Friday, 8 December 2017

conservation laws - Where isn't there a quantum number associated with the Lorentz boost?


In Schwartz's QFT, there is a problem (3.2) on p. 42 asking to calculate the conserved current Kμνρ associated with the Lorentz transformation xμΛμνxν. It's expression is



Kμνρ=Tμ[ρxν]

where, the square brackets imply the antisymmetrization.


One can define the conserved quantities Qj=d3x K00j,

which induce the boosts. Being conserved, one means that dQjdt=0.


However, Schwartz also stated that dQjdt=0 is consistent with iQjt=[Qj,H] !


Now, let us look at the definition of Qj more carefully. You will find out that Qj is a function of t only. So Qjt=dQjdt. But [Qj,H]0! What is going on?




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