Dirac equation in the algebra of physical space


The Dirac equation, as the relativistic equation that describes
spin 1/2 particles in quantum mechanics, can be written in terms of the Algebra of physical space, which is a case of a Clifford algebra or geometric algebra
that is based on the use of paravectors.
The Dirac equation in APS, including the electromagnetic interaction, reads
Another form of the Dirac equation in terms of the Space time algebra was given earlier by David Hestenes.
In general, the Dirac equation in the formalism of geometric algebra has the advantage of
providing a direct geometric interpretation.

Relation with the standard form

The spinor can be written in a null basis as
such that the representation of the spinor in terms of the Pauli matrices is
The standard form of the Dirac equation can be recovered by decomposing the spinor in its right and left-handed spinor components, which are extracted with the help of the projector
such that
with the following matrix representation
The Dirac equation can be also written as
Without electromagnetic interaction, the following equation is obtained from
the two equivalent forms of the Dirac equation
so that
or in matrix representation
where the second column of the right and left spinors can be dropped by defining the
single column chiral spinors as
The standard relativistic covariant form of the Dirac equation in the Weyl
representation can be easily identified
such that
Given two spinors and in APS and
their respective spinors in the standard form as and
, one can verify the following identity
such that

Electromagnetic gauge

The Dirac equation is invariant under a global right rotation applied
on the spinor of the type
so that the kinetic term of the Dirac equation transforms as
where we identify the following rotation
The mass term transforms as
so that we can verify the invariance of the form of the Dirac equation.
A more demanding requirement is that the Dirac equation should be
invariant under a local gauge transformation of the type
In this case, the kinetic term transforms as
so that the left side of the Dirac equation transforms covariantly as
where we identify the need to perform an electromagnetic gauge transformation.
The mass term transforms as in the case with global rotation, so, the form
of the Dirac equation remains invariant.

Current

The current is defined as
which satisfies the continuity equation

Second order Dirac equation

An application of the Dirac equation on itself leads to the second order Dirac equation

Free particle solutions

Positive energy solutions

A solution for the free particle with momentum and positive energy is
This solution is unimodular
and the current resembles the classical proper velocity

Negative energy solutions

A solution for the free particle with negative energy and momentum
is
This solution is anti-unimodular
and the current resembles the classical proper velocity
but with a remarkable feature: "the time runs backwards"

Dirac Lagrangian

The Dirac Lagrangian is

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