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
18.2Questions 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
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
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 (
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, δij, 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