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, ;).
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!