Chapter 3 Chapter 3 Group Theory and the Analysis of Periodic Electromagnetic
3.2 Symmetries of Classical and Quantum Systems
What is a symmetry? We all know from everyday experience that the symmetries of an object correspond to those vantage points from which the object looks iden- tical. Similarly, the symmetries of a physical system can be described as the group of observation points from which the underlying structure of a system "looks" iden- tical. Mathematically, symmetries describe a set of reference frames1 in which the governing equations of motion of the dynamical variables are identical. As the state of the system is the same for all observers, the symmetries then constrain the possible solutions of the dynamical variables2. These constraints can lead to conservation laws (conserved currents )3, and to degeneracies (in frequency or energy) of the different states of the system. Symmetry plays a very important role in physics. It is an ac- tive role. In many cases physicists are led to new theories by postulating a required symmetry of nature4.
The representation of a symmetry may take different forms depending upon the model used to describe the system. This can be a source of much confusion, and one must remember that the form of a symmetry transformation is only valid in context.
This is most evident in quantum theory in which a symmetry is represented as an operator on a Hilbert space which commutes with the Hamiltonian operator. In the following a brief description of the dynamics for classical and quantum theories is given, and an attempt is made to relate the different representations of symmetry transformations.
In classical mechanics one deals with particles localized in space and time. The dynamical variables (degrees of freedom) are the position (qi(t)) and velocity (qi(t)) of each particle. The dynamics of the system are found by extremizing (usually minimizing) the "action" of the paths taken by the particles. The action, S, is the
1 Here we mean reference frame in the most general sense, not just coordinate transformations but also transformations of the fields or particles.
20f course no more than the equations of motion do themselves.
3This is the basis of Noether's theorem.
4Take for example Einstein and his Special Theory of Relativity in which each observer in an inertial reference frame must be using the same physics (equations of motion) to describe nature.
integral along the path5 of the Lagrangian, L(qi(t), qi(t)). An equivalent formulation is possible in which a change of variables is made and the system is described by the particle's position and canonical momentum, pi(t) = &L/&(qi)· The dynamics of the system can then be defined in terms of the Hamiltonian, H(qi,Pi) piqi - L.
In classical field theory every point in space-time6 is considered to be an indepen- dent degree of freedom. The fields are defined as (scalar, vector, spin or, etc.) functions on the space-time manifold. Take for example a scalar field in space-time, <f>(xµ)· The Lagrangian is now an integral over space of a Lagrangian density, .C(<f>(xµ), &µ</>(xµ)).
The action, S, is then the integral of the Lagrangian over a path parametrized by time, t:
ta+T
S= I
dtL(t) = (3.1)ta
S is a functional which measures the "length" of a path that can be taken by the dynamical variables. The classical path, as described by the classical equations of motion, is then determined by extremizing the action S. The symmetries of the system are transformations which leave the equations of motion invariant. More generally, symmetries are transformations which leave the action invariant up to a surface term 7, or equivalently the Lagrangian density invariant up to a 4-divergence OµJP(x)B.
In quantum mechanics the picture is much different. The system is described by a Hilbert space. The dynamical evolution of the Hilbert space states is governed by the Hamiltonian operator which is proportional to the generator of time translations.
Focusing on the Hilbert space states as the fundamental objects, we then qualify symmetries as those operators which commute with the Hamiltonian operator (i.e., leave the equations of motion of the Hilbert states unchanged). The transformation
5 A path in this case is defined by the evolution in time of the particles.
60r whatever manifold the system lives on.
7 An integral over the boundary of the space-time volume in eq. (3.1).
8See [65].
of the Hamiltonian from a function of the dynamical variables to an operator on a Hilbert space may be regarded as simple mathematical trickery. The quantum picture and the classical picture of the symmetries of a system are reconciled by use of the Feynman path integral method as described below.
Feynman9 believed in the concept that all paths are responsible for the dynamics of a system. He used this idea to formulate the Feynman path integral approach. In the Feynman path integral method, all paths are to be treated as equal participants in the system evolution and their "length" is measured (as in classical mechanics) by the action, S. The full dynamics of the system is then governed by the interference pattern formed by all the paths evolving together and is described mathematically as follows:
(3.2) (3.3) where ¢>(x)lt0 = </>a(x) is the initial field distribution, ¢>(x)lt0+r = ¢>h(x) is the final field distribution, U([¢>a(x); ta], [</>b(x); ta+ T]) is the time evolution functional, and P([</>a(x); ta] -+ [¢>h(x); ta
+
T]) is the probability that the system will evolve from<f>a(x) -+ </>b(x) as t =ta-+ ta+ T. In the Feynman path integral, the evolution of the system asymptotically approaches classical motion for large scale systems (n -+ 0).
A symmetry transformation within this framework is then found to be equivalent to the invariance of the action up to a surface term as in the classical case. One can also show that the Hamiltonian, defined as a function of the dynamical variables, satisfies the following relationship:
in
88rU(¢>(xa), ¢>(xb); T)=
HU(¢>(xa), ¢>(xb); T). It is in this sense that one can treat the Hamiltonian (H) equivalently as a linear operator responsible for time translation on an underlying vector (Hilbert) space. Symmetry operators on this underlying Hilbert space are then those operators which commute9This was based on an earlier idea of Dirac's.
with the Hamiltonian operator. The two pictures, although mathematically different, are nonetheless equivalent in their predictions.
In the next section we will study the symmetries of a (classical) electromagnetic field in a macroscopic dielectric. The symmetries of the system will correspond to those transformations of the fields which leave Maxwell's equations, the equations of motion, invariant.