What is the relevance of gauge symmetries in a gauge theory?

https://arxiv.org/abs/1107.4664

Simon Friederich (2012) writes that the ontology of gauge theories has received considerable attention from philosophers in recent years. There is a debate about spontaneous symmetry breaking: is it a genuine phenomenon of nature or is it just imagination of humans.

The author of this blog has had a feeling that Baron Munchausen has written the "derivation" of a gauge field from a global symmetry of a lagrangian.

For example, in the Dirac equation we can multiply a solution by any complex number whose absolute value is 1, and we get a new solution. We say that there in the lagrangian is a gauge symmetry, whose group is U(1), which are the rotations of the complex plane around the origin.

Turning a global gauge symmetry into a local symmetry

Let us then rotate our coordinate axes of the complex plane by a different amount at different points of spacetime, but in a continuous way. If we had a solution of the Dirac equation, that solution is spoiled by the rotations. To restore the lagrangian to its original value, we introduce a 4-vector potential A. If we choose A in a suitable way and replace the partial derivatives in the lagrangian with gauge covariant derivatives

       D_μ = ∂ / ∂x_μ - A_μ,

then we can restore the original value of the lagrangian.

We say that we have turned the global symmetry into a local symmetry by introducing A. We are free to rotate the local complex plane, as long as we compensate with a suitably chosen A.

We then claim that the new lagrangian describes an electron under a potential A.

A philosophical question is, if in nature there existed a "global symmetry" which was turned into a "local symmetry", or is this just a story which we construct in our human minds?

As we have noted in various blog entries in the past year, the new lagrangian with a 4-vector potential A does not describe the interaction of an electron correctly because it fails to take into account the change of the inertial mass of the electron under a static electric field potential. It looks like the story with symmetries is just human imagination, and furthermore, the story is incorrect.

The role of degrees of freedom in a field

Let us think about waves in a tense string. A degree of freedom in the string is the vertical position of a point in the string, at various spacetime locations. The position is given as a real number z(t, x).

We start the construction of a wave equation from this degree of freedom.

Let us imagine that the string has an infinite length. The vertical elevation of whole string is not important for waves. What counts is the differences of the vertical position at various spacetime points. There is a global symmetry

       z(t, x) -> z(t, x) + C

for the solutions, where C is a constant real number.

We could now let C vary according to the spacetime point (t, x), and introduce a gauge field A to compensate for the difference in the lagrangian of our string.

The existence of a global symmetry was the result of two things:

1. We chose a real number z as the degree of freedom.

2. We wrote a typical wave equation where only the differences in the value of the degree of freedom z matter.

Rather than saying that we constructed A from a global gauge symmetry, we could say that we constructed the gauge field A from the degree of freedom which we chose.

We were lucky to find a suitable degree of freedom which describes the string well. The existence of a global symmetry is an automatic consequence from this degree of freedom and the normal way how waves behave. The global symmetry is not interesting in itself.

Can a scalar field grow its energy as the universe expands?

Here is an update to a question we have been looking at this fall. This text is has been copied from a discussion at:

https://www.physicsforums.com/threads/does-a-fields-vacuum-density-violate-conservation-of-energy.977279/

The fields which we know from our everyday experience, for example, the photon field and the electron-positron field, have a constant number of quanta as the universe expands, if we ignore reactions with other particles.

 
            photon
A ~~~~~~~~~~~~~~> B
   <--------------------------------->
      expanding universe

A photon will gain a redshift when it moves in an expanding universe, because if it is emitted by an event A and absorbed in another event B far away, then B "sees" A receding at a great speed - B sees the photon redshifted. If we take the energy of the photon to be what B sees, then the photon has lost energy.

The redshift mechanism reduces the kinetic energy of a electron, too, when it moves in an expanding universe.

What about a scalar field which some people claim, is causing the acceleration of the expansion right now?

The first thing to note is that a field whose energy increases in an expanding universe is "exotic matter", that is, it has a negative pressure and negative gravity. It breaks various energy conditions of General Relativity. We do not know such matter from our everyday experience. It is highly speculative to assume that such matter could exist.

What about the energy content in the hypothetical scalar field? Is that energy contained in quanta of some kind? If yes, does the number of such quanta increase as the universe expands?

Our everyday experience is that energy in a weakly interacting system is, indeed, divided into well-defined quanta which can carry mass and kinetic energy.

The question is harder in a strongly interacting system, say, a crystal. Is the vibrational energy in a crystal divided into quanta of some kind? Certainly not in any unique way.

