Non-Lie and discrete symmetries of the Dirac equation (original) (raw)
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A complete set of symmetry operators for the Dirac equation
Ukrainian Mathematical Journal, 1991
A complete set of symmetry operators of arbitrary finite order admitted by the Dirac equation is found. The algebraic structure of this set is investigated and subsets of symmetry operators that form bases of Lie algebras and superalgebras are isolated.
Il Nuovo Cimento B, 1996
Summary Surprising symmetries in the (j, 0) ⊕ (0,j) Lorentz-group representation space are analysed. The aim is to draw the reader’s attention to the possibility of describing the particle world on the grounds of the Dirac «doubles». Several tune points of the variational principle for this kind of equations are briefly discussed.
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1977
In works [1-6] the canonical-transformation method has been proposed for the investigations of the group properties of the differential equations of the quantum mechanics. This method essence in that the system of differential equation is first transformed to the diagonal or Jordan form and then the invariance algebra of the transformed equation is established. The explicit form of this algebra basis elements for the starting equations is found by the inverse transformation.
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A version of the Dirac equation is derived from first principles using a combination of quaternions and multivariate 4-vectors. The nilpotent form of the operators used allows us to derive explicit expressions for the wavefunctions of free fermions, vector bosons, scalar bosons; Bose-Einstein condensates, and baryons;annihilation, creation and vacuum operators; the quantum field integrals; and C, P and T transformations;
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We point out that the anticommutation properties of the Dirac matrices can be derived without squaring the Dirac hamiltonian, that is, without any explicit reference to the Klein-Gordon equation. We only require the Dirac equation to admit two linearly independent plane wave solutions with positive energy for all momenta. The necessity of negative energies as well as the trace and determinant properties of the Dirac matrices are also a direct consequence of this simple and minimal requirement.
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The Clifford algebra of physical space and Dirac theory
European Journal of Physics, 2016
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term by in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.