An Eventful Journey to Unification of All the Fundamental Forces

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• Contents

In order to appreciate the advances in physical science that have taken place during the last fifty years it is worthwhile retracing the history of their development. Nowadays we are all agreed that four fundamental forces exist, so let us trace how such a conclusion was reached. I should add that there are some suggestions of a fifth force, connected with dark matter, but they are not supported at all convincingly by experiment, so I will not dip my feet into that pool.

The idea of a force field between two interacting bodies grew from the law of conservation of energy and the concept of a potential. The notion of energy is ubiquitous today and is taught from an early age, but we must realise that it had a long gestation period before it achieved its present preeminent status. Nowadays we recognise many different forms of energy (all with intrinsic ability to do work): mechanical, heat, electrical, chemical etc. It is an article of scientific gospel that energy can never be created nor destroyed but only converted from one form into another. It owes its modern formulation to Mayer and Helmholtz. Eventually, when Einstein established the connection of rest energy E to mass m through his famous equation E = mc², all the quantitative measurements in chemistry and atomic and nuclear physics were transparently explained…

In this chapter we will try to explain how disturbances in fields can be resolved into components (or “modes”) and how they propagate; thus we cannot avoid injecting some mathematics into the discussion. More intricate details about such decompositions have been hived off to the Appendix for readers with appropriate mathematical background. I shall start with a simple example which illustrates a lot of points and progress from there to the four basic force fields. Inevitably I will be obliged to introduce some formalism but most of it will be elementary trigonometry…

Quantum theory was launched at the turn of the twentieth century by Planck’s introduction of the fundamental unit h ≡ 2πħ. One of the outstanding puzzles at that time was why classical physics failed to explain the spectrum of a hot blackbody despite accurate measurement of its characteristics. Only at long wavelengths was there any resemblance to the classical Rayleigh–Jeans law, but at higher frequencies the classical predictions failed miserably and, worse still, led to an unacceptable infinite value for the total energy of a radiant body…

Before going any further we need to appreciate a few things. Communication is the transmission of information or exchange of property between one system and another. In quantum field theory this is achieved by the exchange of at least one force field quantum. Such disturbance quanta Φ(x) in and of the medium (even the vacuum) obey field equations of one sort or another. When they satisfy the wave equation □Φ = 0 they lead to transmission with the speed of light and correspond to particles of zero mass; the communication is as fast as possible but is not at infinite speed. In turn it leads to a ‘potential’ V (r) ∝ 1/r between two systems separated by r which is ‘long range’ as in electricity: it continues to be felt at large r even though it has been weakened greatly with distance. See Section A.2 for the explanation…

The last five chapters, augmented by the Appendix, have been in the nature of a refresher course on quantum fields for those of you who have come across that topic. And for those of you unfamiliar with the subject, those chapters have tried to convey the basic ideas about particles and their associated fields. It is now time to introduce concepts which underlie the entire motivation for this monograph and how they manage to produce a unified picture of all the known natural forces.

The critical question was raised in the last section as to how many attributes are required to describe the known elementary constituents. As always we have to be guided by experiment but must also pay heed to economy. If we invoke an abundance of different ζ then we are simply undermining the entire property rationale. Thus with the establishment of QCD as the communicator of strong force it is quite obvious that we require three distinct colours, normally taken as red, green and blue, which we will attach to ζ¹,ζ² and ζ³ respectively; we cannot do with less. On the other hand, electroweak forces indicate that the leptonic pair (ν⁰, e⁻) are two components of a weak isodoublet, and this has nothing to do with QCD, so let us include two more ζ: namely the pair (ζ⁰,ζ⁴). As to what quantum numbers are carried by each of these properties, that is a separate issue which we will deal with when we come to the gauge field mediators and their couplings but we can already guess what they are. Furthermore the connection with handedness of the weak isodoublet is really important and will be covered later since it underpins the Standard Model.

During the last few years of his life, Einstein strove to unify electro-magnetism with gravity in a geometrical way. He was much taken by the idea of Kaluza (further developed by Klein) that a fifth bosonic [25] coordinate x⁵ added to the four spacetime coordinates (x⁰, x¹, x², x³) will achieve that end, but Einstein was never completely satisfied with the idea, as I shall sketch. Einstein knew that he had to achieve that unification first before turning to the other forces of nature, of which he was well aware, but which he studiously ignored initially…

Before going further I ought to mention other physicists who have shared aspects of the philosophy behind this monograph. Foremost amongst these has been Casalbuoni and his collaborators [56], who have also considered blends of Fermi oscillators, instead of using anti-commuting coordinates, but effectively with the same sort of algebra. They have attempted to obtain a unified description of quarks and leptons just by using three coloured creation operators plus an additional colourless one. To get generations they include a further set m of Fermi oscillators and are led to Lie groups of type O(10) × O(2m). However they did not go so far as geometrising their work nor unifying it with gravity…

This and the next chapter contain some hairy mathematics, so gird your loins. I will try my best to explain what the various algebraic expressions and equations mean so as not to lose you. For practitioners of GR it will be a walk in the park and I hope that familiarity does not breed contempt…

This section is concerned with extending the geometrical picture of electromagnetism just described to all the other known forces: the denouement of an eventful journey. We will start with QCD, marry it with electromagnetism before tackling the electroweak standard picture with its peculiar handedness. Finally we shall be combining them all into one full-scale ‘where–when–what’ package. So hold on to your horses as the route brings in some bumpy mathematics.

Most physicists would agree that the theory of quantum fields and particles has reached something of an impasse. It has become too ‘comfortable’, so to speak, in so far as the experimental results that pour out plentifully from the particle colliders accord pretty well with the Standard Model predictions. It has been a disheartening time for budding new theorists searching for a grand overarching scheme that encompasses the known gauge theories. The reason is that none of the currently popular ideas — and some of them have been quite brilliant — such as SUSY, strings, unseen dimensions, etc., has gained any experimental support. Other grander concepts such as universes with separate landscapes, anthropic arguments, holographic perspectives are in my view more theological than physical since they cannot be checked or don’t lead to new fundamental progress. Yet we are confronted by a host of puzzling facts like matter–antimatter asymmetry, the ratios of coupling constants, the nonzero but miniscule cosmological constant magnitude, the absence of magnetic monopoles, etc. They all demand sensible explanation. But none of the proposals forwarded thus far is utterly convincing or is supported by compelling observational evidence…

The following sections are included:

• Fourier series and integrals
• Potentials and Green functions
• Field interactions and perturbation theory
• Spontaneous symmetry breaking and mass generation
• Fermions and the Dirac equation
• Veneziano dual resonance model

The following sections are included:

• Glossary
• Bibliography