Lectures on Quantum Field Theory. Ashok Das

17.4Solving the renormalization group equation

       17.5Callan-Symanzik equation

       17.6References

       18Nielsen identities and gauge independence of physical parameters

       18.1The problem

       18.2Questions associated with the effective potential

       18.3Gauge independence of the fermion mass

       18.3.1Fermion mass

       18.3.2Pole of the fermion Green’s function

       18.3.3Interpolating gauge

       18.3.4Nielsen identity for QED

       18.3.5Gauge dependence of the pole of the propagator

       18.3.6Infrared divergence and gauge dependence of fermion mass

       18.4References

       19Basics of global supersymmetry

       19.1Graded Lie algebras

       19.1.1Representations

       19.2Supersymmetric quantum mechanics

       19.3Supersymmetric field theories

       19.3.1Wess-Zumino theory

       19.3.2Vector multiplet

       19.3.3Supersymmetric Higgs model

       19.4Superspace

       19.5References

       Appendices

       Appendix: More on fermions

       A1Fermions in 4 dimensions

       A2Spinors in D space-time dimensions

       A3References

       Appendix: Gauge invariant potential and the Fock-Schwinger gauge

       B1Gauge invariant potential

       B2Fock-Schwinger gauge

       B3References

       Index

      CHAPTER 1

       Relativistic equations

       1.1Introduction

      As we know, in single particle, non-relativistic quantum mechanics, we start with the Hamiltonian description of the corresponding classical, non-relativistic physical system and promote each of the observables to a Hermitian operator. The time evolution of the quantum mechanical system (state), in this case, is given by the time dependent Schrödinger equation which has the form

      Here ψ(x, t) represents the wave function of the system which corresponds to the probability amplitude for finding the particle at the coordinate x at a given time t and the Hamiltonian, H, has the generic form

      with p denoting the momentum of the particle and V (x) representing the potential through which the particle moves. (Throughout the book we will use a bold symbol to represent a three dimensional vector.)

      This formalism is clearly non-relativistic (non-covariant) which can be easily seen by noting that, even for a free particle, the dynamical equation (1.1) takes the form

      In the coordinate basis, the momentum operator has the form

      so that the time dependent Schrödinger equation, in this case, takes the form

      This equation is linear in the time derivative while it is quadratic in the space derivatives. Therefore, space and time are not treated on an equal footing in this case and, consequently, the equation cannot have the same form (covariant) in different Lorentz frames. A relativistic equation, on the other hand, must treat space and time coordinates on an equal footing and remain form invariant in all inertial frames (Lorentz frames). Let us also recall that, even for a simple fundamental system such as the Hydrogen atom, the ground state electron is fairly relativistic (

for the ground state electron is of the order of the fine structure constant). Consequently, there is a need to generalize the non-relativistic quantum mechanical description to relativistic systems. In this chapter, we will study how we can systematically develop a quantum mechanical description of a single relativistic particle and the difficulties associated with such a description.

1.2Notations

      Before proceeding any further, let us fix our notations. We note that in the three dimensional Euclidean space, which we are all familiar with, a vector is labelled uniquely by its three components. (We denote three dimensional vectors in boldface.) Thus,

      where x and J represent respectively the position and the angular momentum vectors of a particle (system) while A stands for any arbitrary vector. In such a space, as we know, the scalar product of any two arbitrary vectors is defined to be

      where repeated indices are assumed to be summed. The scalar product of two vectors is invariant under rotations of the three dimensional space which is the maximal symmetry group of the Euclidean space that leaves the origin invariant. This also allows us to define the length of a vector simply as

      The Kronecker delta, δ_ij, in this case, represents the metric of the Euclidean space and is trivial (in the sense that all the nonzero components are positive and simply unity). Consequently, it does not matter whether we write the indices “up” or “down”. Let us note from the definition of the length of a vector in Euclidean space that, for any nontrivial vector, it is necessarily positive definite, namely,

      When we treat space and time on an equal footing and enlarge our three dimensional Euclidean manifold to the four dimensional space-time manifold, we can again define vectors in this manifold. However, these would now consist of four components. Namely, any point