Bargmann–Wigner equations

In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles of arbitrary spin j, an integer for bosons (j = 1, 2, 3 ...) or half-integer for fermions (j =  ¹⁄₂,  ³⁄₂,  ⁵⁄₂ ...). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields.

They are named after Valentine Bargmann and Eugene Wigner.

Paul Dirac first published the Dirac equation in 1928, and later (1936) extended it to particles of any half-integer spin before Fierz and Pauli subsequently found the same equations in 1939, and about a decade before Bargman, and Wigner.Eugene Wigner wrote a paper in 1937 about unitary representations of the inhomogeneous Lorentz group, or the Poincaré group. Wigner notes Ettore Majorana and Dirac used infinitesimal operators applied to functions. Wigner classifies representations as irreducible, factorial, and unitary.

In 1948 Valentine Bargmann and Wigner published the equations now named after them in a paper on a group theoretical discussion of relativistic wave equations.

For a free particle of spin j without electric charge, the BW equations are a set of 2j coupled linear partial differential equations, each with a similar mathematical form to the Dirac equation. The full set of equations are

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