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Posted by on Dec 6, 2014 in Quantum Mechanics A to L | 0 comments

C. A Different Dirac Algebra

The Usual Dirac Algebra

Paul Dirac’s genius was to realize that the Klein-Gordon equation, which conserves energy and mass, would perhaps show more structure if it were a first order differential equation in space-time rather than the second order Klein-Gordon equation. This led to spin and anti-matter for starters. The trick he used required that a set of four matrices (called the gamma matrices) had to anti-commute, (i and j take values of 0, 1, 2, 3)

Where I4 is a 4×4 identity matrix and the metric tensor given by,

This means that there is one time variable, the +1, and three spatial variables, the -1.  Good old space-time again.

It is not necessary to go into the properties of the gamma-matrices.  They are well known. However their representation in terms of the Pauli spin matrices is important for what is to follow, so here they are,

There are two spins here: one for matter and the other for antimatter.  From the Dirac equation, spin emerges as a Lorentz invariant defined by the operators {I2123}.

A Different Dirac Algebra

At the end of the first blog (A) in this series, I said that there is a different way to define the gamma matrices for a spin ½.  That is one of the spatial matrices, say γ2, is replaced by (i is the imaginary number)

Why not?  The matrices still anti-commute, so by definition they form a Dirac algebra. It has a different metric tensor,

which means that there are two time variables and two spatial variables.  That is certainly different.  It represents a flat structure with only two spatial dimensions, maybe an anyon?

But two times?  The first is the usual linear time that differs in different inertial frames.  The second is a rotational time which rotates in the plane of the 2D flat space.  This is a phase time or a frequency and accounts for the different relative rotations of 2D objects in different inertial frames.

So what does this different Dirac algebra lead to?  First spin is now defined by {I21,23} .  Second, (opps), the corresponding Dirac equation is neither hermitian nor Lorentz Invariant. It turns out that is not an issue because the solutions lead to Mirror Matterand both mirror states must exist simultaneously: a reflection must have an object

Those Mirror states lead to a 2-dimensional Dirac equation. There is fun with parity too.

At the level of a spin ½ there are only two possibilities which make physical sense {I2,σ1,σ2,σ3} and {I2,σ1,2,σ3}. Adding more imaginary numbers gets rid of space and that is no good.  You might say adding one imaginary number also does not make sense, but it does. I’ll show that next time.




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