Definition (36). An infinite group is a group where the order/number of elements is not finite. We distinguish two main types of groups (but there are more classes out there…):
1) Discrete groups: their elements are a numerable set. Invariance under a discrete group provides multiplicative conservation laws. Elements are symbolized as for a discrete group.
2) Continuous groups: their elements are not numerable, since they depend “continuously” on a finite number of parameters (real, complex,…):
Note that the number or paraters can be either finite or infinite in some cases. The number of parameters define the so-called “dimension” of the group. Please, don’t confuse group order with its dimension. Group order is the number of elements, group dimension is the number of parameters we do need to characterize/generate the group! Invariance under a continuous group has some consequences (due to the Noether’s theorems):
1) Invariance under a finite dimensional r-parametric continuous group provides conservation laws.
2) Invariance under an infinite dimensional continuous group (parametrized by some set of “functions”) provides some relationships between field equations called “dependencies” or “noether identities” in modern language.
Definition (37). Composition rule/law for a group. Let be a continuous group and two elements , then
and we define the composition law of a continuous group as the function that gives and similarly
Theorem (Lie). Every continuous group is a Lie Group. It means that whenever you have a group where the composition rule is given, as the inverse element, then the group elements are differentiable functions (analytic in the complex case) on its arguments.
Some examples of Lie groups (some of them we have already quoted in this thread):
1) The euclidean real space or the hermitian complex space with ordinary vector addition form (in any of that two cases) a n-dimensional noncompact abelian Lie group.
2) The general linear (Lie) group of non-singular matrices over the real number or the complex numbers is a Lie group or .
3) The special linear group or the complex analogue of square matrices with determinant equal to one.
4) The orthogonal group over the real numbers, matrices with real entries is a dimensional Lie group.
5) The special orthogonal group is the subgroup of the orthogonal group whose matrices have determinant equal to one.
6) The unitary group of complex unitary matrices, . Its dimension is equal to over the complex numbers. SU(n) is the dimensional subgroup formed by unitary matrices with determinant equal to one.
7) The symplectic group .
8) The group of upper triangular matrices is a group with dimension .
9) The Lorentz group and the Poincaré group. The are non-compact Lie groups (Poincaré is non-compact due to the fact that the Lorentz subgroup is non-compact). Their dimensions in 4D spacetime are 6 and 10 dimensions respectively.
10) The Standard Model “gauge” (Lie) group is a group formed with direct group (in the group sense) of three groups and it has dimension . The dimensions of the gauge groups in the Standard Model is in direct correspondence with the numbers of gauge bosons: 1 massless photon, 3 vector bosons for the electroweak interactions, and 8 gluons for the quantum chromodynamics (QCD).
11) The exceptional Lie groups , the so called Cartan exceptional groups. Their dimensions are respectively 14, 52, 78, 133 and 248.
The continuous group made of matrices (finite and infinite matrices/operators) play an important role in Physics. Moreover, as Lie groups depend continuously on their arguments AND their dependence is generally differentiable, it makes sense to take derivatives in the group elements. In fact, this fact allow us to define the idea of group generator.
Definition(38). Group generator. If is a continuous (therefore differentiable; remember that continuity implies differentiability but the converse is not necessarily true), then we define the generators of the group in the following (hermitian) way:
Theorem (Lie). Let us choose some one-parameter continuous group and K its generator. Then, the following facts hold:
i) K fully determines the group .
ii) Group elements are obtained using “exponentiation” of generators. That is,
The “proof” involves a group parametrization and an expansion as a series. We have and . Therefore,
This differential equation has one and only one solution for every K-value. The general solution of this equation is the exponential:
Taking into account the initial conditions (elements nearby of any group element are the identity element) we have the desired result for every . Q.E.D.
