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:

The derivatives in 6 are all with respect to spacetime variables, so act only on ; the spinor components are constants with respect to these derivatives. The first row in 6 is therefore

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.

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