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Ukrainian Journal of Physics

Towards an Observable Test of Noncommutative Quantum Mechanics

Shi‐Dong Liang, Tiberiu Harko

Published Nov 25, 2019• 8 citations
Publisher

Abstract

The conceptual incompatibility of spacetime in gravity and quantum physics implies the existence of noncommutative spacetime and geometry on the Planck scale. We present the formulation of a noncommutative quantum mechanics based on the Seiberg–Witten map, and we study the Aharonov–Bohm effect induced by the noncommutative phase space. We investigate the existence of the persistent current in a nanoscale ring with an external magnetic field along the ring axis, and we introduce two observables to probe the signal coming from the noncommutative phase space. Based on this formulation, we give a value-independent criterion to demonstrate the existence of the noncommutative phase space.

DOI
10.15407/ujpe64.11.983

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The conceptual incompatibility of spacetime in gravity and quantum physics implies the existence of noncommutative spacetime and geometry on the Planck scale. We present the formulation of a noncommutative quantum mechanics based on the Seiberg–Witten map, and we study the Aharonov–Bohm effect induced by the noncommutative phase space. We investigate the existence of the persistent current in a nanoscale ring with an external magnetic field along the ring axis, and we introduce two observables to probe the signal coming from the noncommutative phase space. Based on this formulation, we give a value-independent criterion to demonstrate the existence of the noncommutative phase space.
The noncommutative space provides a framework to understand phenomena in Planck scale physics. However, there is no any direct experimental evidence to demonstrate the existence of noncommutative space. We propose an experimental scheme based on the Aharonov-Bohm effect in the nano-scale quantum mechanics to probe the phenomena in the noncommutative phase space. By the Seiberg-Witten map, the free electrons of the nano-scale ring in noncommutative phase space can be mapped equivalently to the quantum mechanical (Heisenberg's algebra) phase space with an extra effective magnetic flux. We introduce two variables related to the persistent current in the ring to probe the noncommutative phase space effect. We give a value-independent criterion to detect the existence of the noncommutative phase space. Namely the answer for existence or nonexistence of the noncommutative phase space depends only on the trend of the curves, rather than the values of the observation data. It can be expected to catch the noncommutative phase-space effect by this scheme.
Noncommutative phase space plays an essential role in particle physics and quantum gravity at the Planck scale. However, direct experimental evidence or observation to demonstrate the existence of noncommutative phase space is still lacking. We study a quantum ring in noncommutative phase space based on the Seiberg-Witten map and give the effective magnetic potential and field coming from the noncommutative phase space, which induces the persistent current in the ring. We introduce two variables as two signatures to detect the noncommutative phase space and propose an experimental scheme to detect the noncommutative phase space as long as we measure the persistent current and the external magnetic flux.
We study the time-dependent Aharonov–Bohm effect on the noncommutative space. Because there is no net Aharonov–Bohm phase shift in the time-dependent case on the commutative space, therefore, a tiny deviation from zero indicates new physics . Based on the Seiberg–Witten map we obtain the gauge invariant and Lorentz covariant Aharonov–Bohm phase shift in general case on noncommutative space. We find there are two kinds of contribution: momentum-dependent and momentum-independent corrections. For the momentum-dependent correction, there is a cancellation between the magnetic and electric phase shifts, just like the case on the commutative space. However, there is a non-trivial contribution in the momentum-independent correction. This is true for both the time-independent and time-dependent Aharonov–Bohm effects on the noncommutative space. However, for the time-dependent Aharonov–Bohm effect, there is no overwhelming background which exists in the time-independent Aharonov–Bohm effect on both commutative and noncommutative space. Therefore, the time-dependent Aharonov–Bohm can be sensitive to the spatial noncommutativity. The net correction is proportional to the product of the magnetic fluxes through the fundamental area represented by the noncommutative parameter θ , and through the surface enclosed by the trajectory of charged particle. More interestingly, there is an anti-collinear relation between the logarithms of the magnetic field B and the averaged flux Φ / N ( N is the number of fringes shifted). This nontrivial relation can also provide a way to test the spatial noncommutativity. For B Φ / N ∼ 1 , our estimation on the experimental sensitivity shows that it can reach the 10 GeV scale. This sensitivity can be enhanced by using stronger magnetic field strength , larger magnetic flux, as well as higher experimental precision on the phase shift.
Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a non-linearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.
Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a nonlinearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.
The Aharonov–Bohm effect on the noncommutative plane is considered. Developing the path integral formulation of quantum mechanics, we find the propagation amplitude for a particle in a noncommutative space. We show that the corresponding shift in the phase of the particle propagator due to the magnetic field of a thin solenoid receives certain gauge invariant corrections because of the noncommutativity. Evaluating the numerical value for this correction, an upper bound for the noncommutativity parameter is obtained.