How do we know generator of U(1) is operator of electric charge?

[ndung200790]

Please teach me this:
How do we know the generator of U(1) symmetry is the operator of electric charge.Is this observation(the value defined by the operator) being the reserved observation deducing from Noether theorem(considering U(1) symmetry).And also how we know this Noether reserved observation were electric charge?
(I have known that from U(1) symmetry we can deduce the Dirac-Maxwell Lagrangian,but I do not understand the relation between the generator and ''physical'' electric charge)
Thank you very much for your kind helping.

 

[tommyli]

The simple answer is that local U(1) symmetry gives rise to conservation of charge by Noether's theorem, yes. But you have to be careful.. the U(1) occurring in the standard model is not electric charge but weak charge which is a combination of electric charge and isospin. The operator of electric charge is generator of U(1) + third component of isospin. This allows the photon to be the only massless particle after spontaneous symmetry breaking. Hope this helps.

 

[tom.stoer]

Have a look at

Quantum Mechanics of Gauge Fixing
F. Lenz, H.W.L. Naus, K. Ohta, M. Thies
Univ Erlangen Nurnberg, Inst Theoret Phys, Staudtstr 7, D 91058 Erlangen, Germany and Univ Tokyo, Inst Phys, Tokyo 153, Japan
Abstract: In the framework of the canonical Weyl gauge formulation of QED, the quantum mechanics of gauge fixing is discussed. Redundant quantum mechanical variables are eliminated by means of unitary transformations and Gauss′s law. This results in representations of the Weyl-gauge Hamiltonian which contain only unconstrained variables. As a remnant of the original local gauge invariance global residual symmetries may persist. In order to identify these and to handle infrared problems and related "Gribov ambiguities," it is essential to compactify the configuration space. Coulomb, axial, and light-cone representation of QED are derived. The naive light-cone approach is put into perspective. Finally, the Abelian Higgs model is studied; the unitary gauge representation of this model is derived and implications concerning the symmetry of the Higgs phase are discussed.

 

[ndung200790]

Why we must add third component of isospin but not add the first and second component into generator of U(1)?

 

[tom.stoer]

Why we must add third component of isospin but not add the first and second component into generator of U(1)?

?

Isospin has nothing to do with electric charge; it is a new quantum number.

The electric charge is related to a local gauge symmetry, therefore you have a 'local generator' which is related to the Gauss law and from which a 'global generator' i.e. charge can be derived (but the global generator misses some features of the local one).

Isospin is a global symmetry (no gauge symmetry), therefore there is no local symmetry, no Gauss law, no related gauge field etc.; that means for isopsin there is only a global generator. Isospin is a special case SU(2) for a larger symmety structure SU(N) where N counts quarks flavors (u,d,s,c,...). For SU(2)-isospin you have three generators which formally look like Pauli-matrices (which have been introduced as three generators for SU(2)-spin).

 

[tommyli]

Why we must add third component of isospin but not add the first and second component into generator of U(1)?

The electroweak interaction has a gauge group U(1) x SU(2). Within this gauge group you must be able to find the gauge group of electromagnetism, which is U(1). Electromagnetism cannot simply correspond to the U(1) subgroup in the product, it so happens that it corresponds to the subgroup of simultaneous U(1) rotations and SU(2) rotations about the z-axis, in other words the gauge group of electromag is represented by rotations e^{i\alpha} e^{-i\alpha \tau_z} where \tau_z is the third component of isospin.

 

[ndung200790]

By the way,please teach me why electric charge also relate with Baryon and Strangeness numbers?

 

[ndung200790]

It seems to me that is Gell-Mann formulation about electric charge.