Symmetry

1While the concept of symmetry is not one that has been touched in the previous QED model, it is a concept of some importance in the general description of quantum field theory. Therefore, it seems worthy of some introduction at this point. In the field of physics, symmetry has come to imply some form of ‘invariance’ or lack of change, typically when subject to a coordinate transform. This concept is particularly important in quantum physics, because it would seem that all the laws of nature reflect some form of fundamental symmetry. At the classical level of reality, we have an intuitive idea of an object being symmetric, e.g. under rotation, although we might also understand that this attribute of physical symmetry may only be an approximation. For example, if we examine a classical object under a microscope, we might quickly realise that physical imperfections are not replicated in perfect symmetry. Of course, these imperfections in the definition of classical symmetry may have no real impact on the physics of classical everyday objects, because classical results might be said to implicitly represent an approximation. However, at the quantum level of reality, the mathematical description of symmetry of quantum objects, e.g. electrons, ceases to be an approximation. So, in contrast to a classical object, a quantum object, such as an electron, has a very limited number of states of existence. In plain English, we might simply say that an electron has to remain perfectly symmetric under any transform change.

But how does such an idea change the physics?

At one level, we might simply assume that if a classical object only has approximate symmetry, then a more exact symmetry would only facilitate the need for a small increase in accuracy in the underlying physics. However, the assumption that exact symmetry in quantum objects would only make a small difference, i.e. based in the accuracy of the approximation, appears to be incorrect. In fact, when a quantum object is defined in terms of its symmetry, it also implies that the rules of classical physics cease to be applicable and the system must be modelled using the more complex, and typically far less intuitive, rules of quantum physics. So while we have not been too specific about the details of symmetry, it might still be generally understood as to why the idea becomes more important in quantum mechanics. When physical systems make the transition from classical to quantum objects, the idea of symmetry changes from an approximation to an exact description, which also has implications on the mathematics. In fact, it is often argued that the underlying nature of a quantum objects becomes so closely associated with its mathematical descriptions that it becomes the only meaningful description, i.e. the idea suggests that physical objectivity may become a meaningless concept; although such an idea is still subject to much philosophical debate.

An Outline of Gauge Theory:

In the current context, ‘gauge invariance’ corresponds to ‘gauge symmetry’ that can be associated to the property of an unobserved field, which when subject to different configurations results in identical observable quantities. A theory that supports such a property is called a ‘gauge theory, while a transformation from one such field configuration to another is associated with a ‘gauge transformation’. So gauge invariance is a type of symmetry that describes the property of a field to undergo a transformation that still results in the same identical observable quantities.

We can also introduce the idea of gauge symmetry in terms of classical electromagnetism. For example, an electric field can result in an observable quantity called the potential difference, i.e. voltage, which is a relative measure typically defined with respect to planet Earth. Of course, it would be possible to choose another ‘zero’ reference to measure the relative potential [V] of the electric field. However, given that the potential [V] can be a parameter of Maxwell's equations, so must any other measure of potential after the appropriate gauge transformation. Therefore, the laws of physics governing electricity and magnetism, as defined by Maxwell equations, are invariant under gauge transformation, such that Maxwell's equations must have the attribute of  gauge symmetry.

So this kind of  ‘invariance’ under a transformation also comes under the general description of symmetry, but as outlined, this attribute becomes increasingly important in any quantum description. In fact,  the suggestion is that gauge symmetry becomes so important in the ‘field’ of quantum mechanics that we might wish to consider its role in unifying quantum mechanics with electromagnetism, i.e. quantum electrodynamics. In this context, gauge symmetry must apply to both electromagnetic waves and electron waves from which it might be inferred that both of these symmetries have to be related. Therefore, if some gauge transformation is applied to electron waves, then a corresponding transformation must be applicable to the potentials that describe electromagnetic waves. As such, it appears that a gauge theory can be linked to a field theory, where the fields and potentials are described by a symmetry group, i.e. the gauge group. As highlighted, classical electrodynamics exhibits gauge symmetry because it is invariant under redefinition of the electrostatic potential, which ultimately aligns to the conservation of electric charge. However, in quantum theory, the phase of the wave function describing a system can be changed without altering the outcome provided that all wave functions are changed in the same way, everywhere in space. So, in a gauge theory there has to be a group of transformations of the field variables, i.e. the ‘gauge transformations’, which leave the physics of the quantum field unchanged and it is this requirement that defines the ‘gauge invariance’ and the symmetry governing its equations. Theoretical physicist adopted the idea of gauge invariance because it appear to be fundamental to so many ‘particle’ interactions, i.e. gravitational, electromagnetic, strong, and weak, such that it has become a building block of the standard model.

An Application of Symmetry:

So, over time, the idea of symmetry has become accepted as an important concept, which was then applied to the weak and the electromagnetic interactions of hadrons; and later, extended to encompass low energy, strong interactions. This said, the initial development of this idea appeared to have reached a restrictive limit, around 1967, when some of its predictions were seen to be in conflict with experimental data. Therefore, during much of the 1950’s and 1960’s, it may be fair to say that progress in classifying, and understanding, the ever-increasing number of hadrons only reflected the general idea of symmetry. So, while symmetry did become a fundamental concept of modern particle physics, it is probably true to say that it was initially only used as a tool to help the organisation and classification of particles. However, the theoretical idea of symmetry was to receive support in the form of two developments in the second half of the 1950’s:

  • Lee and Yang realised that parity is not conserved in the weak interactions

  • Yang and Mills extension of spin symmetry to a local symmetry, analogous to the gauge invariance in QED.

Yang and Mills essentially introduced a new framework to describe elementary particles, which is still used in the foundation of most elementary particle theories. However, while it is claimed that its predictions have been successfully tested, its mathematical foundations are still another matter of debate. However, the perceived success of the Yang-Mills theory lies in its ability to describe the strong interactions of elementary particles in terms of what is often termed the ‘mass gap’ in which all quantum particles have positive masses, even though its wave-like properties travel at the speed of light.  Later, it was noted that in quantum field theories, symmetries could be realized in different ways. For example, it was possible to have the Lagrangian [L] invariant under some symmetry, although this symmetry could not be maintained in the vacuum ground state. In this case, it was said that symmetry was ‘spontaneously broken’. When symmetry is broken in the global context, the boson is described as having zero mass and spin. If the symmetry is broken in the local context, the ‘global’ boson disappears and is replaced by a gauge boson with mass, which is then associated with the ‘Higgs mechanism. In 1967, Steven Weinberg and Abdus Salam independently proposed a gauge theory of the weak interactions that unified the electromagnetic and the weak interactions and made use of the Higgs mechanism. Their model incorporated an idea previously forwarded by Sheldon Glashow, in 1961, as to how a gauge theory might be used to describe weak interactions in which the weak force  was mediated by gauge bosons. Originally, Glashow's approach had been questioned because of its dependency on gauge bosons with mass and the suspicion that such a theory would not support renormalisation. Only later, in 1972, did Gerard ’t Hooft proved that this was not the case and so gauge theory and the mathematical framework of symmetry in QFT continued to play a major role in the development of modern particle physics.