Why is goldstone's theorem incorrect in gauge theories?

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Discussion Overview

The discussion revolves around the validity of Goldstone's theorem in the context of gauge theories and spontaneous symmetry breaking in quantum field theory (QFT). Participants explore the conditions under which Goldstone's theorem applies and the implications of gauge symmetries on these conditions.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that Goldstone's theorem requires a continuous global symmetry and a vacuum state that is not annihilated by the conserved charge, which gauge theories do not satisfy.
  • Others argue that local gauge invariance allows for the "gauge-ing away" of would-be massless Goldstone bosons, leading to massive gauge bosons instead.
  • A participant references Sidney Coleman's statement about gauge fields "eating up" the massless Goldstone bosons, suggesting that the scalar degrees of freedom become the longitudinal polarization of vector gauge bosons.
  • One participant introduces the idea that additional conditions may exist for Goldstone's theorem, citing the BCS model of superconductivity as an example where Goldstone bosons do not appear due to the non-local nature of the Hamiltonian.
  • Another participant questions the assertion that gauge symmetry violates the conditions of Goldstone's theorem, suggesting that local symmetries still encompass global symmetries and thus should maintain the same conserved currents and charges.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Goldstone's theorem in gauge theories, with some asserting that gauge symmetries invalidate the theorem while others maintain that they do not. The discussion remains unresolved regarding the specific conditions under which Goldstone's theorem holds true.

Contextual Notes

Participants mention the need for additional conditions for Goldstone's theorem, such as the locality of the Hamiltonian, and reference specific models like the BCS model and the Anderson Higgs mechanism, indicating that the discussion may be limited by these contextual factors.

LedPhoton
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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
 
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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!
 

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