I'm trying to understand the connection between Noether charges and symmetry generators a little better. In Schwartz QFT book, chapter 28.2, he states that the Noether charge Q generates the symmetry, i.e. is identical with the generator of the corresponding symmetry group. His derivation of this goes as follows: Consider the Noether charge
Q=∫d3xJ0(x)=∫d3x∑mδLδ˙ϕmδϕmδα
which is in QFT an operator and using the canonical commutation relation [ϕm(x),πn(y)]=iδ(x−y)δmn,
[Q,ϕn(y)]=−iδϕn(y)δα.
From this he concludes that we can now see that "Q generates the symmetry transformation".
Can anyone help me understand this point, or knows any other explanation for why we are able to write for a symmetry transformation eiQ, with Q the Noether charge (Which is of course equivalent to the statement, that Q is the generator of the symmetry group)?
To elaborate a bit on what I'm trying to understand: Given a symmetry of the Lagrangian, say translation invariance, which is generated, in the infinite dimensional representation (field representation) by differential operators ∂μ. Using Noethers theorem we can derive a conserved current and a quantity conserved in time, the Noether charge. This quantity is given in terms of fields/ the field. Why are we allowed to identitfy the generator of the symmetry with this Noether charge?
Any ideas would be much appreciated
Answer
Consider an element g of the symmetry group. Say g is represented by a unitary operator on the Hilbertspace Tg=exp(tX)
Also, this answer and links therein ought to help you further.
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