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10

If the theory is invariant under translations in space, then linear momentum is conserved by Noether's theorem. If the theory is quantum, conservation holds only on the level of the expectation values (because that's the only meaningful level where you can talk about momentum as a number that's conserved in time), but it still holds. There is no way out. ...

-1

The real converse of the first theorem is the second one. Your formulation of the converse of the first theorem is too literal and thus valid as a particular case under additional conditions.

2

Noether theorem tells you that if you can find a (one parameter) group of infinitesimal transformations $\alpha$ and $\beta$ such that: $$t'=t+\alpha\epsilon$$ $$q'^\mu=q^\mu+\beta^\mu\epsilon$$ and your lagrangian is invariant under this group of transformations, then the quantity ...

0

The conformal charge in CFTs is a special case of any other standard treatment of classical field theory (e.g. chapter 1 of Peskin & Schroeder). In flat space with $(1, n)$ signature and spacetime $(t,x_1, ..x_n)$ coordinates, we consider a conformal killing vector field with components $\epsilon^\mu$. We assume our field theory yields some ...

2

It is pretty simple just use the following formula, $$\int d^3 x e^{i(p+q)x} = (2\pi)^3\delta(p+q)$$ and thus on integrating $d^3 q$ you will have $\sqrt{2E(p)2E(q)} = E(p)$ in the downstairs, and then it's pretty straightforward.

0

Energy has the capacity to do work. When you feel "full of energy", chemical energy stored in your muscles can be converted into mechanical energy, to enable you to run, or lift small cars above you head. Different types of energy are stored in different ways. Gravitational energy is inherent in any object which the gravitational forced can move downwards ...

2

Comments to the question (v5): If an action functional $S$ is invariant under a Lie algebra $L$ of symmetries, the corresponding Noether currents & charges do not always form a representation of the Lie algebra $L$. There could be (classical) anomalies. In some cases such (classical) anomalies appear as central extensions, cf. e.g. Ref. 1-3 and this ...

4

Comments to the question (v2): Noether's (first) Theorem is really not about Lie groups but only about Lie algebras, i.e., one just needs $n$ infinitesimal symmetries to deduce $n$ conservation laws. If one is only interested in getting the $n$ conservation laws one by one (and not so much interested in the fact that the $n$ conservation laws often ...

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