By W. Siegel

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The phenomenon of symmetry breaking, and the related mechanisms of mass generation, can also be seen at the classical level. In perturbative quantum field theory, classical field theory is simply the leading term in the perturbation expansion. Note that “global” (time-, and usually space-independent) symmetries can eliminate a variable, but not its time derivative. , eliminate) the position of the center of mass of a system at some initial time, but not its time derivative, which is just the total momentum, whose conservation is a consequence of that same symmetry.

Comprehensive; style similar to Bjorken and Drell. 6 B. de Wit and J. Smith, Field theory in particle physics, v. : No Yang-Mills or Higgs (but wait till v. ). I. B. : Numerous examples of QED calculations. P. : Original treatment of quantum field theory as we know it today, but from mechanics; includes reprints of original articles (1949). 37 Modern, but somewhat specialized Basics, plus thorough treatment of an advanced topic 9 J. Zinn-Justin, Quantum field theory and critical phenomena, 4th ed.

In the last identity the ordering is important only in the quantum mechanical case: In general, the difference between classical and quantum mechanics A. COORDINATES 43 comes from the fact that in the quantum case operator reordering after taking the commutator results in multiple commutators. Infinitesimal symmetry transformations are then written as δA = i[G, A], A′ = A + δA where G is the “generator” of the transformation. More explicitly, infinitesimal generators will contain infinitesimal parameters: For example, for translations we have G = ǫi pi ⇒ δxi = i[G, xi ] = ǫi , δpi = 0 where ǫi are infinitesimal numbers.