Download Quantum physics and wave optics as geometric phases

Wave optics and quantum physics as geometric phases
Quantum physics
In this work we show that basic quantum commutation relations are equivalent to the existence of
a geometric phase. We propose phase-space interferometers that would serve to measure this
phase. This offers a new approach from first principles to the foundations of the quantum physics
providing a different perspective on the relation between quantum and classical physics.
In few years geometric phases have gained a preeminent status in physics. Being originally
discovered in the framework of the quantum theory, the question of the possible quantum origin
of some geometric phases has been well discussed. However, much less attention has received
the role they can play in the foundations of the quantum theory. In this work we show that basic
commutation relations sustaining quantum mechanics are fully equivalent to a geometric phase
arising after cyclic evolutions in phase space q, p. This trajectory is made of a succession
(discrete or continuous) of phase-space translations where the only states change is its global
position on the phase-space plane q, p.
On the one hand, if we assume that the operators qˆ , pˆ satisfy the standard commutation relations
qˆ, pˆ   i then the total cyclic transformation is proportional to de identity, being the constant of
proportionality a phase e i where  is the area of the circuit enclosed in units of  . This is a
geometric phase since  depends only on the area enclosed irrespective of the form of the curve.
The conclusion is that the quantum commutation relations imply the existence of a geometric
phase. To complete the equivalence we have also demonstrated that the reciprocal is also true: if
the total cyclic transformation is a geometric phase then qˆ, pˆ   0 .
As it might be expected, this geometric phase disappears when considering the classical limit.
This is because typical phase-space areas in the classical domain are large compared to  . In
such a case any relatively slight deviation or imprecision of the area implies large and
uncontrolled variations of the phase  . This unavoidably implies the effective disappearance of
any observable effect derived from the existence of this phase (usually in the form of
interferometric phenomena which are rather sensitive to phase randomness).
The existence of this phase can be verified experimentally. As for any other phase, the natural
framework to reveal and measure it is interference. We have to superpose the state experiencing
the cyclic transformation with another fixed state acting as a phase reference. Since we are
combining two states that have followed different trajectories in phase space (rather than in
ordinary configuration space) this is a phase-space interferometer. We have proposed simple and
experimentally feasible arrangements for these phase-space interferometers achievable with
current technologies. This can be carried out for light as well as for material systems.
Quantum mechanics as a geometric phase: phase-space interferometers
A. Luis , J. Phys. A: Math. Gen. 34, 7677 (2001)
Wave optics
This analysis can be extended to wave optics. Following an analogous reasoning it can be seen
that there a similar geometric phase for all classical electromagnetic waves. This phase arises
when the wave experiences a cyclic transformation made of spatial displacements combined with
displacements of the propagation direction. Te main property is this phase distinguishing it from
previously discovered geometrical phases is that it takes place by cyclic transformations taking
place in a plane, instead of taking place in curved spaces (typically the sphere).
We have shown very simple interferometric schemes that reveal the existence of this phase.
This geometrical phase distinguish geometrical from wave optics since we have demonstrated
that it disappear in the limit of geometrical optics.
Geometric phase in a flat space for electromagnetic scalar waves
A. Luis, Opt. Lett. 31, 2471-2473 (2006)