Foldy‐Wouthuysen Transformations and Related Problems

Vries, E. de

doi:10.1002/prop.19700180402

Cited by 89 publications

(72 citation statements)

References 120 publications

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“…(4) as the initial and formal expression for the Hamiltonian in the FW representation in terms of the operators E and O. The transformed Hamiltonian was represented in precisely this form in [10,11,16,23].…”

Section: Comparison Of Results Obtained By Different Methods Of the Smentioning

confidence: 99%

Comparative analysis of direct and “step-by-step” Foldy-Wouthuysen transformation methods

Silenko

2013

Theor Math Phys

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Relativistic methods for the FoldyWouthuysen transformation of the step-by-step type already at the first step give an expression for the Hamilton operator not coinciding with the exact result determined by the Eriksen method. The methods agree for the zeroth and first orders in the Planck constant terms but do not agree for the second and higher-order terms. We analyze the benefits and drawbacks of various methods and establish their applicability boundaries.

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Section: Comparison Of Results Obtained By Different Methods Of the Smentioning

confidence: 99%

Comparative analysis of direct and “step-by-step” Foldy-Wouthuysen transformation methods

Silenko

2013

Theor Math Phys

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“…When A is time-independent and V = 0 , we obtain a (diagonalized) form similar to one derived in deVries 24 …”

Section: General Solutionmentioning

confidence: 69%

Analytic representation of the square-root operator

Gill¹,

Zachary²

2005

J. Phys. A: Math. Gen.

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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 Schrödinger 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.When the magnetic field is constant, we obtain an additional singularity, which could be interpreted as particle absorption and emission. The physical picture that emerges is that, in addition to the confined singularities and the additional attractive (repulsive) term, the effective mass of the composite acquires an oscillatory behavior.We also derive an alternate relationship between the Dirac equation (with minimal coupling) and the square-root equation that is much closer than the one obtained via the FoldyWouthuysen method, in that there is no change in the wave function. This is accomplished by considering the scalar potential to be a part of the mass. This approach leads to a new KleinGordon equation and a new square-root equation, both of which have the same eigenvalues and eigenfunctions as the Dirac equation. Finally, we develop a perturbation theory that allows us to extend the range of our theory to include suitable spacetime-dependent potentials.2

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“…The improvement of the FW transformation [6] is the task of many authors from 1950 until today, see, e. g., the recent publications [115][116][117][118][119][120][121][122][123][124][125][126][127][128]. Nevertheless, this transformation for free case of non-interacting spin 1/2 particle-antiparticle doublet is not changed from 1950 (the year of first publication) until today.…”

Section: Discussionmentioning

confidence: 99%

“…The first way is based on the application of the corresponding Foldy-Wouthuysen transformation directly to the canonical field equation (102). Note that there is a nonzero chance to find the appropriate transformation among a big number of so-called Foldy-Wouthuysen transformations for an arbitrary spin [115][116][117][118][119][120][121][122][123][124]. Unfortunately for our approach, many authors considered for the spin s=1 case the 4-vector potential (see, e. g., [124]) and not 3-component wave function -3, 11, 12, 84, 89].…”

Section: The Example Of Covariant Field Equationmentioning

confidence: 99%

“…Dirac's doubts were overcome in [6,7,14]. Today the significance of the Schrödinger-Foldy equation is confirmed by more then a hundred publications about FW (see, e. g., [31,45,50,[115][116][117][118][119][120][121][122][123][124][125][126][127][128] and the references therein) and the spinless Salpeter equations [8, 13, 20-30, 32, 33, 35, 36], which have wide-range application in contemporary theoretical physics.…”

Section: Briefly On the Different Ways Of The Dirac Equation Derivationmentioning

confidence: 99%

See 1 more Smart Citation

On the Relativistic Canonical Quantum Mechanics and Field Theory of Arbitrary Spin

Simulik¹

2017

ujpa

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The new relativistic equations of motion for the particles with arbitrary spin and nonzero mass, suggested by author in years 2014-2016, are under consideration. The complete version of brief paper in J. Phys: Conf. Ser., 670 (2016) 012047(1-16) is given. The axiomatic level description of the relativistic canonical quantum mechanics of an arbitrary mass and spin has been given. The 64-dimensional Cl R (0,6) algebra in terms of Dirac gamma matrices has been suggested. The interpretation of the 28-dimensional gamma matrix representation of SO(8) algebra over the field of real numbers is given. The link between the relativistic canonical quantum mechanics of the arbitrary spin and the covariant local field theory in the form of extended Foldy-Wouthuysen transformation has been found. Different methods of the Dirac equation derivation have been reviewed. The manifestly covariant field equation for an arbitrary spin, that follows from the corresponding equation of relativistic canonical quantum mechanics, has been considered. The found equations are without redundant components. The partial examples for spin s=3/2 and s=2 are presented. The covariant local field theory equations for spin s = (3/2,3/2) particle-antiparticle doublet and spin s = (2,2) particle-antiparticle doublet have been introduced. The Maxwell and slightly generalized Maxwell-like equations containing mass member have been considered as well.

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Foldy‐Wouthuysen Transformations and Related Problems

Cited by 89 publications

References 120 publications

Comparative analysis of direct and “step-by-step” Foldy-Wouthuysen transformation methods

Comparative analysis of direct and “step-by-step” Foldy-Wouthuysen transformation methods

Analytic representation of the square-root operator

On the Relativistic Canonical Quantum Mechanics and Field Theory of Arbitrary Spin

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