- This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators.
|Quantum field theory|
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.
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.
Statement of the 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
which follow the pattern;
for r = 1, 2, ... 2j. (Some authors e.g. Loide and Saar use n = 2j to remove factors of 2. Also the spin quantum number is usually denoted by s in quantum mechanics, however in this context j is more typical in the literature). The entire wavefunction ψ = ψ(r, t) has components
and is a rank-2j 4-component spinor field. Each index takes the values 1, 2, 3, or 4, so there are 42j components of the entire spinor field ψ, although a completely symmetric wavefunction reduces the number of independent components to 2(2j + 1). Further, γμ = (γ0, γ) are the gamma matrices, and
is the 4-momentum operator.
The operator constituting each equation, (−γμPμ + mc) = (−iħγμ∂μ + mc), is a 4 × 4 matrix, because of the γμ matrices, and the mc term scalar-multiplies the 4 × 4 identity matrix (usually not written for simplicity). Explicitly, in the Dirac representation of the gamma matrices:
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 equations are Lorentz covariant,
- all components of the solutions ψ also satisfy the Klein–Gordon equation, and hence fulfill the relativistic energy–momentum relation,
- second quantization is still possible.
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. In other words, it is not possible to make the change Pμ → Pμ − eAμ, where e is the electric charge of the particle and Aμ = (A0, A) is the electromagnetic four-potential. An indirect approach to investigate electromagnetic influences of the particle is to derive the electromagnetic four-currents currents and multipole moments for the particle, rather than include the interactions in the wave equations themselves.
Lorentz group structure
where each Dr is an irreducible representation. This representation does not have definite spin unless j equals 1/2 or 0. 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.
The representations D(j, 0) and D(0, j) can each separately represent particles of spin j. A state or quantum field in such a representation would satisfy no field equation except the Klein-Gordon equation.
Formulation in curved spacetime
where ηij = diag(−1, 1, 1, 1) is the Minkowski metric. For the Latin indices here, i, j = 0, 1, 2, 3. In curved spacetime they are similar:
A covariant derivative for spinors is given by
The covariant derivative transforms like ψ:
With this setup, equation (1) becomes:
- Two-body Dirac equation
- Generalizations of Pauli matrices
- Wigner D-matrix
- Weyl–Brauer matrices
- Higher-dimensional gamma matrices
- Joos–Weinberg equation, alternative equations which describe free particles of any spin
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