Suppose that we work in the Minkowski spacetime. The universe is not expanding. We want to excite the Higgs scalar field which has the famous Mexican hat potential. We use a huge particle collider to produce a vast density of Higgs particles. In that way we store a lot of energy in the Higgs field. Can we say that the excited Higgs field has its energy stored in quanta of some kind? It is a strongly interacting system.

We expect the excited Higgs field to release just the same amount of energy as we pumped into it. What if the universe is expanding at the same time? Can the released energy be bigger than the pumped energy?

That would be surprising. Rather, we would expect the Higgs field lose some of the kinetic energy of the Higgs particles as the universe expands.

The hypothesis that a scalar field can grow its energy in an expanding universe is highly speculative. It is at odds with what we know about other fields.

The Higgs 1964 paper is correct

In the previous two blogs we had doubts if the Goldstone boson really is eliminated when we couple the gauge field A to the Higgs field. How can a massive field A "simulate" a Goldstone boson which is supposed to move at the speed of light?

It turns out that the lagrangian is fundamentally changed by the addition of the coupling. The field φ_1 becomes "redundant" in the sense that only the value of

        B = A - gradient(φ_1)

is relevant in the equation, and we can freely choose φ_1 without affecting the physics of other fields. We say that φ_1 has become redundant.

How is it possible that φ_1 was very relevant before the coupling, but we can discard it after the coupling is done?

/\/\/\/\/\  a
/\/\/\/\    b

A somewhat similar thing happens if we have two springs A and B which exert a constant push force F. Let the length of A be a and the length of B be b. If we squeeze the springs separately, then both a and b are relevant in the lagrangian of the system.

/\/\/\/\/\/\/\/\/\/\/\ a + b

But if we attach the springs together, then only the value of a + b is relevant, not a or b any more. We can freely choose a and b, as long as a + b stays the same.

The Goldstone boson of the uncoupled Higgs model simply does not exist in the coupled model. It would still exist if the coupling were very weak. But the coupling is strong enough, so that the Goldstone field is not relevant at all.

That is what "eating" the Goldstone boson means. The value B ate both the field A and the Goldstone field.

What if the gauge field A is coupled to a fermion?

Let the field A in the Higgs 1964 paper be coupled to a fermion, say, to the electron.

Let us replace A by B in the lagrangian of the fermion, let that be L(B). Let us find a solution ψ for the wave function of the fermion in L(B).

We can make a solution for the old lagrangian with A, by multiplying ψ with a phase factor:

        exp(i e φ_1) ψ

for an arbitrary φ_1. What did we show? We showed that we still can choose φ_1 arbitrarily and have a solution, without affecting the physics seen by outside observers. An observer cannot measure the absolute phase factor of ψ. The Goldstone field stays redundant even when A is coupled to a fermion.

Some analysis on the Higgs 1964 paper Broken Symmetries and the Masses of Gauge Bosons

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.13.508

A freely readable copy of the Peter Higgs 1964 paper (just 2 pages) is available at the link.

The paper introduces two scalar fields and a gauge field A.

A somewhat analogous system would have the electromagnetic 4-potential A as the gauge field, and an electrically charged boson particle field as the scalar fields. An electron is a spinor field in the Dirac equation.

We know that if we have a free electromagnetic 4-potential A (of electromagnetic waves), then we can add the gradient of an arbitraty differentiable function f to A, and nothing changes in the physics. It is called a gauge symmetry.

Our August 24, 2019 post showed that there is no gauge symmetry if an electric charge is present: the inertial mass of the charge varies if its electric potential changes relative to far-away space. The electric potential of far-away space defines a preferred frame.

The perturbation in the Higgs paper

The fields are classical at this point. Higgs introduces a Mexican hat potential which causes the scalar fields to assume a constant non-zero value at a minimum energy configuration. We have defined the coordinates of the scalar fields such that the constant value of the two scalar fields is

        (0, v).

That is, the minimum energy happens when the fields have that constant value everywhere in the Minkowski space.

Higgs studies a small perturbation of the field values

        (Δφ_1, Δφ_2)

around the minimal energy value (0, v).

We have not seen a proof anywhere that the perturbation series converges. Thus, we do not know if the quadratic approximation by Higgs approximates the original system of equations.

The equations are rather complicated. It might be hard to prove the convergence.

One may ask if it matters at all if the perturbation series converges. The perturbation equations by Higgs look nice and we do not really need to care about the original equations.

Introducing the "Unitary gauge": why does the Goldstone boson disappear?

Higgs defines a new 4-vector potential B by subtracting the gradient of Δφ_1 from A and writes equivalent equations using B. We say that he moved to the "unitary gauge" for the 4-vector potential.

He writes that the new equations describe a massive scalar boson and a massive gauge boson.

