- 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 (or BW equations or BWE) are relativistic wave equations which 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. The spin quantum number is usually denoted by s in quantum mechanics, however in this context j is more typical in the literature (see references).
They were proposed by Valentine Bargmann and Eugene Wigner in 1948, using Lorentz group theory, and building on the work of those who pioneered quantum theory within the first half of the twentieth century.
- 1 Origin from the Dirac equation
- 2 BW equations
- 3 Induced matrices
- 4 Lorentz group structure
- 5 Lagrangian
- 6 Formulation in curved spacetime
- 7 See also
- 8 References
- 9 External links
Origin from the Dirac equation
For reference, the Dirac equation is summarized below. It is the basis for building relativistic wave equations with wavefunctions of higher spin.
where Ψ = Ψ(r, t) is a rank-1 4-component spinor field, a function of the particle's position r and time t, with components ψα = ψα(r, t) in which α is a bispinor index that takes values 1, 2, 3, 4. Further, γμ = (γ0, γ) are the gamma matrices, and
is the 4-momentum operator. The operator constituting the entire 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 Dirac equation (1) can be written as a coupled set of equations:
Uncharged massive particles
which follow the pattern;
for r = 1, 2, ... 2j. Again, the operator (−γμPμ + mc) is a 4 × 4 matrix. The wavefunction Ψ = Ψ(r, t) has components
and is now a rank-2j 4-component spinor field, usually symmetric in all bispinor indices, but not necessarily; for example, the spin-0 case is antisymmetric. 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).
The above matrix operator contracts with one bispinor index of Ψ at a time (analogous but not equivalent to matrix multiplication), so some properties of the Dirac equation also apply to the BW equations:
- the equations are Lorentz covariant,
- all components of the solutions Ψ also satisfy the Klein–Gordon equation, and hence fulfil the relativistic energy–momentum relation,
- second quantization is still possible, but the equations become much more complicated, methods of propagators and S-matrices have been developed, without using a Lagrangian (see below).
The components for a totally symmetric wavefunction are explicitly:
where the indices are selected so that: α2 ≤ α3 ≤ ... α2j.
Unlike the Dirac equation, which can incorporate the electromagnetic field via minimal coupling (1C), 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. 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.
where the notation [2j] denotes the 2j induced spinor or matrix (defined in the next section). Each of ψ1,2 and ψ3,4 has 2j + 1 independent components.
These can be recombined:
which upon expanding by the binomial theorem, then factorizing;
shows that each component of the BW wavefunction also satisfies the Klein–Gordon equation, uniquely. Conversely, the solutions to the Klein–Gordon equation satisfy the BW equations but are not unique.
Modified gamma matrices
for r = 1, 2 ... 2j, these equations (2) can also be written:
The γrμ matrices have dimension 42j × 42j. The equations are linear, so adding (3) with respect to the r values gives:
where the factor of 1/2j is inserted because the matrix elements ±1, ±i are added 2j times. Subtracting (3), one r from the next r + 1; the wavefunction satisfies:
for r = 1, 2 ... 2j − 1.
Introducing a 2(2j + 1) × 2(2j + 1) matrix;
The induced matrices arise from the spinor transformation:
The 2j induced matrix arises by expanding:
for m = −j, −j + 1, ... j − 1, j, simplifying, then writing the set of equations in matrix form.
Two reasons for introducing the induced matrices is the simple correspondence between induced matrices and powers of eigenvalues, and ease of diagonalization.
If A is a 2 × 2 matrix, the 2j induced matrix A[2j] has eigenvalues λ1j + mλ2j − m for the same m values as above.
If the transformation A → B−1AB holds, then B[2j] will diagonalize A[2j].
Use in the BW formalism
where matrix indices on the left side are understood to be m, m′ = −j, −j + 1 ... j. The mm′ element of the (2j + 1) × (2j + 1) matrix contains the energy–momentum operators and are given by:
The matrix (σ • p)[2j] has eigenvalues ±|p|2j. The degeneracy of the eigenvalues are as follows:
+|p|[2j] −|p|[2j] Integer spin (j + 1)-fold j-fold Half-integer spin (j + ½)-fold (j + ½)-fold
Lorentz group structure
Under a proper orthochronous Lorentz transformation x → Λx in Minkowski space, all one-particle quantum states ψjσ of spin j with spin z-component σ locally transform under some representation D of the Lorentz group:
where D(Λ) is some finite-dimensional representation, i.e. a matrix. Here ψ is thought of as a column vector containing components with the allowed values of σ. The quantum numbers j and σ as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of σ may occur more than once depending on the representation. Representations with several possible values for j are considered below.
The irreducible representations are labeled by a pair of half-integers or integers (A, B). From these all other representations can be built up using a variety of standard methods, like taking tensor products and direct sums. In particular, space-time itself constitutes a 4-vector representation (1/, 1/) so that Λ ∈ D'(1/2, 1/2). To put this into context; Dirac spinors transform under the (1/, 0) ⊕ (0, 1/) representation. In general, the (A, B) representation space has subspaces that under the subgroup of spatial rotations, SO(3), transform irreducibly like objects of spin j, where each allowed value:
occurs exactly once. In general, tensor products of irreducible representations are reducible; they decompose as direct sums of irreducible representations.
