*Anthony Duncan*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199573264
- eISBN:
- 9780191743313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573264.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter examines the additional rich structure introduced when a local quantum field theory displays a local gauge symmetry. It shows how such symmetries require a generalization of the ...
More

This chapter examines the additional rich structure introduced when a local quantum field theory displays a local gauge symmetry. It shows how such symmetries require a generalization of the canonical Lagrangian/Hamiltonian formalism discussed in Section 12.3 of Chapter 12 in order to handle the presence of constraints entailed by the presence of local symmetries. The chapter is organized as follows. Section 15.1 introduces the concept of a local symmetry with a simple example from classical mechanics. Section 15.2 describes the Dirac constrained Hamiltonian theory, and the Faddeev–deWitt functional quantization method for such systems. The quantization of gauge theories using this functional (path-integral) method is then explained, first using abelian gauge theory in Section 15.3, where the technical complications are minimal. In Section 15.4 the extension to non-abelian gauge theories is performed, again using path-integral methods applied to the constrained Hamiltonian, leading to the Feynman rules for general (unbroken) non-abelian gauge theories. Section 15.5 explores the existence of quantum anomalies in the chiral currents of internal global symmetries. It shows that the classical current conservation implied by Noether's theorem may be violated by quantum effects, yielding a non-vanishing divergence of the Noether current explicitly proportional to Planck's constant. Section 15.6 focuses on the features of spontaneous symmetry breaking in the presence of local gauge symmetry. The chapter then explains the famous ‘Higgs phenomenon’ in the context of the electroweak sector of the Standard Model and outlines the derivation of the Feynman rules for a general spontaneously broken local gauge theory.Less

This chapter examines the additional rich structure introduced when a local quantum field theory displays a local gauge symmetry. It shows how such symmetries require a generalization of the canonical Lagrangian/Hamiltonian formalism discussed in Section 12.3 of Chapter 12 in order to handle the presence of constraints entailed by the presence of local symmetries. The chapter is organized as follows. Section 15.1 introduces the concept of a local symmetry with a simple example from classical mechanics. Section 15.2 describes the Dirac constrained Hamiltonian theory, and the Faddeev–deWitt functional quantization method for such systems. The quantization of gauge theories using this functional (path-integral) method is then explained, first using abelian gauge theory in Section 15.3, where the technical complications are minimal. In Section 15.4 the extension to non-abelian gauge theories is performed, again using path-integral methods applied to the constrained Hamiltonian, leading to the Feynman rules for general (unbroken) non-abelian gauge theories. Section 15.5 explores the existence of quantum anomalies in the chiral currents of internal global symmetries. It shows that the classical current conservation implied by Noether's theorem may be violated by quantum effects, yielding a non-vanishing divergence of the Noether current explicitly proportional to Planck's constant. Section 15.6 focuses on the features of spontaneous symmetry breaking in the presence of local gauge symmetry. The chapter then explains the famous ‘Higgs phenomenon’ in the context of the electroweak sector of the Standard Model and outlines the derivation of the Feynman rules for a general spontaneously broken local gauge theory.

*Anthony Duncan*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199573264
- eISBN:
- 9780191743313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573264.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter deals with the intricacies of interacting field theories, emphasizing very general aspects common to all local quantum field theories (LQFTs). The basic concept here is that of the ...
More

This chapter deals with the intricacies of interacting field theories, emphasizing very general aspects common to all local quantum field theories (LQFTs). The basic concept here is that of the interpolating Heisenberg field in terms of which the dynamics of the theory is specified, but which may be connected in various ways to the actual physical particle states. At this point, a characteristic feature of LQFTs becomes apparent: namely, the absence of any preferred, one-to-one connection between particles and fields. The discussion of field theory in the Heisenberg picture is first carried out in an ‘heuristic’ fashion, ignoring some important mathematical fine points; and then from a rigorous axiomatic point of view, starting with the Wightman axioms (spectral and field); and proceeding, via the Haag–Ruelle formulation of scattering theory, to the asymptotic formalism of Lehmann, Symanzik, and Zimmermann. The latter is treated in some detail, as it is central to subsequent discussions of the nature of the state space of field theory. The chapter also discusses the spectral properties of field theory, and the connection between the internal dynamics as specified by the interpolating fields and the phenomenological content of the theory as encapsulated in the asymptotic particle states and the S-matrix.Less

