# Why do we need gauge symmetry and anomaly freedom?

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In the past, everything was quite clear - non-renormalizable theories are unable to make any predictions, given that they need an infinite number of parameters, thus, a reasonable theory has to be renormalizable.

A massive gauge field would give a non-renormalizable theory, while gauge symmetry gives a renormalizable theory.  So, massive gauge fields are not possible, except they have been constructed with some trick and are, fundamentally, something different, as with the Higgs mechanism.  Then, if the gauge theory is anomalous, it is also non-renormalizable.  So, the gauge theory has to be non-anomalous.  That's fine and not problematic, given that we have with the SM a non-anomalous gauge group and can fake massive gauge fields with the Higgs mechanism.

Then came Wilson and told us that non-renormalizable theories are not a problem at all, they are fine as effective field theories.  An effective field theory has one important unknown parameter, the critical length, and below that critical length the theory has to be replaced by another one. All the infinity of the parameters defining the general Lagrangian of such a theory can be restricted by the condition that they all have to have a comparable order at the critical length.  Once this is assumed, one can consider the large distance limit, and all those non-renormalizable terms go to zero very fast. The higher their order, the higher the suppression with increasing length.  The highest order terms are the renormalizable ones.  In comparison with them, all the non-renormalizable ones will become irrelevant in comparison with the renormalizable ones if the critical length is small enough.  The only exception is gravity, because there is no renormalizable theory, thus, the lowest order non-renormalizable one is what remains - even if it has to be, for the same reasons, very weak.  This approach de-facto solves not only the problem with quantization of gravity (as an effective field theory, it works fine), replacing it with the problem to find a theory for distances below the critical length. It also by the way explains why gravity is so weak in comparison with the other forces.  Wilson has got a Nobel for this, justified.

But it follows that there is no longer a need to have renormalizable theories. We could as well have non-renormalizable ones.

Say, a simple massive gauge theory would be non-renormalizable, but the essentially not that different theory using the Higgs mechanism is renormalizable.   What is the difference?  The reasonable guess is that it is some non-renormalizable component of the massive gauge theory which becomes irrelevant at large distances.  That means, we could use massive gauge fields as they are, in the effective field theory approach the non-renormalizable components will become irrelevant.

The same for anomalous gauge fields.  They are non-renormalizable?  Fine, that means, they will disappear themselves, without any further need to suppress them.

But, once they disappear automatically, this would be a natural way to extend the SM.  If one uses only anomalous gauge fields to extend the gauge group at the critical length, one does not even have to invent anything to suppress them.  The serious problem of all other extensions of the standard model - how to suppress the additional fields to make them invisible at large distances - simply disappears into thin air as long as the extensions are anomalous.

There is a particular example of such an extension - $U(3)_C\times U(2)_L \times U(1)_R$, with two U(1) fields added. The color group acts, then, only on the color degrees of freedom, and the other two only on the electroweak degrees of freedom without any dependence on color and baryon/lepton charge. So, this simplifies the SM giving the charges a much simpler structure.

But this thread is not about this particular speculative proposal, but about the justification for using the Higgs mechanism (instead of simply using massive gauge fields) and restricting oneself to non-anomalous gauge fields, given that in a Wilsonian effective field theory all this is unproblematic.

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The Higgs field also addresses the issue of how elementary particles gain mass. That being said one would have denying the the Higgs boson detection at CERN and other LHC'S. The details can be studied through the CKM and PMNS mixing matrix. Obviously it would be too lengthy to go through the numerous Yukawa couplings in regards to quarks, electron and neutrinos. However the mass term interactions provides the term needed for the electroweak symmetry breaking.

Edited by Mordred

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What is the advantage of the mass terms related with the Higgs field in comparison with the straightforward mass term?

I do not plan to question the Higgs boson detection, but I want to understand what has been really detected, what was the advantage or necessity of proposing such a model.

The necessity of obtaining a renormalizable theory is bogus, given effective field theory.  What could replace it?

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Are you sure you fully understand renormalization of a group ? You don't require the Higgs field for renormalization for starters. That's not it's primary function.

The Renormalization group equations is what provides scale invariance at different energy levels. This has nothing to do with the Higgs. Obviously the other reason is to prevent unwanted infinities.

Any infinite quantity will contain finite quantities. The Renormalization compactifies the group to a finite group. Yes you can use Wilson loops to do the same. QFT has the IR and UV cutoffs, while String theory uses the Dirichlet and Neumann boundaries.

Now in terms of the Higgs Hierarchy problem which is often confused with renormalization in point of detail is independent of renormalization.

Quote

The Hierarchy problem is independent of the renormalization scheme. It is sometimes argued that in dimensional regularization there are no quadratic divergences since the 1/poles correspond to logarithmic divergences. This is fallacious. The Hierarchy problem isn’t about the cancellation of divergences, it is about the separation of the electroweak and uv scales

As you can see from that paper the Higgs isn't about cancelling divergences which is the primary function of renormalization.

