The topic today in this group theory thread is “sixtors and representations of the Lorentz group”.
Consider the group of proper orthochronous Lorentz transformations and the transformation law of the electromagnetic tensor . The components of this antisymmetric tensor can be transformed into a sixtor or and we can easily write how the Lorentz group acts on this 6D vector ignoring the spacetime dependence of the field.
Under spatial rotations, transform separately in a well-known way giving you a reducible representation of the rotation subgroup in the Lorent orthochronous group. Remember that rotations are a subgroup of the Lorentz group, and it contains Lorentz boosts in additionto those rotations. In fact, in the space of sixtors and they are thus a reducible representation, a direct sum group representation. That is, rotations leave invariant subspaces formed by and invariant. However, these two subspaces mix up under Lorentz boosts! We have written before how transform under general boosts but we can simplify it without loss of generality and for some matrices . So it really seems that the representation is “irreducible” under the whole group. But it is NOT true! Irreducibility does not hold if we ALLOW for COMPLEX numbers as coefficients for the sixtors/bivectors (so, it is “tricky” and incredible but true: you change the numbers and the reducibility or irreducibility character does change. That is a beautiful connection betweeen number theory and geometry/group theory). It is easy to observe that using the Riemann-Silberstein vector
and allowing complex coefficients under Lorent transformations, such that
i.e., it transforms totally SEPARATELY from each other () under rotations and the restricted Lorentz group. However, what we do have is that using complex coefficients (complexification) in the representation space, the sixtor decomposes into 2 complex conjugate 3 dimensional representaions. These are irreducible already, so for rotations alone transformations are complex orthogonal since if you write
with and . Be aware: here is an imaginary angle. Moreover, transforms as follows from the following equation:
Remark: Rotations in 4D are given by a unitary 4-vector such as and the rotation matrix is given by the general formula
If you look at this rotation matrix, and you assign with , the above rotations are in fact the same transformations of the electric and magnetic parts of the sixtor! Thus the representation of the general orthochronous Lorentz group is secretly complex-orthogonal for electromagnetic fields (with complex coefficients)! We do know already that
are the electromagnetic main invariants. So, complex geometry is a powerful tool too in group theory! :). The real and the imaginary part of this invariant are also invariant. The matrices of 2 subrespresentations formed here belong to the complex orthogonal group . This group is a 3 dimensional from the complex viewpoint but it is 6 dimensional from the real viewpoint. The orthochronous Lorentz group is mapped homomorphically to this group, and since this map has to be real and analytic over the group such that, as Lie groups, . We can also use the complex rotation group in 3D to see that the 2 subrepresentations must be inequivalent. Namely, pick one of them as the definition of the group representation. Then, it is complex analytic and its complex parameter provide any equivalent representation. Moreover, any other subrepresentation is complex conjugated and thus antiholomorphic (in the complex sense) in the complex parameters.
Generally, having a complex representation, i.e., a representation in a COMPLEX space or representation given by complex valued matrices, implies that we get a complex conjugated reprentation which can be equivalent to the original one OR NOT. BUT, if share with original representation the property of being reducible, irreducible or decomposable. Abstract linear algebra says that to any representation in complex vector spaces there is always a complex conjugate representation in the complex conjugate vector space . Mathematically, one ca consider representations in vector spaces over various NUMBER FIELDS. When the number field is extended or changed, irreducibility MAY change into recubibility and vice versa. We have seen that the real sixtor representation of the restricted Lorentz group is irreducible BUT it becomes reducible IF it is complexified! However, its defining representation by real 4-vectors remains irreducible under complexification. In Physics, reducibility is usually referred to the field of complex numbers , since it is generally more beautiful (it is algebraically closed for instance) and complex numbers ARE the ground field of representation spaces. Why is this so? There are two main reasons:
1st. Mathematical simplicity. is an algebraically closed filed and its representation theory is simpler than the one over the real numbers. Real representations are obtained by going backwards and “inverting” the complexification procedure. This process is sometimes called “getting the real forms” of the group from the complex representations.
2nd. Quantum Mechanics seems to prefer complex numbers (and Hilbert spaces) over real numbers or any other number field.
