# Why is goldstone's theorem incorrect in gauge theories?

• LedPhoton
In summary, the Goldstone theorem holds only if certain conditions are met - without them, the theory would not be consistent.
LedPhoton
Hello, I am currently studying spontaneous symmetry breaking in qft. Several textbooks I've read prove Goldstone's theorem under supposing that
1) There exists a continuous global symmetry under which the Lagrangian is invariant.
2) The vacuum state is not annihilated by the conserved charge(or, alternatively, a field has a non-zero vacuum expectation).

Later it is said that theories with a gauge symmetry do not satisfy these hypothesis and so the goldstone theorem is invalid. In fact, a massive boson appears and not a massless one.
My question is how does a gauge symmetry violate the two hypothesis. Since it is a local symmetry, it also contains the global symmetry(the transformation is independent of spacetime) and so it should have the same conserved currents and charges.
I am guessing this is why Higgs won the nobel prize xD
Thank you

LedPhoton said:
Hello, I am currently studying spontaneous symmetry breaking in qft. Several textbooks I've read prove Goldstone's theorem under supposing that
1) There exists a continuous global symmetry under which the Lagrangian is invariant.
2) The vacuum state is not annihilated by the conserved charge(or, alternatively, a field has a non-zero vacuum expectation).
These are the necessary conditions for spontaneous semmetry breaking.

Later it is said that theories with a gauge symmetry do not satisfy these hypothesis and so the goldstone theorem is invalid.
Which book says that? This is incorrect. Without those conditions the symmetry does not get hidden and the massless gauge (vector) bosons stay massless. However, local gauge invariance allows us the freedom to gauge away "the would be massless Goldstone (scalar) bosons" by simply redefining the fields in the theory by making a clever gauge transformations.
In fact, a massive boson appears and not a massless one.
Have you not heard Sidney Coleman famous saying :The gauge fields have "eaten up" the masselss Goldstone's bosons and become massive. The scalar degrees of freedom become the longitudinal polarization of the vector gauge bosons.
My question is how does a gauge symmetry violate the two hypothesis.
They don't.
Since it is a local symmetry, it also contains the global symmetry(the transformation is independent of spacetime) and so it should have the same conserved currents and charges.
And they do have the same currents and charges.Sam

I think in general there are many more conditions for Goldstones theorem. Physically the most important one is that of the hamiltonian being sufficiently local. For example the BCS model of superconductivity does not contain Goldstone bosons because the reduced hamiltonian considered by BCS is too non-local. That was quite a lucky coincidence as a true superconductor also has no Goldstone bosen. However in the latter situation this is due to the Anderson Higgs mechanism.
An interesting read on that topic is the book "Symmetry breaking" by Franco Strocchi.

Ok, I think I understand. Thank you!

## 1. What is Goldstone's theorem and how does it relate to gauge theories?

Goldstone's theorem states that in a spontaneously broken symmetry, massless scalar particles called Goldstone bosons must exist. In gauge theories, this theorem is used to explain the existence of massless particles such as photons.

## 2. Why is Goldstone's theorem considered incorrect in gauge theories?

In gauge theories, the Higgs mechanism is used to give mass to particles without breaking any symmetries. This mechanism does not require the existence of Goldstone bosons, making Goldstone's theorem incompatible with gauge theories.

## 3. Can Goldstone's theorem be modified to be applicable in gauge theories?

Yes, there have been attempts to modify Goldstone's theorem to make it applicable in gauge theories. One such modification is the Higgs-Kibble mechanism, which incorporates Goldstone bosons into the Higgs field and explains the origin of mass in gauge theories.

## 4. Are there any experimental evidence supporting the incorrectness of Goldstone's theorem in gauge theories?

Yes, the discovery of the Higgs boson in 2012 by the Large Hadron Collider (LHC) at CERN provided strong evidence for the Higgs mechanism and the incorrectness of Goldstone's theorem in gauge theories.

## 5. How does the incorrectness of Goldstone's theorem impact our understanding of particle physics?

The incorrectness of Goldstone's theorem has led to the development of the Higgs mechanism, which is a crucial component of the Standard Model of particle physics. This discovery has greatly enhanced our understanding of how particles acquire mass and has opened up new possibilities for future research in this field.

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