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Group theory is the mathematical theory that we use to describe symmetries. Almost everything in modern physics has its origin in some symmetry consideration and for this reason, group theory is used everywhere in modern physics.
We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs these operations. Such a super-mathematics is the Theory of Groups.
Sir Arthur Stanley Eddington
In the 1970’s, high energy physicists pursued Lie algebra theory as a valuable tool to characterize all the gauge interactions. These are now understood to be SU(3) for the strong force (which describes the interacations between quarks, which are the constituents of hadrons such as the proton), and SU(2) × U(1) for both the weak and electromagnetic interactions of quarks and leptons (such as the electron). This is an important feature of the standard model of particle physics [YM,We,Sa,Gl]. Grand unification was an effort to combine these symmetries as subgroups of a unifying group such as SU(5). Superstring unification provides an alternative mechanism to combine symmetries.
The interpretation of fundamental force laws in terms of group theory is now commonly understood in terms of E. Noether’s theorem which identifies the elements of the Lie algebra with the charges conserved in the interactions [No]. The symmetry is used to label the physical states: the eigenvalues of the Cartan subalgebra are the quantum numbers of the elementary particles. The quantum numbers are known as charge, spin, hypercharge, isospin, etc. depending on which group is being considered. Modern high energy theorists effectively think of the elementary particles of the strong, weak and electromagnetic forces as irreducible representations of the direct product of the Poincar´e group and SU(3) × 2 SU(2) × U(1). Selection rules derived from the conserved charges limit allowed transitions of quantum numbers. Explicit solutions of four-dimensional quantum field theory transition amplitudes, however, are known only in a perturbative expansion, i.e. they are not known exactly. This weak coupling perturbative approximation is extremely useful in electroweak theory, but the strong coupling problem of why the quarks are confined inside the hadrons remains more elusive
A man who is tired of group theory is a man who is tired of life. Sidney Coleman
An important distinction is between continuous groups and finite groups.
Important Finite Groups
The combination of charge conjugation $C$, parity $P$ and time reversal $T$, called $CPT$ is a fundamental symmetry of nature.
However, each individual symmetry is broken. The discovery that $C$, $P$, $T$, and also the combination $CP$ are broken was one of the biggest surprises in modern physics and there is still no theoretical explanations.
Internal Symmetry Groups
In addition to the discrete groups related to the Poincare group there are many additional groups that are of interested in modern physics.
The "atoms" of all groups are called "simple groups" since all other groups can be written as products of these groups.
All simple finite groups are known and a nice essay on the classification of them can be found here. The standard reference is the Atlas of Finite Group by John Horton Conway.
Discrete groups are used in crystallography and also in flavor models that try to solve the flavor puzzle.
Important Continuous Groups
Internal Symmetry Groups
SO(n) | SU(n) | |||||||||||||||||||||||||||
... | ||||||||||||||||||||||||||||
E8 | SO(18) | |||||||||||||||||||||||||||
E6 | ||||||||||||||||||||||||||||
SO(10) | ||||||||||||||||||||||||||||
Pati-Salam | SU(5) | |||||||||||||||||||||||||||
SU(3) | ||||||||||||||||||||||||||||
SU(2) | ||||||||||||||||||||||||||||
U(1) | ||||||||||||||||||||||||||||
The "atoms" of all groups are the so-called simple groups. Every other group can be written as a product of the simple groups. All simple groups are known and this classification tells us what groups we can use in physics.
On the left-hand side, some of the most important groups that are used in physics are shown. Most of them are important in the context of grand unified theories. The full classification is shown here:
Spacetime Symmetry Groups
Diffeomorphism Group | ||||||||||||||||||||||||||||
Conformal Group | ||||||||||||||||||||||||||||
DeSitter Group | Anti DeSitter Group | |||||||||||||||||||||||||||
Poincare Group | ||||||||||||||||||||||||||||
Galilei Group | ||||||||||||||||||||||||||||
For a long time, physicists assumed that the Galilei group is the correct spacetime symmetry group. However, then it was discovered that the speed of light is constant and this meant that instead of the Galilei group, the Poincare group had to be considered.
Nowadays, we know additionally that the cosmological constant is non-zero. This, in turn, means that instead of the Poincare group, we have to use the deSitter group.
A classification of all possible "kinematical" group by deforming the known groups was done in
In addition, the conformal group and the diffeomorphism group are important. These groups cannot be understood as deformations of the other groups. However, all the other groups are subgroups of them.
The diffeomorphism group is the most general spacetime group and consists of all sufficiently smooth transformations. The conformal group is the correct spacetime symmetry group for massless particles.
