the relations in (2.134), it is now straightforward to obtain
The two relations in (2.135) are known as the Fierz rearrangement identities which are very useful in calculating cross sections. In deriving these identities, we have assumed that the spinors are ordinary functions. On the other hand, if they correspond to anti-commuting fermion operators, the right-hand sides of the identities in (2.135) pick up a negative sign which arises from commuting the fermionic fields past one another.
Let us note that using the explicit forms for Γ(a) and Γ(a) in (2.123) and (2.125) respectively, we can write the first of the Fierz rearrangement identities in (2.135) as (assuming the spinors to be ordinary functions and not anti-commuting fermion fields which will introduce an overall negative sign, for example, in commuting ψ2 past
Since this is true for any matrices M, N and any spinors, we can define a new spinor
which is often calculationally simpler. Thus, for example, if we choose
then using various properties of the gamma matrices derived earlier as well as (2.111) and (2.114), we obtain from (2.137)
This is the well known fact from the weak interactions that the V − A form of the weak interaction Hamiltonian proposed by Sudarshan and Marshak is form invariant under a Fierz rearrangement (the negative sign is there simply because we are considering spinor functions and will be absent for anti-commuting fermion fields).
2.7References
1.J. D. Bjorken and S. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York (1964).
2.A. Das, Lectures on Quantum Mechanics, Hindustan Publishing, New Delhi, India and World Scientific, Singapore (2011).
3.A. Das and S. Okubo, Lie Groups and Lie Algebras for Physicists, Hindustan Publishing, New Delhi and World Scientific, Singapore (2014).
4.C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York (1980).
5.S. Okubo, Real representations of finite Clifford algebras. I. Classification, Journal of Mathematical Physics 32, 1657 (1991).
6.L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York (1968).
7.E. C. G. Sudarshan and R. E. Marshak, Proceedings of Padua-Venice conference on mesons and newly discovered particles, (1957); Physical Review 109, 1860 (1958).
CHAPTER 3
Properties of the Dirac equation
3.1Lorentz transformations
In three dimensions, we are well acquainted with rotations. For example, we know that a rotation of coordinates around the z-axis by an angle θ can be represented as the transformation
where R represents the rotation matrix such that
Here we are using a three dimensional notation, but this can also be written in terms of the four vector notation we have developed earlier. The rotation around the z-axis in (3.2) can also be written in matrix form as
so that the coefficient matrix on the right hand side can be identified with the rotation matrix R in (3.1), namely,
Thus, we see from (3.4) that a finite rotation around the 3-axis (z-axis) or in the 1-2 plane is denoted by an orthogonal matrix, R
where we have identified θ = ϵ = infinitesimal. We observe here that the matrix representing the infinitesimal change under a rotation (namely,
Under a Lorentz boost along the x-axis, we also know that the coordinates transform as (boost velocity β = v since c = 1, otherwise,
such that
