Given any physical system, we can perform certain “operations” or “transformations” with it. Some examples are well known: rotations, traslations, scale transformations, conformal transformations, Lorentz transformations,… The ultimate quest of physics is to find the most general “symmetry group” leaving invariant some system. In Classical Mechanics, we have particles, I mean point particles, and in Classical Field Theories we have “fields” or “functions” acting on (generally point) particles. Depending on the concrete physical system, some invariant properties are “interesting”.
Similarly, we can leave the system invariant and change the reference frame, and thus, we can change the “viewpoint” with respect to we realize our physical measurements. To every type of transformations in space-time (or “internal spaces” in the case of gauge/quantum systems) there is some mathematical transformation acting on states/observables. Generally speaking, we have:
1) At level of states:
2) At level of observables:
These general transformations should preserve some kind of relations in order to be called “symmetry transformations”. In particular, we have conditions on 3 different objects:
A) Spectrum of observables:
These operators must represent observables that are “identical”. Generally, these operators must be “isospectral” and they will have the same “spectrum” or “set of eigenvalues”.
B) In Quantum Mechanics, the probabilities for equivalent events must be the same before and after the transformations and “measurements”. In fact, measurements can be understood as “operations” on observables/states of physical systems in this general framework. Therefore, if
where is a set of eigenvectors of O, and
where is a set of eigenvectors of O’, then we must verify
C) Conservation of commutators. In Classical Mechanics, there are some “gadgets” called “canonical transformations” leaving invariant the so-called Poisson brackets. There are some analogue “brackets” in Quantum Mechanics: the commutators are preserved by symmetry transformations in the same way that canonical transformations leave invariant the classical Poisson brackets.
These 3 conditions constrain the type of symmetries in Classical Mechanics and Quantum Mechanics (based in Hilbert spaces). There is a celebrated theorem, due to Wigner, saying more or less the mathematical way in which those transformations are “symmetries”.
Let me define before two important concepts:
Antilinear operator. Let A be a linear operator in certain Hilbert space H. Let us suppose that and . An antilinear operator A satisfies the condition:
Antiunitary operator. Let A be an antilinear opeartor in certain Hilbert space H. A is said to be antiunitary if it is antilinear and
Any continuous family of continuous transformations can be only described by LINEAR operators. These transformations are continuously connected to the unity matrix/identity transformation leaving invariant the system/object, and this identity matrix is in fact a linear transformation itself. The product of two unitary transformations is unitary. However, the product of two ANTIUNITARY transformations is not antiunitary BUT UNITARY.
Wigner’s theorem. Let A be an operator with eigenvectors and another operator with eigenvectors . Moreover, let us define the state vectors:
Then, every bijective transformation leaving invariant
can be represented in the Hilbert space using some operator. And this operator can only be UNITARY (LINEAR) or ANTIUNITARY(ANTILINEAR).
This theorem is relative to “states” but it can also be applied to maps/operators over those states, since for the transformation of operators. We only have to impose
Due to the Wigner’s theorem, the transformation between operators must be represented by certain operator , unitary or antiunitary accordingly to our deductions above, such that if , then:
This last relation is valid vor every element in a set of complete observables like the basis, and then it is generally valid for an arbitrary vector. Furthermore,
There are some general families of transformations:
i) Discrete transformations , both finite and infinite in order/number of elements.
ii) Continuous transformation . We can speak about uniparametric families of transformations or multiparametric families of transformations . Of course, we can also speak about families with an infinite number of parameters, or “infinite groups of transformations”.
Physical transformations form a group from the mathematical viewpoint. That is why all this thread is imporant! How can we parametrize groups? We have provided some elementary vision in previous posts. We will focus on continuous groups. There are two main ideas:
a) Parametrization. Let be a family of unitary operators depending continuously on the parameter . Then, we have:
b) Taylor expansion. We can expand the operator as follows:
There is other important definitiion. We define the generator of the infinitesimal transformation , denoted by , in such a way that
Moreover, must be an hermitian operator (note that mathematicians prefer the “antihermitian” definition mostly), that is:
There is a fundamental theorem about this class of operators, called Stone theorem by the mathematicians, that says that if is a generator of a symmetry at infinitesimal level, then determines in a unique way the unitary operator for all value . In fact, we have already seen that
So, the Stone theorem is an equivalent way to say the exponential of the group generator provides the group element!
We can generalize the above considerations to finite multiparametric operators. The generator would be defined, for a multiparametric family of group elements . Then,
There are some fundamental properties of all this stuff:
1) Unitary transformations form a Lie group, as we have mentioned before.
2) Generators form a Lie algebra. The Lie algebra generators satisfy
3) Every element of the group or the multiparametric family can be written (likely in a non unique way) such that:
4) Every element of the multiparametric group can be alternatively written in such a way that
where the parameters are functions to be determined for every case.
What about the connection between symmetries and conservation laws? Well, I have not discussed in this blog the Noether’s theorems and the action principle in Classical Mechanics (yet) but I have mentioned it already. However, in Quantum Mechanics, we have some extra results. Let us begin with a set of unitary and linear transformations . These set can be formed by either discrete or continuous transformations depending on one or more parameters. We define an invariant observable Q under G as the set that satisfies
Moreover, invariance have two important consequences in the Quantum World (one “more” than that of Classical Mechanics, somehow).
1) Invariance implies conservation laws.
Given a unitary operator , as , then
If we have some set of group transformations , such the so-called hamiltonian operator is invariant, i.e., if
Then, as we have seen above, these operators for every value of their parameters are “constants” of the “motion” and their “eigenvalues” can be considered “conserved quantities” under the hamiltonian evolution. Then, from first principles, we could even have an infinite family of conserved quantities/constants of motion.
This definifion can be applied to discrete or continuous groups. However, if the family is continuous, we have additional conserved constants. In this case, for instance in the uniparametric group, we should see that
and it implies that if an operator is invariant under that family of continuous transformation, it also commutes with the infinitesimal generator (or with any other generator in the multiparametric case):
Every function of the operators in the set of transformations is also a motion constant/conserved constant, i.e., an observable such as the “expectation value” would remain constant in time!
2) Invariance implies (not always) degeneration in the spectrum.
Imagine a hamiltonian operator and an unitary transformation such as . If
1) is also an eigenvalue of H.
2) If and are “linearly independent”, then is (a) degenerated (spectrum).
1st step. We have
2nd step. If and are linarly independent, then and thus
If the hamiltonian is invariant under a transformation group, then it implies the existence (in general) of a degeneration in the states (if these states are linearly independent). The characteristics features of this degeneration (e.g., the degeneration “grade”/degree in each one of these states) are specific of the invariance group. The converse is also true (in general). The existence of a degeneration in the spectrum implies the existence of certain symmetry in the system. Two specific examples of this fact are the kepler problem/”hydrogen atom” and the isotropic harmonic oscillator. But we will speak about it in other post, not today, not here, ;).
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.