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I an attempt to evade the Goldstone Theorem, it is argued in Gilbert and Klein and Lee's paper that in a non-relativistic field there exists a preferred direction which can be used to evade Goldstone's Theorem. Such a preferred direction exists in some condensed matter systems like a ferromagnet. There exists an external preferred direction given by the externally applied field.

My question is on the definition on Non-relativistic field theory in Condensed Matter Physics. Does non-relativistic fields means one which has a preferred direction and hence is not Lorentz invariant? This definition is not the same as one where the velocity is much smaller than the speed of light, which is the usual definition of non-relativistic limit.

References: http://prl.aps.org/abstract/PRL/v12/i25/p713_1
http://prl.aps.org/abstract/PRL/v12/i10/p266_1

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    $\begingroup$ Preferred direction in space or spacetime? $\endgroup$ – Michael Brown Aug 26 '13 at 1:03
  • $\begingroup$ In spacetime. We basically have a unit time like vector $n_\mu$ which specifies the direction. $\endgroup$ – Omkar Aug 26 '13 at 7:02
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You should not worried too much about these papers. The way to evade the Goldstone theorem is the Higgs mechanism, which discusses local gauge transformation and invariance of the theory. This can be done both in relativistic and non-relativistic formalism (where it is called a Anderson-Higgs mechanism).

More details -- even discussing the paper by Gilbert and the Anderson's contribution, too -- in the Nobel Commitee document from the 2013 Nobel Prize.

In particular, the two last paragraphs of page 7 discuss the two papers you refer to:

Anderson’s ideas were not pursued much by particle physicists, who instead tried to even further strengthen Goldstone’s theorem. However in a paper [reference] from March 1964, Abraham Klein and Benjamin W. Lee, inspired by the comment of Anderson took up the question if one would be able to sidestep Goldstone’s theorem in a relativistically invariant theory. They set up a non-relativistic version of the arguments by Goldstone, Abdus Salam and Weinberg. They then followed it through and argued that also in this case there should be a massless scalar mode. Finally they showed the flaw in the argument and stated that the same would be valid in a relativistic theory.

Their arguments were immediately criticised by Walter Gilbert (Nobel Prize in chemistry,1980) [reference]. He showed that the arguments of Klein and Lee could be given a seemingly relativistic form by introducing a constant vector $n^{\mu}=\left(1,0,0,0\right)$. He could then show how terms can cancel each other leading to the absence of a massless pole. In the relativistic case there is only one vector at hand, the momentum, $k^{\mu}$, and the cancellation cannot occur. This was the situation up to the summer of 1964 when the breakthrough finally came.

Then the next section discusses the appearance of the Higgs and Englert & Brout papers, discussing how to evade the Goldstone's theorem.

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  • $\begingroup$ Ya but the work of Higgs was based on the work of Gilbert and Klein and Lee. You can see the references mentioned in the Higgs 1964 papers. $\endgroup$ – Omkar Oct 18 '13 at 6:16
  • $\begingroup$ @Omkar It was not based on the two papers you refer to, it contradicts them (say quickly). Please see the edit of the answer, and the document I previoulsy cited. $\endgroup$ – FraSchelle Oct 21 '13 at 12:08

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