QuantumPhysics.dvi

9 Symmetries in Quantum Physics

Symmetry, as wide or as narrow as you may define its meaning,

is one idea by which man through the ages has tried to

comprehend and create order, beauty and perfection.

– Hermann Weyl –

Symmetries can manifest themselves in different ways.

• A symmetries of an objectis the oldest of the two concepts. The circle, the regular

hexagon, square, or triangle, for example, represent some of the deepest symbols of civiliza-

tion, mainly because of their high degree of symmetry. The most elementary definition of

symmetry then is that an object is symmetric if it will appear the sameif viewed from a

transformed perspective. For all the above planar figures, theirshapes will appear unchanged

after certain planar rotations about their centers.

• A symmetry of the laws of Natureis a more recent concept, which can be traced back to

the Renaissance. Galileo realized that the laws governing classical mechanics do not change

if we transform our observational frame in time or in space – in a uniform manner. This

property is referred to as Galilean relativity; it includes the symmetries of time translation

as well as space translation and rotations. Einstein considerably extended the range of

applicability of the relativity principle to special and then general relativity. In the latter,

the laws of physics are unchanged under any transformation of our observational frame.

9.1 Symmetries in classical mechanics

Recall that in classical mechanics we encountered two types of symmetries, following the

model of the above discussion.

• Symmetries of the solutions of the Euler Lagrange equations

(such as, for example, symmetries of shapes of orbits);

• Symmetries of the Euler Lagrange equations themselves

(i.e. symmetries of the laws of Nature).

From the modern perspective on physics, the symmetries of the laws of Nature are considered

to be primordial. For example, translation and rotation invariance are ingrained in the basic

laws of Nature, and are associated with the conservation of energy, momentum and angular

momentum. The symmetries of the solutions, such as the symmetries of orbits, are viewed

to be less fundamental. In fact, while the orbits of most planets areroughly circular, closer

inspection reveals that they are really closer to elliptical, with further deviations from the

elliptical form. Thus the symmetries of the solutions are for the most only approximate and