Symmetry

by landonlehman

To me, part of what makes physics beautiful and profound is the fundamental role played by symmetry.  This is a vast subject, and becomes even more vast if symmetry breaking is included (e.g. electroweak symmetry breaking).  The idea of symmetry allows us to remove unnecessary complications from the analysis of a system.  For example, the basic idea that I get the same result for an experiment performed on the ground floor of a building and an experiment performed on the third floor of a building means I can remove the factor of which floor contained the experiment from the analysis (of course the experiment must not be a very precise measurement of the acceleration due to Earth’s gravity).

I recently started reading Landau’s Mechanics, and I have been struck by the simple way that he uses symmetries to derive the basic conservation laws in classical mechanics.  Of course Noether’s theorem associates symmetries with conservation laws, but at this point Landau has not proved Noether’s theorem.  So his derivations are not 100% mathematically precise, but rather make use of more physical arguments.

Here is a list:

  • homogeneity of time leads to conservation of energy
  • homogeneity of space leads to conservation of linear momentum
  • isotropy of space leads to conservation of angular momentum

These three conservation laws define the seven additive integrals of the motion for a closed system.  Isotropy of time does not lead to a specific conservation law, but just means that the motion of a system obeying the laws of classical mechanics is (in principle) reversible.  Landau also uses the homogeneity of space and time and Galilean invariance to justify the familiar form of the Lagrangian for a free particle in an inertial frame:

L = \frac{1}{2}m v^2 .

The derivation of these conservation laws from basic symmetries of space and time is amazing.  But it is even more amazing that this same connection between symmetries and conservation laws carries over into quantum mechanics.

  • symmetry under space translation (homogeneity of space) leads to conservation of linear momentum
  • time-translation invariance (homogeneity of time) leads to conservation of energy
  • invariance under spatial rotations (isotropy of space) leads to conservation of (total) angular momentum

Now, note that conservation has a slightly different meaning than in classical mechanics.  Since “energy” (the Hamiltonian) and momentum are operators, conservation just means that the expectation value of the operator is time-independent.  Also, isotropy of time is not a fundamental symmetry – it is violated by weak interactions.  Finally, these are not the only symmetries – there are more symmetries in quantum mechanics than there are in classical mechanics.  As Weinberg puts it: “…many if not all of the operators representing observables in quantum mechanics are the generators of symmetries.” (Lectures on Quantum Mechanics, pg. 72).

I can grasp the importance of symmetries at this point in my study, but I still feel like I do not understand them at a truly fundamental level.  I am sure I will understand more when I study the Standard Model, which is full of symmetries.

A quote from Feynman to close things off:

“…a fact that most physicists still find somewhat staggering, a most beautiful and profound thing, is that, in quantum mechanics, for each of the rules of symmetry there is a corresponding conservation law; there is a definite connection between the laws of conservation and the symmetries of physical laws.”

Richard Feynman, The Feynman Lectures on Physics, Vol. 1, Ch. 52

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