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Parity violation, the asymmetry between left and right, is one of the fundamental properties of the quantum vacuum of the Standard Model. This effect is strong at high energy on the order of the electroweak scale, but is almost imperceptible in low-energy condensed matter physics. At this scale the left and right particles are hybridised and only the left-right symmetric charges survive. An analog of parity violation exists in superfluid ³He-A alongside related phenomena such as chiral anomaly and macroscopic chiral currents. The fermionic charge of right-handed particles minus that of left-handed ones is conserved at the classical level but not if quantum properties of the physical vacuum are taken into account. This charge can be transferred to the inhomogeneity of the vacuum via the axial anomaly in the process of helical instability. The inhomogeneity which absorbs the fermionic charge arises as a hypermagnetic field configuration in the Standard Model and as vortex texture in ³He-A. This allowed the experimental simulation of magnetogenesis (generation of hypermagnetic field) in ³He-A. Chern–Simons energy term in the Standard Model and in ³He-A is also discussed, where the effective chemical potential for chiral fermions is provided by counterflow velocity: relative velocity of motion of normal component of the liquid with respect to the superfluid one.

This chapter discusses the physics of quantized Dirac fields with detailed treatment of Dirac equation, representations of gamma matrices, products of gamma matrices, relativistic covariance (boosts, rotations, and invariants), helicity, gauge transformations, chirality, solution of the Dirac equation (Dirac representation, chiral representation, two-component helicity eigenstate spinors, and massless field), quantization, symmetry transformation of states (space-time translations and Lorentz transformations), C, P, and T transformations, wave packets, and Fierz transformations.

This chapter discusses in greater detail some of the topics introduced in Chapter 2. It covers magnetosonic waves; magnetohydrodynamic shocks; self-gravitating systems and the virial theorems; magnetostatic equilibrium and force-free fields; magnetic helicity; stability; the effects of dissipation and reconnection; and macroscopic dissipation.

The term ‘dynamo action’ appears in the literature describing two related but distinct processes. When the velocity v of a conducting fluid is driven by the non-magnetic forces in the direction opposite to that of the Lorentz force, the rate of working (j × B/c) · v is negative, implying an input of energy into the magnetic field; and in the high magnetic Reynolds number domain, the input on balance exceeds the Ohmic dissipation, so that an already existing field is amplified. This chapter discusses the ‘self-exciting dynamo’ problem, which demands that there be no sources of the magnetic field B other than the currents generated by fluid motions in the presence of B. It covers laminar kinematic dynamos; the Parker model; turbulent dynamos; kinematic models of the turbulent dynamo; non-linear dynamical feedback; fundamental problems; the role of magnetic helicity in the dynamo problem; numerical simulations; and dynamo action guided by a strong pre-existing field.

In addition to the Alfv èn wave, field-freezing and the consequent Lorentz force yield fast and slow magnetoacoustic waves, and magnetohydrodynamic shocks. The integral scalar and tensorial virial theorems are a reliable guide to the effect of large-scale, frozen-in magnetic fields on self-gravitating systems. Magnetostatic equilibria include, as a sub-class, domains with locally force-free fields. Magnetic helicity is a measure of a topological property of a field, which in a dissipative system may be destroyed more slowly than magnetic energy. The MHD energy principle is a valuable tool for the study of the dynamical stability of MHD equilibria. For systems with a zero-order velocity, e.g. rotating systems, one has normally to go straight to the perturbed dynamical and kinematical equations. In some MHD problems, instability can result when Ohmic dissipation leads to field reconnection. In general, one must consider fields with both poloidal and toroidal components.

We present the phenomenology of the weak interactions in a historical perspective, from Fermi’s four-fermion theory to the V−A current×current interaction. The experiments of C.S. Wu, which established parity violation, and M. Goldhaber, which measured the neutrino helicity, are described. We study in turn the leptonic, semi-leptonic and non-leptonic weak interactions. We introduce the concept of the conserved vector current and the partially conserved axial current and show that the latter is the result of spontaneously broken chiral symmetry with the pion the corresponding pseudo-Goldstone boson. We study Gell–Mann’s current algebra and derive the Adler–Weisberger relation. Strangeness changing weak interactions and the Cabibbo theory are described. We present a phenomenological analysis of CP-violation in the neutral kaon system and we end with the intermediate vector boson hypothesis.