Important Concepts
Recommended Resources
Three books that explain group theory (but do not proof things) with a focus on how it's used in modern physics are
Good books that focus a on mathematical aspects (and also include proofs) are:
Definition
A group $(G, \circ)$ is a set $G$, together with a binary operation $\circ$ defined on $G$, that satisfies the following axioms
For Group Theory in Quantum Mechanics see http://math.ucr.edu/home/baez/lie/lie.html
The case I have chosen to treat in this paper concerns the question as to whether the group concept should be extended to, or even subsumed by, the groupoid concept. Over a period stretching from at least as long ago as the early nineteenth century, the group concept has emerged as the standard way to measure the degree of invariance of an object under some collection of transformations.4 The informal ideas codified by the group axioms, an axiomatisation which even Lakatos thought unlikely to be challenged, relate to the composition of reversible processes revealing the symmetry of a mathematical entity. Two early manifestations of groups were as the permutations of the roots of a polynomial, later re-interpreted as the automorphisms of the algebraic number field containing its roots, in Galois theory, and as the structure-preserving automorphisms of a geometric space in the Erlanger Programme. Intriguingly, it now appears that there is a challenger on the scene. In some situations, it is argued, groupoids are better suited to extracting the vital symmetries. And yet there has been a perception among their supporters— who include some very illustrious names—of an unwarranted resistance in some quarters to their use, which is only now beginning to decline.The importance of mathematical conceptualisation by David Corfield
Consider two descriptions of the same phenomenon: (1) Space is homogeneousand isotropic. (2) Space is invariant under translations and rotations of coor-dinates. Statements in the logical form of (1) were exclusive in pre-twentiethcentury physics. Statements in the form -of (2) dominate twentieth centuryphysics; quantum mechanics contains various representations of the same phy-sical state and rules for transforming among them. Description (1) appearsclean; it describes nature without explicit conventional and experientialnotions. Description (2) is all contaminated; it invokes conventional coordi-nates and intellectual transformations of the coordinates. The coordinates areusually interpreted as perspectives of observations; so they are somehowrelated to human subjects. However, physicists agree that (2) is more objective,for it uncovers the hidden presuppositions of (1) and neutralizes their undesir-able effects. They retrofit the conceptual structure embodied in (2) into classi-cal mechanics to make it more satisfactory. The statement (1) is often interpreted in a framework of things; (2) can beinterpreted in the framework of objects. The object framework includes thething framework as a substructure and further conveys the epistemological ideathat the things are knowable through observations and yet independent ofobservations. The two frameworks exemplify two different views of the world From "How is Quantum Field Theory possible" by Auyang
Resources:
There are groups that have nothing to do with symmetries like, for example, braid groups:
„Once we have the notion “group,” we can look for other examples. It turns out there are many examples of groups that have nothing to do with symmetries, which was our motivation to introduce the concept of a group in the first place. This is actually a typical story. The creation of a mathematical concept may be motivated by problems and phenomena in one area of math (or physics, engineering, and so forth), but later it may well turn out to be useful and well adapted to other areas. It turns out that many groups do not come from symmetries“
Love and Math by Frenkel
To help your intuition further: Lie groups are "almost always" matrix groups as follows. There is a corollary to a difficult theorem known as Ado's theorem that every Lie algebra can be realized as a Lie algebra of square matrices. The same is not true of Lie groups: not every Lie group can be represented as a group of matrices but it is almost true (a consequence of the Peter-Weyl theorem is that every compact group can be realized as a group of square matrices). Certainly, since we can find a square matrix realization for every Lie algebra, we can build a matrix Lie group with that algebra as its Lie algebra through the matrix exponential function; then we find that matrix group's universal cover and this is where we sometimes fail to get a matrix group. This is not typical and the first Lie groups that were not also matrix groups (so called metaplectic groups) weren't found until 1937. These oddballs are all covering groups of noncompact groups.
By exponentiating the Lie algebra elements, which can always can be written as matrices, we get representations of the corresponding universal covering group that belongs to this Lie algebra.
There are two views about symmetry, one of which Michael was expressing. I'll rephrase it as 'symmetry is not fundamental in the structure of physics'. It's just that as you let the renormalization group evolve, you get attracted to fixed points which are often more symmetric than where you started. This particular approach is taken to an extreme by Holga Nielson, who would like to start with nothing, no symmetry principles, not even any logical principles, and somehow get to a fixed point in which everything emerges. My view of this approach is, 'garbage in, beauty out'. I really don't like that point of view.
David Gross in "Conceptual Foundations of Quantum Field Theory" , Edited by Cao
This line of research goes by the name "random dynamics".
What Nielsen imagines is that the whole cosmos is just at the point of a phase transition between two phases. He and his colleagues, such as Don Bennett, try to demonstrate that many of the observed properties of the elementary particles arise simply from this fact, independently of whatever the fundamental laws of physics are. They want to say that, just as bubbles are universally found in liquids that are boiling, the fundamental particles we observe may be simply universal consequences of the universe being balanced at the point of a transition between phases. If so, their properties may to a large extent be independent of whatever fundamental law governs the world.
The Life of the Cosmos by Lee Smolin
His starting point was the theory of algebraic equations (such as the quadratic, or second-degree, equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially.
But Galois also showed that solutions to such equations must be linked by symmetry. For example, the solutions to x5 = 1 are five points on a circle when plotted onto a graph comprised of real numbers along one axis and imaginary numbers on the other. He showed that even when such equations cannot be solved, he could still glean a great deal of information about the solutions from studying such symmetries.https://www.nature.com/articles/d41586-018-03423-x