The Goldstone boson, which travels at the speed of light, disappeared. Where did it go?

The Goldstone boson is not directly visible in the new equations. But it still exists in the system. A mathematical manipulation of the equations does not change the physics. There still exist waves that carry energy at the speed of light.

What are the "real" physical fields described by the equations? We cannot answer that question by defining a new variable and removing an old variable.

Does the Higgs change of a variable prove that there is no speed-of-light energy transfer in the system? Or does it just hide the obvious speed-of-light waves behind complex equations?

A way to study this is to keep using the old field A and not move to B. The physics have to be the same. Is there some mechanism in the equations which couples the light-speed waves tightly to slower waves and prevents energy from being transferred at the speed of light?

Light within a medium moves slower than the light speed in the vacuum. If an electron is shot at a high speed into the medium, it will move faster than the local speed of light, and will emit Cherenkov radiation. Could something similar happen in the Higgs model? A Goldstone wave would lose energy to the gauge field and soon disappear? Since a Goldstone boson can have a low energy, it cannot produce massive gauge bosons. In the Higgs paper there are no massless gauge bosons, but in the Standard Model there is the photon.

The Goldstone bosons of the Higgs field may quickly decay to photons, and would not be noticed in the LHC instruments.

The analogue of light within a medium

Light moves slower in a medium, for example, inside glass than in vacuum.

There are charges in glass which can oscillate in synchrony with the light waves.  The massless photon field is coupled to a massive "field" of the charges. The photon "acquires a mass" through the interaction, and moves thus slower than the light speed in vacuum.

This effect is clearly analogous to the model in the Higgs 1964 paper.

When a wave of light enters glass, the wave quicly loses some energy to the oscillation of the charges. The end result is an oscillation wave moving slower than light in vacuum.

We may imagine that a photon has been converted to a quantum of an oscillation wave. A photon became a phonon.

Does a photon "exist" within glass? If we have no way of creating a photon inside the glass without at the same time creating the corresponding oscillation wave, then we might say that a photon cannot exist. It has been "eaten" by the oscillation wave.

We may imagine that charges in the glass are attached to the solid matter with springs. There are rubber bands between adjacent charges.

The electromagnetic field has the role of the gauge field.

The lagrangian in the Higgs paper

It is easiest to analyze the model in the Higgs paper by looking at the lagrangian.

We first analyze the fields as classical.

It is best to keep the original variables. Let

       φ_1 = 0

       φ_2 = v

approximately, where v is constant and non-zero. We will look at small deviations of the φ_i fields from those values.

Let us first forget A. Then a disturbance in φ_1 moves at the speed of light.

A disturbance in φ_2 moves much slower because a deviation from v stores a lot of energy in the lagrangian formula. It corresponds to a massive particle.

Let us then couple A to the system. We keep φ_2 a constant v. If we let φ_1 = 0, we see that a deviation of A from 0, A^2 stores energy in the lagrangian. That means that A is massive and a disturbance in A moves slowly.

We have now coupled A. What happens if we first set A = 0 in all space, and then disturb φ_1 at a point in space?

The lagrangian shows an interaction between φ_1 and A. Obviously, energy will start leaking from φ_1 to A immediately. The disturbance in φ_1 keeps moving at the speed of light, and it keeps leaking energy to A for its whole trip. The energy stored in A moves much slower.

Let us then use quantum mechanics. A disturbance in φ_1 corresponds to a massless boson. The energy E of a quantum may be very small.

The mass of a quantum of A may be larger than E. Then the quantum of φ_1 cannot decay into a quantum of A.

The reasoning above suggests that a massless Goldstone boson cannot be eaten by a massive gauge boson.

Can we excite φ_1 without exciting the field A at the same time? In the electroweak interaction, the gauge field A interacts with many different particles. We can probably excite A separately from the Goldstone boson field. Then the Goldstone boson field cannot be merged to the gauge field A.

The coupling "constant" (not really a constant) between φ_1 and A in the lagrangian is

       2e φ_2 = 2ev.

If we set the constant e very small, then we clearly have two separate fields, φ_1 and A which interact very little. How fast does energy leak to A?

Decoupling the Goldstone bosons?

If the only equations of our physical system are the ones given in the Higgs paper, then moving to the new variable B can bee seen as decoupling the free degree of freedom in A and the Goldstone field.

Once one has solved B, one is free to choose any A and the Goldstone field φ_1, as long as 

       B = A - 1 / (ve) * gradient(φ_1).

This requires that all the communication to these fields go through the "interface" given by B. Then we may say that we decoupled a degree of freedom.

However, in more complicated lagrangians, one can generally communicate with A in many ways. Then we cannot do a decoupling in the way given.

