From Classical to Quantum Fields. Free Fields

We apply the canonical and the path integral quantisation methods to scalar, spinor and vector fields. The scalar field is a generalisation to an infinite number of degrees of freedom of the single harmonic oscillator we studied in Chapter 9. For the spinor fields we show the need for anti-commutation relations and introduce the corresponding Grassmann algebra. The rules of Fermi statistics follow from these anti-commutation relations. The canonical quantisation method applied to the Maxwell field in a Lorentz covariant gauge requires the introduction of negative metric states in the Hilbert space. The power of the path integral quantisation is already manifest. In each case we expand the fields in creation and annihilation operators.

Higgs fields induced by Yang–Mills type Lagrangians on gauge-natural prolongations of principal bundles

We address some new issues concerning spontaneous symmetry breaking. We define classical Higgs fields for gauge-natural invariant Yang–Mills type Lagrangian field theories through the requirement of the existence of canonical covariant gauge-natural conserved quantities. As an illustrative example, we consider the ‘gluon Lagrangian’, i.e. a Yang–Mills Lagrangian on the [Formula: see text]-order gauge-natural bundle of [Formula: see text]-principal connections, and canonically define a ‘gluon’ classical Higgs field through the split reductive structure induced by the kernel of the associated gauge-natural Jacobi morphism.

Electromagnetic spin creates torsion

It is shown the intrinsic spin, and only the spin, of the electromagnetic field creates torsion. The struggle raged for decades: How to reconcile the facts that photons have spin, but minimal coupling breaks gauge invariance and therefore must be abandoned, leaving us with the unphysical situation in which spin does not create torsion. By generalizing the gauge freedom of the torsion field, a covariant, gauge-invariant description is found whereby the electromagnetic spin creates a torsion field. In fact, it is shown if electromagnetic gauge invariance holds, torsion must be present.

Geometric aspects of interpolating gauge-fixing in Chern–Simons theory

In this paper we investigate an interpolating gauge-fixing procedure in (4l + 3)-dimensional Abelian Chern–Simons theory. We show that this interpolating gauge is related to the covariant gauge in a constant anisotropic metric. We compute the corresponding propagators involved in various expressions of the linking number in various gauges. We comment on the geometric interpretations of these expressions, clarifying how to pass from one interpretation to another.

Spontaneous breaking of (2 + 1)-dimensional Lorentz symmetry by an antisymmetric tensor

One kind of spontaneous (2 + 1)-dimensional Lorentz symmetry breaking is discussed. The symmetry breaking pattern is SO(2, 1) [Formula: see text] SO(1, 1). Using the coset construction formalism, we derive the Goldstone covariant derivative and the associated covariant gauge field. Finally, the two-derivative low-energy effective action of the Nambu–Goldstone bosons is obtained.

The Weizsäcker-Williams distribution of linearly polarized gluons (and its fluctuations) at small x

The conventional and linearly polarized Weizsäcker-Williams gluon distributions at small x are defined from the two-point function of the gluon field in light-cone gauge. They appear in the cross section for dijet production in deep inelastic scattering at high energy. We determine these functions in the small-x limit from solutions of the JIMWLK evolution equations and show that they exhibit approximate geometric scaling. Also, we discuss the functional distributions of these WW gluon distributions over the JIMWLK ensemble at rapidity Y ~ 1/αs. These are determined by a 2d Liouville action for the logarithm of the covariant gauge function g2tr A+(q)A+(-q). For transverse momenta on the order of the saturation scale we observe large variations across configurations (evolution trajectories) of the linearly polarized distribution up to several times its average, and even to negative values.