Wednesday, 22 May 2019

quantum field theory - Are there any theorems that support the commutation relations in QFT?


I am learning Quantum Field Theory. I am confused about the commutation relations, which says that the field and its conjugate momentum don't commute. Are there any theorems that support the commutation relations in QFT? Or why the relation is correct? Simply due to experimental test?



Answer




It is just a way to get from a classical to what we call a quantum theory. In classical mechanics you have the poisson brackets governing the dynamical behaviour of your system. If you leave the Hamiltonian/Lagrangian the same and change this poisson bracket to a commutator of operators you obtain a new theory. This is than basically labeled "Quantum". There is of course a whole array of material on why this way of quantizing a theory is identical to other ways like for example the path integral formulation.


Of course in the end the only way to know that this gives you something "correct" is experimental. In fact you can create a lot of theories which are consistent but ultimately probably don't give a description of nature. It just so turns out that experiments confirm up to some point that the theories which do describe nature in particular QED are theories which are in the sense Quantum that its components, the fields and their conjugate moment, fullfill canonical commutation or anticommutation relations.


As a side node the mathematics behind QFT is partially not rigorous mathematics at all. Completely rigorous mathematical QFT has already a problem with operator valued distributions, which fields supposedly are, because they commute to the delta function. In this sense you have trouble even defining a field, but just because the mathematical tools aren't there doesn't necessarily mean the theories are invalid.


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