Noncommutative quantum mechanics—a perspective on structure and spatial extent

Rohwer, Christian M.; Zloshchastiev, Konstantin G.; Gouba, Laure; Scholtz, F. G.

doi:10.1088/1751-8113/43/34/345302

Cited by 26 publications

(46 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In our case the microscopical theory of the background BEC can be regarded as the quantum gravity itself so there is a hope that the non-axiomatic approach based on logarithmic wave equation will do its job here as well. As for the underlying microscopical theory then the presence of two length scales, 0 and Ω , points out at the possibility that the noncommutative-space extension of quantum mechanics (NCQM) is a strong candidate -and, indeed, the objects which resemble the Cooper pairs (and can be viewed as the dipoleorder approximation of a fluid element) do arise there naturally [43,44]. Another approach would be to leave the spatial commutators intact but instead treat the (bare) condensate particles and Euclidean space as the underlying entities, and construct the microscopical theory in the spirit of the conventional non-relativistic theories of superfluidity and superconductivity, and then use the maps like (37) and (43) to translate the results into the language of a physical (relativistic) observer.…”

Section: Discussionmentioning

confidence: 99%

Untitled

Zloshchastiev¹

2011

Acta Phys. Pol. B

View full text Add to dashboard Cite

Our primary task is to demonstrate that the logarithmic nonlinearity in the quantum wave equation can cause the spontaneous symmetry breaking and mass generation phenomena on its own, at least in principle. To achieve this goal, we view the physical vacuum as a kind of the fundamental Bose-Einstein condensate embedded into the fictitious Euclidean space. The relation of such description to that of the physical (relativistic) observer is established via the fluid/gravity correspondence map, the related issues, such as the induced gravity and scalar field, relativistic postulates, Mach's principle and cosmology, are discussed. For estimate the values of the generated masses of the otherwise massless particles such as the photon, we propose few simple models which take into account small vacuum fluctuations. It turns out that the photon's mass can be naturally expressed in terms of the elementary electrical charge and the extensive length parameter of the nonlinearity. Finally, we outline the topological properties of the logarithmic theory and corresponding solitonic solutions.

show abstract

Section: Discussionmentioning

confidence: 99%

Untitled

Zloshchastiev¹

2011

Acta Phys. Pol. B

View full text Add to dashboard Cite

show abstract

“…In this paper we have taken some first steps towards a head-on approach to quantum mechanics on noncommutative spaces, an approach that has been demanded and studied to some extent in the literature [34,35]. The novelty lies in the attempt to express wavefunctions purely in terms of operator-valued coordinates, rather than in terms of c-valued functions that are multiplied together by means of a star product.…”

Section: Discussionmentioning

confidence: 99%

On the Noncommutative Eikonal

Isidro

Córdoba

Rivera-Rebolledo

et al. 2011

Int. J. Geom. Methods Mod. Phys.

View full text Add to dashboard Cite

show abstract

“…The right hand sector describes another property of the system. A more detailed discussion of this interpretation can be found in [21]. This interpretation is also borne out by calculating the position representation of this state, e.g., (w|z, n) = e …”

Section: Computation Of Variance Matrix In the Non-commutative Casmentioning

confidence: 93%

Thermal effective potential in two- and three-dimensional non-commutative spaces

Devi

Ghosh

Chakraborty

et al. 2013

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

The issue of thermal correlation functions and the associated effective statistical potential in twodimensional Moyal space, arising in the twisted approach to implement rotational symmetry, has been revisited in an operatorial formulation where no explicit star product is used initially. The corresponding results using Moyal and Voros star products are then easily obtained by taking the corresponding overlap with Moyal and Voros bases. in contrast to the Moyal case where the concept of distance and, in particular, the relative separation between a pair of particles remain ambiguous when the Moyal star product is used, the Voros basis is more physical and the inter-particle distance can be introduced unambiguously. The forms of the correlation function and the effective potential are found to be same as the Moyal case except that the thermal wavelength undergoes a non-commutative deformation, ensuring that it has a lower bound of the order of √ θ. It is shown that in a suitable basis (called here quasi-commutative basis) in the multiparticle sector the thermal correlation function coincides with the commutative result both in the Moyal and Voros cases along with the restoration of the Pauli principle, except that in the Voros case the thermal wavelength, again, gets a non-commutative correction. Finally, we extend our result to three-dimensional non-commutative space and compute the correlation function and effective potential using both twisted and quasi-commutative bases in the Moyal and Voros cases. We find that there is SO(3) → SO(2) symmetry breaking in the effective potential, which also violates the Pauli principle, even for a pair of free particles, despite the fact that a deformed co-product is used to construct twisted symmetric/anti-symmetric basis. However, this SO(3) symmetry, along with Pauli principle, is restored once we use the quasi-commutative bases.

show abstract

Noncommutative quantum mechanics—a perspective on structure and spatial extent

Cited by 26 publications

References 23 publications

Untitled

Untitled

On the Noncommutative Eikonal

Thermal effective potential in two- and three-dimensional non-commutative spaces

Contact Info

Product

Resources

About