The case of Poincaré symmetry
There is a important symmetry group in (relativistic, quantum) Physics. This is the Poincaré group! What is the Poincaré group definition? There are some different equivalent definitions:
i) The Poincaré group is the isometry group leaving invariant the Minkovski space-time. It includes Lorentz boosts around the 3 planes (X,T) (Y,T) (Z,T) and the rotations around the 3 planes (X,Y) (Y,Z) and (Z,X), but it also includes traslations along any of the 4 coordinates (X,Y,Z,T). Moreover, the Poincaré group in 4D is a 10 dimensional group. In the case of a ND Poincaré group, it has parameters/dimensions, i.e., the ND Poincaré group is dimensional.
ii) The Poincaré group formed when you add traslations to the full Lorentz group. It is sometimes called the inhomogenous Lorentz group and it can be denoted by ISO(3,1). Generally speaking, we will generally have , a D-dimensional () Poincaré group.
The Poincaré group includes as subgroups, the proper Lorentz transformations such as parity symmetry and some other less common symmtries. Note that the time reversal is NOT a proper Lorentz transformation since the determinant is equal to minus one.
Then, the Poincaré group includes: rotations, traslations in space and time, proper Lorentz transformations (boosts). The combined group of rotations, traslations and proper Lorentz transformations of inertial reference frames (those moving with constant relative velocity) IS the Poincaré group. If you give up the traslations in space and time of this list, you get the (proper) Lorentz group.
The full Poincaré group is a NON-COMPACT Lie group with 10 “dimensions”/parameters in 4D spacetime and in the ND case. Note that the boost parameters are “imaginary angles” so some parameters are complex numbers, though. The traslation subgroup of the Poincaré group is an abelian group forming a normal subgroup of the Poincaré group while the Lorentz grou is only a mere subgroup (it is not a normal subgroup of the Poincaré group). The Poincaré group is said, due to these facts, to be a “semidirect” product of traslations in space and time with the group of Lorentz transformations.
The case of Galilean symmetry
We can go back in time to understand some stuff we have already studied with respect to groups. There is a well known example of group in Classical (non-relativistic) Physics.
The Galilean group is the set or family of non-relativistic continuous space-time (yes, there IS space-time in classical physics!) transformations in 3D with an absolute time. This group has some interesting subgroups: 3D rotations, spatial traslations, temporal traslations and proper Galilean transformations ( transformations leaving invariant inertial frames in 3D space with absolute time). Thereforem the number of parameters of the Galilean group is 3+3+1+3=10 parameters. So the Galileo group is 10 dimensional and every parameter is real (unlike Lorentz transformations where there are 3 imaginary rotation angles).
The general Galilean group can be written as follows:
Any element of the Galileo group can be written as a family of transformations . The parameters are:
i) , an orthogonal (real) matrix with size . It satisfies , a real version of the more general unitary matrix .
ii) is a 3 component vector, with real entries. It is a 3D traslation.
iii) is a 3 component vector, with real entries. It gives a 3D non-relativistic (or galilean) boost for inertial observers.
iv) is a real constant associated to a traslation in time (temporal traslation).
Therefore, we have 10 continuous parameters in general: 3 angles (rotations) defining the matrix , 3 real numbers (traslations ), 3 real numbers (galilean boosts denoted by ) and a real number (traslation in time). You can generalize the Galilean group to ND. You would get parameters, i.e, you would obtain a dimensional group. Note that the total number of parameters of the Poincaré group and the Galilean group is different in general, the fact that in 3D the dimension of the Galilean group matches the dimension of the 4D Poincaré group is a mere “accident”.
The Galilean group is completely determined by its “composition rule” or “multiplication operation”. Suppose that:
Then, gives the composition of two different Galilean transformations into a new one. The composition rule is provided by the following equations:
Why is all this important? According to the Wigner theorem, for every continuous space-time transformation should exist a unitary operator acting on the space of states and observables.
We have seen that every element in uniparametric groups can be expressed as the exponential of certain hermitian generator. The Galilean group or the Poincaré group depends on 10 parameters (sometimes called the dimension of the group but you should NOT confuse them with the space-time dimension where they are defined). Remarkably, one can see that the Galilean transformations also act on “spacetime” but where the time is “universal” (the same for every inertial observer). Then, we can define
These generators, for every parameter , will be bound to dynamical observables such as: linear momentum, angular momentum, energy and many others. A general group transformation for a 10-parametric (sometimes said 10 dimensional) group can be written as follows:
We can apply the Baker-Campbell-Hausdorff (BCH) theorem or simply expand every exponential in order to get
The Lie algebra will be given by
and where the structure constants will encode the complete group multiplication rules. In the case of the Poincaré group Lie algebra, we can write the commutators as follows:
Here, we have that:
i) are the generators of the traslation group in spacetime. Note that as they commute with theirselves, the traslation group is an abelian subgroup of the Lorentz group. The noncommutative geometry (Snyder was a pioneer in that idea) is based on the idea that and more generally even the coordinates are promoted to noncommutative operators/variables/numbers, so their own commutator would not vanish like the Poincaré case.
ii) are the generators of the Lorent group in spacetime.
