Free Quantum Fields

The notion of free quantum field is thoroughly discussed in the linearised setting associated with the choice of a detector. The discussion requires attention to certain details that are often overlooked in the standard literature. Explicit expressions for generic fields, Dirac fields, gauge fields and ghost fields are laid down, as well the ensuing free-field expressions of important functionals. The relations between super-commutators of free fields and propagators, and the canonical super-commutation rules, follow from the above results.

Multi-particle Spaces

The fundamental algebraic notions needed in many-particle physics are exposed. Spaces of free states containing an arbitrary number of particles of many types are introduced. The operator algebra generated by absorption and emission operators is studied as a natural generalisation of standard exterior algebra. The link between the discrete and the distributional formalisms is provided by the spaces of finite linear combinations of semi-densities of Dirac type.

Infinitesimal Deformations of ECD Fields

The standard notion of Lie derivative is extended in order to include Lie derivatives of spinors, soldering forms, spinor connections and spacetime connections. These extensions are all linked together, and provide a natural framework for discussing infinitesimal deformations of Einstein-Cartan-Dirac fields in the tetrad-affine setting.

First-order Theory of Fields with Arbitrary Spin

By exploiting the previously exposed results in 2-spinor geometry, a general description of fields of arbitrary spin is exposed and shown to admit a first-order Lagrangian which extends the theory of Dirac spinors. The needed bundle is the fibered direct product of a symmetric ‘main sector’—carrying an irreducible representation of the angular-momentum algebra—and an induced sequence of ‘ghost sectors’. Several special cases are considered; in particular, one recovers the Bargmann-Wigner and Joos-Weinberg equations.

Classical Gauge Field Theory

After a sketch of Lagrangian field theory on jet bundles, the notion of a gauge field is introduced as a section of an affine bundle which is naturally constructed without any involvement with structure groups. An original approach to gauge field theory in terms of covariant differentials (alternative to the jet bundle approach) is then developed, and the adaptations needed in order to deal with general theories are laid out. A careful exposition of the replacement principle allows comparisons with approaches commonly found in the literature.

Scattering Matrix Computations

The basic thechniques for the computations of scattering amplitudes are illustrated within the context exposed in the previous chapter. In particular, the appearence of propagators is demonstrated.

Electroweak Geometry and Fields

A formulation of electroweak field theory which is equivalent to the usual one, but completely dispenses with structure groups, is presented. The notions of Higgs field and symmetry breaking are treated within this approach, and the various terms of the electroweak Lagrangian are worked out.

Spinors and Minkowski Space

A partly original approach to spinor geometry, showing how a 2-dimensional vector space, without any further assumpions, generates by natural constructions the fundamental algebraic structures needed to deal with spacetime geometry and particles with spin. Several related notions are expressed in a concise, intrinsic form.

Detectors

The notion of detector is brought forward as the ground for laying down the ideas of particle physics in relation to quantum field theory. Several notions related to the existence of a preferred time are discussed in such context.

Generalized Maps

Spaces of generalised sections (also called section-distributions) are introduced, and their fundamental properties are described. Several special cases are considered, with particular attention to the case of semi-densities; when a Hermitian structure on the underlying classical bundle is given, these determine a rigged Hilbert space, which can be regarded as a basic notion in quantum geometry. The essentials of tensor products in distributional spaces, kernels and Fourier transforms are exposed.