Generalizations of the Klein-Gordon and the Dirac equations from non-standard Lagrangians. (English) Zbl 1330.35364
Summary: We present various generalizations of the Klein-Gordon and Dirac formalisms based on non-standard Lagrangians within the framework of the calculus of variations characterized by a power-law Lagrangian \(\mathsf{L}^{1+\gamma}\), \(\gamma\) being a free parameter. In the case of the bosonic scalar field, the modified dispersion relation has been derived and based on this, it was observed that for a particular choice of non-standard Lagrangians, the new field theory forbid the presence of massless particles. Besides, the Klein-Gordon equation is modified and becomes similar to the Barut equation which is a second order equation in the \((1/2,0)\oplus (0,1/2)\) representation of the Lorentz group, which explains the splitting of leptons. For the case of the spinor scalar field, the Barut-like equation was derived from a non-standard Lagrangian as well. For some specific class of non-standard Lagrangians, the modified dispersion is modified and prohibits the presence of massless particles.
MSC:
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35Q40 | PDEs in connection with quantum mechanics |
Keywords:
non-standard Lagrangians; modified dispersion relation; modified Klein-Gordon equation; modified Dirac equation; Barut equationReferences:
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