User:Maschen/Bargmann-Wigner equations

Following Griener's RQM (only have first edition)...

Multicomponent spinors in the particle's rest frame
The solutions of the Dirac equation spinorial wave functions, whose components are usually arranged into column vectors


 * $$ \psi = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix} \,, \quad \psi^{(+)} = \begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} \,, \quad \psi^{(-)} = \begin{pmatrix} \psi_3 \\ \psi_4 \end{pmatrix} $$

The superscript (+) corresponds to (upper) components of the "positive energy" wave function (α = 1, 2), similarly the superscript (−) corresponds to (lower) components of the "negative energy" wave function (α = 3, 4). The plane wave solutions to the massive free particle Dirac equation in the rest frame of the (spin-1/2) particle take the form


 * $$ \psi_\alpha^{(+)} = \omega^{(+)}_\alpha (0) e^{-i E_0 t /\hbar} \,,\quad \psi_\alpha^{(-)} = \omega^{(-)}_\alpha (0) e^{+i E_0 t /\hbar} $$

where the index α takes the values 1, 2, 3, 4, the rest energy of the particle is E0 = m0c2, and


 * $$ \omega_1(0) = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \,,\quad \omega_2(0) = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} \,,\quad \omega_3(0) = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \,,\quad \omega_4(0) = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} $$

are constant spinors. Notice the β entry of ωα is the Kronecker delta, (ωα)β = δαβ. The spinors ωα(p) for a boosted particle will be obtained from an appropriate transformation later.

A positive energy wave function for a spin-j particle can be obtained by the tensor product of ωα(+) with itself (2j + 1) times, motivated by the fact that given (2j + 1) subsystems of spin-1/2 particles, their composite system is (2j + 1) spin-1/2 particles with the total spin of the system taking the spin projection values m = j, j − 1, ..., −j + 1, −j. Explicitly (suppressing momentum arguments of ω)


 * $$ \omega_{\alpha_1\alpha_2 \ldots \alpha_{2j} }^{(+)} = \omega_{\alpha_1}^{(+)} \omega_{\alpha_2}^{(+)} \ldots \omega_{\alpha_{2j}}^{(+)} $$

Since the same spinor is tensor multiplied by itself (2j + 1) times, it is a symmetric spinor in all indices, and a lot of components will coincide. For example,


 * There is one component ω11...1, corresponding to a spin projection m = j,
 * There are (2j)!/(2j − 1)!1! = 2j components with (2j − 1) indices taking the value 1 and one index taking the value 2, e.g. ω21...1, ω12...1, ..., ω12...1, corresponding to a spin projection m = j − 1,
 * There are (2j)!/(2j − 2)!2! = (2j − 1)j components with (2j − 2) indices taking the value 1 and two indices taking the value 2, e.g. ω22...1, ω21...2...1, ..., ω1...22, corresponding to a spin projection m = j − 2,

In all there are 2j + 1 independent components of ω.

In general, the number of ways of choosing k indices out of 2j to be 2 (the rest are 1) is given by the binomial coefficient (2j)!/(2j − k)!k!. It will be useful to use the spin projection quantum number m corresponding to spin j by taking (j + m) indices to take the value 1 and k = (j − m) indices to take the value 2, and enumerate the independent components by m as follows:


 * $$\omega^{(+)}(0,m) = \omega^{(+)}_{ \underbrace{11\cdots 1}_{j+m} \underbrace{22 \cdots 2}_{j-m}}(0)$$

Spin operator for multicomponent spinors (z direction only)
The z-component spin operator is


 * $$\Sigma^z_{\alpha_1\bar{\alpha}_1 \alpha_2\bar{\alpha}_2 \ldots \alpha_{2j}\bar{\alpha}_{2j}} = \frac{\hbar}{2} \left( \Sigma^z_{\alpha_1\bar{\alpha}_1} \delta_{\alpha_2\bar{\alpha}_2} \ldots \delta_{\alpha_{2j}\bar{\alpha}_{2j}} + \delta_{\alpha_1\bar{\alpha}_1} \Sigma^z_{\alpha_2\bar{\alpha}_2} \ldots \delta_{\alpha_{2j}\bar{\alpha}_{2j}} + \cdots + \delta_{\alpha_1\bar{\alpha}_1} \delta_{\alpha_2\bar{\alpha}_2} \ldots \Sigma^z_{\alpha_{2j}\bar{\alpha}_{2j}} \right) $$

where ħ is Planck's reduced constant, all indices each take values 1, 2, 3, 4, and


 * $$\Sigma^z = \begin{pmatrix} \sigma_z & 0 \\ 0 & \sigma_z \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} $$

in which σz is the z-component (or third) Pauli matrix.

One finds


 * $$\Sigma^z \omega^{(+)}(0,m) = \hbar m \omega^{(+)}(0,m) \,, $$

in words, the ω(+)(0, m) is an eigenvector of the z-component spin Σz, with corresponding eigenvalue ħm, exactly as it should be. This applies for all allowed values of m.

The "negative energy" solutions ω(−)(0, m) are similarly dealt with as with positive solutions. The quotation marks indicate there is no real negative energy involved. The modern interpretation is that the "negative energy" wavefunctions correspond to an antiparticle with positive energy.

Multicomponent spinors in a boosted frame
Assuming standard configuration, transform the ω(+) (and separately ω(−)) to a boosted frame with velocity −v along the negative x directions of the frames (this corresponds to relative velocity v in the original frame in the previous section).

