Exact Foldy-Wouthuysen transformation for a Dirac theory revisited

Abstract

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]. ...

External sources in a minimal and nonminimal CPT-odd Lorentz violating Maxwell electrodynamics

“…The energy (30) Now, we consider a system composed by two parallel Dirac strings located a distance a ⊥ apart. We choose a coordinate system where the first string lies along the z axis, with internal magnetic flux Φ 1 and the second one, with Φ 2 lying along the line a ⊥ = (a 1 , a 2 , 0).…”

“…Most of these investigations have been performed in the framework of the Standard Model Extension (SME) [1][2][3], also called minimal SME. We mention for instance, investigations which concern the photon CPT-even sector [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], photon CPT-odd sector [20][21][22][23][24][25][26][27], fermion sector [28][29][30][31][32],…”

Maxwell electrodynamics modified by a CPT -odd dimension-five higher-derivative term

“…Studies based on the SME have been carried out to look for Lorentz-violating (LV) effects and to develop a precision program that may allow us to examine the limitation of Lorentz symmetry in various physical interactions. In this sense, a large number of investigations has been realized in the context of the fermion sector [4][5][6], CPT symmetry violation [7], the electromagnetic CPT -odd sector [8][9][10], the electromagnetic CPTeven sector [11,12], photon-fermion interactions [13][14][15], and radiative corrections [16][17][18]. Phenomenological and theoretical developments focusing on LV contributions of mass dimensions 3 and 4 have been continuously undertaken in the latest years.…”