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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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