scholarly journals Hilbert space representation of the minimal length uncertainty relation

1995 ◽  
Vol 52 (2) ◽  
pp. 1108-1118 ◽  
Author(s):  
Achim Kempf ◽  
Gianpiero Mangano ◽  
Robert B. Mann
Keyword(s):  
Hilbert Space ◽  
Uncertainty Relation ◽  
Minimal Length ◽  
Space Representation

On the Heisenberg algebra in Bargmann–Fock space with natural cutoffs

2016 ◽  
Vol 13 (05) ◽  
pp. 1650054 ◽  
Author(s):  
K. Nozari ◽  
M. Roushan
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
General Framework ◽  
Fock Space ◽  
Heisenberg Algebra ◽  
Minimal Length ◽  
Space Representation ◽  
Maximal Momentum

We construct a generalized Hilbert space representation of quantum mechanics in [Formula: see text]-dimensions based on Bargmann–Fock space with natural cutoffs as a minimal length, minimal momentum and maximal momentum in order to have a general framework for Heisenberg algebra in Bargmann–Fock space.

scholarly journals Measurement theory in classical mechanics

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
So Katagiri
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Uncertainty Relation ◽  
Measurement Theory ◽  
Classical Mechanics ◽  
The State ◽  
Measurement Device ◽  
Von Neumann ◽  
Relative Interpretation ◽  
Classical And Quantum Mechanics

Abstract We investigate measurement theory in classical mechanics in the formulation of classical mechanics by Koopman and von Neumann (KvN), which uses Hilbert space. We show a difference between classical and quantum mechanics in the “relative interpretation” of the state of the target of measurement and the state of the measurement device. We also derive the uncertainty relation in classical mechanics.

scholarly journals (2+1)-Dimensional Duffin-Kemmer-Petiau Oscillator under a Magnetic Field in the Presence of a Minimal Length in the Noncommutative Space

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Bing-Qian Wang ◽  
Zheng-Wen Long ◽  
Chao-Yun Long ◽  
Shu-Rui Wu
Keyword(s):  
Magnetic Field ◽  
Probability Density ◽  
Momentum Space ◽  
Jacobi Polynomials ◽  
Minimal Length ◽  
Wave Functions ◽  
Noncommutative Space ◽  
Space Representation ◽  
Energy Eigenvalues ◽  
Special Cases

Using the momentum space representation, we study the (2 + 1)-dimensional Duffin-Kemmer-Petiau oscillator for spin 0 particle under a magnetic field in the presence of a minimal length in the noncommutative space. The explicit form of energy eigenvalues is found, and the wave functions and the corresponding probability density are reported in terms of the Jacobi polynomials. Additionally, we also discuss the special cases and depict the corresponding numerical results.

scholarly journals On the Minimal Length Uncertainty Relation and the Foundations of String Theory

2011 ◽  
Vol 2011 ◽  
pp. 1-30 ◽  
Author(s):  
Lay Nam Chang ◽  
Zachary Lewis ◽  
Djordje Minic ◽  
Tatsu Takeuchi
Keyword(s):  
General Relativity ◽  
Quantum Theory ◽  
String Theory ◽  
Momentum Space ◽  
Vacuum Energy ◽  
Uncertainty Relation ◽  
Minimal Length

We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of nonperturbative string theory.

scholarly journals Nonrelativistic anti-Snyder model and some applications

2017 ◽  
Vol 32 (02n03) ◽  
pp. 1750009 ◽  
Author(s):  
C. L. Ching ◽  
C. X. Yeo ◽  
W. K. Ng
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Bound States ◽  
Homogeneous Magnetic Field ◽  
Maximum Energy ◽  
Minimal Length ◽  
Landau Levels ◽  
Space Representation ◽  
Free Particles ◽  
Ordinary Quantum

In this paper, we examine the (2[Formula: see text]+[Formula: see text]1)-dimensional Dirac equation in a homogeneous magnetic field under the nonrelativistic anti-Snyder model which is relevant to doubly/deformed special relativity (DSR) since it exhibits an intrinsic upper bound of the momentum of free particles. After setting up the formalism, exact eigensolutions are derived in momentum space representation and they are expressed in terms of finite orthogonal Romanovski polynomials. There is a finite maximum number of allowable bound states [Formula: see text] due to the orthogonality of the polynomials and the maximum energy is truncated at [Formula: see text]. Similar to the minimal length case, the degeneracy of the Dirac–Landau levels in anti-Snyder model are modified and there are states that do not exist in the ordinary quantum mechanics limit [Formula: see text]. By taking [Formula: see text], we explore the motion of effective massless charged fermions in graphene-like material and obtained a maximum bound of deformed parameter [Formula: see text]. Furthermore, we consider the modified energy dispersion relations and its application in describing the behavior of neutrinos oscillation under modified commutation relations.

scholarly journals Hilbert space representations of decoherence functionals and quantum measures

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Stan Gudder
Keyword(s):  
Hilbert Space ◽  
Closed Subspace ◽  
Complex Hilbert Space ◽  
Space Representation ◽  
Quantum Operator ◽  
Quantum Measure ◽  
Usual Quantum ◽  
Vector Valued ◽  
Quantum Formulation ◽  
Decoherence Functional

AbstractWe show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.

Sign in / Sign up

Export Citation Format

Share Document