6 hours ago, Schmelzer said:

What is the advantage of the mass terms related with the Higgs field in comparison with the straightforward mass term?

The straight forward SM model coupling constants could not account for the neutrino mass terms for one. Originally the SM model predicted them to be massless. However it was found they do indeed have mass.

I am going to save a lot of typing and latex by simply providing the following article. To address why the standard coupling constants were insufficient. At all energy scales.

Edited by Mordred

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15 hours ago, Mordred said:

As you can see from that paper the Higgs isn't about cancelling divergences which is the primary function of renormalization.

The straight forward SM model coupling constants could not account for the neutrino mass terms for one. Originally the SM model predicted them to be massless. However it was found they do indeed have mass.

I am going to save a lot of typing and latex by simply providing the following article.

Thanks, the second article has answered one of the questions, namely why this coupling of the Higgs to fermions to get the mass terms:  The standard mass term simply destroys the gauge invariance of the chiral gauge fields. Fine.

Unfortunately it leaves unanswered the original question why we need a gauge invariant theory at all, given that we do not need renormalizability any more once we have learned how to handle non-renormalizable theories as effective field theories, but from the start assumes the gauge invariance as obligatory.

It was my understanding that the primary reason for having gauge invariance is that gauge theories are renormalizable but massive gauge theories not.  So, I have understood the Higgs only as a trick to gain gauge invariance, and not as a tool to do actual renormalization or so.

The point with the neutrino masses I don't understand. Of course, initially there were none, they were later added.  But as far as I understand, one uses now the same formalism (simply with another mass matrix) as for leptons, as for quarks. Not?

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Well quite frankly I find gauge symmetry and Gauge invariance incredibly useful. So I really don't see what the issue is.

Perhaps the theory that works for you is loop quantum gravity. If you really want to understand particle mass cross section with regards to couplings study the CKMS and PMNS matrix.

Particularly the particle family generations.

Edited by Mordred

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My question is not if gauge symmetry is useful.  Symmetries are technically always useful, even if they are only approximate symmetries.

The issue is if there are today, in the light of the Wilsonian approach (means, of understanding non-renormalizable theories as effective field theories, not Wilsonian lattice gauge theory) any problems with massive gauge theories.

Why this interests me is because of my own theories, which have problems similar to lattice gauge theories:  The vector gauge fields do not create problems, here Wilsonian lattice gauge theory gives a nice lattice gauge theory with exact gauge symmetry on the lattice.  This does not work for chiral gauge theories. All what has been proposed to solve this looks completely artificial to me, but there is a quite simple solution:  The lattice theory for the chiral gauge fields simply does not have an exact gauge symmetry on the lattice. What would be the straightforward consequence?  The large distance approximation of chiral gauge fields would not be gauge invariant, in particular it could contain mass terms.  But this is also what we observe: chiral gauge fields are massive.   So we have a nice correspondence between observation (vector gauge fields massless, chiral gauge fields massive) and simple lattice theories (vector gauge fields allow exact lattice gauge symmetry following Wilson, chiral gauge theories not).

But the mainstream does not like this. Instead, they consider this to be a big problem that there is no exact lattice gauge symmetry for chiral gauge fields, only to use later the Higgs mechanism to create a massive gauge field out of these gauge fields with exact lattice gauge symmetry.

Something I can understand, given that massive gauge theory is non-renormalizable and many physicists have not recognized that this is no longer a problem.  But beyond this, I cannot see any justification to look for exact gauge symmetry for chiral gauge fields.

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Well you might not consider divergences leading to infinite mass terms a problem. Others feel different yes there are methods to have effective cutoffs prior to Infinities as well as alternates to renormalization the divergences can often lead to other factors being involved.

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Hm, if there are others who feel different, have they some arguments why, say, a simple lattice regularization on a large cube, which reduces the number of degrees of freedom to a finite number, nonetheless leads to big problems if the theory is not renormalizable?  Or is there nothing but a feeling?

The issue is not the renormalization procedure itself, Wilson has shown that it is useful for understanding condensed matter theory, even if it is a regular theory without divergences given that its atomic structure defines a sort of lattice regularization of the  continuous theory.

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I would think the contention lies more in the direction of the desire for a full TOE. A large body feels that as we have been so successful at unifying the other three forces into an effective gauge theory then we should also be able to do so with GR.

In essence a full fetched renormalized quantum gravity. The problem being is that GR is well behaved as a low energy field theory but once you start hitting blackholes and other singularity conditions you get a range of energies where GR cannot fully describe within finite loop corrections.

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Posted (edited)
On 11/5/2019 at 10:54 PM, Schmelzer said:

But this thread is not about this particular speculative proposal, but about the justification for using the Higgs mechanism (instead of simply using massive gauge fields) and restricting oneself to non-anomalous gauge fields, given that in a Wilsonian effective field theory all this is unproblematic.

