The Thermodynamics of Quantum Yang–Mills Theory

Theory and Applications

Ralf Hofmann

• Publisher: WSPC
• Year: 2016
• eBook ISBN: 9789813100503
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

This latest edition enhances the material of the first edition with a derivation of the value of the action for each of the Harrington–Shepard calorons/anticalorons that are relevant for the emergence of the thermal ground state. Also included are discussions of the caloron center versus its periphery, the role of the thermal ground state in U(1) wave propagation, photonic particle–wave duality, and calculational intricacies and book-keeping related to one-loop scattering of massless modes in the deconfining phase of an SU(2) Yang–Mills theory. Moreover, a derivation of the temperature–redshift relation of the CMB in deconfining SU(2) Yang–Mills thermodynamics and its application to explaining an apparent early re-ionization of the Universe are given. Finally, a mechanism of mass generation for cosmic neutrinos is proposed.Advanced students, postdocs and researchers in theoretical physics and mathematics, as well as experimentalists.Selfdual Field Configurations; Calorons; Thermal Ground State; Inert Adjoint Scalar Field; Caloron Overlap; Effective Coupling; Attractor Solution; Quantum of Action; Electric-Magnetically Dual Interpretation; Unitary-Coulomb Gauge; Mandelstam Variables; Constraints; Radiative Corrections; Caloron Center; Caloron Periphery; Wave Propagation; Photons; Monopole Condensate; Preconfining Phase; Center-Vortex Loops; Hagedorn Transition; Polarization Tensor; Black-Body Anomaly; Cosmic Microwave Background (CMB); Unexplained Extragalactic Emission; Planck-Scale Axion; CMB Temperature-Redshift Relation; Cosmic Neutrinos Key Features:This is the first research book of its kind with essentially nonperturbative and analytical approach to Yang–Mills thermodynamicsAn overview on relevant topological field configurations and interconnection between deep theory and physical predictions in the realm of low-temperature photon gases

PART 1

Theory

Chapter 1

The Classical Yang–Mills Action

1.1 Historical Remarks

The extraction of the specific gauge principle of electromagnetism from accumulated and theoretically distilled experience, the discovery that generalizations of this principle provide in a fundamental way for dynamical physical law, and subsequent successful exploitations in the realm of particle physics belong to the greatest triumphs of the human mind.

The evolution of gauge theory started to take its course when, during the first half of the nineteenth century, M. Faraday and others undertook their insightful experiments on static electric and magnetic phenomena, on electromagnetic induction, and on light propagation in vacuum and media. These results constituted the experimental basis for the paper “A dynamical theory of the electromagnetic field” authored by J. C. Maxwell in 1864. An important result of Maxwell’s theory was the prediction that light is nothing but the wavelike propagation of the electromagnetic field. Twenty years later H. Hertz could confirm this consequence of the theoretical unification of electricity and magnetism experimentally by verifying that periodically accelerated electric charges emit electromagnetic waves.

It was A. Einstein who in 1905 understood that the covariance of Maxwell’s equations is reconcilable with G. Galilei’s principle of relativity only for those coordinate changes between uniformly moving frames that leave the speed of light unchanged. The respective Lorentz-covariant form

of Maxwell’s equations, j_ν the electromagnetic current four-vector and F_μν

∂_μA_ν−∂_νA_μ the antisymmetric, rank-two, electromagnetic field-strength tensor, makes an invariance of these equations under the transformation

most explicit (φ a Lorentz-invariant and real function of space and time). The quantity A_μ (μ = 0, 1, 2, 3) is known as the gauge potential.

That the symmetry of Eq. (1.2) is actually associated with the local action of the unitary group U(1) was understood in the frame-work of quantum mechanics, where it became clear that the dynamical four-current j_ν can be written as a bilinear of a quantum mechanical wave function ψ (principle bundle). Demanding the covariance of the equation obeyed by ψ under local phase changes (U(1) rotations) makes the gauge potential A_μ a connection¹ on the manifold of the gauge group. Notice that the transformation law of Eq. (1.2) can be rewritten in the form

where Ω is an element of the group U(1): Ω

e^iφ.

The concept of covariance or invariance of dynamical equations under gauge transformations is generalizable to non-Abelian groups. It was W. Pauli who first formulated such a generalization and together with his collaborators Barker and Gulmanelli tried to investigate its physical consequences. When, during a Princeton seminar in early 1954, C.-N. Yang reported about his work with R. L. Mills on a local four-dimensional field theory with SU(2) gauge invariance Pauli was in the audience and very explicitly expressed his dislike. According to A. Pais [von Meyenn (1999)] Yang described the situation as follows: “Pauli asked, ‘What is the mass of this field B_μ′. I said we did not know. Then I resumed my pres...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Preface of Second Edition
• Preface of First Edition
• Contents
• Part 1 Theory
• Part 2 Applications
• Acknowledgments
• Author Index
• Subject Index