UPDATE June 27, 2022: Suppose that we define the Higgs field as a "helix":
φ(x) = (r, α),
where r >= 0 is real and the angle α is any real number.
The "value" of φ in an equation is the complex number defined by r and α, but after one round around the origin of the complex plane we do not return to the same point. It is like an Archimedes' screw.
We require the Higgs field to be continuous in spacetime. Could it be that then there will be no "crystal boundaries" nor defects?
But in this case there is a singularity at φ(0, 0).
Another attempted solution: require that the vector field defined by φ never contains a source. We can ensure that it stays that way by putting the Higgs potential infinite for φ(x) = 0. Is there any reason why the Higgs potential should be finite there?
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v exp(i α),
where v is the radius of the groove, or the circular valley, of the Mexican hat potential, and α is an angle from the real axis.
The field rolls to some angle α in the groove of the Mexican hat potential.
We in this blog are somewhat worried of the following possibilities:
A. Even in the minimum energy state, spacetime seems to contain information: the angle α.
B. If the field rolls down from an arbitratry state in a large spatial volume, then the angle α will differ from place to place. There will probably be something like boundaries of "crystals" or defects in the state of the field. Could there even be singularities?
Kien Nguyen (2009) has written a simple introduction to symmetry breaking.
Problem A: the information in "empty space"
We may define empty space, or the vacuum, as the lowest energy state of the classical fields.
For most fields, that means that they are identically zero everywhere.
However, the Higgs field is strange: it is not zero, and also seems to contain information, the angle α.
This would be ugly, but it looks like that we cannot in any way find out the value of α. We can only see that α is the same throughout space.
If α is not the same constant everywhere, then the electroweak 4-potential A is not zero everywhere, and we should see some kind of "matter" in space.
Only differences in α are observable.
In the paper of Kien Nguyen the angle α is eliminated altogether, and the 4-potential A will do its job.
Thus, do we have a broken symmetry or not?
Not in empty space. But if the space is not empty? Then the existence of matter breaks the symmetry.
Ward Struyve (2011) argues that there is no symmetry breaking.
He writes that Peter Higgs himself in his 1966 paper presented an interpretation where the symmetry is not broken.
Problem B: are singularities possible in the Higgs field? No
For low energies, the Higgs field is approximately determined by the angle α at each location.
Since α can vary from place to place, it may happen that we will have something which is analogous to crystal boundaries or defects.
We are saved from singularities, because the potential of the Higgs field at zero, or at the center of the Mexican hat, is finite.
There could still exist crystal boundaries. We need to check the work of Tom Kibble. He has written about defects.
Conclusions
Empty space is still "empty", even with the Higgs field. There is no genuine symmetry breaking where the Higgs field in empty space would determine a preferred direction in the complex plane.
The Higgs field is "scalar": it does not determine any direction in spacetime.
We do not like singularities. Fortunately, singularities cannot form.
We do not like that the Higgs field in empty space has a non-zero value nor that it has a charge which fills all the space. The Higgs field is different from all the other fields in that respect. But we have not found a way to build a model where the value of the Higgs field is zero: we cannot find a way to give a mass to the W and Z bosons if the field is zero.
Have astronomers found anything which might be crystal boundaries in the Higgs field? The large-scale structure of galaxy clusters looks like "filaments". Could they come from defects in the Higgs field?
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