What fields "exist" if some fields are decoupled?

One may argue that the field φ_1 does not exist at all if we use the new equations with B. Then the claim that there was a symmetric potential for φ_i and the symmetry was broken is vacuous. The system where the symmetry broke does not exist.

If we claim that φ_1 really existed, and exists right now, then we probably can excite φ_1 separately from A, and the Goldstone boson exists.

Conclusion

If the scalar fields existed, and do exist now, and the fields fell into an energy minimum of the Mexican hat, then the Goldstone boson does exist, contrary to the claim in the Higgs 1964 paper. The gauge field cannot eat the Goldstone boson.

Maybe it is hard to excite the Goldstone field, and that is the reason why the boson has not been found.

It might also be that φ_1 does not exist at all. There was no symmetry breaking nor a Mexican hat potential. There is just the field φ_2 and its potential curve which has the minimum at a non-zero value v.

Can one really "eat" the Goldstone bosons of the Higgs field after a symmetry breaking?

https://www.theorie.physik.uni-muenchen.de/lsfrey/teaching/archiv/sose_09/rng/higgs_mechanism.pdf

When the Higgs field finds its minimum at a certain vacuum expectation value, the field can still oscillate around that minimum.

One direction x of the oscillation has a potential term of a form

       V(x) = k x^2,

where x is the displacement from the minimum.

Other directions y of oscillation have the respective

       V(y) = 0.

Each such direction y corresponds to the massless Klein-Gordon equation, and a massless scalar particle, a Goldstone boson.

A Brout-Englert-Higgs trick is to do away with the motion in the directions y by "rotating" the local frame so that all the displacements y stay zero. This requires adjusting the gauge fields in such a way that they "simulate" the effect that a Goldstone field would have had on the value of the lagrangian. The trick is described as "eating" the Goldstone bosons.

If a gauge field is only coupled through its derivatives to other fields, then one has a large freedom to transform the gauge field, and the physics stays the same. An example is the electromagnetic field, where one has a very large freedom to modify the 4-vector potential (= the gauge field). However, when the gauge field is coupled to, e.g., the Higgs field, then transformations are very much restricted.

Some people seem to think that one can carelessly transform the gauge fields and do away with the Goldstone bosons, and still keep the physics the same. That is a mistake.

http://philsci-archive.pitt.edu/10962/1/Sebastien_Rivat_-_Spontaneous_symmetry_breaking_-_2.pdf

Sebastien Rivat has observed that there is a problem in eating the Goldstone bosons.

As we wrote in our blog on August 24, 2019, the gauge symmetry of the electric potential does not work if an electron is present: the inertial mass of the electron depends on its potential difference relative to far-away space. The interaction with the electron spoils the gauge symmetry. A similar thing happens in the above analysis with the Higgs field: the interaction with the Higgs field spoils the gauge symmetry in the gauge fields.

It is not clear to us yet, what implications the various problems in gauge symmetry have for the standard model.

https://physicstoday.scitation.org/doi/full/10.1063/PT.3.2196

Eating up the Goldstone bosons was the revolution of 1964 which earned a Nobel prize for Englert and Higgs.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.13.508

Above is a link to a freely readable copy of the original Higgs paper published in October 1964. The paper is just two pages.

Peter Higgs first defines two real scalar fields which interact with a vector field A.

In equation (3) he defines a vector field B which is based on A and the Goldstone field

       Δφ_1.

He claims that B in equation (4) describes a vector field whose quanta have a non-zero mass.

But a disturbance in the Goldstone field moves at the speed of light. How can we model it with a new field B where the quantum moves slower than light?

Also, the vector field A is usually understood as an external field. For example, in the Dirac equation with the minimal coupling, A is an external field. Does it make sense to define a field B which is a mix of the particle fields (the scalar fields) and the vector field?

The Aharonov-Bohm effect

https://en.wikipedia.org/wiki/Aharonov–Bohm_effect

The Aharonov-Bohm effect shows that the absolute value of the electromagnetic vector potential A has observable effects. It is not just the electric field E and the magnetic field B which affect the observable behavior of electrons.

One cannot carelessly transform the electromagnetic vector potential A and still keep the observable physical behavior of the system same. The coupling between the electron field and the electromagnetic field spoiled the gauge symmetry.

Similarly, the coupling of the Higgs field to the gauge field probably spoils the gauge symmetry. The trick by Higgs and others of eating the Goldstone bosons by moving the contribution of the Goldstone field to the gauge field probably does not work.

Why have we not observed the massless Goldstone bosons? Would they show up in the LHC accelerator?

Or is the Z and W boson mass creation mechanism very different from what the standard model claims?