2.1. Group Theoretical Derivation
To derive the Dirac equation with chiral symmetry, we begin by following Wigner [
13,
14] and identifying particles with the induced irreducible representations (irreps) of the Poincaré group
, with
being a non-invariant Lorentz group of rotations and boosts and
an invariant subgroup of spacetime translations. The condition that the Dirac spinor wavefunction transforms as one of the irreps of
extended by parity can be written as the following matrix eigenvalue equation [
12].
where
= 0, 1, 2 and 3 and
is the Dirac spinor wavefunction with
and
being two-component bispinors. Here
and
Y are five
which nontrivially mix the spinor components. To determine form of these matrices, we constrain them to satisfy covariant conditions under the Lorentz transformations of
, and find the resulting Poincaré invariant chiral Dirac equation (CDE) [
12] to be given by
where
is the chiral angle. This angle appears as a necessary degree of freedom as it is unconstrained by the covariant transformation constraints.
Because this method of deriving the CDE is based explicitly on the symmetries of Minkowski spacetime, we may be certain the resulting equation is Poincaré invariant. Furthermore, The CDE correctly accounts for all four sets of states (spins up and down, matter and antimatter), and its solutions have important physical implications [
12]. Being Poincaré invariant and local, the CDE satisfies the two basic characteristics required for the equation to be fundamental. Two other characteristics are gauge invariance and the existence of Lagrangian. Since we consider only free elementary particles, gauge invariance is not discussed here. We now demonstrate that the CDE can be formally derived from Lagrangian formalism, which is sufficient to call the CDE the fundamental equation of physics.
2.2. Derivation from Lagrangian Formalism
The Lagrangian formalism is a powerful and independent way to derive a dynamical equation. The Lagrangian for the Dirac equation (
in Equation (
3)) is very well-known and presented in textbooks (e.g., [
15,
16]) without derivation. In fact, the Lagrangian was not a part of the Dirac’s original paper where the equation first appeared [
1]. An interesting attempt to obtain the Dirac Lagrangian is presented and discussed in [
17]. Let us briefly review the main points of this attempt and then use them to obtain the Lagrangian for Equation (
3).
In case
, Equation (
3) reduces to the Dirac equation
which describes a free, massive, non-chiral and spin 1/2 relativistic elementary particle [
1,
15,
16,
17].
To obtain the Lagrangian density for this equation, we follow [
17] and require that the Lagrangian is a hermitian, single-valued proper scalar or pseudo-scalar in
and
. Since
has the double-valued properties under rotations, the terms in the Lagrangians must have even numbers of
. The simplest proper scalar is
, where
is the Dirac adjoint. Now, the construction of a scalar kinetic term has to be done with caution as
requires saturation of the index
, which cannot be done by another derivative since the result would be a second-order equation. Therefore, the Dirac matrices
are used to saturate the index
, and write the kinetic term as
. Since the physical units of the kinetic term than
, the latter must be multiplied by an inverse length dimension, which in natural units is mass. Then, the Dirac Lagrangian can be written in the following form
This is a fully symmetric form of the Lagrangian, which shows that when evaluated along a stationary path the Dirac Lagrangian vanishes [
17]. Both the Dirac equation and its Lagrangian are Poincaré invariant.
Using the above procedure, the Lagrangian for the CDE (see Equation (
3)) can also be obtained and written as
Similarly to
, the Lagrangian is also fully symmetric, hermitian and single-valued proper scalar, and it vanishes when evaluated along a stationary path. Moreover, the CDE and its Lagrangian are Galilean invariant. By substituting
into the Euler-Lagrange equation for variations with respect to
, the CDE given by Equation (
3) is obtained. This method of deriving the CDE is independent from the group theory derivation and serves to demonstrate that the equation satisfies a least-action principle requisite of any fundamental theory. We is derivation based on projection operators that is now presented.
2.3. Derivation from Orthogonal Idempotents
We may define a set of projection operators operating on an
N-dimensional complex vector space with any set of
N-by-
N orthogonal idempotent matrices satisfying
The total number of projection operators of a given vector space is maximally equal to the dimensions of the space considered. Therefore, for
, we may expand the most general operators acting on a spinor in terms of the Pauli matrices and the two-by-two identity matrix. Let
The total number of projection operators of a given vector space is maximally equal to the dimensions of the space considered. Therefore, for
, we may expand the most general operators acting on a spinor in terms of the Pauli matrices and the two-by-two identity matrix. Let
Solving this we find two projection operators for our symmetry group whose degrees of freedom may be parameterized in terms of the unit vector
. We write these projection operators succinctly as
For a fixed
, these operators allow us to define two types of objects in our two-dimensional vector space. For any such element
we may define
. It necessarily follows that
It is now a simple matter to extend these projection operators to projections in
-dimensional vector spaces. In general we may write
for
. It is easy to see that these projection operators satisfy our orthogonal idempotent constraints. We then define our set of eigenvectors in a similar manner
By restricting our considerations to the vector space of spinors we are able to give these abstract considerations physical significance. Recall that the rotation operator corresponding to a rotation of
about the axis defined by
for the vector space of two-component spinors takes the form
It follows that eigenstates of our projection operators are physically invariant under rotations about , differing only by a phase. We therefore identify as projecting out the portion of the state vector with spin parallel (+) or anti-parallel (−) to the -axis. By choosing , where is the three-momentum of the particle, we find to be the helicity projection operators. This most neatly encapsulates the experimental observance of binary spin states in mathematical terms.
We now wish to use our projection operator methodology to classify states of positive and negative energies, e.g., matter/anti-matter. The inclusion of an additional two-valued quantum property necessitates (at minimum) a four-dimensional vector space. We therefore construct the projection operators of the form
where we have introduced the vector
about which we will have more to say shortly, for the time being
is simply a set of three complex numbers and satisfies
.
Next we define the operand
The corresponding generalization of Equation (
11) yields
Exploiting the fact that
we may construct the equations
We now identify the set of gamma matrices in the chiral representation
satisfying the Clifford algebra
Taking the usual definition
as the matrix which anticommutes with all
, let us write
which give
Rewriting the operators of Equations (
15) and (
16) in terms of the gamma matrices, we find
It is our goal to identify the vector
with our physical quantities. Scaling these operators from the left with
, we now restrict
and
to be of equal-magnitude (this is equivalent to the on-mass-shell asssumption). We then obtain equivalent equations to Equations (
15) and (
16) of the form
It is now possible to make the identifications with our physical quantities explicit. For the vector
, we identify:
The identification of () is chosen to align with our definition of gamma matrices though equivalent linear combinations of the vector may be found through unitary transformations. It may appear curious upon first inspection that the vector is seemingly compelled to take on imaginary values in two components and real values in the third. It is, however, a simple matter to absolve ourselves of this inhomogeneity by performing a Wick-rotation of the energy axis in the complex plane and thereby considering the four vectors of Minkowski spacetime in purely Euclidean terms. In this way the connection between our projection operators as a basis of and the Lorentz group may be made explicit. These simplifications notwithstanding, we will continue consider real-valued energies in Minkowski spacetime.
Substituting our terms of Equation (
25), into Equations (
23) and (
24) we obtain
It is clear that Equations (
26) and (
27) are the momentum space analogues of the CDE for positive and negative energies and therefore are equivalent to plane wave solutions of Equation (
3). The eigenstates therefore satisfy
The method of projection operators for deriving fundamental equations has potential to extend beyond the vector space of bi-spinors. While outside the scope of the present work, the possibility of extending the concepts and methodologies presented here to investigate the algebraic structure inherent in three particle flavors and their mass spectrum remains a tantalizing possibility.