In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non-zero mass and 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.
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.
[2] Wigner notes Ettore Majorana and Dirac used infinitesimal operators applied to functions.
Explicitly, in the Dirac representation of the gamma matrices:[1] where σ = (σ1, σ2, σ3) = (σx, σy, σz) is a vector of the Pauli matrices, E is the energy operator, p = (p1, p2, p3) = (px, py, pz) is the 3-momentum operator, I2 denotes the 2 × 2 identity matrix, the zeros (in the second line) are actually 2 × 2 blocks of zero matrices.
The above matrix operator contracts with one bispinor index of ψ at a time (see matrix multiplication), so some properties of the Dirac equation also apply to the BW equations: Unlike the Dirac equation, which can incorporate the electromagnetic field via minimal coupling, the B–W formalism comprises intrinsic contradictions and difficulties when the electromagnetic field interaction is incorporated.
[6][7] An indirect approach to investigate electromagnetic influences of the particle is to derive the electromagnetic four-currents and multipole moments for the particle, rather than include the interactions in the wave equations themselves.
One may perform a Clebsch–Gordan decomposition to find the irreducible (A, B) terms and hence the spin content.
This redundancy necessitates that a particle of definite spin j that transforms under the DBW representation satisfies field equations.
In curved spacetime they are similar: where the spatial gamma matrices are contracted with the vierbein biμ to obtain γμ = biμ γi, and gμν = biμbiν is the metric tensor.
A covariant derivative for spinors is given by with the connection Ω given in terms of the spin connection ω by: The covariant derivative transforms like ψ: With this setup, equation (1) becomes: Relativistic wave equations: Lorentz groups in relativistic quantum physics: