There are no Goldstone bosons in two dimensions
 Sidney Coleman
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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).
 Goldstone, J., Salam, A., Weinberg, S.: Phys. Rev.127, 965 (1962).
 Actually, the righthand 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.
 As stated, for example. In: Streater, R., Wightman, A.: TCP, Spin and Statistics, and All That. New York: R. A. Benjamin 1964.
 This is the procedure of Kastler, Robinson, and Swieca (Ref. 3). The desired generalization has been proved by L. Landau (private communication).
 Mermin, N. D., Wagner, H.: Phys. Rev. Letters17, 1133 (1966).
 Englert, F., Brout, R.: Phys. Rev. Letters13, 321 (1964).
 Higgs, P.: Phys. Letters12, 132 (1964).
 Guralnik, G., Hagen, C., Kibble, T.: Phys. Rev. Letters13, 585 (1964).
 Higgs, P.: Phys. Rev.145, 1156 (1966).
 Kibble, T.: Phys. Rev.155, 1554 (1967).
 Private communication (through A. Wightman).
 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 _{μ} P ^{μ}, the usual twodimensional 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 twodimensional kinematics that, despite the fact that this is a normalizable twoparticle state in Fock space, it is still an eigenstate ofP _{μ} P ^{μ}. Consistent with the remarks above, though, the field : \(\bar \psi \psi\) :, whose twopoint function one might expect to possess a deltafunction singularity because of the existence of these states, has in fact zero amplitude for creating these states from the vacuum.
 Title
 There are no Goldstone bosons in two dimensions
 Journal

Communications in Mathematical Physics
Volume 31, Issue 4 , pp 259264
 Cover Date
 19731201
 DOI
 10.1007/BF01646487
 Print ISSN
 00103616
 Online ISSN
 14320916
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Sidney Coleman ^{(1)}
 Author Affiliations

 1. Joseph Henry Laboratories, Princeton University, Princeton, New Jersey, USA