Field Theory

The following sections are included:

• Dedication
• Preface to the First Edition
• Preface to the Second Edition
• Preface to the Third Edition
• Contents

The following sections are included:

• Particles and fields
• Metric and other notations
• Functionals
• Review of quantum mechanics
• References

The following sections are included:

• Basis states
• Operator ordering
• Feynman path integral
• The classical limit
• Equivalence with the Schrödinger equation
• Free particle
• References

The following sections are included:

• Path integral for the harmonic oscillator
• Method of Fourier transform
• Matrix method
• The classical action
• References

The following sections are included:

• Euclidean rotation
• Time ordered correlation functions
• Correlation functions in definite states
• Vacuum functional
• Anharmonic oscillator
• References

The following sections are included:

• Fermionic oscillator
• Grassmann variables
• Generating functional for fermions
• Feynman propagator
• The fermion determinant
• References

The following sections are included:

• Supersymmetric oscillator
• Supersymmetric quantum mechanics
• Shape invariance
• Example
• Supersymmetry and singular potentials
  • Regularized superpotential
  • Alternate regularization
• References

The following sections are included:

• WKB approximation
• Saddle point method
• Semi-classical methods in path integrals
• Double well potential
• References

The following sections are included:

• Instantons
• Zero modes
• The instanton integral
• Evaluating the determinant
• Multi-instanton contributions
• References

The following sections are included:

• Systems with many degrees of freedom
• Relativistic scalar field theory
• Feynman rules
• Connected diagrams
• References

The following sections are included:

• The classical field
• Effective action
• Loop expansion
• Effective potential at one loop
• References

The following sections are included:

• Symmetries of the action
• Nöther’s theorem
  • Example
• Complex scalar field
• Ward identities
• Spontaneous symmetry breaking
• Goldstone theorem
• References

Gauge theories are very fundamental in our present understanding of physical forces and, in this chapter, we will study how such theories are described in the path integral formalism. Gauge theories are defined to be theories with a local symmetry and the symmetry is based on some relevant symmetry group. As we have seen in the last chapter, in general, symmetries have important consequences and in the case of a local symmetry, the consequences are even more powerful. However, along with beautiful structures, local invariances also bring in difficulties. For theories with simple local symmetries such as Maxwell’s theory, these difficulties can be handled with ease, but problems become quite severe when the relevant local symmetries are more complex such as a non-Abelian symmetry. In this case, the path integral description needs to be analyzed carefully. (Similar difficulties also manifest in the canonical quantization of such theories.) In this chapter, we will start with Maxwell’s theory where the difficulties can be simply handled and then we will go into a detailed discussion of non-Abelian gauge theories within the context of the path integral formalism.

We have seen earlier that a continuous global symmetry in a quantum field theory leads to a current which is conserved and the Ward identities of the theory reflect this conservation law. Sometimes, however, the classical (tree level) conservation of the current is violated by quantum corrections (loop effects). This can happen, for example, if the regularization, used to define divergent amplitudes in a quantum field theory, does not respect the symmetry that leads to the conservation law. In this case, the divergence of the current density no longer vanishes and one says that there is an anomaly in the conservation law (or that the current is anomalous). In turn, this modifies the tree level Ward identities of the theory to new ones that are known as anomalous Ward identities of the theory (which reflect the modified conservation law). Anomalies are quite important from a physical point of view. Global anomalies such as the chiral anomaly have direct physical consequences. On the other hand, an anomaly in a gauge theory (namely, in the conservation of the gauge current) can render the theory inconsistent and, therefore, needs to be taken care of before one can make sense of such a theory. In this chapter, we will study some issues associated with this phenomenon.

The following sections are included:

• Statistical mechanics
• Critical exponents
• Harmonic oscillator
• Fermionic oscillator
• References

The following sections are included:

• One dimensional Ising model
• The partition function
• Two dimensional Ising model
• Duality
• High and low temperature expansions
• Quantum mechanical model
• Duality in the quantum system
• References

The idea of proper time method goes back to Fock who had noticed that sometimes it is useful to express the propagator or the Green’s function as an integral over an additional auxiliary variable. Schwinger used this observation and developed the idea fully to what is now known as the proper time method or the proper time formalism. This is quite useful in studying many properties of quantum field theories, in particular, gauge theories since it, equivalently, introduces a gauge invariant regularization. It can be used at zero temperature as well as at finite temperature. The proper time formalism was the precursor of the zeta function regularization and the heat kernel method which we will discuss in the next chapter. These latter methods are quite useful in the covariant study of quantum field theories on nontrivial backgrounds (with or without boundaries) as well as in the study of Casimir energy.

The following sections are included:

• Riemann zeta function
  • Euler’s zeta function
  • Riemann’s zeta function
• Zeta function regularization
  • Partition function for the bosonic oscillator
• Heat kernel method
  • Bosonic propagator in flat space-time
• Expansion of the heat kernel
• Schwinger model in curved space-time
• References

The following section is included:

• Index