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There are no Goldstone bosons in two dimensions. (English) Zbl 1125.81321

Summary: In four dimensions, it is possible for a scalar field to have a vacuum expectation value that would be forbidden if the vacuum were invariant under some continuous transformation group, even though this group is a symmetry group in the sense that the associated local currents are conserved. This is the Goldstone phenomenon, and Goldstone’s theorem states that this phenomenon is always accompanied by the appearance of massless scalar bosons. The purpose of this note is to show that in two dimensions the Goldstone phenomenon can not occur; Goldstone’s theorem does not end with two alternatives (either manifest symmetry or Goldstone bosons) but with only one (manifest symmetry).

MSC:

81V45 Atomic physics
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[1] Goldstone, J., Salam, A., Weinberg, S.: Phys. Rev.127, 965 (1962). · Zbl 0106.20601
[2] Actually, the right-hand side of Eq. (2) is intolerably crudely defined. For a proper definition, see Kastler, D., Robinson, D., Swieca, A.: Commun. math. Phys.3, 151 (1966). The use of the proper definition does not affect the proof given in the text, except by making some of the equations look more complicated. · Zbl 0177.41303
[3] As stated, for example. In: Streater, R., Wightman, A.: TCP, Spin and Statistics, and All That. New York: R. A. Benjamin 1964.
[4] This is the procedure of Kastler, Robinson, and Swieca (Ref. 3). The desired generalization has been proved by L. Landau (private communication).
[5] Mermin, N. D., Wagner, H.: Phys. Rev. Letters17, 1133 (1966).
[6] Englert, F., Brout, R.: Phys. Rev. Letters13, 321 (1964).
[7] Higgs, P.: Phys. Letters12, 132 (1964).
[8] Guralnik, G., Hagen, C., Kibble, T.: Phys. Rev. Letters13, 585 (1964).
[9] Higgs, P.: Phys. Rev.145, 1156 (1966).
[10] Kibble, T.: Phys. Rev.155, 1554 (1967).
[11] Private communication (through A. Wightman).
[12] This is an old observation. Schroer, B.: Fortschr. der Physik11, 1 (1963) and Wightman, A.: in High Energy Electromagnetic Interactions and Field Theory, ed. by Levy, M.: New York: Gordon and Breach 1967. This statement should not be taken to mean that there are no zero mass scalar particles in two dimensions. Indeed, if one defines ”particle” in the usual way, as a normalizable eigenstate ofP {\(\mu\)} P {\(\mu\)}, the usual two-dimensional theory of massless Dirac fields contains massless scalar particles; these are states of one fermion and one antifermion, both in normalizable states moving to the left. It is a peculiarity of massless two-dimensional kinematics that, despite the fact that this is a normalizable two-particle state in Fock space, it is still an eigenstate ofP {\(\mu\)} P {\(\mu\)}. Consistent with the remarks above, though, the field : \(\bar \psi \psi\) :, whose two-point function one might expect to possess a delta-function singularity because of the existence of these states, has in fact zero amplitude for creating these states from the vacuum.
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