The importance of is understood from the Maxwell equations as well. In vacuum, without sources or charges, the full Maxwell equations read
These equations are Lorentz covariant and reducibility is essential there. It is important to note that
implies that we can choose ONLY one of the components of the sixtor, or , or one single component of the sixtor is all that we need. If in the induction law there were a plus sign instead of a minus sign, then both representations could be used simultaneously! Furthermore, Lorentz covariance would be lost! Then, the Maxwell equations in vacuum should satisfy a Schrödinger like equation due to complex linear superposition principle. That is, if and are solutions then a complex solution with complex coefficients should also be a solution. This fact would imply invariance under the so-called duality transformation
However, it is not true due to the Nature of Maxwell equations and the (apparent) absence of isolated magnetic charges and currents!
Representation theory is the part of Group Theory which is used in the main applications. Matrices acting on the members of a vector space are assigned to every element of a group. The connections between particle physics and representation theory is “natural”. It was noted by Eugene Wigner firstly, and the properties of elementary particles and the representation theory are due to some special groups, the so-called Lie groups and Lie algebras. The deep connection between Lie groups and Lie algebras with particle physics is observed in the different quantum states of an elementary particle, as a consequence of the irreducible representations of the Poincaré group. We are going to study this a little bit.
Definition (29). Representation. Group representation is every linear map D in G, . That is, a linear representation of a group is a group homomorphism onto the general linear group due to the elements of the group. Informally speaking, it is a rule that assings to every group element a square matrix.
Here V is certain vector space, sometimes called “the space of the representation”, and is the “general linear group” of non-singular matrices defined over V.
Definition (30). Equivalent representations. Two representations and are said to be equivalent representations of G, if there exists some matrix , an isomorphism, such as
Definition (31). Character. The character of an element , given the representation
is the number
Of course, the character is related to the trace of the matrix, so in the case of infinite groups we have to be more careful and precise with the definition of “trace”. But this point is not relevant or important in the present discussion. Clearly, the character could be interpreted as an homomorphism between the group and the group of real (complex) numbers.
Definition (32). Character of a representation. The character of a representation is the of numbers
Property: if and are two equivalent representations of G, then their characters are the same
Remark (I): The space a representation IS the vectorial space V where the matrices (belonging to the general linear group) act.
Remark(II): The dimension of a representation IS the dimension of the vectorial space V.
Remark (III): Do you know some examples of group representations we have seen in fact already? Remember that the trace of a matrix verifies that and and so on. This is the cyclic property of the trace.
Given two group representations, of , we can build other representations from them in at least two simple ways:
Definition (33) . Direct sum representation. The direct sum is defined over the space . The dimension of the direct sum:
where in block for we have
Definition (34). Tensor product representation. Given two representations of G, we can build the tensorial product representation
as a tensor product of matrices.
The dimension of the tensor product representation is equal to:
Question: Do you know the basic properties of tensorial products in vector spaces? Review their properties and work out some examples yourself.
Definition (35). Invariant subspace of a representation is a susbspace (or even the whole space itself) such as
We distinguish two types of representations according to the existence or not of non-trivial invariant subspaces with respect to them:
1st. Irreductible representations. These class of representations are those such as V does not contain any invariant subspace, except V itself. That is, irreducible representations are “single” pieces/boxes from which group representations are made of. They are the “atoms” of group representations.
2nd. Reducible representations. In this case, V contains some invariant subspaces different to itself.
3rd. Fully reducible representation is any representation, if exists, such as there is some invariant subspace , such that there exists another invariant subspace that satisfies
The study of the group representations is based on the study of the irreducible representations (irreps.), since reducible representations can be contructed from irreducible representations (reducible representations would be “molecules” made of irreducible representations).
Some properties of group representations are interesting:
1) Let G be a finite group. Then, all representation on an inner product space are equivalent to an unitary representation.
2) Let G be a group and let D be an unitary representation. Then, if the representation is reducible, then it is fully reducible.
3) Every reducible representation of a finite group is fully reducible.
In summary: a finite group has always an unitary representation and if if is reducible, it is fully reducible.
See you in other group theory 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.