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

Silenko, Alexander J.

doi:10.1007/s11232-013-0086-1

Cited by 36 publications

(60 citation statements)

References 42 publications

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“…The resulting transformed Hamiltonian in the FW representation is exact for terms proportional to the zeroth and first powers of the Planck constant and also for terms proportional to 2 and describing contact interactions. It is given by [17,18,23]…”

Section: Electromagnetic Interactions Of a Dirac Particle In A Romentioning

confidence: 99%

Manifestations of the rotation and gravity of the Earth in high-energy physics experiments

2016

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The inertial (due to rotation) and the gravitational fields of the Earth affect the motion of an elementary particle and its spin dynamics. This influence is not negligible and should be taken into account in high-energy physics experiments. Earth's influence is manifest in perturbations in the particle motion, in an additional precession of the spin, and in a change of the constitutive tensor of the Maxwell electrodynamics. Bigger corrections are oscillatory and their contributions average to zero. Other corrections due to the inhomogeneity of the inertial field are not oscillatory but they are very small and may be important only for the storage ring electric dipole moment experiments. Earth's gravity causes the Newton-like force, the reaction force provided by a focusing system, and additional torques acting on the spin. However, there are no observable indications of the electromagnetic effects due to Earth's gravity.

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Section: Electromagnetic Interactions Of a Dirac Particle In A Romentioning

confidence: 99%

Manifestations of the rotation and gravity of the Earth in high-energy physics experiments

2016

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“…In the last section we derive the bound state of the theory, given by Eq. (26). It worth mentioning that the possibility of the weakness of CPT-Lorentz terms to be compensated by the presence of a strong magnetic field.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…This was explicitly shown on [19]. In this paper we show another case where it happens.In [26] the author performs in a very didactic way the formal comparison between the two methods. He also described which is the most efficient method for each possible applications.…”

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confidence: 99%

Exact Foldy-Wouthuysen transformation for a Dirac theory revisited

2019

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The Exact Foldy-Wouthuysen transformation (EFWT) method is generalized here. In principle, it is not possible to construct the EFWT to any Hamiltonian. The transformation conditions are the same but the involution operator has a new form. We took a particular example and constructed explicitly the new involution operator that allows one to perform the transformation. We treat the case of the Hamiltonian with 160 possible CPT-Lorentz breaking terms, using this new technique. The transformation was performed and physics analysis of the equations of motion is shown.Another possible phenomenological approach to this problem can be constructed step by step by searching for new terms in the Hamiltonian that describes this situation. Thinking this way, it makes sense the appearance of some terms in the equations of motion that could give a mix between an external known field with sufficient enough big amplitude to compensate the fact that the CPT-Lorentz terms have small amplitudes.The idea is the same shown in [15], where the strong magnetic field could, in principle, change the trajectory of the Dirac particle that interacts with gravitational waves. It is important to take into account the corrections, made with canonical FW, to these results that were shown in [16]. In [17], the massive linearized gravity was studied and some possible experiments that could measure indirect effects of gravitational waves on Dirac fermions were indicated. However, solving the Dirac equation for the general case is not a simple procedure [18]. It is well known in literature that working with the EFWT is a more prominent approach to interpret a Dirac Hamiltonian than the canonical transformation [19]. But this is true not only for the fact that it can give us new terms, but it is a faster and more economic (in terms of algebraic calculation) procedure [15,20,21,22]. One can see this transformation as a generalization of the usual FWT.Let us perform a comparison on the two procedures. It is possible to see that in the usual FWT the multiplication on each step (on each order on 1/m) by the term that makes the Hamiltonian even, generates a maximum of 1 + 2n even terms, where n represents the number of terms of the previous Hamiltonian (see, for example, pages 48-51 in [23]). The maximum number of terms in the nth-Hamiltonian is straightforward obtained by the fact that this is an expansion in power series of an operator. The factor 2 on 1 + 2n expression is obtained in case when it does not commute with all original terms.On the other hand, the EFWT impose the multiplication of all terms of the Hamiltonian by themselves. Analogous arguments give us the maximum of 1+2n 2 on the expanded Hamiltonian. If the parameter of expansion here is also taken to be 1/m, one can see that the possibility of having new terms in comparison with the usual method is greater. In many particular known cases [21,24,25], the anti-commutators on both cases are such that the results are the same! But it is not the general case. This was explicitly shown on [19]. ...

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“…The Hamiltonian transformed to the FW representation is exact for terms proportional to the zeroth and first powers of the Planck constant and also for terms describing contact interactions and proportional to 2 . It is given by [8,11]…”

Section: Phenomenological Description Of Strong Interaction With the mentioning

confidence: 99%

“…A possibility to restrict ourselves to a consideration of local interactions (because of short-range nuclear forces) allows us to realize advantages of this approach. Developed methods of the relativistic FW transformation [7][8][9] (see also [10][11][12] and references therein) ensure fulfilling this transformation in arbitrarily strong external fields. While this approach has not been used for strong interactions, it has proven itself as a powerful tool for a description of electromagnetic [11,13,14], gravitational [15][16][17][18][19][20][21], and weak [22,23] interactions of single particles.…”

Section: Introductionmentioning

confidence: 99%

Phenomenological description of spin effects in electromagnetic and strong interactions of quarks

Silenko

Teryaev

2017

EPJ Web Conf.

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Abstract. Phenomenological description of interactions of relativistic quarks by the Dirac equation with the Cornell potential is given. The general form of the initial equation containing the vector and scalar parts of the Cornell potential is used at the arbitrary connection between these parts. The Hamiltonian in the Foldy-Wouthuysen representation is derived in the general form with allowance for the electromagnetic interactions. Unlike precedent investigations, it is relativistic and exact for terms of the zeroth and first powers in the Planck constant and also for such terms of the second power which describe contact interactions. General quantum mechanical equations of motion for the momentum and the spin are derived and the classical limit of the Hamiltonian and the equations of motion are found for the first time. A connection between the angular velocity of the quark spin precession and the force acting on it is determined. The energy of the spin-orbit interaction is rather high (of the order of 100 MeV). The terms describing the spin-orbit and contact interactions have opposite signs for the scalar and the vector parts of the Cornell potential. The evolution of the quark helicity and the spin-spin interaction of the quarks are also calculated.

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

Cited by 36 publications

References 42 publications

Manifestations of the rotation and gravity of the Earth in high-energy physics experiments

Manifestations of the rotation and gravity of the Earth in high-energy physics experiments

Exact Foldy-Wouthuysen transformation for a Dirac theory revisited

Phenomenological description of spin effects in electromagnetic and strong interactions of quarks

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