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Chapter III Dirac Field Lecture 3 Books Recommended: Lectures on Quantum Field Theory by Ashok Das Advanced Quantum Mechanics by Schwabl Postulates of special theory of relativity: (1) All the physical laws are same in all inertial frames of reference which are moving with constant velocity relative to each other. There is no absolute or universal frame of reference. (2) The speed of light in vacuum is the same in every inertial frame Lorentz Transformations (along x-axis) Rotation in dimension ---(1) Where, rotation matrix R (about Z axis) is -----------(2) Thus, ----(3) Rotation matrix R is orthogonal and have unit determinant For infinitesimal rotation ----(4) Anti-symmetric matrix. Lorentz Transformations Under Lorentz boost, coordinates transform from one Inertial frame to other as Such that -----(1) --------(2) Above eq is written as In above ω is known as rapidity. ------(3) In last eq. ------(4) Such that As Thus Thus, we write the Lorentz transformation as -------(5) Where, Lorentz transformation matrix ----(6) We write ---(7) Covariant co-ordinate will transform as Matrix representing Lorentz transformation are orthogonal ---(8) And have unit determinant as was the case of rotation Matrix in 3-D about z-axis. Infinitesimal Lorentz transformation - --(9) Where --(10) Matrix in Eq (10) is anti-symmetric ----(11) •Lorenz transformations leave the length of vector Invariant ----(12) Covariance of Dirac Equation Consider dynamical Eq ------(13) Where L is operator. Above Eq will be covariant if -------(14) Consider Dirac Eq ----(15) Under Lorentz tranfromation ----(16) If Eq (15) takes form ---(17) Then it will be form invariant. Under L.T. Transformed wave function is written as ----(18) Where is a 4 by 4 matrix used to Represent the effect of LT on wave function. It must possess inverse ------(19) From (16) we can write ---(20) We write ---(21) Comparing (21) with (15), for invarianve we should have -----(22) Define --------(23) We have ----(24) Consider the infinitesimal Lorentz transformation --------(25) Where Expanding S matrix ----(26) Where -----(27) Also ----(28) Where ----(29) From (22), we write -----(30) Recall -----(31) From (30), identify ------(32) Thus, -----(33) which is RHS of (30). Thus, for infinitesimal transformation ---(34) Indentify as generator of LT