Non-standard question about the Standard Model

[Antiphon]

(I have a question about the Standard Model in spite of the topic title.
It seems the right place to ask it after perusing the other forum topics.
Also, I had erroneously posted this to the "Advanced Physics Forms" where
the crickets are still chirping. It's not my intention to cross post into
different forums.)


I have an engineer friend who is not well versed in tensors or QM but he
has a solid handle on most of the basic concepts of modern physics.
He posed an interesting question and I'd like to pose it to the forum in
turn, because I didn't have a good answer.

He said to me that Maxwell's equations began as a description of a (pair of)
classical fields and later they were quantized and eventually became
a fully quantum theory, QED.

His question to me was this; has anyone ever considered the particle
fields of the Standard Model (let's stick to say the electroweak Bosons)
and written them out as continuous fields in the "classical limit?"

As an example, one could start with the "photon" and "work back to"
classical E and B fields and their sources (continuous charge densities and
motions).

How would one do this for the Electroweak bosons W+/- and Z?

I suppose we should end up with a set of equations resembling Proca's with a new set of values for epsilon, and mu, maybe three polarization states instead of two, sub-c wave propagation, etc etc. Is this right?

Does anyone know how many fields we would even end up with? Would they Lorentz transform into one another like E and B? etc etc.

Is anyone aware of any work done on this (homework, published research, or otherwise?)

 

[cuallito]

Well the weak force isn't that strong outside of the quantum limit so it wouldn't really be too usefull to make an "updated" set of maxwells equations for it.

 

[Haelfix]

Well you definitely can work backwards starting from the full QFT eom of a photon, truncate them to first order (eg ignore radiative effects) and get Maxwells equations in some sort of limit (say hbar --> 0).

And in fact you can do that for the weak and strong force too (eg write down the classical SU(3) 'Maxwell color' equations for instance).

This is completely useless in practise (b/c the strong force is mostly important in quantum regimes see eg confinement) and the classical eom solutions are of no practical importance, but some textbooks have it written down somewhere usually as a homework problem.

Still, its a bit of a cheat of course, b/c we know these particular physical forces have classical limits and exist. We normally start from a well defined classical solution (or at least we guess the action) and then derive the quantum behaviour.

It turns out there are examples of truly quantum theories with no classical limit (or possibly many to one) so its not always obvious if or how to go backwards.

 

[xepma]

I'm just repeating what Haelfix is saying, but I just can't resist ;)

Well, if you ignore symmetry breaking of the SU(2)xU(1) gauge group you end up with a 'regular' Yang-Mills theory for the W and Z bosons (or SU(3) if you consider gluons). These theories are a generalization of ordinary U(1) theory (i.e. electromagnetism). This means that you have a B and an E field, and these fields satisfy a nonlinear version of Maxwell's equations (namely: the Bianchi Identity and the Yang-Mills equation). These are indeed classical equations of motion for these fields!

However, solutions of these equations of motion are hardly useful. The reason being is that for the strong and electroweak force quantum effects are very important. This is also a reflection of the non-linear nature of these theories (gluons self-interact - photons do not) and the relatively strong coupling constant.

 

[javierR]

Sorry for any redundancies, but I typed this yesterday but couldn't submit it...since I took the time to do it, here it is:

The Standard Model Lagrangian itself is classical before quantization, so we can get the generalizations of Maxwell's equations from that already. The other interactions (weak and strong) involve self-couplings of the gauge fields, whereas in electromagnetism we have one field that doesn't carry charge. In particular, one of the new terms in the wave equation for the weak fields will involve a mass as you noted (at high energies, the fields will be massless though). The wave equations for the weak and strong fields are therefore not of free-waves, and they do not have the type of long-range propagating waves like those of EM or gravitational radiation.

One can also start with quanta of the quantum field theory and compose classical field distributions using coherent states (of the quanta). An electromagnetic field background can be considered to be coherent states of photons. One problem with Yang-Mills field theory is that general solutions to the field equations are unknown due to the self-coupling terms. But there are special solutions such as instantons, which are topological field configurations.

As mentioned in the previous responses, the short-range weak and strong interactions mean that we end up looking at these fields over short distance scales so there is a limit to how much we can treat the background fields as semi-classical in practice.

As for the number of fields, they simply appear in the Standard Model Lagrangian (which gives the field equations). Aside from the EM field, there are 3 weak fields and 8 strong (QCD) fields. Each of these has an associated Lorentz fieldstrength $$F_{\mu\nu}$$ like the EM field has, and so one can indeed identify the generalizations of electric and magnetic fields that sit inside, and they transform into each other under Lorentz transformations.

 

[tom.stoer]

There are some ideas regarding classical solutions (instantons, merons, magnetic monopoles, center-Vortices) of Yang-Mills equations in QCD in order to explain confinement. What I have seen so far is that these field configurations do not CAUSE confinement directly but instead indicate certain symmetry structures (related to the center symmetry of SU(N)) which responsible for confinement.

In (classical) electrodynamics the Maxwell equations are important for two reasons:
1) the describe to a very good approximation the field configurations even AFTER quantization; see for example the lamb-shift which is the QED correction to an energy level of an atom
2) special solutions of these equations (plane waves) can be used directly to quantize the electromagnetic field

In Yang-Mills theories this is no longer the case
1) there is no reason why the classical solutions mentioned above should be good approximations to the quantum field configurations; in most cases they are NOT!
2) the known classical solutions cannot be used for the quantization
3) the classical field configurations (e.g. plane waves) used for the quantization are not solutions of the classical field equations

 

[Bob_for_short]

Well you definitely can work backwards starting from the full QFT eom of a photon, truncate them to first order (eg ignore radiative effects) and get Maxwells equations in some sort of limit (say hbar --> 0).
.
.
.
It turns out there are examples of truly quantum theories with no classical limit (or possibly many to one) so its not always obvious if or how to go backwards.

Even in QED working correctly backward is tricky. For example, instead of truncating and considering the limit h_bar --> 0, it is better to consider the inclusive QED picture where soft radiation effects are taken into account (the only non-zero QED result for scattering). Then for the charge motion one obtains classical equations in a potential field (Rutherford cross section, for example) and coherent radiation for EMF.