geometric phase

In plain terms

A quantum state has an overall phase that no measurement can see, which is why states differing only by a phase are counted as the same state. That makes it surprising that any phase should matter, and the resolution is that a difference of phase between two paths does matter, as interference can see it.

Take a state slowly around a loop and return it to the ray it started from. Part of the phase it picks up is the familiar one, accumulated at a rate set by the energy, and subtracting that part leaves a remainder. The remainder doesn't depend on how fast you went or on how you chose to keep track of the overall phase along the way, and it doesn't vanish. It depends only on the loop traced out in the space of states, which is why it's called geometric.

Curvature is the right picture. The space of states is curved, and carrying a direction around a loop on a curved surface brings it back rotated by an amount equal to the curvature enclosed, exactly as a vector carried around a triangle on a sphere comes back turned. The geometric phase is that rotation, and the enclosed curvature is the area the loop bounds measured by the state space's own geometry.

In CSD

The geometric phase is an observable consequence of the sector geometry the programme builds on, so proving it belongs with the Fubini-Study and Kahler layers rather than beside them. The corpus defines the connection form along a curve of unit vectors, proves that the geometric phase is unchanged by any differentiable change of the overall phase, and then proves that it's the holonomy of the connection: the horizontal lift, the one whose connection form vanishes, returns rotated by exactly the geometric phase.

The curvature formula comes next, with the phase written as the integral of the state space's own two-form over a surface the loop bounds, proved through the two-dimensional divergence theorem. The identification of that two-form is what closes the circle: the pullback of the Fubini-Study form along a family of states is computed chart-free and bundled as a genuine differential form, so the curvature is the Kahler form of the state space and the geometric phase is half the integral of the pullback. Two independent derivations agree on the cone example. The adiabatic setting, where Berry's original argument lives, is handled for a rigidly rotated Hamiltonian exactly, and the general adiabatic theorem is an open row.

Mathematically

connectionForm psi t = Im <psi t, psi-dot t> along a curve of unit vectors, dynamicalPhase its integral, geometricPhase psi T phi = phi - dynamicalPhase for a cyclic curve with psi T = e^(i phi) psi 0. connectionForm_rephase shows the form shifts by theta-prime under psi to e^(i theta) psi, and geometricPhase_rephase is gauge invariance, so the phase is a function of the closed curve of rays. connectionForm_horizontalLift and horizontalLift_cyclic are the holonomy statement, and geometricPhase_of_schrodinger evaluates it for a self-adjoint generator as phi + T times the conserved energy.

geometricPhase_eq_neg_integral_curvature (GeometricPhaseCurvature.lean) is the curvature form, minus the double integral of 2 Im <d_s psi, d_t psi> over the polar-form disc. fsPullbackForm_apply (ProjectiveSpaceGeometricPhase.lean) computes the pullback of the Fubini-Study form along a smooth family without charts, fsPullbackDifferentialForm bundles it, and geometricPhase_eq_half_integral_fsPullbackForm is the identification. The general Kato-Born-Fock adiabatic theorem is BACKLOG row 88 and is not claimed.

The name

Michael Berry published the phase in 1984 at Bristol, in the adiabatic setting, and the result was immediately recognised as something that ought to have been noticed decades earlier. It had been, in fact, as Shivaramakrishnan Pancharatnam had described the optical version in 1956 and Christopher Longuet-Higgins and colleagues had met a sign of it in molecular physics across the following decade.

Barry Simon reinterpreted the phase as holonomy straight away, in a paper that appeared before Berry's own, and Yakir Aharonov and Jeeva Anandan removed the adiabatic assumption in 1987, leaving the general statement about closed curves of rays that the corpus proves.

Related entries

• Aharonov-Bohm effectAn electron that never enters a magnetic field can still be shifted by it, as the enclosed flux changes the interference between paths, which shows that the field's potential carries physics the field strength alone doesn't.
• Kahler formOn the space of quantum states, distance, area and a quarter-turn coming from the complex numbers lock together, and that lock is what makes the volume of the space nobody's choice.
• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.
• Fisher-Rao metricThe natural way to measure distance between quantum states agrees with the statistician's way to measure distance between the probability distributions they produce, exactly and with no constant left to fix.
• Bloch sphereThe states of the simplest quantum system form an ordinary ball, which is a lovely accident of low dimension and the single most misleading picture in quantum mechanics.