The study of objects' symmetries has many applications in physics and chemistry, for example, in the relativity theory and quantum theory, crystallography, and spectroscopy.
Crystals and molecules can be described using the number and type of symmetry operations that can be performed on them. (Valid symmetry operations are those that can be performed without changing their appearance, as in the example above, and the number and type of these operations depend on the geometry of the object to which the operations are applied.)
For the study of the evolution of physical systems, space-time symmetries have an important objective validity because it is the same for all observers, and does not depend on a particular perspective, being invariant in relation to the transformation group of the reference points. This condition of direct observation of spatiotemporal invariance principles is the starting point of the epistemological aspect of symmetries, and a prerequisite for the very possibility of discovering the natural laws of conservation of quantities.
The mathematics of symmetries is group theory: a quantitative expression and the formalism that describes the types of symmetries and the restrictions they have, demonstrating the symmetry transformations through mathematical operations. In these, something or some quantity is changed but the result remains the same, allowing us to learn about the nature of the object.
In most recent theories, symmetry has proved to be the principle that sustains the physical universe. Both theories of relativity and quantum mechanics involve notions of symmetries in a fundamental way. In today's scientific research, physics is studied in terms of the characteristics of approximate and exact symmetries. Even elementary physics has gained new interpretations when viewed in the light of symmetries in their equations. The symmetries found between theories represent the properties existing in nature or characterize a structure of the physical world.
The particles that make up all matter in our universe are governed by "internal symmetries" - symmetries that do not depend on space or time but govern the behavior and interactions of the particles themselves. These symmetries determine a large part of the structure we see in microphysics.
Physicists are trying to discover even deeper symmetries in the physical world - deeper symmetries in subatomic particles and deeper symmetries in space and time. In particle physics, symmetry can be used to derive conservation laws and to determine which interactions of particles can occur and which cannot (called "forbidden" interactions). Symmetries can predict new phenomena as they explain why certain events occur but not others, by tracing the form of the laws that govern them.
In the opinion of many scientists, it seems that at the most fundamental level, for an unknown reason, Nature prefers symmetrical beauty and is incredibly inventive in many forms of beauty.
If this is true, it is an important guiding principle for understanding the physical world, and a tool for exploring the laws of natural interactions. Perhaps, rather than regulating every law of physics individually, the universe obtained its set of symmetries that naturally led to the laws suitable for the emergence of life.
Symmetries are sometimes hidden, and in some sense, they are not even symmetries, it's just a convenient way of describing the system. They also have a normative role, applied as constraints on physical theories according to the gauge principle. As an invariance requirement for a transformation group, it restricts the form that a theory can take and limits the types of quantities that may play a role in this theory, as well as the form of their fundamental equations.
The symmetries of the principle of gauge invariance are equivalents much useful to uncomplicate the parameters of an equation, simplifying the number of redundant degrees of freedom, a very practical benefit for complicated calculations. Complex mechanisms and interactions can be better understood through the study of the gauge symmetries, that fixes the dynamics. Instead of looking at the interaction given by arbitrary terms, one can just specify the field content and the transformation group.
Accepting gauge invariance as a "yardstick" and as a law of symmetry is theoretically very inviting since the physical quantities have a relational structure that can be measured as being gauge variables everywhere. In addition, gauge variables describe the connections with which the systems couple. According to the law of gauge symmetry, the symmetries can be measured using different criteria: a particular choice of gauge demonstrates a specific property, for example, the rotation of the particle; and the nuclear forces of particle interactions can be derived from other gauge invariances.
Gauge invariance is not a metaphysical redundancy of our mathematics, it is an indication of the relational character of the fundamental observables in physics. It refers to the ubiquitous relational properties between entities, such as relative velocity, relative location, relative orientation in the inner space, and so on. Gauge interactions reveal the relational structure of our universe because Nature is described by certain relative quantities that refer to more than one object.
Why gauge theory? Why are they so successful and useful?
Certainly, because they are helping us to classify everything that exists. Two apparently different objects or phenomenologies prove to be indeed the same as soon as we are able to discover a gauge that connects them (eg, electricity and magnetism). And conversely, when a valid object and gauge is given, we are capable of predicting the transformations of the initial object.
Symmetries have such a widespread generality that they can be used in pure theoretical physics to explain the structures of the physical laws and the occurrence of certain events, following the symmetries or asymmetries of the situation, like temporal symmetry, charge, parity, and so on.
Gauge symmetries and other types of symmetries reflect deep ontological properties of the laws of Nature and of physical reality. The information on fundamental physical knowledge comes to us through methodological and epistemological derivatives such as symmetries.
Careful caliber fixation can greatly simplify calculations, but this becomes progressively more difficult as the physical model becomes more realistic. Its application to quantum field theory (where quantum interactions between particles are described in terms of the interaction between the corresponding underlying fields, or "perturbations" of free fields) shows the best example of a paradigm of symmetry interactions. This is a methodological argument for the study of symmetries that can explain and justify gauge fixation.
The development of a group theory of physical symmetries is a way towards a unified (theoretical) description of the fundamental forces of nature (gravitational, weak, electromagnetic and strong), in terms of their underlying symmetry groups, with the consequent possibility of unifying the different quantifications by means of a unification of the corresponding transformation groups.
Physics' symmetries offer some ontological, epistemological and methodological interpretation possibilities. Taking a stand will depend on the preferred approach in philosophy of science, including such topics as realism, the laws of nature, the relationship between mathematics and physics, the nature of theoretical entities, and so on. To understand the status and the meaning of physical symmetries is a challenge for physicists and philosophers.
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