Theorem (Lie). A multiparametric Lie group (N-dimensional) is a Lie group G with functions , and generators obtained by exponentiation. That is:
Check (Easy simplified proof): Using the previous result, we have to fix only all the parameters . Then, a simple “empatic mimicry” of the previous one dimensional provides:
and finally, iterating the process N-times we get
The generators of any Lie group satisfy some algebraic and important relations. In the case of dealing with matrix or operator groups, the generators are matrices or operator theirselves. These mathematical relations can be written in terms of (ordinary) algebraic commutators. There is a very important theorem about this fact:
Theorem. First Lie theorem. Lie group generators form a closed commutator algebra under “matrix/operator” products. That is:
without distinction of lower and upper “labels”.
There the commutator of two matrices/operators is defined to be and the contants or are the so-called structure constants of the Lie group. The structure constants of a Lie group are:
1) Antisymmetric with respect to the first two indices (or the paired ones, , with our notation).
2) Characteristic of the group but they do change, in a particular way, if we form linear combinations of the Lie group generators.
There is a nice formula called Baker-Campbell-Hausdorff identity that relates group exponentials and group commutators. It is specially important in the theory of Lie groups and Lie algebras:
The Baker-Campbell-Hausdorff (BCH) formula. For any matrix/operator A,B, under certain very general conditions, we have:
In the case that the matrices A and B do commute, then we recover the usual ordinary exponentiation of “elements”:
A beautiful and simple application of the BCH formula is the next feature which allows us to write ANY member of a Lie group as the exponential of a sum of the Lie group generators. Let us write the group elements, firstly, as
and let us write the group generators as . Then, we have
where the parameters are related to the parameters in a simple continuous way
The specific form of this realtion can be expanded and computed/calculated term by term using the BCH formula, as given before.
See you in the next blog post of this group theory thread!
Schur’s lemmas are some elementary but very useful results in group theory/representation theory. They can be also used in the theory of Lie algebras so we are going to review these results in this post (for completion).
FIRST SCHUR’S LEMMA. If and are 2 finite-dimensional irreducible representations of G, and if A is certain linear map (generally a matrix) from to such as it commutes with the action of the group (for any element ), i.e.,
then at least one of the following consequences holds:
1) A is invertible, and then the representation are necessarily equivalent.
SECOND SCHUR’S LEMMA. If is a complex matrix of order that commutes with every matrix from an irreducible representation
Then, A must be a scalar matrix multiple of the identity matrix, i.e., .
Schur’s lemmas and their corollaries are used to prove the so-called (Schur’s) orthogonality relations and to develop the basics of the representation theory of finite groups. Schur’s lemmas admits some generalizations to Lie groups and some other kind of structures, such as Lie algebras and other sets of operators or matrices. Therefore, it is important to understand Schur’s lemmas!
Consequences of these lemmas:
1) Irreducible representations of an abelian finite group are one dimensional.
2) Orthonormality of group representations. Let be a finite group, and two irreducible representations of G. Then,
3) Completeness of irreducible representations. Let be a finite group and the set of every irreducible representation of G. Then,
4) The number of irreducible representations of a finite group is finite and they can be computed with the aid of the following formula:
We can give some elementary examples of irreducible representations of abelian (finite) groups.
1) For , then and so there are two and only two irreducible representations, both one dimensional:
2) For , then and there are 3 irreducible representations:
3) For we have two cases: and . And then we have:
3.1) The case of the cyclic group with 4 unidimensional representations
3.2) The case of the Klein group where , and with . The representations of this group, in the unidimensional case, are given by:
Of course, you can build the matrix representation of the above group representations as a nice homework. :).
See you in another blog post!
Other important concepts and definitions in group theory!
Definition (22). Normal or invariant group. Let be a subgroup of other group G. We say that is a normal or invariant subgroup of G if the following condition holds:
Proposition. Let be a subgroup of G. is invariant if and only if (iff) is a union of conjugacy classes of .
Check: H is normal iff . Thus, H is normal iff whenever it contains an element , then it also contains the conjugacy class of , so another way to say this is that H is a union of conjugacy classes.