If we study the Galilean group, there are some interesting commutation relationships fo the corresponding generators (rotations and traslations). There are 6 “interesting” operators:
These equations provide
The case of the traslation group
In Quantum Mechanics, traslations are defined in the space of states in the following sense:
Let us define two linear operators, and associated, respectively, to initial position and shifted position. Then the transformation defining the traslation over the states are defined by:
Furthermore, we also have
The case of the rotation group
What about the rotation group? We must remember what a rotation means in the space . A rotation is a transformation group
The matrix associated with this transformation belongs to the orthogonal group with unit determinant, i.e., it is an element of . In the case of 3D space, it would be . Moreover, the ND rotation matrix satisfy:
The rotation matrices in 3D depends on 3 angles, and they are generally called the Euler angles in some texts. . Therefore, the associated generators are defined by
Any other rotation matric can be decomposed into a producto of 3 uniparametric rotations, rotation along certain 2d planes. Therefore,
where the elementary rotations are defined by
Rotation around the YZ plane:
Rotation around the XZ plane:
Rotation around the XY plane:
Using the above matrices, we can find an explicit representation for every group generator (3D rotation):
and we also have
where the is the completely antisymmetry Levi-Civita symbol/tensor with 3 indices. There is a “for all practical purposes” formula that represents a rotation with respect to some axis in certain direction . We can make an infinitesimal rotation with angle , due to the fact that rotation are continuous transformations, it commutes with itself and it is unitary, so that:
In the space of physical states, with some arbitrary vector
Here, the operators are the infinitesimal generators in the space of physical states. The next goal is to relate these generators with position operators through commutation rules. Let us begin with
Using this last result, we can calculate for any 2 vectors :
or equivalent, in component form,
These commutators complement the above commutation rules, and thus, we have in general
In summary: a triplet of rotation operators generates “a vector” somehow.
The case of spinning particles
In fact, these features provide two different cases in the case of a single particle:
i) Particles with no “internal structure” or “scalars”/spinless particles. A good example could it be the Higgs boson.
ii) Particles with “internal” degrees of freedom/structure/particles with spin.
In the case of a particle without spin in 3D we can define the angular momentum operator as we did in classical physics (), in such a way that
Note that the “cross product” or “vector product” in 3D is generally defined if as
or by components, using the maginal word XYZZY, we also have
Remember that the usual “dot” or “scalar” product is
Therefore, the above operator defined in terms of the cross product satisfies the Lie algebra of .
By the other hand, in the case of a spinning particle/particle with spin/internal structure/degrees of freedom, the internal degrees of freedom must be represented by some other operator, independently from . In particular, it must also commute with both operators. Then, by definition, for a particle with spin, the angular momentum will be a sum with two contributions: one contribution due to the “usual” angular momentum (orbital part) and an additional “internal” contribution (spin part). That is, mathematically speaking, we should have a decomposition
If , the spin operator, satisfies the above commutation rules (in fact, the same relations than the usual angular momentum), we must impose
The case of Parity P/Spatial inversions
This special transformation naturally arises in some applications. From the pure geometrical viewpoint, this transformation is very simple:
In coordinates and 3D, the spatial inversion or parity is represented by a simple matrix equals to minus the identity matrix
This operator correspods, according to the theory we have been studying, to some operator P (please, don’t confuse P with momentum) that satisfies
and where are the usual position and momentum operators. Then, the operator
is invariant by parity/spatial inversion P, and thus, this feature can be extended to any angular momentum operator like spin S or angular momentum J. That is,
The Wigner’s theorem implies that corresponding to the operator P, a discrete transformation, must exist some unitary or antiunitary operator. In fact, it shows that P is indeed unitary
If P were antiunitary we should get
Then, the parity operator P is unitary and . In fact, this can be easily proved from its own definition.
If we apply two succesive parity transformations we leave the state invariant, so . We say that the parity operator is idempotent. The check is quite straightforward
Therefore, from this viewpoint, there are (in general) only 2 different ways to satisfy this as we have :
i) . The phase is equal to modulus . We have hermitian operators
Then, the effect on wavefunctions is that . That is the case of usual particles.
ii) The case . The phase is equal to modulus . This is the case of an important class of particles. In fact, Steven Weinberg has showed that where F is the fermion number operator in the SM. The fermionic number operator is defined to be the sum where L is now the leptonic number and B is the baryonic number. Moreover, for all particles in the Standard Model and since lepton number and baryon number are charges Q of continuous symmetries it is possible to redefine the parity operator so that . However, if there exist Majorana neutrinos, which experimentalists today believe is quite possible or at least it is not forbidden by any experiment, their fermion number would be equal to one because they are neutrinos while their baryon and lepton numbers are zero because they are Majorana fermions, and so would not be embedded in a continuous symmetry group. Thus Majorana neutrinos would have parity equal to . Beautiful and odd, isnt’t it? In fact, if some people are both worried or excited about having Majorana neutrinos is also due to the weird properties a Majorana neutrino would have under parity!
The strange case of time reversal T
In Quantum Mechanics, temporal inversions or more generally the time reversal is defined as the operator that inverts the “flow or direction” of time. We have
And it implies that . Therefore, the time reversal operator satisfies
In summary: T is by definition the “inversion of time” so it also inverts the linear momentum while it leaves invariant the position operator.
Thus, we also have the following transformation of angular momentum under time reversal:
Time reversal can not be a unitary operator, and it shows that the time reversal T is indeed an antiunitary operator. The check is quite easy:
This equation matches the original definiton if and only if (IFF)
Time reversal is as consequence of this fact an antiunitary operator.
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.