and combination coefficients in the gauge generator are the functions of time, canonical variables and constraint multipliers, the extended canonical Noether identities (ECNI) are deduced. Finally, a counter-example of Dirac's conjecture is discussed based on the phase-space canonical Noether identities (CNI) and ECNI, and the result shows that the Dirac's conjecture is still invalid in this circumstance.