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- Thread starter alphaone
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Thanks that is what I thought.

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samalkhaiat

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

Look up S Coleman theorm:

If, for any 4-vector [tex]J_{a}[/tex] ,

[tex] Q = \int d^{3}x J^{0}[/tex]

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

[tex]\partial_{a}J^{a} = 0[/tex]

sam

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

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samalkhaiat

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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.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 [tex]Q|0> = 0[/tex] ?

When

[tex]\delta \Phi = [iQ ,\Phi][/tex]

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

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