This chapter deals with the intricacies of interacting field theories, emphasizing very general aspects common to all local quantum field theories (LQFTs). The basic concept here is that of the interpolating Heisenberg field in terms of which the dynamics of the theory is specified, but which may be connected in various ways to the actual physical particle states. At this point, a characteristic feature of LQFTs becomes apparent: namely, the absence of any preferred, one-to-one connection between particles and fields. The discussion of field theory in the Heisenberg picture is first carried out in an ‘heuristic’ fashion, ignoring some important mathematical fine points; and then from a rigorous axiomatic point of view, starting with the Wightman axioms (spectral and field); and proceeding, via the Haag–Ruelle formulation of scattering theory, to the asymptotic formalism of Lehmann, Symanzik, and Zimmermann. The latter is treated in some detail, as it is central to subsequent discussions of the nature of the state space of field theory. The chapter also discusses the spectral properties of field theory, and the connection between the internal dynamics as specified by the interpolating fields and the phenomenological content of the theory as encapsulated in the asymptotic particle states and the S-matrix.

*Anthony Duncan*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199573264
- eISBN:
- 9780191743313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573264.003.0017
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the technical tools needed for analyzing the perturbative renormalizability of a specific local quantum field theory. It presents proof of cutoff-insensitivity, using ...
More

This chapter discusses the technical tools needed for analyzing the perturbative renormalizability of a specific local quantum field theory. It presents proof of cutoff-insensitivity, using traditional graphical methods and from the point of view of effective Lagrangian theory. The chapter first examines in detail the structure of the cutoff dependence of general multi-loop Feynman integrals appearing in the perturbative expansion of amplitudes in a local quantum field theory. The occurrence of divergent integrals (and subintegrals) in such loop amplitudes is associated with cutoff-dependent contributions, which have a very simple momentum dependence. This latter fact is then exploited to demonstrate the equivalence of the set of subtractions needed to remove the leading cutoff dependence of an arbitrary Feynman amplitude (reducing it to the inverse power-dependence of the type seen above), resulting in a reparameterization of a set of coupling and mass parameters appearing in the Lagrangian of the theory. The intimate connection of reparameterization/subtraction is the essence of the proof of cutoff-insensitivity for perturbatively renormalizable theories.Less

This chapter discusses the technical tools needed for analyzing the perturbative renormalizability of a specific local quantum field theory. It presents proof of cutoff-insensitivity, using traditional graphical methods and from the point of view of effective Lagrangian theory. The chapter first examines in detail the structure of the cutoff dependence of general multi-loop Feynman integrals appearing in the perturbative expansion of amplitudes in a local quantum field theory. The occurrence of divergent integrals (and subintegrals) in such loop amplitudes is associated with cutoff-dependent contributions, which have a very simple momentum dependence. This latter fact is then exploited to demonstrate the equivalence of the set of subtractions needed to remove the leading cutoff dependence of an arbitrary Feynman amplitude (reducing it to the inverse power-dependence of the type seen above), resulting in a reparameterization of a set of coupling and mass parameters appearing in the Lagrangian of the theory. The intimate connection of reparameterization/subtraction is the essence of the proof of cutoff-insensitivity for perturbatively renormalizable theories.

*Anthony Duncan*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199573264
- eISBN:
- 9780191743313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573264.003.0014
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter examines the role of global symmetries in local quantum field theories. It shows that exact global symmetries are rare — indeed, if we take gravitational effects into account, probably ...
More

This chapter examines the role of global symmetries in local quantum field theories. It shows that exact global symmetries are rare — indeed, if we take gravitational effects into account, probably non-existent. Nevertheless, approximate global symmetries play an enormously important role in modern field theory. The appearance of massless Goldstone particles once an exact global symmetry is spontaneously broken is of enormous importance in modern field theory, and proof of the Goldstone theorem embodying this phenomenon is given. Dynamical aspects of spontaneous symmetry-breaking (SSB) are examined, and it is shown that the essence of SSB resides in the energetics of the theory in the infrared (i.e., at long distances).Less

This chapter examines the role of global symmetries in local quantum field theories. It shows that exact global symmetries are rare — indeed, if we take gravitational effects into account, probably non-existent. Nevertheless, approximate global symmetries play an enormously important role in modern field theory. The appearance of massless Goldstone particles once an exact global symmetry is spontaneously broken is of enormous importance in modern field theory, and proof of the Goldstone theorem embodying this phenomenon is given. Dynamical aspects of spontaneous symmetry-breaking (SSB) are examined, and it is shown that the essence of SSB resides in the energetics of the theory in the infrared (i.e., at long distances).