dorsal/arxiv
View SchemaAnalytic Representation of The Square-Root Operator
| Authors | T. L. Gill, W. W. Zachary |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0405150 |
| URL | https://arxiv.org/abs/quant-ph/0405150 |
| DOI | 10.1088/0305-4470/38/11/010 |
Abstract
In this paper, we use the theory of fractional powers of linear operators to construct a general (analytic) representation theory for the square-root energy operator of relativistic quantum theory, which is valid for all values of the spin. We focus on the spin 1/2 case, considering a few simple yet solvable and physically interesting cases, in order to understand how to interpret the operator. Our general representation is uniquely determined by the Green's function for the corresponding Schrodinger equation. We find that, in general, the operator has a representation as a nonlocal composite of (at least) three singularities. In the standard interpretation, the particle component has two negative parts and one (hard core) positive part, while the antiparticle component has two positive parts and one (hard core) negative part. This effect is confined within a Compton wavelength such that, at the point of singularity, they cancel each other providing a finite result. Furthermore, the operator looks like the identity outside a few Compton wavelengths (cutoff). To our knowledge, this is the first example of a physically relevant operator with these properties.
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"abstract": "In this paper, we use the theory of fractional powers of linear operators to\nconstruct a general (analytic) representation theory for the square-root energy\noperator of relativistic quantum theory, which is valid for all values of the\nspin. We focus on the spin 1/2 case, considering a few simple yet solvable and\nphysically interesting cases, in order to understand how to interpret the\noperator. Our general representation is uniquely determined by the Green\u0027s\nfunction for the corresponding Schrodinger equation. We find that, in general,\nthe operator has a representation as a nonlocal composite of (at least) three\nsingularities. In the standard interpretation, the particle component has two\nnegative parts and one (hard core) positive part, while the antiparticle\ncomponent has two positive parts and one (hard core) negative part. This effect\nis confined within a Compton wavelength such that, at the point of singularity,\nthey cancel each other providing a finite result. Furthermore, the operator\nlooks like the identity outside a few Compton wavelengths (cutoff). To our\nknowledge, this is the first example of a physically relevant operator with\nthese properties.",
"arxiv_id": "quant-ph/0405150",
"authors": [
"T. L. Gill",
"W. W. Zachary"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/38/11/010",
"title": "Analytic Representation of The Square-Root Operator",
"url": "https://arxiv.org/abs/quant-ph/0405150"
},
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