Clifford Algebra and generalizing Dirac equaution

[Traruh Synred]

http://arxiv.org/abs/1001.2485
---
The above paper is about a possible two-time formulation of physics. It is by serious people.

To understand it I'm trying to generalized the Dirac eqn. to 3+2 dimensions with signature (++---)

I found the following (now closed post) useful:
https://www.physicsforums.com/threa...f-dirac-spinor-in-arbitrary-dimensions.244296

Apparently the Dirac matrices would still be 4x4 in 5 dimensions.

Is this the case?

I also guess that there's not another linear combination that could be used to distinguish the two times, so you don't get cross terms when the you square up the Dirac eqn. to find the space-timel part of the plane wave solution.

With 4x4 matrices obeying Dirac orthogonality condition

(g_a*g_b+g_b*g-a=g_ab*delta_b=0)

Are there more than 4 independent matrices? I don't think I can use gamma_5, it already got a role.

 

[jambaugh]

There's a book by Porteus "Clifford Algebras and the Classical Groups" which gives a full account of the mathematics of the clifford algebras. Their matrix representations forms a periodic structure based on the metric signature and dimension.

Let's see: for SO(3,2) you get that the algebra is isomorphic to either [itex]\mathbb{C}(4)[/itex] the algebra of 2x2 complex matrices, or [itex]\mathbb{R}(4)\oplus\mathbb{R}(4)[/itex]. Which depends on whether your grade one elements are three roots of -1 and two of +2 or vice versa.

Now these clifford algebras define the "pinor" group for the full O(3,2) representation. Restricting to the spin group rep, SO(3,2) you would get the subalgebra [itex]\mathbb{R}(4)[/itex] of 4x4 Real matrices. That's the algebra that expresses only the even grade elements of the full clifford algebra.

To construct a basis there's a straightforward way to build up the algebra using the usual pauli matrices for [itex]\mathbb{C}(2)[/itex]. Let
[tex] \sigma_1=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right),\quad \sigma_3=\left(\begin{array}{cc} 1 & 0 \\ 0& -1\end{array}\right)[/tex]
and then [itex]\sigma_3\sigma_1 =\mathbf{J}= i\sigma_2 = -\sigma_1\sigma_3[/itex]. These three form a set of three anti-commuting real matrices. (We want to be conscious of whether we use real or imaginary entries.)

Let's pick our five matrices so that [itex]\gamma_1^2 = \gamma_2^2 = 1, \quad \gamma_3^2 = \gamma_4^2 = \gamma_5^2 = -1[/itex].
We build up using tensor products noting that a tensor product of two 2x2 matrices is a 4x4 matrix.

[itex] \gamma_1 = \sigma_1\otimes \mathbf{1}[/itex]
[itex] \gamma_2 = \sigma_3 \otimes \mathbf{1}[/itex]
[itex] \gamma_3 = \mathbf{J}\otimes\sigma_1[/itex]
[itex] \gamma_4 = \mathbf{J}\otimes\sigma_3[/itex]
[itex] \gamma_5 = i\mathbf{J}\otimes \mathbf{J}[/itex]

One begins with the identity for all factors to the right, then use two of the three anti-commuting generators in turn. Then fix at the third generator as one moves to the next factor. If needed you can use the third generator for the very last gamma matrix. Here I inserted an imaginary unit to keep the signature so [itex]\gamma_5^=-1[/itex] as planned. Had we chosen the opposite signature we could drop this [itex]i[/itex] and have all real matrices. We also want to focus on the top element [itex]\gamma_{[12345]}=\gamma_1\gamma_2\gamma_3\gamma_4\gamma_5[/itex].

[tex]\gamma_1\gamma_2 = -\mathbf{J}\otimes \mathbf{1},\quad \gamma_1\gamma_2\gamma_3 = \mathbf{1}\otimes\sigma_1,\quad \gamma_1\gamma_2\gamma_3\gamma_4 = -\mathbf{J}\otimes\mathbf{J},\quad \gamma_1\gamma_2\gamma_3\gamma_4\gamma_5 = -i\mathbf{1}\otimes\mathbf{1}[/tex]
So with my construction the top element is the central element [itex]-i[/itex]. Complex conjugation of all elements gives an equivalent rep.

If I had left off the i factor for the last clifford generator we'd have three roots of -1 and two roots of 1. But to express the full algebra faithfully the top element should be non-trival and have its own conjugation operation. Use a third factor and define [itex]\gamma_5 = \mathbf{J}\otimes\mathbf{J}\otimes\sigma_3[/itex] and the top element will be [itex]\mathbf{1}\otimes\mathbf{1}\otimes\sigma_3[/itex] with a +1 block and a -1 block hence the [itex]\mathbb{R}(4)\oplus\mathbb{R}(4)[/itex] equivalence.You should be able to find quite a bit of work on Dirac's equation in higher dimensions of arbitrary signature. The mathematics is pretty straightforward, the physical interpretation o.t.o.h...

 

[Traruh Synred]

Thanks that's very useful!

I have managed to work out a two-time version of Dirac using the traditional Dirac [itex]\gamma^5[/itex] as the matrix for the second time. I think this likely is equivalent to your suggestion, but I'll have to work it out. I have derived plane-wave solutions. With wrapped up extra time this can give tachyon and, even, imaginary energies. Doubles the number of solutions to the equation ([itex]\pm E_2[/itex]).