The reason is because it is based on the Yang-Mills gauge theory.  It is the only theory in quantum mechanics which was able to accurately describe electrodynamics by assuming that a field applies a force by exchanging a virtual photon.  It has a strong basis in the formulation to be capable of describing the affects of SR in all directions at once, instead of just one single instance of a photon moving from the front of a train from the middle, for instance.  It also describes the photons going to the back of the train and to the roof and floor, etc., all at the same time.  The paths of photon  predictions are probabilistic in nature.

Edited by Conjurer

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QED is, of course, a fine working theory.  Gauge theories were favored some time because they are renormalizable and it was thought that non-renormalizable theories make no sense, are unable to make nontrivial physical predictions.  But after Wilson we know that such theories make sense as effective field theories.

Compatibility of the field equations with SR is also not a problem at all.

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On 1/5/2020 at 1:13 AM, Schmelzer said:

QED is, of course, a fine working theory.

The part I have trouble accepting in QED is how it assumes that fields interact with the exchange of a virtual photon.  Most of how this translates to other fields, like electronics, seems to be wrong in describing their operations.  Since it has been a fine working theory in quantum mechanics, they assume that it makes that assumption correct.  It ended up putting an end to work being done on if there was another type of fundamental particle in the hidden variables of quantum theory.

There is a very small remote possibility that it is actually due to something else taking place, or there is a fundamental particle that makes up all other particles in a TOE.  That is why I have been raising questions about it.  It could be possible that describing it that way just so happens to be accurate for what they are measuring.  There is no reasoning as to why particles interacting that way could induce a field to create a force of nature.

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2 hours ago, Conjurer said:

The part I have trouble accepting in QED is how it assumes that fields interact with the exchange of a virtual photon.

That is not an assumption, it is the way the mathematics describes the interactions.

2 hours ago, Conjurer said:

Most of how this translates to other fields, like electronics, seems to be wrong in describing their operations.

Can you provide an example where quantum theory does not correctly describe electronics?

2 hours ago, Conjurer said:

It ended up putting an end to work being done on if there was another type of fundamental particle in the hidden variables of quantum theory.

There is lots of research into various extensions to quantum theory, many of which require new particles. So I don’t believe this to be true.

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41 minutes ago, Conjurer said:

The part I have trouble accepting in QED is how it assumes that fields interact with the exchange of a virtual photon.  Most of how this translates to other fields, like electronics, seems to be wrong in describing their operations.  Since it has been a fine working theory in quantum mechanics, they assume that it makes that assumption correct.

This is a problem of inadequate choices of informal descriptions.

The best way to "understand" this is to consider all these Feynman diagrams simply as formal denotations for particular terms in the particular approximation without any deeper meaning.

The language was developed for scattering processes, where free particles fly around, hit each other, and then fly away again. Other things, like static field configurations, were irrelevant. If we consider particles flying away independently, when we have to care for the EM field only about light - transversal EM waves.  But there are also other EM fields, which do not consist of light freely flying around, namely static EM fields between static sources.  When all the particles hit each other, these forces clearly will be important too, even if they don't consist of photons which can fly away freely.  So, the approximation formulas will contain the corresponding terms too.  And these things have been named "virtual particles".  A denotation which confuses many people, but if one accepts that it is only a bad name given to a particular term in a particular approximation, it is not really a problem.

Don't forget that photons are essentially quantum effects of the EM field, in a way completely similar to the phonons in condensed matter theory.  These are simply discrete energy levels. Don't invest too much in taking them seriously as real particles. (In semiclassical gravity it becomes quite obvious that the fields are more fundamental than the particles - the notion of particles, as well as the vacuum, changes there together with the gravitational field, while the field degrees of freedom don't change.)

If you consider things different from scattering theory, this approximation becomes quite irrelevant, and all the methods developed to handle them may appear useless.

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8 hours ago, Strange said:

Can you provide an example where quantum theory does not correctly describe electronics?

They say that it means that a voltage goes across a line at the speed of light due to this, but it does not.  It has a change in frequency that is less than c.  Then a voltage across a line is not related to a magnetic field in most setups.

8 hours ago, Strange said:

There is lots of research into various extensions to quantum theory, many of which require new particles. So I don’t believe this to be true.

It put a end in the search, in a traditional sense, as a regular quantum theory of particles.  It caused the search for it to branch off into other areas like symmetry in extra dimensions and quantum gravity.  Then none of the new theories have been successful.  If they cannot be successful starting in a new direction, then it implies that they may be taking the wrong direction with it.

5 hours ago, Schmelzer said:

This is a problem of inadequate choices of informal descriptions.

Probably so, it has been a while since I have studied it.  It seems like there is a big gap in our understanding of it if it just means that it can be solved just by saying that there is an interaction at the speed of light, because that is the fastest that anything could have an interaction at.  It seems like there should be a lot more to it than that, if anyone ever wishes to have any type of intuition with working on it.  I thought I may have, but it went out the window when I got to this part of QED.

Edited by Conjurer

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