Observations On Chiral Symmetries Of Clifford Algebra

[Dubbelosix]

 

Using two new mathematical tools [math](\gamma_3,\gamma^3,A_i, A^i)[/math] you can find an internal symmetry between two spin states, which is called their Chirality given by [math]\gamma^5[/math]. The relationship to the Chirality of a particle was simply [math]\gamma_3 \mathbf{A}^i[/math] or [math]\gamma^3 \mathbf{A}_i[/math] but they became the definition of the four common gamma matrices with the use of a unit psuedoscalar [math]i[/math]. 

 

 

If you hit this with the four common gamma matrices which give the Dirac basis using a unit pseudoscalar, [math]i \gamma^0 \gamma^1 \gamma^2 \gamma^3[/math] yields the appropriate spin designated from a pilot state. Chirality is provided through the Dirac base 

 

[math]i \gamma^0 \gamma^1 \gamma^2 \gamma^3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix}[/math]

 

The gamma^3 and gamma_3 matrices I use symbolize a flip in the spin sign [math](\gamma^3, \gamma_3)[/math] - the is achieved through two new matrices:

 

 

[math]\mathbf{A}^i = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix}[/math] 

 

 

and 

 

 

[math]\mathbf{A}_i = \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/math]

 

 

So in this work, when you see an index change, we use it in the sense that it has changed the signs around; same goes for the notation that follows in the work. This is different to normal index changes in the gamma matrices through the use of contravariant or covariant metric (ie [math]\gamma_i = g_{ij} \gamma^i[/math]). We have implied a new kind of rule and we'll see how it generates symmetries in the matrices. Lubos Motl once explained that Of course that we may define γ^5 and γ_5 to be either the same thing or the same thing with the opposite sign, to suit any conventions

 

 

To show exactly how it does this, you can see the diagonal matrices acting on the gamma(3) matrices yields the spin up or spin down solutions to the gamma(5) chirality matrices:

 

 

[math]\gamma_3 \mathbf{A}^i = i\gamma^0 \gamma^1 \gamma^2 \gamma^3 = \gamma^{5}[/math]

 

[math]\gamma^3 \mathbf{A}_i = i\gamma_0 \gamma_1 \gamma_2 \gamma_3 = \gamma_{5}[/math]

 

 

and the two corresponding anticommutator relationships are

 

 

[math]<\ \gamma_3 \mathbf{A}^i, \gamma^{5}\ > = (\gamma^3 \mathbf{A}^i) \gamma^{5} + \gamma^{5}(\gamma^3\mathbf{A}^i)[/math]

 

[math]<\ \gamma^3 \mathbf{A}_i, \gamma_{5}\ > = (\gamma_3 \mathbf{A}_i) \gamma_{5} + \gamma_{5}(\gamma_3\mathbf{A}_i)[/math]

 

 

All the common gamma matrices share properties together, here is an example:

 

 

[math]\gamma^{3} \gamma^1 \gamma^0 = \begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1\\1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} = \gamma_{3}[/math]

 

 

And the product matrix of [math]\gamma^3\gamma_3[/math] is Hermitian even in using [math]\mathbf{A}^i\mathbf{A}_i[/math] which tended towards a poitive diagonal matrix. For those involving the Chirality formula re-written for our new hidden variable matrices

 

Another identity that has been noted is:

 

[math]\gamma^3\gamma_3 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i} \cdot i \gamma_0 \gamma_1 \gamma_2 \gamma_3 \mathbf{A}_{i} = \mathbf{I}_4[/math]

 

 

 

When you separate the left handedness from the right handedness in the equations above you find it's Eigenvalues satisfying [math]\pm 1[/math] because of [math](i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i})^2 = \mathbf{I}_4[/math] This is the same as saying [math](\gamma^5)^2 = \mathbf{I}_4[/math], The term [math]i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i}[/math] should also anticommute with the four gamma matrices

 

[math]<\i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i}, \gamma^{\mu}> = (i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i}) \gamma^{\mu}  + \gamma^{\mu} (i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i})[/math]

 

 

Take the superpositioning state now

 

[math]\gamma_3\mathbf{A}^i + \gamma^3\mathbf{A}_i[/math]

 

This is a superpositioning of a chirality using

 

[math]\gamma_3 \mathbf{A}^i = i\gamma^0 \gamma^1 \gamma^2 \gamma^3 = \gamma^{5}[/math]

 

[math]\gamma^3 \mathbf{A}_i = i\gamma_0 \gamma_1 \gamma_2 \gamma_3 = \gamma_{5}[/math]

 

or simply as

 

[math]\gamma_3 \mathbf{A}^i + \gamma^3 \mathbf{A}_i = \gamma^5 + \gamma_5[/math]

 

In full matrix form for the entries is just

 

[math]\begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/math]

 

 

 

CONCLUSION

 

The use of [math](A_i,A^i)[/math] are new matrices which are not covered by the many matrices of Dirac/Clifford algebra. I consider the possibility that they may play a fundamental role in the objective reality of spin, with two solutions of spin up and spin down. If so, they might be a type of hidden variable, in which it the two matrices dictate whether a system should have a spin up or spin down. 

Edited April 5, 2017 by Dubbelosix