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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.

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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