✎ Group Theory [The Physics Travel Guide]

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====== Group Theory ======

Group theory is the mathematical theory that we use to describe [[basic_tools:symmetry|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.

----

<blockquote>
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.

<cite>Sir Arthur Stanley Eddington</cite>
</blockquote>

<blockquote>
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

<cite>https://arxiv.org/pdf/hep-th/9601117.pdf</cite>
</blockquote>

<blockquote>A man who is tired of group theory is a man who is tired of life. <cite> Sidney Coleman</cite></blockquote>

An important distinction is between continuous groups and finite groups.

* __Continuous groups__ describe symmetries that can be parametrized by a continuous parameter. Continous groups are also called __Lie groups__. 
  * __Finite groups__ cannot be parametrized by such a continuous parameter.

**Important Finite Groups**
<WRAP group>

<WRAP half column>
__Spacetime Symmetry Groups __
<diagram>
||||| AA|| |AA=[[advanced_notions:cpt_symmetry|CPT]]
||||,| -|^|- |-|-|. | | | |
||| AA||||BB |AA=[[advanced_notions:cp_symmetry|CP]]|BB=[[advanced_notions:t_symmetry|T]]
| |,| -|^|- |. | | | |
| AA||BB |AA=[[advanced_notions:c_symmetry|C]]|BB=[[advanced_notions:ü_symmetry|P]]
</diagram>

__Internal Symmetry Groups __

</WRAP>

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.

A nice essay on the classification of all simple finite groups can be found [[https://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups|here]]. The standard reference is the Atlas of Finite Group by John Horton Conway
</WRAP>
</WRAP>

**Important Continuous Groups**

__Internal Symmetry Groups __

<diagram>
|||||||X2||X3||X2=SO(n)|X3=SU(n)|
||||||||!||||!|
|||||||...|||!|
||||||||!||||!|
|A1||||A2|||!|A1=E8|A2=SO(18)
||!||||||!||||!|
||!||||||!||||!|
|C1|||||!||||!|C1=E6
||`|-|-|v|-|-|'||||!|
||||D1|D1=SO(10)|||||||!|
||,|-|'|`|-|-|-|-|-|v|'|
|E1|||||||E2||E1=Pati-Salam|E2=SU(5)
|`|-|-|-|-|v|-|-|-|-|'|
| |||| XX |||||XX=SU(3)
| ||||| ! ||||| 
| |||| XX |||||XX=[[advanced_tools:group_theory:su2|SU(2)]]
| ||||| ! ||||| 
| |||| XX |||||XX=U(1)
</diagram>

</WRAP>

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.

* To get an idea how this classification works see [[http://jakobschwichtenberg.com/classification-of-simple-lie-groups/|Classification of all Simple Lie Groups]] by J. Schwichtenberg 
  * and for the mathematical details see [[http://www.mat.univie.ac.at/~cap/files/wisser.pdf|this article]].

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 [[theories:speculative_theories:grand_unified_theories|grand unified theories]]. The full classification is shown here:

-->Classification of all Groups in Terms of Dynkin Diagrams#
{{ :advanced_tools:group_theory:dynkinclassification.jpg?nolink |}}

<--

</WRAP>

----

__Spacetime Symmetry Groups __<diagram>
|||| |AAA| ||||  |AAA=Diffeomorphism Group
| |,|-|-|-|'|
| B1 | | |||  | |B1=[[advanced_tools:group_theory:conformal_group|Conformal Group]]
| , | ' |
| C1 | | C2 ||||||||C1=[[advanced_tools:group_theory:deSitter|DeSitter Group]]|C2=Anti DeSitter Group
|`| -|v|-|'|
||D1|| |D1=[[advanced_tools:group_theory:poincare_group|Poincare Group]]
|||!||
||E1|| |E1=Galilei Group
</diagram>

</WRAP>

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 [[advanced_tools:group_theory:group_contraction|deforming]] the known groups was done in

* Bacry, H., Lévy-Leblond, J.-M.: Possible Kinematics. J. Math. Phys. 9, 1605–1614 (1968)
  * Bacry, H., Nuyts, J.: Classification of ten-dimensional kinematical groups with space isotropy. J. Math. Phys. 27, 2455–2457 (1986)

In addition, the conformal group and the diffeomorphism group are important. These groups cannot be understood as [[advanced_tools:group_theory:group_contraction|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.

</WRAP>

----

**Important Concepts**

* [[advanced_tools:group_theory:group_contraction]]

* The **Basic idea** is "//Numbers measure size, groups measure symmetry.//" (Groups and Symmetry by Mark A. Armstrong)
  * A nice read on the role of group theory in physics is [[http://ricardoheras.com/wp-content/uploads/2014/02/Symmetry-in-Physics-Wigners-legacy.pdf|Symmetry in physics: Wigner's legacy]] by Gross
<tabbox Student>

**Recommended Resources**

- http://jakobschwichtenberg.com/why-group-theory/
  - http://jakobschwichtenberg.com/motivation-for-the-group-theory-axioms/
  - http://jakobschwichtenberg.com/naive-introduction-lie-theory/
  - http://jakobschwichtenberg.com/short-introduction-motivation-representation-theory/
  - http://jakobschwichtenberg.com/lie-algebra-able-describe-group/
  - http://jakobschwichtenberg.com/adjoint-representation/

