Wednesday 13 June 2018

quantum field theory - "QFT is simple harmonic motion taken to increasing levels of abstraction"



"QFT is simple harmonic motion taken to increasing levels of abstraction."




This is my memory of a quote from Sidney Coleman, which is probably in many textbooks.


What does it refer to, if he meant something specific?


If he did not, which is far most likely, then rather than asking a list question, if somebody can point to an example of how we move from SHM to, presumably some example of fields interacting, producing or destroying a particle.


I don't think I can go any further with asking a question, because of the restrictions when asking a specific recommendation, but if someone is able to say, "wait until you get to chapter X of Zee, Tong, P & S (please don't say Weinberg) and enlightenment will follow", that would be very much appreciated.


My apologies if I have mangled the quote, nobody has heard of it or I am trying to run way ahead of myself. No need to hold back on telling me that last part.



Answer



If you quantize the oscillator you get a natural particle interpretation. If instead you extend to a field theory the oscillator becomes a classical Klein-Gordon field, with frequency become mass. If you now quantize that you again get a particle interpretation, but the field is a linear combination of ladder operators so particle number becomes another observable. (This is related to how relativity prevents you sticking to a theory of one particle.) Bogoliubov transformations provide different "perspectives" on particle number, in analogy with changes in a classical oscillator's choice of phase space coordinates.


If you now extend the theory to multiple particle species you can couple your oscillators to denote interactions that create or destroy particles, in analogy with energy transferred between classical oscillators. If you let your Lagrangian gain terms that make the EOMs nonlinear, you have nonzero VEVs as in the Higgs effect. This is analogous to anharmonic oscillation.


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