Dimensions of Hilbert Spaces confusion

[tom.stoer]

samalkhaiat, thanks for the excellent summary, but I think you miss some important points.

The Gauß law is identical to the generator of gauge transformations only modulo integration by parts, i.e. surface terms; of course they vanish on the compact 3-torus.

In QCD on compact spaces it is natural that physical states are color-neutral states. And I don't see why there should be a problem at all if the theory predicts strict color-neutrality.

So there may be obstacles on R³ but not on T³.

 

[samalkhaiat]

samalkhaiat, thanks for the excellent summary, but I think you miss some important points.

Well, I missed a lot of important points. I already said I would.

The Gauß law is ... i.e. surface terms; of course they vanish on the compact 3-torus.

Do they? Can you show me how you can get rid of the “Schwinger” type terms in the commutator algebra that come from the anomalous divergence of the source current? As far as I know, with careful calculations one ends up with
$$[ i G_{ a } ( x ) , G_{ b } ( y ) ] = f_{ a b c } G_{ c } ( x ) \delta^{ 3 } ( x – y ) - i S_{ a b } ( A ; x , y ) \ \ (1)$$
It is known fact that whenever a dynamical current, i.e. a source current in a gauge theory, possesses a anomalous divergence, then also the equal-time commutators for its components must contain anomalous terms. This is because a divergence can be represented by commutator with the translation generators $P^{ \mu }$ and the commutator must be anomalous since the divergence is. Since $P^{ 0 } = H$ contains $J^{ \mu }_{ a }$, one expects that the equal-time commutator $[ J^{ \mu }_{ a } , J^{ 0 }_{ b } ]$ is anomalous. This commutator is one source of anomaly in (1). The other coming from the commutator of $( D_{ j } \delta / \delta A_{ j } )_{ a }$ with $J^{ 0 }_{ b }$. This commutator induces infinitesimal gauge transformation on any explicit dependence of $J^{ 0 }_{ b }$ on the gauge field.

In QCD on compact spaces it is natural that physical states are color-neutral states. And I don't see why there should be a problem at all if the theory predicts strict color-neutrality.

That does not help me to determine how much spin do the quarks and gluons contribute to the proton.

 

[strangerep]

See also the following review

[2] Lavelle, M. and McMullan, D. Phys. Rep. 279, 1(1997).

It's so cool that you mention these authors! Their papers are really interesting. (I've been working with some of their other papers that deal with IR issues and gauge invariance in QED.)

For the benefit of other readers, that paper is freely available here.

 

[tom.stoer]

Can you show me how you can get rid of the “Schwinger” type terms in the commutator algebra that come from the anomalous divergence of the source current? As far as I know, with careful calculations one ends up with
$$[ i G_{ a } ( x ) , G_{ b } ( y ) ] = f_{ a b c } G_{ c } ( x ) \delta^{ 3 } ( x – y ) - i S_{ a b } ( A ; x , y ) \ \ (1)$$

With gauge-invariant regularization these Schwinger terms do vanish for gauge currents. (I don't know whether this works for chiral gauge theories)

That does not help me to determine how much spin do the quarks and gluons contribute to the proton.

Of course it doesn't. I don't see why the local color gauge algebra shall be able to do that. For the color-neutral axial current things may change, but that's not a gauge current.

 

[samalkhaiat]

It's so cool that you mention these authors! Their papers are really interesting. (I've been working with some of their other papers that deal with IR issues and gauge invariance in QED.)

For the benefit of other readers, that paper is freely available here.

They understand the role played by Poincare’ generators in gauge field theories. And that is good enough for me. Plus, one of them is a good teacher.

 

[samalkhaiat]

With gauge-invariant regularization these Schwinger terms do vanish for gauge currents. (I don't know whether this works for chiral gauge theories)

I think, I said that the anomalies come from the source current not the gauge field current.

I don't see why the local color gauge algebra shall be able to do that.

That is a very long story. See any paper by E. Leader on the “proton angular momentum controversy”. Or read page 49-51 in the above mentioned review by Lavelle & McMullan.