V.N.Rodionov
Starting with the modified Dirac equations for free massive particles with the γ5-extension of the physical mass \(m\to m_1+\gamma_5m_2\), we consider equations of relativistic quantum mechanics in the presence of an external electromagnetic field. The new approach is developing on the basis of existing methods for study the unbroken PT symmetry of Non-Hermitian Hamiltonians. The paper shows that this modified model contains the definition of the mass parameter, which may use as the determination of the magnitude scaling of energy M. Obviously that the transition to the standard approach is valid when small in comparison with M energies and momenta. Formally, this limit is performed when \(M\to\infty\), which simultaneously should correspond to the transition to a Hermitian limit \(m2\to0\). Inequality \(m\leq M\) may be considered and as the restriction of the mass spectrum of fermions considered in the model. Within of this approach, the effects of possible observability mass parameters: \(m_1, m_2, M\) are investigated taking into account the interaction of the magnetic field with charged fermions together with the accounting of their anomalous magnetic moments.
http://arxiv.org/abs/1404.0503
High Energy Physics – Theory (hep-th); High Energy Physics – Phenomenology (hep-ph); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
V.N.Rodionov
The modified Dirac equations for the massive particles with the replacement of the physical mass \(m\) with the help of the relation \(m\rightarrow m_1 + \gamma_5 m_2\) are investigated. It is shown that for a free fermion theory with a \(\gamma_5\) mass term, the finiteness of the mass spectrum at the value \(m_{max}= {m_1}^2/2m_2\) takes place. In this case the region of the unbroken \(\cal PT\)-symmetry may be expressed by means of the simple restriction of the physical mass \(m\leq m_{max}\). Furthermore, we have that the areas of unbroken \(\cal PT\)-symmetry \(m_1\geq m_2\geq 0\), which guarantees the reality values of the physical mass \(m\), consists of three different parametric subregions: i) \(0\leq m_2 < m_1/\sqrt{2}\), \,\,ii) \(m_2=m_1/\sqrt{2}=m_{max},\) \,\,(iii) \(m_1/\sqrt{2}< m_2 \leq m_1\). It is vary important, that only the first subregion (i) defined mass values \(m_1,m_2,\) which correspond to the description of traditional particles in the modified models, because this area contain the possibility transform the modified model to the ordinary Dirac theory. The second condition (ii) is defined the “maximon” – the particle with maximal mass \(m=m_{max}\). In the case (iii) we have to do with the unusual or “exotic” particles for description of which Hamiltonians and equations of motion have no a Hermitian limit. The formulated criterions may be used as a major test in the process of the division of considered models into ordinary and “exotic fermion theories”.
http://arxiv.org/abs/1309.0231
High Energy Physics – Theory (hep-th); High Energy Physics – Phenomenology (hep-ph); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
V.N.Rodionov
The modified Dirac equations for the massive particles with the replacement of the physical mass \(m\) with the help of the relation \(m\rightarrow m_1+ \gamma_5 m_2\) are investigated. It is shown that for a fermion theory with a \(\gamma_5\)-mass term, the limiting of the mass specter by the value \( m_{max}= {m_1}^2/2m_2\) takes place. In this case the different regions of the unbroken \(\cal PT\) symmetry may be expressed by means of the restriction of the physical mass \(m\leq m_{max}\). It should be noted that in the approach which was developed by C.Bender et al. for the \(\cal PT\)-symmetric version of the massive Thirring model with \(\gamma_5\)-mass term, the region of the unbroken \(\cal PT\)-symmetry was found in the form \(m_1\geq m_2\) \cite{ft12}. However on the basis of the mass limitation \(m\leq m_{max}\) we obtain that the domain \(m_1\geq m_2\) consists of two different parametric sectors: i) \(0\leq m_2 \leq m_1/\sqrt{2}\) -this values of mass parameters \(m_1,m_2\) correspond to the traditional particles for which in the limit \(m_{max}\rightarrow \infty\) the modified models are converting to the ordinary Dirac theory with the physical mass \(m\); ii)\(m_1/\sqrt{2}\leq m_2 \leq m_1\) – this is the case of the unusual particles for which equations of motion does not have a limit, when \(m_{max}\rightarrow \infty\). The presence of this possibility lets hope for that in Nature indeed there are some “exotic fermion fields”. As a matter of fact the formulated criterions may be used as a major test in the process of the division of considered models into ordinary and exotic fermion theories. It is tempting to think that the quanta of the exotic fermion field have a relation to the structure of the “dark matter”.
http://arxiv.org/abs/1303.7053
Quantum Physics (quant-ph); High Energy Physics – Phenomenology (hep-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
V. N. Rodionov
The quantum-field model described by non-Hermitian, but a \({\cal PT}\)-symmetric Hamiltonian is considered. It is shown by the algebraic way that the limiting of the physical mass value \(m \leq m_{max}= {m_1}^2/2m_2\) takes place for the case of a fermion field with a \(\gamma_5\)-dependent mass term (\(m\rightarrow m_1 +\gamma_5 m_2 \)). In the regions of unbroken \(\cal PT\) symmetry the Hamiltonian \(H\) has another symmetry represented by a linear operator \( \cal C\). We exactly construct this operator by using a non-perturbative method. In terms of \( \cal C\) operator we calculate a time-independent inner product with a positive-defined norm. As a consequence of finiteness mass spectrum we have the \(\cal PT\)-symmetric Hamiltonian in the areas \((m\leq m_{max})\), but beyond this limits \(\cal PT\)-symmetry is broken. Thus, we obtain that the basic results of the fermion field model with a \(\gamma_5\)-dependent mass term is equivalent to the Model with a Maximal Mass which for decades has been developed by V.Kadyshevsky and his colleagues. In their numerous papers the condition of finiteness of elementary particle mass spectrum was introduced in a purely geometric way, just as the velocity of light is a maximal velocity in the special relativity. The adequate geometrical realization of the limiting mass hypothesis is added up to the choice of (anti) de Sitter momentum space of the constant curvature.
http://arxiv.org/abs/1207.5463
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)