dorsal/arxiv
View SchemaHow to Quantize Phases and Moduli!
| Authors | H. A. Kastrup |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0109013 |
| URL | https://arxiv.org/abs/quant-ph/0109013 |
Abstract
A typical classical interference pattern of two waves with intensities I_1, I_2 and relative phase phi = phi_2-phi_1 may be characterized by the 3 observables p = sqrt{I_1 I_2}, p cos\phi and -p sin\phi. They are, e.g. the starting point for the semi-classical operational approach by Noh, Fougeres and Mandel (NFM) to the old and notorious phase problem in quantum optics. Following a recent group theoretical quantization of the symplectic space S = {(phi in R mod 2pi, p > 0)} in terms of irreducible unitary representations of the group SO(1,2) the present paper applies those results to that controversial problem of quantizing moduli and phases of complex numbers: The Poisson brackets of the classical observables p cos\phi, -p sin\phi and p > 0 form the Lie algebra of the group SO(1,2). The corresponding self-adjoint generators K_1, K_2 and K_3 of that group may be obtained from its irreducible unitary representations. For the positive discrete series the modulus operator K_3 has the spectrum {k+n, n = 0, 1,2,...; k > 0}. Self-adjoint operators for cos phi and sin phi can be defined as ((1/K_3)K_1 + K_1/K_3)/2 and -((1/K_3)K_2 + K_2/K_3)/2 which have the theoretically desired properties for k > or = 0.5. The approach advocated here solves, e.g. the modulus-phase quantization problem for the harmonic oscillator and appears to provide a full quantum theoretical basis for the NFM-formalism.
{
"annotation_id": "5b66d601-4e38-4b79-9eb4-a9ab49fdb1f1",
"date_created": "2026-03-02T18:01:45.897000Z",
"date_modified": "2026-03-02T18:01:45.897000Z",
"file_hash": "f78a4a734ae82d7906f6aefdc1a146a8c956863d60fb0fdd7bb7adc39ef92e3b",
"private": false,
"record": {
"abstract": "A typical classical interference pattern of two waves with intensities I_1,\nI_2 and relative phase phi = phi_2-phi_1 may be characterized by the 3\nobservables p = sqrt{I_1 I_2}, p cos\\phi and -p sin\\phi. They are, e.g. the\nstarting point for the semi-classical operational approach by Noh, Fougeres and\nMandel (NFM) to the old and notorious phase problem in quantum optics.\nFollowing a recent group theoretical quantization of the symplectic space S =\n{(phi in R mod 2pi, p \u003e 0)} in terms of irreducible unitary representations of\nthe group SO(1,2) the present paper applies those results to that controversial\nproblem of quantizing moduli and phases of complex numbers: The Poisson\nbrackets of the classical observables p cos\\phi, -p sin\\phi and p \u003e 0 form the\nLie algebra of the group SO(1,2). The corresponding self-adjoint generators\nK_1, K_2 and K_3 of that group may be obtained from its irreducible unitary\nrepresentations. For the positive discrete series the modulus operator K_3 has\nthe spectrum {k+n, n = 0, 1,2,...; k \u003e 0}. Self-adjoint operators for cos phi\nand sin phi can be defined as ((1/K_3)K_1 + K_1/K_3)/2 and -((1/K_3)K_2 +\nK_2/K_3)/2 which have the theoretically desired properties for k \u003e or = 0.5.\nThe approach advocated here solves, e.g. the modulus-phase quantization problem\nfor the harmonic oscillator and appears to provide a full quantum theoretical\nbasis for the NFM-formalism.",
"arxiv_id": "quant-ph/0109013",
"authors": [
"H. A. Kastrup"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.MP",
"physics.atom-ph"
],
"title": "How to Quantize Phases and Moduli!",
"url": "https://arxiv.org/abs/quant-ph/0109013"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "95c76082-5e29-42cf-9929-a32b65f1a388",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}