The representation for the BW equations is the choice:
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.
For the JW equations the choice is:
This representation has definite spin j. It turns out that a spin j particle in this representation satisfy field equations too. These equations are very much like the Dirac equations. It is suitable when the symmetries of charge conjugation, time reversal symmetry, and parity are good.
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.
Lorentz covariant tensor description of Weinberg-Joos states
The six-component spin-1 representation space, DJW=D(1,0)⊕D(0,1) can be labeled by a pair of anti-symmetric Lorentz indexes, [α,β], meaning that it transforms as an antisymmetric Lorentz tensor of second rank B[α,β], i.e.
The j-fold Kronecker product T[α1,β1]...[αj,βj] of B[α,β]
decomposes into a finite series of Lorentz irreducible representation spaces according to,
and necessarily contains a D(j,0)⊕D(0,j) sector. This sector can instantly be identified by means of a momentum independent projector operator P(j,0), designed on the basis of C(1), one of the Casimir elements (invariants) of the Lie algebra of the Lorentz group, which are defined as,
where Mμν are constant quadratic (2j1+1)(2j2+1) × (2j1+1)(2j2+1) matrices defining the elements of the Lorentz algebra within the D(j1,j2)⊕D(j2,j1) representations. The Capital Latin letter labels indicate  the finite dimensionality of the representation spaces under consideration which describe the internal angular momentum (spin) degrees of freedom.
The representation spaces D(j1,j2)⊕D(j2,j1) are eigenvectors to C(1) in (8B) according to,
In terms of C(1), the projector operator, P(j,0), is defined as 
Here, Πkl denotes the operation of successive multiplication, λ(1)(j,0) is the C(1) eigenvalues of the D(j,0)⊕D(0,j) sector, while λ(1)(jk,jl) are the C(1) eigenvalues of all the remaining (dummy) sectors, (jk,jl) ⊕ (jl,jk), of the hosting tensor in (8A). Such projectors can be employed to search through T[α1,β1]...[αj,βj] for D(j,0)⊕D(0,j), and exclude all the rest. Relativistic second order wave equations for any-j are then straightforwardly obtained in first identifying the D(j,0)⊕D(0,j) sector in T[α1,β1]...[αj,βj] in (8A) by means of the Lorentz projector in (8C) and then imposing on the result the mass shell condition.
This algorithm is free from auxiliary conditions. The scheme also extends to half-integer spins, s=j+1/2 in which case the Kronecker product of T[α1,β1]...[αj,βj] with the Dirac spinor, D(1/2,0)⊕D(0,1/2) , has to be considered. The choice of the totally antisymmetric Lorentz tensor of second rank, B[αi,βi], in the above equation (8A) is only optional. It is possible to start with multiple Kronecker products of totally symmetric second rank Lorentz tensors, Aαi,βi. The latter option should be of interest in theories where high-spin D(j,0)⊕D(0,j) Joos-Weinberg fields preferably couple to symmetric tensors, such as the metric tensor in gravity.
An Example.  The (3/2,0)⊕(0,3/2) transforming in the Lorenz tensor spinor of second rank, ψ [μν]=[(1,0)⊕(0,1)]⊗[(1/2,0)⊕(0,1/2)].
The Lorentz group generators within this representation space, denoted by [MATSμν][α β][γ δ], are
where 1[αβ][γδ] stands for the identity in this space, 1S and MSμν are the respective unit operator and the Lorentz algebra elements within the Dirac space, while γμ are the standard gamma matrices . The [MATμν][αβ][γδ] generators express in terms of the generators in the four-vector,
Then, the explicit expression for the Casimir invariant C(1) in (8B) takes the form,
and the Lorentz projector on (3/2,0)⊕(0,3/2) is given by,
In effect, the (3/2,0)⊕(0,3/2) degrees of freedom, denoted by
are found to solve the following second order equation,
Expressions for the solutions can be found in.
One method proposed by Kamefuchi and Takahashi in 1966 was to expand the wavefunctions in terms of 4 × 4 matrices with a required symmetry (conserved properties of the quantum system), then substitute back into the BW equations to yield field equations with that symmetry. From then a Lagrangian can be found by working backwards from the Euler–Lagrange field equations.
D.S. Kaparulin, S.L. Lyakhovich, and A.A. Sharapov take this fundamental approach by starting from symmetries directly, by means of a Poincaré invariant Lagrange anchor. A Lagrange anchor geometrically defines a mapping between fiber bundles, comprising vector bundles, tangent bundles, and the configuration space for the quantum fields. This is less restrictive than a variational formulation (based on the principle of least action) to obtain the equations for the quantum fields.
Formulation in curved spacetime
where ηij = diag(−1, 1, 1, 1) is the Minkowski metric. For the Latin indices here, i, j = 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 (2) becomes:
- Two-body Dirac equation
- Generalizations of Pauli matrices
- Wigner D-matrix
- Weyl–Brauer matrices
- Bargmann–Michel–Telegdi equation
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