This chapter covers the new on-shell methods that have been developed over the past twenty years for computing scattering amplitudes in quantum field theory. These methods break free from the traditional approach of Feynman diagrams. The chapter covers a subset of topics, setting up the basic kinematics, spinor helicities, spinor products, and the calculation of tree amplitudes.

The Dirac equation for an electron is introduced. This chapter describes the gamma-matrices necessary for formulating this and also the concepts of chirality and helicity. The basis states are called spinors and the chapter describes two types: Dirac spinors and Weyl spinors.

The phenomenon of spin-rotation coupling provides the key to the determination of the kernel. Imagine an observer rotating in the positive sense about the direction of propagation of an incident plane monochromatic electromagnetic wave of positive helicity. Using the locality postulate, the field as measured by the rotating observer can be determined. If the observer rotates with the same frequency as the wave, the measured radiation field loses its temporal dependence. By a mere rotation, observers could in principle stay at rest with respect to an incident positive-helicity wave. To avoid this possibility, we assume that a basic radiation field cannot stand completely still with respect to an accelerated observer. This basic principle eventually leads to the determination of the kernel and a nonlocal theory of accelerated systems that is in better agreement with quantum mechanics than the standard theory based on the hypothesis of locality.

Decomposition of the perturbations over FRW into scalar, vector and tensor perturbations. Physical and unphysical degrees of freedom. Gauge-invariant metric perturbations, Bardeen variables. Gauge-invariant perturbations of the energy-momentum tensor

This chapter describes the description of the proton as a bound state of partons. After a review of the properties of parton distribution functions, it introduces the evidence for a component of the proton responsible for its binding. It introduces the model of strong interactions as mediated by a spin 1 gluon and presents the evidence for this model from event shapes in electron-positron annihilation to hadrons.

This chapter works out the theory of electron-positron annihilation to muon pairs as a model for electron-positron annihilation to quarks. It explains that this naive model provides a good description of observed properties of the process of electron-positron annihilation to hadrons.

Starting from the spinor representation of the Lorentz group,Weyl spinors and their transformation properties are derived. The Dirac equation and the properties of its solutions are discussed. Graßmann numbers and the gener-ating functional for fermions are introduced. Weyl and Majorana fermions are examined.

The unitarity of the S-matrix is used to derive the optical theorem. The connection between Green functions and scattering amplitudes given by the LSZ reduction formula is derived. The trace and the helicity method are developed and applied to the calculation of QED processes. The emission of soft photons and gravitons is discussed. In an appendix, the connection between S-matrix elements, Feynman amplitudes and decay rates or cross-sections is derived.

A computational approach to rotations and Lorentz transformation is presented. The discussion starts with the mathematical properties of the rotation and the proper orthochronous Lorentz groups. Rotation and Lorentz transformations are implemented by exponentiating generators of SU(2). This approach allows rotations of states in a finite-dimensional Hilbert space to be carried out with the same machinery used to rotate ordinary vectors and it allows Lorentz boosts of Dirac spinors to be carried out with the same machinery used to boost four-vectors. Rotations and boosts can be applied to other objects with simple transformation properties. A computational approach to the helicity formalism used in scattering and decay theory is presented, and a library devoted to the manipulation of objects in spacetime is introduced.

Turbulence is an immense and controversial subject. Chapters 4, 5, and 6 present some ideas from turbulence theory that are relevant to flow in the oceans and atmosphere. In this chapter, we examine the connections between vorticity and turbulence. From ocean models that omit inertia, we turn to flows in which the inertia is a dominating factor. Vorticity is of central importance, and, in the case of threedimensional motion, we must take its vector character fully into account.