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I was having algebraic trouble verifying orthonormality relationships for spinor solutions to the Dirac free particle equation, and initially started preparing these notes to post a question to physicsforums. However, in the process of doing so, I spotted my error. A side effect of making these notes is that I got a nice summary of some of the relationships, and it was a good starting point for some personal notes expanding on the content of these chapters.
Context for the original question.
In Desai’s QM book , the non-covariant form of the free particle equation is developed as
where each block in the matrix above is two by two. Recall that
For spin up and spin down states, the positive energy solutions are found to be
and the negative energy states associated with are found to be
The z-axis spin up state and spin down state are also used to find one specific set of states for the positive energy solutions
and negative energy solutions
(the book uses for both the negative energy states, but I’ve used here for the negative states for consistency with the covariant equation solutions).
Later a complete set of states are identified as solutions to the covariant Dirac equations , , where as follows
Note very carefully the sign change above. This is important, since without that we do not have a zero inner product between all and states. Spelled out explicitly, these states for the z-axis spin up case are
In order to construct a covariant current conservation relationship a quantity, the Dirac adjoint, was defined as
This Dirac adjoint can be used to form an inner product of the form
It’s claimed in the text that we have , , and . Let’s verify all these relationships.
Verify the non-covariant solutions.
A non-relativistic approximation argument was used to determine the solutions 2.6a, but we can verify that these hold generally by substitution. For example, for the positive energy z-axis spin up state we have
Here the relationship between the free particle’s energy and momentum has been used, so we have a zero as desired, and no non-relativistic approximations are required. We can show this generally too, without requiring the specifics of the z-axis spin up or down solutions. This is actually even easier. For the positive energy solutions 2.4 we have
where the identity has been used. For the negative energy solutions 2.5 we have
Is there something special about the z-axis orientation?
Why was the z-axis spin orientation picked? It doesn’t seem to me that there would be any reason for this. For y-axis spin, recall that our eigenstates are
Our positive energy states should therefore be
It is straightforward to verify that these are solutions. We find for example that
as expected. What’s the general solution? For
Should we wish to consider an arbitrarily oriented spin, expressing in spherical coordinates also makes sense
and we find (with and for and respectively)
Substitution back into 2.4, and 2.5 is then easy. Expressing these with the angles expressed as sums and differences is strongly suggested. With , and this gives
This is probably about as tidy as things can be made for the general case.
Expanding the current equation.
We can expand the current for a general spin up or spin down state with respect to either the positive energy or negative energy solutions.
Those (normalized) solutions are respectively
For the th component of the positive energy solution current we have
Similarly for a negative energy solution we have
We can expand the inner term of both easily
so that we have for the positive and negative energy solutions currents of
This finds the velocity dependence noted in section 33.4, but does not require taking any specific spin orientation, nor any specific momentum direction.
Unpacking the covariant equation.
Pre-multiplication of the covariant Dirac equation by should provide a space-time split of the Dirac equation. Let’s verify this
This recovers 2.1 as expected.
Two by two form for the covariant equations.
If we put the covariant Dirac equations in two by two matrix form we get
This form makes it easy to verify that our solutions are
It’s curious to consider these part of a basis for a single equation. I suppose that all together they are actually eigenstates of the equation
which have the form of the Klein-Gordan equation.
Orthonormality for the vectors is easy to show, and we can do so without requiring any specific spin orientation
It’s also easy for vectors
For the cross terms we have
Resolution of identity.
It’s claimed that an identity representation is
This makes some sense, but we can see systematically why we have this negative sign. Suppose that we have a basis for which we have (rather than the strict orthonormality condition ). Consider the calculation of the Fourier coefficients of a state
For , so that the coefficient is
The coordinate representation of this state vector with respect to this basis is thus
Shuffling things around, employing the somewhat abusive seeming Dirac ket-bra operator notation, we find the general identity operation takes the form
so that the identity itself has the form
This is the sum of all the ket-bras for which the braket is one, minus the sum of all the ket-bras for which the braket is negative, showing that the form of the claimed identity is justified.
We can also verify this directly by computation, and find
We can pull the summation into the matrices and note that (the two by two identity), so that we are left with
Similarly, we find
summing the two (noting that ) we get the block identity matrix as desired.
We’ve also just calculated the projection operators. Let’s verify that expanding the covariant form in the text produces the same result
Now compare to 3.39, and 3.40, which we rewrite using as
Lorentz transformation of Dirac equation.
Equation (35.107) in the text is missing the positional notation to show the placement of the indexes, and should be
where the solution is
This does have the form I’d expect, a bivector, but we can show explicitly that this is the solution without too much trouble. Consider the commutator
Would this be any easier to prove without utilizing the dot and wedge product identities? I used a few of them, starting with
In matrix notation we would have to show that the anticommutator commutes with any to make the first cancellation. We can do so by noting
That’s enough to get us on the path to how to prove this in matrix form
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.