J. Anandan

Geometric Phases for Mixed States in Interferometry

We provide a physical prescription based on interferometry for introducing the total phase of a mixed state undergoing unitary evolution, which has been an elusive concept in the past. We define the parallel transport condition that provides a connection form for obtaining the geometric phase for mixed states. The expression for the geometric phase for mixed state reduces to well known formulas in the pure state case when a system undergoes noncyclic and unitary quantum evolution.

Electromagnetic effects in the quantum interference of dipoles

Abstract The phase shift in the interference of a magnetic or electric dipole due to the electromagnetic field is obtained relativistically and non-relativistically. This is analogous to the Aharonov-Bohm effect, but is richer because of its non-Abelian character. The force and torque on the particle are obtained in the classical limit. The field equations in the presence of the quantum dipole are obtained at low energy.

Non-adiabatic non-abelian geometric phase

Abstract The non-integrable geometric phase factor is generalized to a non-abelian phase factor which in the adiabatic limit is the Wilczek-Zee generalization of the Berry phase. It is then extended to the evolution of mixed states. A theorem of Narasimhan and Ramanan is used to relate the gauge field connection to the generalized geometric connection.

Classical and Quantum Interaction of the Dipole

A unified and fully relativistic treatment of the interaction of the electric and magnetic dipole moments of a particle with the electromagnetic field is given. New forces on the particle due to the combined effect of electric and magnetic dipoles are obtained. Several new experiments are proposed, which include observation of topological phase shifts.

A geometric approach to quantum mechanics

It is argued that quantum mechanics is fundamentally a geometric theory. This is illustrated by means of the connection and symplectic structures associated with the projective Hilbert space, using which the geometric phase can be understood. A prescription is given for obtaining the geometric phase from the motion of a time dependent invariant along a closed curve in a parameter space, which may be finite dimensional even for nonadiabatic cyclic evolutions in an infinite dimensional Hilbert space. Using the natural metric on the projective space, we reformulate Schrödingers equation for an isolated system. This metric is generalized to the space of all density matrices, and a physical meaning is proposed.

Resource Letter GPP-1: Geometric Phases in Physics

This Resource Letter provides a guide to the literature on the geometric angles and phases in classical and quantum physics. Journal articles and books are cited for the following topics: anticipations of the geometric phase, foundational derivations and formulations, books and review articles on the subject, and theoretical and experimental elaborations and applications.

The meaning of protective measurements

Protective measurement, which we have introduced recently, allows one to observe properties of the state of a single quantum system and even the Schrödinger wave itself. These measurements require a protection, sometimes due to an additional procedure and sometimes due to the potential of the system itself The analysis of the protective measurements is presented and it is argued, contrary to recent claims, that they observe the quantum state and not the protective potential. Some other misunderstandings concerning our proposal are also clarified.

Geometric angles in quantum and classical physics

Abstract The geometric angles introduced in the group theoretical treatment of Berrys phase by Anandan and Stodolsky are generalized to the non-adiabatic cyclic evolutions studied by Aharonov and Anandan. This is done in the Schrodinger and Heisenberg representations. In the adiabatic, classical limit, these angles are shown to correspond to Hannays angles. A purely group theoretical description of these angles as determining the holonomy of a connection in a principal fiber bundle over a homogeneous space is given in both quantum and classical physics.

Topological and geometrical phases due to gravitational field with curvature and torsion

Abstract The gravitational phase factor, which may be used to determine the evolution of a wave function in the WKB approximation, is modified to incorporate the Fermi-Walker transport of the wave function when it is accelerated, for example by a wave guide. The modified phase factor is used to obtain geometrical and topological phases in the interference of two coherent beams around a cosmic string containing mass and intrinsic spin. An exact solution for the string from the Einstein-Cartan-Sciama-Kibble gravitational field equations is used to interpret the topological phases as being due to the fluxes of curvature and torsion inside the string.

On the reality of space-time geometry and the wavefunction

The action-reaction principle (AR) is examined in three contexts: (1) the inertial-gravitational interaction between a particle and space-time geometry, (2) protective observation of an extended wave function of a single particle, and (3) the causal-stochastic or Bohm interpretation of quantum mechanics. A new criterion of reality is formulated using the AR principle. This criterion implies that the wave function of a single particle is real and justifies in the Bohm interpretation the dual ontology of the particle and its associated wave function. But it is concluded that the Bohm theory is not dynamically complete because the particle and its associated wave function do not satisfy the AR principle.