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

• ndung200790
In summary, the generator of U(1) symmetry is the operator of electric charge which is a result of local U(1) symmetry giving rise to the conservation of charge by Noether's theorem. The U(1) occurring in the standard model is actually the weak charge, a combination of electric charge and isospin. The electric charge is related to a local gauge symmetry and is derived from the 'local generator' related to the Gauss law. Isospin, on the other hand, is a global symmetry and has only a 'global generator'. This is because the electroweak interaction has a gauge group U(1) x SU(2) and the gauge group of electromagnetism corresponds to simultaneous rotations about the
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.

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.

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.

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

ndung200790 said:
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).

Last edited:
ndung200790 said:
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.

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

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

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

The generator of U(1) is known as the electric charge operator because it has been experimentally observed that it governs the behavior of electrically charged particles. This means that the operator of electric charge is responsible for the interactions between charged particles, such as the repulsion between two positively charged particles or the attraction between a positively charged particle and a negatively charged particle.

## 2. What is U(1) and how is it related to electric charge?

U(1) is a mathematical group that represents the symmetries of the electromagnetic force, which includes electric charge. The U(1) group is associated with the conservation of electric charge, meaning that the total amount of electric charge in a closed system remains constant over time.

## 3. How does the operator of electric charge behave under certain transformations?

The operator of electric charge follows a specific set of rules known as the commutation relations when it undergoes certain transformations. These commutation relations describe how the operator of electric charge interacts with other operators, such as the momentum or angular momentum operators, and how they all behave together under transformations.

## 4. Can the generator of U(1) be used to predict the behavior of electrically charged particles?

Yes, the generator of U(1) can be used to make predictions about the behavior of electrically charged particles. This is because the operator of electric charge is a fundamental part of the equations that describe the electromagnetic force, and these equations have been extensively tested and proven to accurately predict the behavior of charged particles in experiments.

## 5. Are there any other generators besides U(1) that could potentially be the operator of electric charge?

No, U(1) is the only generator that has been experimentally confirmed to be the operator of electric charge. Other generators, such as SU(2) and SU(3), are associated with other fundamental forces in the Standard Model of particle physics, but they do not govern the behavior of electrically charged particles.

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