# No explanation of quantized charge from U(1)?

## Main Question or Discussion Point

I just read the following passage on unification from Howard Georgi:

"The SU(2) x U(1) theory is not particularly beautiful. It is often called
a unification of the weak and electromagnetic interactions, but, in fact,
the unification is partial at best. The problem is the U(1) charge....
[T]his is a charge that commutes with all the other weak and colour
charges, so group theory tells us nothing about it. In particular, be-
cause of the U(1), the theory gives us no explanation of the striking fact
of electric charge quantization."

But I thought the fact that charge has a U(1) symmetry DOES explain why charge is quantized? John Baez seems to say as much here: http://math.ucr.edu/home/baez/qg-spring2003/hypercharge/

Can anyone tell me what it is that Georgi has in mind? Why doesn't the U(1) symmetry of electric charge explain why it is quantized? (And hence what's wrong with Baez's argument?!) Thanks!

## Answers and Replies

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Staff Emeritus
2019 Award
Baez is not talking about a U(1). He is talking about SU(2) x U(1).

Baez is not talking about a U(1). He is talking about SU(2) x U(1).
Yes, in the article as a whole: but where he discusses the case of hypercharge alone (which also has a U(1) sym) it seems his discussion is confined to U(1)...

Staff Emeritus
2019 Award
But he knows he is going to have to link it to an SU(2).

I do not think Baez in this article gives a conclusive proof that quarks and leptons must carry a multiple of the same electric charge.

tom.stoer
Afaik there is no proof for quantized U(1) electric charge besides magnetic monopoles.

But there is a consistence requirements for a weighted sum over certain charges of the standard model to be zero for anomaly cancellation.

samalkhaiat
If the gauge group is compact simple group, then the corresponding gauge theory has a unique coupling constant and naturally leads to discrete charges (eigenvalues of the charge generators). However, a gauge theory of an abelian group (which does not distinguish between the coupling constant and the charges) can couple to fields with arbitrary coefficients in the covariant derivative. So there is a problem with charge quantization in the U(1) gauge theory. Indeed, it is not known for certain why electric charge is quantized.

As far as we know, matter fields with an abelian gauge group have charges that are integral multiples of e/3 where e is the charge of the electron. So, the electric-charge quantization problem is the conflict between the observed (quantized) values of the electric charge and the theoretically unconstrained continuum of allowable electric charges.

Over the years, many physicists have tried to offer explanations for the quantization of electric charge. Dirac [1] in 1931 showed that monopoles would quantize electric charge. Georgi and Glashow [2] and Pati and Salam [3] explained the charges of the fermions by associating the photon with a traceless generator of a larger unifying gauge group. Others [4],[5],[6],[7],[8],[9] have exploited anomaly cancellation; the general principle in here is the following;
If a Lagrangian contains global symmetries which are anomaly-free (and therefore gaugeable) and independent of the hypercharge, then that Lagrangian does not yield electric charge quantization. Thus to analyse charge quantization in such theory one has to find all of its anomaly-free global U(1) symmetries.

Sam

***
[1] P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. Roy. Soc. Lond. A133 (1931) 60.
[2] H. Georgi and S. L. Glashow, Unity of all elementary particle forces, Phys. Rev. Lett.32 (1974) 438.
[3] J. C. Pati and A. Salam, Lepton number as the fourth color, Phys. Rev. D10 (1974)275.
[4] R. Delbourgo and A. Salam, The gravitational correction to pcac, Phys. Lett. B40 (1972)381.
[5] T. Eguchi and P. G. O. Freund, Quantum gravity and world topology, Phys. Rev. Lett.37 (1976) 1251.
[6] L. Alvarez-Gaume and E. Witten, Gravitational anomalies, Nucl. Phys. B234 (1984) 269.
[7] K. S. Babu and R. N. Mohapatra, Is there a connection between quantization of electric charge and a majorana neutrino?, Phys. Rev. Lett. 63 (1989) 938.
[8] K. S. Babu and R. N. Mohapatra, Quantization of electric charge from anomaly constraints and a majorana neutrino, Phys. Rev. D41 (1990) 271.
[9] R. Foot, H. Lew, and R. R. Volkas, Electric charge quantization, J. Phys. G19 (1993) 361. [hep-ph/9209259].

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DrDu
Doplicher Haag Ruelle theory shows quite convincingly how quantized charges lead in a one to one fashion to a global U(1) symmetry, at least in a massive theory.

See, R. Haag, Local quantum physics, Springer, Berlin, 1996

samalkhaiat
Doplicher Haag Ruelle theory shows quite convincingly how quantized charges lead in a one to one fashion to a global U(1) symmetry, at least in a massive theory.

See, R. Haag, Local quantum physics, Springer, Berlin, 1996
The current associated with such “quantum charge operator” does not satisfy Gauss’ law. This means that the U(1) group in question IS NOT a gauge group.
Electrodynamics (QED) is a theory lying beyond the scope of problems treated in those algebraic formulations of QFT. Gauge field theories (QED included) are characterized by the fact that the (quantal) Noether currents, associated with non-vanishing charges generating global transformations, obey Gauss’ law. From this fact, it follows that the fundamental fields do not conform simultaneously with the requirement of LOCALITY and POSITIVE DEFINITENESS of the metric in Hilbert space. In gauge field theories, one can show that the states corresponding to non-zero eigenvalues of the electric charge cannot be obtained by applying local operators to the vacuum state; see [1],[2],[3].
Druhl, Haag, Roberts and Doplicher [4],[5],[6] assumed in the setting of algebraic QFT that a particle state cannot be distinguished from the vacuum by measurements performed in the space-like complement of large enough but bounded regions of Minkowski space. Particles with charge, such as baryon number or strangeness, fit well into their schemes. However, when it comes to gauge theories, their approach (and all other approaches in algebraic QFT) fails miserably. Gauss’ law causes that states with electric charge to be distinguished from the vacuum in the causal complement of any bounded region as it is possible to calculate the electric charge by measuring the total flux through an arbitrary large sphere.
As far as we know there exists no convincing explanation for the observed quantized values of the electric charge. At least not until we see a monopole.

Sam
***

[1] D. Buchholz and K. Fredenhagen, Comm. Math. Phys. 84 (1982) 1.
[2] -------------------------------------. Nucl. Phys. B154 (1979) 226.
[3] J. A. Swieca, Phys. Rev. D13 (1970) 312.
[4] K. Druhl, R. Haag and J. E. Roberts, Comm. Math. Phys. 18 (1970) 204.
[5] S. Doplicher, R. Haag and J. E. Roberts, Comm. Math. Phys. 23 (1971) 199.
[6] --------------------------------------------------------. 35 (1974) 49.

DrDu