User:Maschen/Bargmann-Wigner equations

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

Heuristic construction from Dirac equation

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

$\psi _{\alpha }^{(+)}=\omega _{\alpha }^{(+)}(0)e^{-iE_{0}t/\hbar }\,,\quad \psi _{\alpha }^{(-)}=\omega _{\alpha }^{(-)}(0)e^{+iE_{0}t/\hbar }$

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 ω_α is the Kronecker delta, (ω_α)_β = δ_αβ. The spinors ω_α(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 ω_α⁽⁺⁾ 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 ω_11...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. ω_21...1, ω_12...1, ..., ω_12...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. ω_22...1, ω_21...2...1, ..., ω_1...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

$\Sigma _{\alpha _{1}{\bar {\alpha }}_{1}\alpha _{2}{\bar {\alpha }}_{2}\ldots \alpha _{2j}{\bar {\alpha }}_{2j}}^{z}={\frac {\hbar }{2}}\left(\Sigma _{\alpha _{1}{\bar {\alpha }}_{1}}^{z}\delta _{\alpha _{2}{\bar {\alpha }}_{2}}\ldots \delta _{\alpha _{2j}{\bar {\alpha }}_{2j}}+\delta _{\alpha _{1}{\bar {\alpha }}_{1}}\Sigma _{\alpha _{2}{\bar {\alpha }}_{2}}^{z}\ldots \delta _{\alpha _{2j}{\bar {\alpha }}_{2j}}+\cdots +\delta _{\alpha _{1}{\bar {\alpha }}_{1}}\delta _{\alpha _{2}{\bar {\alpha }}_{2}}\ldots \Sigma _{\alpha _{2j}{\bar {\alpha }}_{2j}}^{z}\right)$

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 σ_z 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 Σ^z, 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

$(-{\hat {E}}+mc)^{2j}\psi _{1,2}^{[2j]}=(-{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }})^{[2j]}\psi _{3,4}^{[2j]}$

(2A)

$({\hat {E}}+mc)^{2j}\psi _{3,4}^{[2j]}=({\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }})^{[2j]}\psi _{1,2}^{[2j]}$

(2B)

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:

$[(E^{2}-(mc^{2})^{2})^{2j}-({\hat {\mathbf {p} }}^{2})^{2j}]{\begin{pmatrix}\psi _{1,2}^{[2j]}\\\psi _{3,4}^{[2j]}\end{pmatrix}}=0$

(3)

which upon expanding by the binomial theorem, then factorizing;

$[(E^{2}-(mc^{2})^{2})-({\hat {\mathbf {p} }}^{2})]\left((E^{2}-(mc^{2})^{2})^{2j-1}+(E^{2}-(mc^{2})^{2})^{2j-2}{\hat {\mathbf {p} }}^{2}+\cdots ({\hat {\mathbf {p} }}^{2})^{2j-1}\right){\begin{pmatrix}\psi _{1,2}^{[2j]}\\\psi _{3,4}^{[2j]}\end{pmatrix}}=0$

(4)

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:

${\begin{pmatrix}a&c\\b&d\\\end{pmatrix}}{\begin{pmatrix}\psi _{1}\\\psi _{2}\\\end{pmatrix}}={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}$ (5)

that is:

${\begin{aligned}a\psi _{1}+c\psi _{2}&=\chi _{1},\\b\psi _{1}+d\psi _{2}&=\chi _{2}.\end{aligned}}$

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 ψ₁^{2j − (k + l)}ψ₂^{k + l} for (k + l) = 0, 1, 2, ..., 2j in one column vector, and all the products for χ₁^{j + m}χ₂^{j − 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 Ψ_n 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 λ₁^{j + m}λ₂^{j − m} for the same m values as above.

Diagonalization

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

Use in the BW formalism

In the above equations (2A), (2B):

$({\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }})^{[2j]}=(i|{\hat {\mathbf {p} }}|)^{2j}e^{-i\pi \mathbf {J} ^{(j)}\cdot \mathbf {n} }$ (6)

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:

${({\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }})_{mm'}^{[2j]}=(-1)^{m'-m}\sum _{r=-\infty }^{\infty }{\frac {(-1)^{r}p_{-}^{j}(-{\hat {p}}_{z})^{j-m'-r}{\hat {p}}_{z}^{j+m-r}(-p_{+})^{m'-m+r}}{r!(j-m'-r)!(j+m-r)!(m'-m+r)!}}{\sqrt {(j+m)!(j-m)!(j+m')!(j-m')!}}}$ (7)

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:

	+|p|[2j]	−|p|[2j]
Integer spin	(j + 1)-fold	j-fold
Half-integer spin	(j + ½)-fold	(j + ½)-fold