Definition (23). Simple group. We say that G is a simple group if there is no invariant subgroups except the trivial, the neutral/identity element .
Definition (24). Semisimple group. We say that G is a semisimple group if there is no any “abelian invariant subgroup” except the trivial element .
Simple groups are always semisimple, but the converse is not true. Simple and semisimple finite groups have been completely classified by mathematicians. We will talk about this later.
Definition (25). Coset. Let be an invariant subgroup. Then we say that
i) The set with is a coset by the left, and
ii) The set sith is a coset by the right.
Theorem. The set
with composition law is a group.
Definition (25). Quotient group. The group defined in the previous theorem is called quotient group and it is generally denoted by .
Remark: Informally speaking, the elements of the quotient group are “the difference” between the elements of G and those in H.
Now, some additional definitions about morphisms, homomorphisms and isomorphisms in group theory.
Definition (26). (Group) Homomorphism. Let and be two different groups. Any map/application/function/functor
is called an homomorphism if it preserve the operations of the respective groups (their “products” or “multiplications”) in the following sense
Definition (27). (Group) Isomorphism. Let be an homomorphism, then is an isomorphism if is “bijective”, or one-to-one correspondence between the elements of and .
Definition (28). Kernel. Let and be two different groups. and a function between them. Then, we define the kernel of f as the following set
and where is the neutral element in the group .
There are two important theorems for group homomorphisms:
Theorem. First theorem of group homomorphisms. Let and be two different groups. If we hav a function being a group homomorphism, then its kernel is an invariant subgroup of .
Property. It is clear from the above theorem that, if f is bijective, then we have the special case in which . Then, the theorem is trivially satisfied!
Theorem. Second theorem of group homomorphisms. Let and be two different groups. If is a group homomorphism, then an aplication exists such as
such as is a group isomorphism.
May the group theory be with you until the next blog post!
I am going to build a “minicourse” thread on Group Theory in this and the next blog posts. I am trying to keep the notes self-contained, since group theory is a powerful tool and common weapon in the hands of many theoretical physicists and mathematicians. I am not consider myself an expert, but I have learned a little bit about group theory from my books, the world wide web and with some notes I own from my Master degree and my career.
Let’s begin. First of all, I would wish to say you that the current axioms of the algebraic structure that mathematicians and physicists known as “group theory” formalize the essence of symmetry! What is symmetry? Well, it is a really good question. I am not going to be too advanced today, but I want to give you some historical and interesting remarks. Symmetry as a powerful tool for physicists likely gegun in the 19th century, with analytical mechanics (perhaps even before, but I am being subjective at this point) and the early works of the foundations of geometry by Riemann, Gauss, Clifford, and many others. Weyl realized in some point during the 20th century, and he launched the so-called Erlangen program, an ambitious and wonderful project based on the idea that geometry is based on the “invariants” objects that some set of transformations own. Therefore, the idea of the Erlangen program was to study “invariant objetcts” under “certain transformations” that we do name “symmetries” today. Symmetries form (in general) a group (although some generalizations can be allowed to this idea, like the so-called quantum groups and other algebraic structures) or some other structure with “beautiful” invariants. Of course, what is a beautiful invariant is on the eyes of the “being”, but mathematics is beautiful and cool. During the 20th century, Emmy Noether derived two wonderful theorems about the role of symmetry and conserved quantities and field equations that have arrived until today. Those theorems impressed Einstein himself to the point to write a famous letter trying to get Noether a position in the German academy ( women suffered discrimination during those times in the University, as everybody knows).
From the pure mathematical viewpoint, a group of symmetries/transformations are “closed” because if you take a symmetry of any object, and then you apply another symmetry, the result will still be a symmetry. This composition property is very important and simple. The identity itself keeps the object “fixed” and it is always a symmetry of the object. Existence of “inverse” transformations (that allow us to recover the original untransformed object) is guaranteed by undoing the symmetry and the associativity that generally comes with the group axioms comes from the fact that symmetries are functions on certain “space”, and composition of functions are associative (generally speaking, since we can invent non-associative stuff as well, but they don’t matter in the current discussion).
