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Relativistic wave equations with fractional derivatives and pseudodifferential operators. (English) Zbl 1007.81043
Summary: We study the class of the free relativistic covariant equations generated by the fractional powers of the d’Alembertian operator \((\square^{1/n})\). The equations corresponding to \(n=1\) and \(2\) (Klein-Gordon and Dirac equations) are local in their nature, but the multicomponent equations for arbitrary \(n>2\) are nonlocal. We show the representation of the generalized algebra of Pauli and Dirac matrices and how these matrices are related to the algebra of \(\text{SU}(n)\) group. The corresponding representations of the Poincaré group and further symmetry transformations on the obtained equations are discussed. The construction of the related Green functions is suggested.

MSC:
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
47G30 Pseudodifferential operators
26A33 Fractional derivatives and integrals
34B27 Green’s functions for ordinary differential equations
15A66 Clifford algebras, spinors
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