Is J_z|0> equal to zero in QFT for spinor field?

[Heirot]

Let's say that we're trying to calculate J_z|0> for spinor field. Now, one would naturally expect that it vanishes as do all of the observables in vacuum state. On the other hand, J_z isn't diagonal in the number basis, so one has J_z|0> ~ |2>, i.e. particle + antiparticle state due to the two creation operators in J_z. So, which one is it? All the books I've seen says that it vanishes. But the direct calculations says it isn't so.

Thanks

 

[fzero]

The angular momentum operators do conserve particle number. Fields are constructed to lie in a definite angular momentum representation. The vacuum state has $$j=m_j=0$$ (a scalar) so any angular momentum operator acting on it gives zero.

 

[Heirot]

How can one state (vaccum) have well defined momenta and angular momenta at the same time?

 

[fzero]

There's no contradiction. The uncertainty relation has a bound of zero when the expectation value of one of the operators vanishes completely.

 

[Heirot]

What about if one uses the plane wave decomposion of spinor field psi and use it in the angular momentum operator? Then one has J_z|0> ~ |2>

 

[fzero]

The angular momentum operators never change the particle number. If $$\vec{J}=\vec{L}+\vec{S}$$ then you use the differential operator representation of $$\vec{L}$$ while $$\vec{S}$$ acts on the internal spin state.

 

[strangerep]

What about if one uses the plane wave decomposion of spinor field psi and use it in the angular momentum operator? Then one has J_z|0> ~ |2>

What expression are you trying to use for J_z ??

 

[Heirot]

I'm trying to derive an operator containing creation and destruction operators for electrons and positrons (b and d). I can't seem to shape it into a form that contains only bTb and dTd combinations (T - hermitian conjugation) which would preserve the particle nubmer. Is there any reference for J in terms of b and d? I could only find it in terms of Dirac field psi.

 

[fzero]

I'm trying to derive an operator containing creation and destruction operators for electrons and positrons (b and d). I can't seem to shape it into a form that contains only bTb and dTd combinations (T - hermitian conjugation) which would preserve the particle nubmer. Is there any reference for J in terms of b and d? I could only find it in terms of Dirac field psi.

The angular momentum operators never depend on creation and destruction operators. Field operators act on the vacuum to create a state of definite spin. The angular momentum operators are defined by

$$J_i = L_i + S_i.$$

The first term is the spatial representation (in units where $$\hbar=1$$)

$$L_i = \frac{1}{2} \epsilon_{ijk} x^j \partial^k,$$

while the internal spin is completely determined by its action on a general spin state:

$$S^2 |s,m_s\rangle = s(s+1) |s,m_s\rangle, ~ S_z |s,m_s\rangle = m_s |s,m_s\rangle , etc.$$

 

[Avodyne]

I'm trying to derive an operator containing creation and destruction operators for electrons and positrons (b and d). I can't seem to shape it into a form that contains only bTb and dTd combinations (T - hermitian conjugation) which would preserve the particle nubmer. Is there any reference for J in terms of b and d? I could only find it in terms of Dirac field psi.

I don't know of an explicit reference, but you should be able to derive it from the expression in terms of the field. It should definitely come out in the form of bTb and dTd terms only.

 

[TriTertButoxy]

I'm trying to derive an operator containing creation and destruction operators for electrons and positrons (b and d). I can't seem to shape it into a form that contains only bTb and dTd combinations (T - hermitian conjugation) which would preserve the particle nubmer. Is there any reference for J in terms of b and d? I could only find it in terms of Dirac field psi.

I remember trying to derive just such a formula a while back, but I ran into several problems -- I don't quite remember what the were. I ultimately gave up on it, and settled on whatever they have in Itzykson+Zuber p147. If you find out how to get the forumla, please let me know.

 

[Heirot]

The problems include the derivatives of the spinors which don't satisfy the orthonormal relation. In that way, the time dependent terms includind bTdT and db do not vanish! Apperently, no books speaks of this problem.

 

[strangerep]

The problems include the derivatives of the spinors which don't satisfy the orthonormal relation. In that way, the time dependent terms includind bTdT and db do not vanish! Apperently, no books speaks of this problem.

Try Eugene Stefanovich's book, available as

http://arxiv.org/abs/physics/0504062

In ch7, pp246-247, you'll find a derivation of the Lorentz generators in terms of
c/a operators. E.g., eqn(7.32) gives the expression for J_z (in the noninteracting
representation).

I'm not sure if this is exactly what you want, but it might give you
some idea of how to derive this sort of stuff.

 

[Heirot]

Unfortunately, this book also deals with angular momentum of a scalar field which can't be easily generalized to a non zero spin.

 

[Heirot]

I remember trying to derive just such a formula a while back, but I ran into several problems -- I don't quite remember what the were. I ultimately gave up on it, and settled on whatever they have in Itzykson+Zuber p147. If you find out how to get the forumla, please let me know.

After a long and very much tedious calculation, I've derived it! Would you like to see the derivation?

 

[TriTertButoxy]

After a long and very much tedious calculation, I've derived it! Would you like to see the derivation?

YES YES! Yes Very MUCH SO! As I said before, I have been trying to derive it for ages, and couldn't and was forced to give up. If you could show me the derivation, I would be indebted to you.

 

[Heirot]

I'll post it as soon as I type it in Latex ;)

 

[naima]

Take your time but PLEASE, don't forget.

thanks.

 

[TriTertButoxy]

I think he forgot

 

[Heirot]

He didn't. Next week...

 

[Heirot]

Sorry for the delay... Merry Christmas!

 

[naima]

Thanks
You too

 

[TriTertButoxy]

This is so awesome! Thanks. Happy new year.