Why are groups important in Physics? Groups are important because they describe the symmetries of the physical laws! I mean, physical laws are “invariant” under some sets of transformations, and that sets of transformations are what we call the symmetries of physics. For instance, we use groups in Classical Physics ( rotations, translations, reflections,…), special relativity theory (Lorentz boosts, Poincarè transformations, rotations,…), General Relativity (diffeomorphism invariance/symmetry) and Quantum Mechanics/Field Theory (the standard model, containing electromagnetism, has gauge invariances from different “gauge” symmetries). We have studied secretly group theory in this blog, without details, when I explained special relativity or the Standard Model. There, the Lorentz group played an important role, an gauge transformations too.
Quantum Mechanics itself showed that matter is made of elementary systems such as electrons, positrons or protons that are “truly” identical, or just very similar, so that symmetry in their arrangement is “exact” or “approximate” to some extent as it is (indeed) in the macroscopic world. Systems or particles should be seen to be described by “functions” (or “fields” as physicists generally call those functions) of position in the space or the spacetime (in the case of relativistic symmetries). These particles are subject to the usual symmetry operations of rotation, reflection or even “charge conjugation” (in the case of charged particles), as well as other “symmetries” like the exchange of “identical particles” in systems composed by several particles. Elementary particles reflect symmetry in “internal spaces”, beyond the usual “spacetime” symmetries. These internal symmetries are very important in the case of gauge theories. In all these cases, symmetry IS expressed by certain types of operations/transformations/changes of the concerned systems, and Group Theory is the branch of Mathematics that had previously been mainly a curiosity withouth direct practical application, …Until the 20th century and the rise of the two theories that today rule the whole descriptions of the Universe: (general/special) relativity theory and Quantum Mechanics/Quantum Field Theory(QFT)/the Standard Model (SM).
Particle Physics mainly uses the part of Group Theory known as the theory of representations, in which matrices acting on the members of certain vector space are the central elements. It allows certain members of the space to be created that are symmetrical, and which can be classified by their symmetries and “certain numbers” (according to the so-called Wigner’s theory). We do know that every observed spectroscopic state of composed particles (such as hadrons, atomic nuclei, atoms or molecules) correspond to such symmetrical functions and representations (as far as we know, Dark Matter and Dark Energy don’t seem to fit in it, yet), and they can be classified accordingly. Among other things, it provides the celebrated “selection rules” that specify which reactions or state transitions are boserved, and which not. I would like to add that there is a common “loophole” to this fact: the existence of some “superselection rules” in Quantum Mechanics/QFT are not explained in a clear way as far as I know.
The connection between particle physics and representation theory, first noted by Eugene Wigner, is a “natural” connection between the properties of elementary particles and the representation theory of Lie groups and Lie algebras. This connection explains that different quantum states of elementary particles correspond to different irreducible representations (irreps.) of the Poincaré group. Furthermore, the properties several particles, including their energy or mass spectra, can be related to representations of Lie algebras that correspond to “approximate symmetries” of the current known Universe.
There are two main classes of groups if we classify them by number of “elements”/”constituents”. Finite groups and infinite groups. Finite groups has a finite number of “members” and they are useful in crystal/solid state physics, molecular spectra, and identical particles systems. They are nice examples of the power and broad applications of group theory. Infinite groups, groups having “infinite” number of elements, are important in gauge theories and gravity (general relativity or its generalizations). An important class of infinite groups are Lie groups, named after the mathematician Sophus Lie. Lie groups are important in the study of differential equations and manifolds since they describe the symmetries of continuous geometries and analytical structures. Lie groups are also a vital ingredient of gauge theories in particle physics. Lie groups naturally appear in quantum mechanics and elementary particle physics (the SM) because their representations share many of the symmetries of those natural systems. Lie groups are very similar to finite groups in many aspects.
