Charge conservation question
Jay R. Yablon
equation is:
d_uJ^u = d_u(psi-bar gamma^u psi) = 0 (1)
so too, the axial / chiral current is conserved:
d_uJ^{5u} = d_u(psi-bar gamma^u gamma^5 psi) = 0 (2)
In weak SU(2) interactions, (2) is no longer true (no axial charge
conservation), but is (1) still true, i.e., is:
d_uJ^iu = d_u(psi-bar T^i gamma^u psi) = 0 (3)
In SU(3) QCD, is it correct to suppose that both (1) and (2) are true
because color is thought to interact equally with left and right-handed
states?
Thanks,
Jay.
____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm
Hans de Vries
> In QED, the electric current J^u is conserved, and the continuity
> equation is:
>
> d_uJ^u = d_u(psi-bar gamma^u psi) = 0 (1)
>
> so too, the axial / chiral current is conserved:
>
> d_uJ^{5u} = d_u(psi-bar gamma^u gamma^5 psi) = 0 (2)
>
> In weak SU(2) interactions, (2) is no longer true (no axial charge
> conservation), but is (1) still true, i.e., is:
>
> d_uJ^iu = d_u(psi-bar T^i gamma^u psi) = 0 (3)
>
> In SU(3) QCD, is it correct to suppose that both (1) and (2) are true
> because color is thought to interact equally with left and right-handed
> states?
>
> Thanks,
>
> Jay.
Hi, Jay
The axial current in U(1) is actually non-conserved.
Most QFT textbook handle this as the "Chiral Anomaly"
See the biography in:
http://physics-quest.org/Book_Chapter_EM2_ChernSimonsSpin.pdf
A good overview is given by Jackiw here in section 2.
It handles both the Abelian and Non Abelian case.
http://arxiv.org/PS_cache/hep-th/pdf/0011/0011274v1.pdf
It can be shown that the axial current plus the so-called
Chern Simons term is co-conserved. The latter can
therefor be considered as the electromagnetic spin-
density of the vacuum. (see my chapter 6 above)
Regards, Hans
--------------------------------------------
http://physics-quest.org/
Jay R. Yablon
"Hans de Vries" <Hans.de....@gmail.com> wrote in message
news:9f177ec7-f9d4-4363...@c36g2000prc.googlegroups.com...
What about the regular current J^u? Is there any situation in which we
need to be careful about setting d_uJ^u=0, or can this be done more or
less with impunity? Isn't J^u *defined* so as to be that which is
conserved and so has the equation d_uJ^u=0?
Jay.
Hans de Vries
> Hi Hans,
>
> What about the regular current J^u? Is there any situation in which we
> need to be careful about setting d_uJ^u=0, or can this be done more or
> less with impunity? Isn't J^u *defined* so as to be that which is
> conserved and so has the equation d_uJ^u=0?
Yes, this is the conservation of charge after all.
The axial current describes chiral motion of charge.
Its time component is zero in the rest-frame. There
is no such thing as "chiral charge". Chiral current
is chiral motion of ordinary charge.
Regards, Hans
Regards, Hans
Mirror Space
NO! There is notion of axial charge - it is for example, Axial-vector
coupling
constant in nucleon current...
andy
Hans de Vries
I presume this is just about the definition of charge....
I'm referring to the time component of a vector as
being the charge. In case of an axial vector the
time component is per definition zero in the rest
frame so there is no axial charge.
Even if we where talking about a magnetic monopole
charge then this would be the time component of
a normal vector which would be an extra vector
dual to the EM potential vector A . It would not be
part of an axial vector.
That said. I'm basically talking U(1) and not SU(3)
Regards, Hans
--------------------------------------------
http://physics-quest.org/
Regards, Hans