II.1 Symmetry and Quantum Mechanics

The concept of symmetry and conservation laws are extraordinarily useful in Quantum Mechanics and the Standard Model of Elementary Particles, mainly when verbalized via Group Theory and Lie algebra, which play essential roles in the mathematical formulation of Quantum Field theories. The forms of the fundamental quantum operators, such as energy, as a partial time derivative, and momentum, as a spatial gradient, become very clear by considering the initial state and then slightly changing some parameters, such as displacements, durations or angles. Additionally, the invariance of certain quantities can be seen by making such changes, which illustrate their conservation.

For example, elementary particles have internal symmetry defined by the baryon number, which is one of the essential properties of hadrons, which are made of three quarks. The baryon number of hadrons is . On the other hand, when this number is , this describes particles such as bosons, leptons or mesons, which are made of one quark and one anti-quark. Because the baryon number is conserved during nuclear interactions, symmetry implies that there must exist another class of particles with a baryon number of , which are the hadrons of antimatter. Other famous symmetries include the conservation of charge, parity and time reversal. Physicists were astonished when it was demonstrated that parity is maximally violated in some weak interactions, and it is still not understood yet why this is the case, or why certain other symmetries, such as  and  are only slightly violated, as will be discussed in section II.2.

After the evident success of Quantum Mechanics in 1920s, Quantum Field Theory tried to combine classical fields with Special Relativity and Quantum Mechanics in one theoretical framework for constructing quantum mechanical models of subatomic particles, as well as quasi-particles in condensed matter physics. Because it is based on Special Relativity, the Poincar group is the basic symmetry of Quantum Field Theory. This means that the laws of relativity cannot be reduced, but they can be extended by other symmetries, which can be deduced from experimental results. The Standard Model, as the current accepted description of subatomic quantum physics, was found in this manner.

In classical physics, a field is a quantity, represented by a number or tensor, that has a value for each point in space and time, but in the quantum theory of fields, a field occupies space, contains energy, and its presence precludes a classical true vacuum. This led physicists to consider electromagnetic and some other fields to be a physical entity. In the beginning, since it was able to combine Special Relativity with Quantum Mechanics, Quantum Field Theory was thought to be fundamental, but due to its continued failures in quantizing General Relativity, it is only a very good effective field theory, to some yet undiscovered more fundamental one. When developed further, the theory often contradicts observation, but its creation and annihilation operators can be empirically tied down, or re-normalized. The essence of the problem is that it is impossible to quantize space-time itself without supposing some background geometry or topology in which it is quantized. In this regard, the Duality of Time Theory seems to be the only way out of this riddle, because it produces a dynamic and self-contained space-time, without any presupposed background. This will be even extremely easier than all current approaches, such as Strings Theory and Loop Quantum Gravity, because the resulting genuinely-complex time-time geometry is originally Euclidean, and continuous, but it becomes intrinsically granular and hyperbolic when its perfect symmetry is broken into the two orthogonal arrows of time, as we introduced Chapter I.

In Quantum Field Theory, particles are the field quanta, or excited states, of the underlying fields, and the interactions among these particles are described by interaction terms among the corresponding underlying quantum fields. Each one of the four fundamental interactions, or forces, is associated with a field with is responsible for some type of particles, so for example: electrons and photons are the quanta of the electromagnetic field, and the graviton is the quanta of the gravitational field. These interactions are conveniently visualized by Feynman diagrams. The main difference between Quantum Mechanics and Quantum Field Theory is that the latter allows for the creation and annihilation of particles, so it is capable of explaining how matter is being created, rather than just describing the outward interactions between objects and particles.

In early 1930s, Eugene Wigner (1902-1995) and Hermann Weyl (1885-1955) were the first to apply the theory of group representations to Quantum Mechanics. They recognized that this would allow solving problems in atomic spectroscopy very easily, as well as addressing other foundational questions, which eventually lead to the new formulation of Quantum Mechanics.

However, at the beginning, the introduction of Group Theory to Quantum Mechanics was not universally welcomed, because it did not seem to be necessary, since anything that can be done by Group Theory can be done without it. However, soon it became clear that Group Theory is like a map that is capable of showing the whole picture and giving more information with very little input. Wigner writes in the preface to the 1959 English edition of hisGroup Theory and Its Application to the Quantum Mechanics of Atomic Spectra:

When the original German version was first published, in 1931, there was a great reluctance among physicists toward accepting group theoretical arguments and the group theoretical point of view. It pleases the author that this reluctance has virtually vanished and that, in fact, the younger generation does not understand the causes and basis for this reluctance. Of the older generation it was probably M. von Laue who first recognized the significance of Group Theory as the natural tool with which to obtain a first orientation in problems of Quantum Mechanics [34].

In accordance with Group Theory, Quantum Mechanics showed that, unlike the macroscopic world which is characterized by various approximate symmetries, the symmetry of elementary particles is exact, because for example electrons are identical. Quantum systems are also described by functions that are subject to the usual symmetry operations, such as rotation and reflection, as well as other symmetrical properties that are manifested in more esoteric spaces. In all these fields, symmetry can be expressed by certain operations covered by groups and their representations, where symmetry matrices are acting on the members of a vector space, which then can be classified according to their symmetry. All the observed spectroscopic states of atoms and molecules correspond to such symmetrical functions, which give the selection rules that specify which transitions are observed. Spectroscopic classifications are essentially identifications of the irreducible representation to which the state belongs.