By the other hand, angular momentum is a very well known and studied in depth example in classical mechanics or quantum physics about the importance of “symmetry”. Symmetry transformations and general momentum theory is in fact “almost” group theory in action. Orbital angular momentum faces with irreducible representations of the rotation group. For instance, in 3 spation dimension we have the group , and the rules for combining them appear “naturally”. In classical mechanics group theory appear through the role of Galileo group and/or the Euler angles for the rigid solid. General angular momentum theory is, in fact, the study of the representation theory of the “Lie algebra” , the algebraic elementary structure behind the or groups. Lie algebras express the structure of certain continuous group in a very powerful framework, and it is very easy to use. In fact, we can use matrices to deal with group representations or not, according to our needs.
While the spacetime symmetries in teh Poincaré group are particularly important, there are also other classes of symmetries that we call internal symmetries. For instance, we have , the color group of QCD, or , the gauge group of the electroweak interactions. An exact symmetry corresponds to the continuous “interchange” of the 3 quark color “numbers”. However, despite the fact that the Poincaré group or the color group are believed to be “exact” symmetries, other symmetries are only approximate in the following sense: flavor symmetry, for instance, is an gropup symmetry corresponding to varying the quark “flavor”. There are 6 quark flavours: up (u), down (d), charm (c), strange (s), botton (b) and top (t). This particular “flavor symmetry” is an approximate symmetry since it is “violated” by quark mass differences and the electroweak interactions. In fact, we do observe experimentally that hadron particles can be neatly divided into gropups that form irreducible representations of the Lie algebra , as first noted by the Nobel Prize Murray Gell-Mann in his “eightfold way” approach, the origin of the modern quark theory.
In summary, we have to remember the main ideas of Group Theory in Physics:
1) Group Theory studies invariant objects under certain classes of transformations called symmetry transformations or symmetries.
2) Group Theory relates geometry with “invariant objects”. And mathematicians have classified and studied the most important and used groups under “minimal assumptions”. That is cool, since we, physicists, have only to use them.
3) Group Theory is very important in several parts of Physics, and specifically, in particle physics and relativity theory.
4) Groups are classified into finite or infinite groups, depending on the number of “elements”/”constituents” in the group. Finite groups have a finite number of members, infinite groups (like Lie groups) have an infinite number of elements.
5) Symmetries in physics can be classified into spacetime symmetries or internal symmetries. Spacetime symmetries act on spacetime coordinates, internal symmetries act on “quantum numbers” like electric charge, color or flavor.
6) Symmetries in physics can be “exact” (as it is the case of the Lorentz/Poincaré group) or “approximate” like the SU(3) flavor symmetry (and some others like Parity, Charge conjugation or Time Reversal).
7) Symmetries can be “continous” or “discrete”. Continuous symmetries are spacetime symmetries or gauge symmetries, and discrete symmetries are parity (also called reflection symmetry), charge conjugation or time reversal. Continous symmetries obey (in general) additive conservation laws while discrete symmetries obey (in general) multiplicative conservation laws.
8) Noether’s theorems relate symmetries with invariance transformations, mathematical identities and conservation laws/field equations. To be more precise, Noether’s theorem (I) relates continuous symmetries with a finite number of generators and conservation laws. Noether’s theorem (II) relates gauge symmetries with an “infinite number” of generators with “certain class of mathematical identities” in the equations of motion for either particles or fields.
9) Representation theory is a part of group theory that can explain the spectroscopy of fundamental objects (atoms, molecules, nuclei, hadrons or elementary particles). It provides some general spectral properties plus some “selection rules”.
10) Group theory applications are found in: solid state physics, molecular spectra, identical particles, angular momentum theory, spacetime symmetries, gauge symmetries.
See you in the next Group Theory blog post.