TY - JOUR

T1 - Noncommutative coordinate picture of the quantum phase space

AU - Kong, Otto C.W.

AU - Liu, Wei Yin

N1 - Publisher Copyright:
© 2021 The Physical Society of the Republic of China (Taiwan)

PY - 2021/6

Y1 - 2021/6

N2 - We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.

AB - We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.

KW - Fundamental picture of quantum mechanics

KW - Noncommutative differential symplectic geometry

UR - http://www.scopus.com/inward/record.url?scp=85103979004&partnerID=8YFLogxK

U2 - 10.1016/j.cjph.2021.03.014

DO - 10.1016/j.cjph.2021.03.014

M3 - 期刊論文

AN - SCOPUS:85103979004

VL - 71

SP - 418

EP - 434

JO - Chinese Journal of Physics

JF - Chinese Journal of Physics

SN - 0577-9073

ER -