We revisit the quantum phase operator $Φ$ introduced by Garrison and Wong. Denoting by $N$ the number operator, we provide a detailed proof of the Heisenberg commutation relation $ΦN - NΦ= iI$ on the natural maximal domain $D(ΦN) \cap D(N Φ)$ as well as the failure of the Weyl commutation relations, and discuss some further interesting properties of this pair.We revisit the quantum phase operator $Φ$ introduced by Garrison and Wong. Denoting by $N$ the number operator, we provide a detailed proof of the Heisenberg commutation relation $ΦN - NΦ= iI$ on the natural maximal domain $D(ΦN) \cap D(N Φ)$ as well as the failure of the Weyl commutation relations, and discuss some further interesting properties of this pair.

arXiv:2008.08935Aug 20, 20201 citation

Einstein’s E = mc2 Derivable from Heisenberg’s Uncertainty Relations

Sibel Başkal, Young S Kim, Marilyn E. Noz

Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the O ( 2 , 1 ) group. This group has three independent generators. The one-dimensional step-up and step-down operators can be combined into one two-by-two Hermitian matrix which contains three independent operators. If we use a two-variable Heisenberg commutation relation, the two pairs of independent step-up, step-down operators can be combined into a four-by-four block-diagonal Hermitian matrix with six independent parameters. It is then possible to add one off-diagonal two-by-two matrix and its Hermitian conjugate to complete the four-by-four Hermitian matrix. This off-diagonal matrix has four independent generators. There are thus ten independent generators. It is then shown that these ten generators can be linearly combined to the ten generators for Dirac’s two oscillator system leading to the group isomorphic to the de Sitter group O ( 3 , 2 ) , which can then be contracted to the inhomogeneous Lorentz group with four translation generators corresponding to the four-momentum in the Lorentz-covariant world. This Lorentz-covariant four-momentum is known as Einstein’s E = m c 2 .Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the O ( 2 , 1 ) group. This group has three independent generators. The one-dimensional step-up and step-down operators can be combined into one two-by-two Hermitian matrix which contains three independent operators. If we use a two-variable Heisenberg commutation relation, the two pairs of independent step-up, step-down operators can be combined into a four-by-four block-diagonal Hermitian matrix with six independent parameters. It is then possible to add one off-diagonal two-by-two matrix and its Hermitian conjugate to complete the four-by-four Hermitian matrix. This off-diagonal matrix has four independent generators. There are thus ten independent generators. It is then shown that these ten generators can be linearly combined to the ten generators for Dirac’s two oscillator system leading to the group isomorphic to the de Sitter group O ( 3 , 2 ) , which can then be contracted to the inhomogeneous Lorentz group with four translation generators corresponding to the four-momentum in the Lorentz-covariant world. This Lorentz-covariant four-momentum is known as Einstein’s E = m c 2 .

arXiv:1911.03818Quantum ReportsNov 9, 201911 citations

On the Heisenberg commutation relation. I

Konrad Schmüdgen

Journal of Functional AnalysisJan 1, 198370 citations

A Theory of the Relative Phase and Number Difference of Two Quantized Harmonic Oscillators

G. Björk, Jonas S derholm, Alexei S. Trifonov +1 more

We present a comprehensive and self-consistent theory of relative-phase measurements and the associated Hermitian relative-phase operator of two harmonic oscillators. We find that since Nature does not favor any particular initial condition of the two oscillators, the relative-phase operator is not unique. We show that the relative-phase eigenstates are maximally entangled. Therefore, most relative-phase operators lack a classical correspondence, even in the high-excitation limit. Furthermore, we find that the relative phase and the excitation number difference are noncommuting, noncanonical observables and we derive a commutation relation.We present a comprehensive and self-consistent theory of relative-phase measurements and the associated Hermitian relative-phase operator of two harmonic oscillators. We find that since Nature does not favor any particular initial condition of the two oscillators, the relative-phase operator is not unique. We show that the relative-phase eigenstates are maximally entangled. Therefore, most relative-phase operators lack a classical correspondence, even in the high-excitation limit. Furthermore, we find that the relative phase and the excitation number difference are noncommuting, noncanonical observables and we derive a commutation relation.

Physica ScriptaJan 1, 20024 citations

Non-Hermitian Heisenberg representation

Miloslav Znojil

arXiv:1505.01036Physics Letters AJun 23, 201526 citations

We revisit the quantum phase operator

Φ

introduced by Garrison and Wong. Denoting by

N

the number operator, we provide a detailed proof of the Heisenberg commutation relation

ΦN - NΦ= iI

on the natural maximal domain

D(ΦN) \cap D(N Φ)

as well as the failure of the Weyl commutation relations, and discuss some further interesting properties of this pair.