9 research outputs found
The geometric tensor for classical states
We use the Liouville eigenfunctions to define a classical version of the
geometric tensor and study its relationship with the classical adiabatic gauge
potential (AGP). We focus on integrable systems and show that the imaginary
part of the geometric tensor is related to the Hannay curvature. The
singularities of the geometric tensor and the AGP allows us to link the
transition from Arnold-Liouville integrability to chaos with some of the
mathematical formalism of quantum phase transitions
The Schwinger action principle for classical systems
We use the Schwinger action principle to obtain the correct equations of
motion in the Koopman-von Neumann operational version of classical mechanics.
We restrict our analysis to non-dissipative systems and velocity-independent
forces. We show that the Schwinger action principle can be interpreted as a
variational principle in these special cases
Projective representation of the Galilei group for classical and quantum-classical systems
A physically relevant unitary irreducible non-projective representation of
the Galilei group is possible in the Koopman-von Neumann formulation of
classical mechanics. This classical representation is characterized by the
vanishing of the central charge of the Galilei algebra. This is in contrast to
the quantum case where the mass plays the role of the central charge. Here we
show, by direct construction, that classical mechanics also allows for a
projective representation of the Galilei group where the mass is the central
charge of the algebra. We extend the result to certain kind of
quantum-classical hybrid systems
Adiabatic driving and parallel transport for parameter-dependent Hamiltonians
We use the Van Vleck-Primas perturbation theory to study the problem of
parallel transport of the eigenvectors of a parameter-dependent Hamiltonian.
The perturbative approach allows us to define a non-Abelian connection
that generates parallel translation via unitary transformation of
the eigenvectors. It is shown that the connection obtained via the perturbative
approach is an average of the Maurer-Cartan 1-form of the one-parameter
subgroup generated by the Hamiltonian. We use the Yang-Mills curvature and the
non-Abelian Stokes' theorem to show that the holonomy of the connection
is related to the Berry phase
Operational classical mechanics: Holonomic Systems
We construct an operational formulation of classical mechanics without
presupposing previous results from analytical mechanics. In doing so, several
concepts from analytical mechanics will be rediscovered from an entirely new
perspective. We start by expressing the basic concepts of the position and
velocity of point particles as the eigenvalues of self-adjoint operators acting
on a suitable Hilbert space. The concept of Holonomic constraint is shown to be
equivalent to a restriction to a linear subspace of the free Hilbert space. The
principal results we obtain are: (1) the Lagrange equations of motion are
derived without the use of D'Alembert or Hamilton principles, (2) the
constraining forces are obtained without the use of Lagrange multipliers, (3)
the passage from a position-velocity to a position-momentum description of the
movement is done without the use of a Legendre transformation, (4) the
Koopman-von Neumann theory is obtained as a result of our ab initio operational
approach, (5) previous work on the Schwinger action principle for classical
systems is generalized to include holonomic constraints