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Register a new userAxial momentum for the relativistic Majorana particle
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Authors
H. Arodz
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The Hilbert space of states of the relativistic Majorana particle consists of normalizable bispinors with real components, and the usual momentum operator $- i abla$ can not be defined in this space. For this reason, we introduce the axial momentum operator, $ - i gamma_5 abla$ as a new observable for this particle. In the Heisenberg picture, the axial momentum contains a component which oscillates with the amplitude proportional to $m/E$, where $E$ is the energy and $m$ the mass of the particle. The presence of the oscillations discriminates between the massive and massless Majorana particle. We show how the eigenvectors of the axial momentum, called the axial plane waves, can be used as a basis for obtaining the general solution of the evolution equation, also in the case of free Majorana field. Here a novel feature is a coupling of modes with the opposite momenta, again present only in the case of massive particle or field.
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This article is a pedagogical introduction to relativistic quantum mechanics of the free Majorana particle. This relatively simple theory differs from the well-known quantum mechanics of the Dirac particle in several important aspects. We present its three equivalent formulations. Next, so called axial momentum observable is introduced, and general solution of the Dirac equation is discussed in terms of eigenfunctions of that operator. Pertinent irreducible representations of the Poincare group are discussed. Finally, we show that in the case of massless Majorana particle the quantum mechanics can be reformulated as a spinorial gauge theory.
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