Conservation of Lepton and Baryon numbers

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I am considerably confused about conservation laws like lepton number (L), baryon number (B) and comparable.

Unlike the conservation laws for energy, momentum, angular momentum and electric charge, the conservations of L and B are not rigorously covered in textbooks. So my questions

- How can I tell from a Lagrangian that it conserves L, B, B-L or so?

- Is there a Noether mechanism leading to these conservations? What are the associated symmetries?

- In QED, from the Feynman diagrams of second quantization it is quite obvious that the electron number is conserved; but this structure emerges from the charge conservation, doesn’t it?

- Electric charge is associated with the phase of the wave function of a particle, and from first quantization one can well see how this works; but the particle number should be associated with the norm of the wave function in first quantization; how can one comprehend that it is conserved?

Thank you very much in advance!
 
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jambaugh
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Here is a partial answer. With the quark model, since Baryons are composites Baryon number conservation when it holds will be a result of quark conservation and color confinement. Baryon number was an observed conserved quantity before the quark model emerged so we still speak in those terms. There have been speculations in the early Grand Unification attempts to extend the color group to SU(4) incorporating leptons as the fourth quark. See Lepton Number as the Fourth Color by Pati and Salam Phys.Rev. D10 (1974) 275-289. It is a classic.

Color conservation is a result of the existing SU(3) gauge symmetry of the strong force, as a Noether conserved quantity. If we can include leptons in the quark family then lepton number is the one quark color that is not confined but it would again be part of this Noether conservation from a (larger) symmetry. There are however difficulties with this view, the principle being it implies a much larger zoo of force carrying lepto-color bosons which are not observed nor implied by combinatoric calculations.

I am out of my depth on the fine details and how the Lagrangian might look so defer to others on this point.

As to your comments and questions about electric charge, you associated the norm of the wave function in first quantization with particle number but that is not correct. The norm of the wave function as a whole is associated with the unit probability, we presume, of finding that particle in some mode when we speak of the probability amplitudes for finding it in specific ones. "Second quantization" is actually quantification, where we begin to consider particle number as a variable in the system. Note that we may conceive of a wholly real wave function with no complex phase, or a more complexly complex wave function (you can rediscover the weak interaction by extending from complex to quaternion amplitudes). It is the existence of complex phase in the description along with its non-obserability that gives us the U(1) gauge symmetry associated with electromagnetism. Note that we non-the-less only consider the underlying description as Real (vs complex or weirder) because we only consider real parametrizations of the respective groups. U(1), SU(2) et all are Real groups. Were we to fully complexify them we would be talking GL(1,C) and SL(2,C) et al.

A way to think of these gauge degrees of freedom are as auxiliary variables we have added to fit the system description within a "flatter" mathematical description. By way of analogy consider how much easier it is to describe say the curved manifold of a 2-sphere or 3-sphere by embedding them in a flat 3-space or 4-space respectively. (In this analogy the additional gauge degree of freedom is the scale parameter identified with the sphere's radius within the embedding space.) You can possibly see the interplay of this in the Kaluza-Klein descriptions of the gauge symmetries, or more precisely in their comparison with the classic group representation counterparts.

Ok let me then return to your last question. In "1st" quantization the 1 particle is the system and thus necessarily the scalar value 1 is conserved and identified with the particle number. As we quantify (so called "2nd quantization") we effectively construct composite systems from the statistics of our particles and the 1 particle system descriptions of each particle type. In simple QED where particles are only distinguished by charge, charge is conserved via Noether in the symmetry we intentionally induce so as to preserve that charge. In QED within a larger EM-Weak-Color theory particles are distinguished by charge, weak-isospin, and color charge, as well as the rank of their representation under these groups. Whether electron number is conserved will be a matter of whether there is an interaction within the allowed dynamics which might change electron number. Note for example that "redness" is conserved but not red-quark number since a red quark can emit a red-anti-green gluon and become a green quark. That it has this gluon available is built into the dynamic. Had we a lepto-red boson we could see electrons turn into quarks by emitting such... and then we would not speak of lepton number conservation but rather lepto-quark number.

A final point. I mentioned representation rank as well. Consider a spin system where you are preserving rotational symmetry. You have all of the components of spin-angular momentum conserved. (Angular momentum as a vector is conserved in the absence of symmetry breaking torques.) You also have the total spin conserved as a function of how representations of the rotation group break up into irreducible sub-representations. Even when you then break the rotational symmetry, say by providing and external B field, you still have conservation of total spin-rank. There is a conservation of information at play here. The system description cannot change dimension, you must conserve probability itself. I don't know to what extent that might come into play with e.g. lepton number conservation when one rejects, say, lepto-color gauge models.
 
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