Reference: References: Robert D. Klauber, Student Friendly Quantum Field Theory, (Sandtrove Press, 2013) – Chapter 4, Problem 4.5.
The Dirac equation in relativistic quantum mechanics can be written as
When written out in its matrix components, this equation is actually 4 differential equations.
Remember that is a 4-d column vector in spinor space rather than a single function, so that the subscript index in indicates which component in spinor space we’re dealing with. These equations have four solutions denoted by for . Note that each is a full 4-component vector in spinor space; that is, the superscript indicates which complete vector we’re dealing with. Thus is the th component of the th vector.
We can write the 4 PDEs as a matrix eigenvalue equation by moving the terms involving to the RHS and factoring out an from the terms remaining on the LHS:
We’ll now look at the four solutions and verify that they satisfy 6. First, we have
where is defined by this equation as the constant multiplied by the 4-d spinor factor. Remember that is a 4-vector product:
Thus the first row of 6 is verified. The other 3 rows can be verified similarly. For row 2:
For row 3:
And for row 4:
The other 3 solutions are
If you really want to, you can verify that these 3 vectors satisfy 6 by grinding through the calculations as above. One point worth noting is that the constant that multiplies all the solutions could be any other constant and still satisfy 6 (since the constant just cancels off both sides). It’s chosen to be to make later calculations easier.