On quantum theory of constrained systems

Aidan Burch

Working within the HPO (History Projection Operator) Consistent Histories formalism, we follow the work of Savvidou on (scalar) field theory [J. Math. Phys. 43, 3053 (2002)] and that of Savvidou and Anastopoulos on (first-class) constrained systems [Class. Quantum Gravt. 17, 2463 (2000)] to write a histories theory (both classical and quantum) of Electromagnetism. We focus particularly on the foliation-dependence of the histories phase space/Hilbert space and the action thereon of the two Poincaré groups that arise in histories field theory. We quantize in the spirit of the Dirac scheme for constrained systems.

DOMENICO PICCA

Scott Lawrence

A quantum-mechanical system comes naturally equipped with a convex space: each (Hermitian) operator has a (real) expectation value, and the expectation value of the square any Hermitian operator must be non-negative. This space is of exponential (e.g.~in volume) dimension, but low-dimensional projections can be efficiently explored by standard algorithms. Such approaches have been used to precisely constrain critical exponents of conformal field theories ("conformal bootstrap") and, more recently, to constrain the ground state physics of various quantum-mechanical systems, including lattice field theories. In this talk we discuss related approaches to systematically constraining the real-time dynamics of quantum systems, which are otherwise obstructed from study by sign problems and the ill-posed nature of analytic continuation.

Claudio Castelnovo

Tudor Stefan Ratiu, Oleg Georgievich Smolyanov

This paper discusses the Hamiltonian aspects of the quantum theory of constrained Hamiltonian systems; in particular, we extend results of [9, 6] to such systems. In this paper, we usually refer to these systems as Hamilton–Dirac systems, because it was Dirac who first defined them (under the name of generalized Hamiltonian systems). Such systems arise in the application of the Legendre transform to unconstrained Lagrangian systems with singular Lagrange function (for which the Legendre transform is noninvertible). It should be emphasized that constrained Hamiltonian systems bear no relation to Lagrangian systems with holonomic constraints; however, them can be regarded as objects dual to Lagrangian systems with nonholonomic constraints. Namely, the Legendre transform turns Lagrangian systems with nonholonomic constraints into unconstrained Hamiltonian systems with singular Hamilton function