Lectures on Quantum Field Theory. Ashok Das

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 with unit determinant (det R = 1). We also note from (3.2) that an infinitesimal rotation around the 3-axis (z-axis) takes the form

      where we have identified θ = ϵ = infinitesimal. We observe here that the matrix representing the infinitesimal change under a rotation (namely, see also (3.4) with θ = ϵ) is anti-symmetric.

      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

      where the Lorentz factor γ is defined in terms of the boost velocity to be

      We recognize that (3.7) can also be written in the matrix form as

      where we have defined

      so that

      Since the range of the boost velocity is given by −1 ≤ β ≤ 1 (namely, |v| ≤ c = 1), we conclude from (3.10) that −∞ ≤ ω ≤ ∞.

      Thus, we note that a Lorentz boost along the x-direction can be written as a matrix relation

      where

      From this, we can obtain,

      which would lead to the transformation of the covariant coordinate vector as

      The matrix representing the Lorentz transformation of the coordinates, (or ), is easily seen from (3.13) or (3.14) to be an orthogonal matrix in the sense that

      where we have used

      From (3.16), we see that the matrix Λ^µ_ν has a unit determinant, much like the rotation matrix R in (3.3). (Incidentally, (3.16)