There is one thing I find odd. It may just be a mistake. If I start with the original Dirac formulation and just add a time w/o it's own [itex]\gamma[/itex]- matrix I get the dispersion relation to be

[itex](E_1+E_2)^2-p^2-m^2=0[/itex]

instead of

[itex]E_1^2+E_2^2-p^2-m^2=0 [/itex].

This is odd as the as the original formulation seems to me to only differ from the explicitly invariant one by multiplying through by Dirac's [itex]\beta[/itex], a.k.a., [itex]\gamma^0[/itex]. I use the explicitly invariant version. The other would lead to quite different solutions though with fewer obvious pathologies.

I appreciate you laying out how it works for me. I'm not so good a group theory and would like have trouble extracting what needed from the book.

-Traruh

 

[Traruh Synred]

Dear SA:

So your set of [itex]\gamma[/itex]-matrices is not the same as mine, but they obey the same anti-commutation relation and thus both correspond to the [itex](++---)[/itex] signature. I assume that just means they are in a different basis. All results will be the same, I think.

 

[Traruh Synred]

<<Link deleted. Distribution of copyrighted material is not allowed of PF.>>

A pdf version of the book you recommended -- free!

 

[jambaugh]

[snip]

A pdf version of the book you recommended -- free!

I do not believe that is a licensed copy and you should remove the link as it is against the forum's terms of use. I rather doubt the author or publisher has released their copyrights.
https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/

Now I'm not a total prude about copyrighted material especially crap entertainment media, but this text is well worth the price and the author well deserves to be compensated for his diligent hard work in producing it.

 

[Traruh Synred]

<<Link deleted. Distribution of copyrighted material is not allowed of PF.>>

A pdf version of the book you recommended -- free!

I see link is already gone. Good!

 

[Traruh Synred]

Dear SA:

After much struggle with signs, I've got solutions for plane-waves for Dirac with two-times. They collapse back to normal 3+1 versions when I 'turn-off' the extra time variable.

I have two questions that I hope you might help with:

1) I used traditional Dirac [itex] \gamma_5 [/itex] for the 5th gamma matrix. This set has the right anti-commutation properties, but is not the same as the matrices you suggested. Is that ok?

2) I find a different normalization for left and right moving waves along the z-direction[a]. Can this be right? I assume this direction is selected for the additional E2 terms because z is the quantization direction. However, this difference in normalization ofr R and L does seem odd, so I wonder if I'm doing something wrong. Maybe gamma5 is not appropriate?
)
-Traruh

The [itex] k_z [/itex] in terms in spinor become [itex](1/(E_2\pm k_z)[/itex] where only [itex] k_z [/itex] flips sign in going from R to L.

 

[jambaugh]

Dear SA:

So your set of [itex]\gamma[/itex]-matrices is not the same as mine, but they obey the same anti-commutation relation and thus both correspond to the [itex](++---)[/itex] signature. I assume that just means they are in a different basis. All results will be the same, I think.

Right, it's a matter of the choice of spinor basis.

 

[jambaugh]

Dear SA:
[...]
1) I used traditional Dirac [itex] \gamma_5 [/itex] for the 5th gamma matrix. This set has the right anti-commutation properties, but is not the same as the matrices you suggested. Is that ok?

Yes that's fine except remember that [itex] gamma_5[/itex] was originally the top element and used to "observe" pairity. You need to use the new top element [itex]\gamma_{12345}[/itex] (or i times it) where before [itex]\gamma_5[/itex] was used.

2) I find a different normalization for left and right moving waves along the z-direction[a]. Can this be right? I assume this direction is selected for the additional E2 terms because z is the quantization direction. However, this difference in normalization ofr R and L does seem odd, so I wonder if I'm doing something wrong. Maybe gamma5 is not appropriate?

Could be because you're using the same gamma_5 for two distinct roles. See my above comment about the top element.

Note that one must also be careful how one generalizes the spinor metric. Dirac used the gamma_0 matrix, as beta, but that was a very basis dependent choice so the higher dimensional generalization is ... unsimple. It's been a while since I played with the physics part of this stuff. However I do recall there being several references out there ... here's the first google hit:http://link.springer.com/book/10.1007/978-94-007-1917-0. Although it does not deal with plural times the mathematical part should translate directly.

 

[secur]

Now I'm not a total prude about copyrighted material especially crap entertainment media, but this text is well worth the price and the author well deserves to be compensated for his diligent hard work in producing it.

So garbage, which tends to rot society, should be freely available to everyone. But good, intelligent material, which tends to elevate society, should be behind a paywall, restricted to the select few.

Doing well is the result of doing good. That's what capitalism is all about.

Please ignore this brief aside, just thought you might appreciate it

 

[jambaugh]

Please ignore this brief aside, just thought you might appreciate it

Did you intentionally pull that from my signature line, or not notice my signature line quotes and paste it in independently?

[Edit]PS Just thinking how my other sig. line quote sounds in this context. hmmm...

 

[secur]

I pirated it from your signature line! A perfect set-up I couldn't resist, glad you appreciate the irony. You also get the serious point, no doubt
.
But please don't let me interrupt this interesting, and productive, thread - although perhaps Traruh Synred's question has been thoroughly answered by now