* See also https://www.quora.com/What-role-do-Lie-groups-and-or-Lie-algebras-play-in-physics/answer/Ron-Maimon?srid=TYwn

Three books that explain group theory (but do not proof things) with a focus on how it's used in modern physics are

* Symmetry and the Standard Model: Mathematics and Particle Physics by Matthew Robinson
  * Symmetries in Fundamental Physics by Kurt Sundermeyer
  * Physics From Symmetry by Jakob Schwichtenberg
  * [[https://www.liealgebrasintro.com/|Lie groups and algebras A simplified introduction to Lie groups and algebras]] by Doug McKenzie 
  * {{ :advanced_tools:groups-symmetry-and-topology-global-symmetry-and-local-symmetry-lecture-notes-physics.pdf |How to Talk to a Physicist: Groups,Symmetry, and Topology}} by Daniel T. Larson

Good books that focus a on mathematical aspects (and also include proofs) are:

* Visual Group Theory by Nathan Carter
  * Naive Lie Theory by John Stilwell
  * An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee

* Another nice introduction is https://www-zeuthen.desy.de/~kolanosk/eep06/skripte/lie.pdf
  * https://www.fizyka.umk.pl/~bgw/rzeszow1.pdf
  * See also {{ :advanced_tools:hos.pdf |Building Blocks of Symmetry by Attila Egri-Nagy}}

**Definition**

A group $(G, \circ)$ is a set $G$, together with a binary operation $\circ$ defined on $G$, that satisfies the following axioms

* Closure: For all $g_1, g_2 \in G$,  $g_1 \circ g_2 \in G$
  * Identity: There exists an identity element $e \in G$ such that for all $g \in G$, $g \circ e = g = e \circ g$
  * Inverses: For each $g \in G$, there exists an inverse element $g^{-1} \in G$ such that  $g  \circ g^{-1}=e = g^{-1} \circ g $.
  * Associativity: For all $g_1, g_2, g_3 \in G$,  $g_1 \circ (g_2 \circ g_3) = (g_1 \circ g_2) \circ g_3$.

----

For Group Theory in Quantum Mechanics see http://math.ucr.edu/home/baez/lie/lie.html

<tabbox Researcher> 
<blockquote>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 [[advanced_tools:category_theory:groupoids|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.<cite>[[http://www.sciencedirect.com/science/article/pii/S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</cite></blockquote>

* One of the best books to get familiar with many of the most important advanced topics in group theory is "Geometrical methods of mathematical physics" by Bernard F. Schutz
  * Other nice advanced textbooks are:
    * Daniel Bump, Lie Groups
    * Shlomo Sternberg, Group Theory and Physics
    * Robert Hermann, Lie Groups for Physicists
    * George Mackey, Unitary Group Representations in Physics, Probability, and Number Theory
    * Brian Hall, Lie Groups, Lie Algebras, and Representations
    * William Fulton and Joe Harris, Representation Theory - a First Course
    * J. Frank Adams, Lectures on Lie Groups

<blockquote>
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
<cite>From "How is Quantum Field Theory possible" by Auyang</cite>
</blockquote>

**Resources:**

* http://www.liegroups.org/
  * https://groupprops.subwiki.org/wiki/Main_Page

-->  What does conjugation mean?#

http://math.stackexchange.com/questions/11971/intuition-behind-conjugation-in-group-theory
<--

--> Can every group be written in terms of matrices?#

<blockquote>
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.

<cite>http://physics.stackexchange.com/a/76191/37286</cite>
</blockquote>

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.

<--

--> What groups are there?#

See http://jakobschwichtenberg.com/classification-of-simple-lie-groups/
 
<--

--> Where does symmetry come from?#

<blockquote>
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.

<cite>David Gross in "Conceptual Foundations of Quantum Field Theory" , Edited by Cao</cite>
</blockquote>

This line of research goes by the name "random dynamics".

<blockquote>
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.

<cite>The Life of the Cosmos by Lee Smolin</cite>
</blockquote>
<--

-->What Information About a Lie Group is Not Encoded in its Lie Algebra?#

http://www.wetsavannaanimals.net/wordpress/what-information-about-a-lie-group-is-not-encoded-in-its-lie-algebra/

<--

--> Why is the Jacobi identity important?#

“[T]he real meaning of the Jacobi identity: it’s there so the adjoint representation of a Lie group, clearly a very basic and fundamental thing, induces a homomorphism in the corresponding Lie algebras that respects Lie brackets.” [[http://www.wetsavannaanimals.net/wordpress/the-adjoint-representation-of-the-lorentz-group/]]

<--

--> How can I show that a group is not simple?#

See [[http://www.math3ma.com/mathema/2015/5/4/4-ways-to-show-a-group-is-not-simple|4 Ways to Show a Group is Not Simple]] by Tai-Danae Bradley

<--

<blockquote>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.<cite>https://www.nature.com/articles/d41586-018-03423-x</cite></blockquote>
----

* See the excellent account in "Felix Klein and Sophus Lie: Evolution of the idea of symmetry in the nineteenth century" by I. M. Yaglom, 
  * Hans Wussing, The genesis of the abstract group concept. A contribution to the history of the origin of abstract group theory.
</tabbox>

{{tag>theories:quantum_theory:quantum_mechanics theories:quantum_theory:quantum_field_theory}}

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