The role of phase space geometry in Heisenberg’s uncertainty relation

Abstract

Aiming towards a geometric description of quantum theory, we study the coherent states-induced metric on the phase space, which provides a geometric formulation of the Heisenberg uncertainty relations (both the position-momentum and the time-energy ones). The metric also distinguishes the original uncertainty relations of Heisenberg from the ones that are obtained from non-commutativity of operators. Conversely, the uncertainty relations can be written in terms of this metric only, hence they can be formulated for any physical system, including ones with non-trivial phase space. Moreover, the metric is a key ingredient of the probability structure of continuous-time histories on phase space. This fact allows a simple new proof the impossibility of the physical manifestation of the quantum Zeno and anti-Zeno paradoxes. Finally, we construct the coherent states for a spinless relativistic particle, as a non-trivial example by which we demonstrate our results.

Generalized coherent states for spinning relativistic particles

“…In previous work [4], we proved that the condition δs 2 ∼ 1 is equivalent to the Heisenberg uncertainty relations. The metric together with the connection allow the determination of the coherent state propagator z|e −iĤt |z ′ by means of a path integral…”

“…Its role is twofold. First it determines the resolution of phase space measurements thus implementing the Heisenberg uncertainty relation [4]. Second, it is a crucial ingredient of the coherent state path integral [5,6], because it defines a Wiener process through which the path integral may be regularised.…”

“…Reference [11] has dealt with the Moyal representation for spinning relativistic particles, on the same state space with our generalised coherent states. Finally, a precursor of our construction for the massive spinless particles may be found in [4].…”

Uncertainty Relations for Some Central Potentials in N-Dimensional Space

“…There has been a continual interest in utilizing HUP in different settings. For example, it has been used in the study of central potentials [7]- [11] and others consider its connection to geometry [12] [13]. Furthermore, it has been generalized to describe a minimal length as a minimal uncertainty in position measurement [14]- [18] through the modification of Heisenberg commutation relation into a generalized form.…”

On Heisenberg Uncertainty Relationship, Its Extension, and the Quantum Issue of Wave-Particle Duality

“…, when the quantum objects are complementarily manifested either as wave or particle ) nor with the wave-particle equivalency in free evolution (that is not necessarily related with free motion but with quantum existence independent of any experiment or observation). Nevertheless, possible generalizations and reformulation of HUR were suggested during the last decade by the modern quantum mathematics [ 10 – 12 ], optics [ 13 ], information theory [ 14 – 16 ], still without establishing the HUR description in the absence of commutation rules [ 17 , 18 ] or versions of Schwarz inequality [ 19 , 20 ].…”