And another one on Lorentz invariance

[alphaone]

It is clear that a conserved current $$\partial_{\mu} J^\mu = 0$$ implies the existence of a conserved charge $$Q= \int d^3x J^0$$. Now I want to go the other way round: Suppose we have a basis of momentum eigenstates, such that these states are also eigenstates of the charge. Then clearly the charge commutes with the energy operator and is thus conserved but can we say anything else about the 4-vector current by for example invoking Lorentz invariance? It would be nice if there was a way to deduce $$\partial_{\mu} J^\mu = 0$$ but I do not see how that is possible

 

[Demystifier]

If I understood you correctly, you would like to derive local conservation from global conservation. I don't think that it is possible. The requirement of local conservation is much stronger than that of global one.

 

[alphaone]

Thanks that is what I thought.

 

[samalkhaiat]

It is clear that a conserved current $$\partial_{\mu} J^\mu = 0$$ implies the existence of a conserved charge $$Q= \int d^3x J^0$$. Now I want to go the other way round

Look up S Coleman theorm:

If, for any 4-vector $$J_{a}$$ ,

$$Q = \int d^{3}x J^{0}$$

is a well defined operator on the H-space, and $$Q|0> = 0$$ (i.e.,the symmetry is manifest), then

$$\partial_{a}J^{a} = 0$$

sam

 

[alphaone]

Thanks for the relpy. That result amazes me and I will definitely look up the theorem! Could you elaborate on what you mean by saying that the symmetry is manifest? Under what conditions can I assume $$Q|0> = 0$$ ? Please let me know.

 

[alphaone]

Could you also name a reference where I can find a proof of the Coleman theorem? Thanks in advance

 

[alphaone]

I now know why we need Q|0>=0 and what it means, but could somebody please tell me where to find the theorem of Sidney Coleman samalkhaiat was talking about. Thanks in advance

 

[samalkhaiat]

Thanks for the relpy. That result amazes me and I will definitely look up the theorem! Could you elaborate on what you mean by saying that the symmetry is manifest?

It means that the symmetry of the Lagrangian is also a symmetry of the ground state. I.e., it is unitarly implimented on at least a dense subset of the Hilbert space including the vacuum.

Under what conditions can I assume $$Q|0> = 0$$ ?

When

$$\delta \Phi = [iQ ,\Phi]$$

does not develop non-vanishing vacuum expectation value.


You can find a simple proof of Coleman's theorem on page 515 of Itzykson & Zuber ; Quantum Field Theory.

regards

sam