Induced matrices
Separating the components of $ψ$, BW equations can be written as a set of coupled equations

where the notation $[2j]$ denotes the $2j$ induced spinor or matrix, and


 * $$ \Psi =

\begin{pmatrix} \psi_{1,2} \\ \psi_{3,4} \\ \end{pmatrix}\,\quad \psi_{1,2} = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \end{pmatrix}\,\quad \psi_{3,4} = \begin{pmatrix} \psi_3 \\ \psi_4 \\ \end{pmatrix}\,.$$

One 2-component spinor $ψ_{1,2}$ describes the spin-1/2 fermion, the other $ψ_{3,4}$ describes the antifermion.

Each of $ψ_{1,2}$ and $ψ_{3,4}$ 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.

Definition
The induced matrices arise from the spinor transformation:

that is:


 * $$\begin{align}

a\psi_1 + c\psi_2 &= \chi_1, \\ b\psi_1 + d\psi_2 &= \chi_2. \end{align}$$

The $2j$ induced matrix arises by expanding:


 * $$(a\psi_1 + c\psi_2)^{j+m}(b\psi_1 + d\psi_2)^{j-m} = \chi_1^{j+m}\chi_2^{j-m}\,,$$

for $m = −j, −j + 1, ... j − 1, j$, simplifying, then writing the set of equations in matrix form.

Expanding by the binomial theorem


 * $$\sum_{k=0}^{j+m} \binom{j+m}{k} (a\psi_1)^{j+m-k}(c\psi_2)^k \sum_{\ell=0}^{j-m} \binom{j-m}{\ell} (b\psi_1)^{j-m-\ell}(d\psi_2)^\ell = \chi_1^{j+m}\chi_2^{j-m}\,,$$


 * $$\sum_{k=0}^{j+m} \sum_{\ell=0}^{j-m} \frac{(j+m)!}{(j+m-k)!k!} \frac{(j-m)!}{(j-m-\ell)!\ell!} a^{j+m-k} b^{j-m-\ell} c^k d^\ell \psi_1^{2j-(k+\ell)} \psi_2^{k+\ell} = \chi_1^{j+m}\chi_2^{j-m}\,,$$

then listing all the products for ψ12j − (k + l)ψ2k + l for (k + l) = 0, 1, 2, ..., 2j in one column vector, and all the products for χ1j + mχ2j − m for m = j, j − 1, ..., −j + 1, −j in one column vector,


 * $$\begin{pmatrix}

. & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \end{pmatrix}\begin{pmatrix} \psi_1^{2j} \\ \psi_1^{2j-1}\psi_2 \\ \psi_1^{2j-2}\psi_2^2 \\ \vdots \\ \psi_1^2\psi_2^{2j-2} \\ \psi_1\psi_2^{2j-1} \\ \psi_2^{2j} \\ \end{pmatrix}=\begin{pmatrix} \chi_1^{2j} \\ \chi_1^{2j-1}\chi_2 \\ \chi_1^{2j-2}\chi_2^2 \\ \vdots \\ \chi_1^2\chi_2^{2j-2} \\ \chi_1\chi_2^{2j-1} \\ \chi_2^{2j} \\ \end{pmatrix}$$

defining


 * $$\Psi_{k+\ell} = \psi_1^{2j-(k+\ell)} \psi_2^{k+\ell} \,,\quad X_m = \chi_1^{j+m}\chi_2^{j-m}\,,$$


 * $$C_{mk\ell} = \frac{(j+m)!}{(j+m-k)!k!} \frac{(j-m)!}{(j-m-\ell)!\ell!} a^{j+m-k} b^{j-m-\ell} c^k d^\ell \,,$$

the formula is


 * $$\sum_{k=0}^{j+m} \sum_{\ell=0}^{j-m} C_{mk\ell} \Psi_{k+\ell} = X_m \,,$$

For k + l = n and keeping n fixed, the coefficient for Ψn is


 * $$\sum_{\ell+k=n} C_{mk\ell} \,,$$

Along the rows of the square matrix, n = k + l varies from 0 to 2j, and m is constant. Down each row, m decreases from j in the first row to −j in the last, and n = k + l is constant. The matrix elements are


 * $$M_{mn} = \sum_{\ell+k=n} C_{mk\ell} \,,$$

and the equation finally takes the compact form


 * $$\sum_n M_{mn} \Psi_n = X_m \,,$$

(Jeffery's paper has different normalizations in the column vectors Ψ and X).

Properties
Two reasons for introducing the induced matrices is the simple correspondence between induced matrices and powers of eigenvalues, and ease of diagonalization.


 * Eigenvalues

If $A$ is a 2 × 2 matrix, the 2j induced matrix $A^{[2j]}$ has eigenvalues $λ_{1}^{j + m}λ_{2}^{j − m}$ for the same $m$ values as above.


 * Diagonalization

If the transformation $A → B^{−1}AB$ holds, then $B^{[2j]}$ will diagonalize $A^{[2j]}$.

Use in the BW formalism
In the above equations ($$), ($$):

where matrix indices on the left side are understood to be $m, m&prime; = −j, −j + 1 ... j$. The $mm&prime;$ element of the $(2j + 1) × (2j + 1)$ matrix contains the energy–momentum operators and are given by:

where $n = p/|p|$ is a unit vector and $J^{(j)} = (J^{(j)}_{1}, J^{(j)}_{2}, J^{(j)}_{3})$ is the vector of the Pauli matrices for spin s.

The matrix $(σ • p)^{[2j]}$ has eigenvalues $±|p|^{2j}$. The degeneracy of the eigenvalues are as follows:


 * {| class="wikitable"

! $+&#124;p&#124;^{[2j]}$ ! $−&#124;p&#124;^{[2j]}$ ! Integer spin ! Half-integer spin
 * $(j + 1)$-fold
 * $j$-fold
 * $(j + ½)$-fold
 * $(j + ½)$-fold
 * }
 * }