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Coleman–Mandula theorem

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In theoretical physics, the Coleman–Mandula theorem is a no-go theorem stating that spacetime and internal symmetries can only combine in a trivial way. This means that the charges associated with internal symmetries must always transform as Lorentz scalars. Some notable exceptions that overcome the no-go theorem are conformal symmetry and supersymmetry. It is named after Sidney Coleman and Jeffrey Mandula who proved it in 1967 as the culimnation of a series of increasingly genearlized no-go theorems investigating how internal symmetries can be combined with spacetime symmetries.[1]

History

In the early 1960s, the global symmetry associated with the eightfold way was shown to successfully described the hadron spectrum for hadrons of the same spin. This led to efforts to expand the global symmetry into a larger symmetry mixing both flavour and spin, an idea similar to that previously considered in nuclear physics by Eugene Wigner in 1937 for an symmetry.[2] This non-relativistic model united vector and pseudoscalar mesons of different spin into a 35-dimensional multiplet and it also united the two baryon decuplets into a 56-dimensional multiplet.[3] The model reasonably successful in describing various aspects of the hadron spectrum, although in hindsight this is merely a consequence of the flavour and spin independence of the force between quarks in quantum chromodynamics. This success led to many attempts to generalize this non-relativistic model into a fully relativistic one.

At the time it was also an open question whether there existed a symmetry in which particles of different masses could belong to the same multiplet. Such a symmetry group could then possibly account for the mass splitting found in mesons and baryons at the time.[4] It was only later understood that this is instead a consequence of the breakdown of the internal symmetry.

These two motivations led to a series of no-go theorems to show that spacetime symmetries and internal symmetries could not be combined in any but a trivial way.[5] The first notable theorem was proved by William McGlinn in 1964,[6] with a subsequent generalization by Lochlainn O'Raifeartaigh in 1965.[7] These efforts culminated with the most general theorem by Sidney Coleman and Jeffrey Mandula in 1967.

Little notice was given to this theorem in subsequent years. As a result, the theorem played no role in the early development of supersymmetry, which instead emerged in the early 1970s from a study of string theory rather than from any attempts to overcome the no-go theorem.[8] Similarly, the Haag–Łopuszański–Sohnius theorem, which is a supersymmetric generalization of the Coleman–Mandula theorem, was derived in 1975 after the study of supersymmetry was well underway.[9]

Theorem

Consider a theory that can be described by an S-matrix and that satisfies the following conditions[1]

The Coleman–Mandula theorem states that the symmetry given by a Lie group of this theory is necessarily a direct product of the Poincaré group and an internal symmetry group.[10] Note that the last technical assumption is unnecessary if the theory is also a quantum field theory and is only needed to apply the theorem in a wider context.

A kinematic argument for why the theorem should hold was provided by Edward Witten.[11] The argument is that Poincaré symmetry is far too strong of a constraint for elastic scattering, leaving only the scattering angle unknown. Hence, any additional spacetime dependent symmetry would overdetermines the amplitudes, making them nonzero only at discrete scattering angles. But this conflicts with the assumption of the analyticity of scattering angles and thus rules out the possibility of such additional symmetries.

Limitations

Conformal symmetry

The theorem does not apply to a theory of massless particles, with these admitting an additional spacetime symmetry called conformal symmetry.[10] In particular, the allowed algebra is the Poincaré algebra together with a dilaton generator and the special conformal transformations generator, giving the conformal algebra.

Supersymmetry

The Coleman–Mandula theorem assumes that the only symmetry algebras are Lie algebras, but relaxing this assumption to also allow for Lie superalgebras leads to an evasion of the theorem. In particular, additional anticommutating generators known as supercharges can be added, with these transforming as spinors under Lorentz transformations. The resulting algebra is known as a super-Poincaré algebra, with the associated symmetry known as supersymmetry. The Haag–Łopuszański–Sohnius theorem is a generalization of the Coleman–Mandula theorem to Lie superalgebras, showing that supersymmetry the only possible such generalization. For a theory of massless particles, the theorem is again evaded by conformal symmetry which can be present in addition to supersymmetry giving a superconformal algebra.

Low dimensions

In a one or two dimensional theory the only possible scattering is forwards and backwards so analyticity of the scattering angles is no longer possible and the theorem no longer holds. Spacetime dependent internal symmetries are then possible, such as in the massive Thirring model which can admit an infinite tower of conserved charges of ever higher tensorial rank.[12]

Quantum groups

Models with nonlocal symmetries whose charges do not act on multiparticle states as if they were a tensor product of one-particle states evade the theorem.[13] Such an evasion is found more generally for quantum group symmetries which avoid the theorem because the corresponding algebra is no longer a Lie algebra.

Other limitations

For other spacetime symmetries besides the Poincaré group, such as theories with a de Sitter background or nonrelativistic field theories with Galilean invariance instead, the theorem no longer applies.[14] It also does not hold for discrete symmetries, since these are not Lie groups, or for spontaneously broken symmetries since these do not act on the S-matrix level and thus do not commute with the S-matrix.[15]

See also

Notes

  1. ^ a b Coleman, S.R.; Mandula, J. (1967). "All Possible Symmetries of the S Matrix". Phys. Rev. 159: 1251–1256. doi:10.1103/PhysRev.159.125.
  2. ^ Wigner, E. (1937). "On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei". Phys. Rev. 51 (2): 106–119. doi:10.1103/PhysRev.51.106.
  3. ^ Wess, J. (2009). "From Symmetry to Supersymmetry". arXiv:0902.2201.
  4. ^ Duplij, S. (2003). Concise Encyclopedia of Supersymmetry. Springer. p. 265–266. ISBN 978-1402013386.
  5. ^ Shifman, M.; Kane, G. (2000). The Supersymmetric World:The Beginnings of the Theory. World Scientific Publishing. p. 184–185. ISBN 978-9810245221.
  6. ^ McGlinn, W.D. (1964). "Problem of Combining Interaction Symmetries and Relativistic Invariance". Phys. Rev. Lett. 12 (16): 467–469. doi:10.1103/PhysRevLett.12.467.
  7. ^ O'Raifeartaigh, L. (1965). "Lorentz Invariance and Internal Symmetry". Phys. Rev. 139 (4B): B1052--B1062. doi:10.1103/PhysRev.139.B1052.
  8. ^ Cao, T.Y. (2004). "19". Conceptual Foundations of Quantum Field Theory. Cambridge University Press. p. 282. ISBN 978-0521602723.
  9. ^ Haag, R.; Łopuszański, J.T.; Sohnius, M. (1975). "All possible generators of supersymmetries of the S-matrix". Nuclear Physics B. 88 (2): 257–274. doi:10.1016/0550-3213(75)90279-5.
  10. ^ a b Weinberg, S. (2005). "24". The Quantum Theory of Fields: Supersymmetry. Vol. 3. Cambridge University Press. p. 12–22. ISBN 978-0521670555.
  11. ^ Zichichi, A. (2012). The Unity of the Fundamental Interactions: 19. Springer. p. 305–315. ISBN 978-1461336570.
  12. ^ Berg, B.; Karowski, M.; Thun, H.J. (1976). "Conserved currents in the massive thirring model". Physics Letters B. 64 (3): 286–288. doi:10.1016/0370-2693(76)90203-3.
  13. ^ Bernard, D.; LeClair, A. (1991). "Quantum group symmetries and non-local currents in 2D QFT". Communications in Mathematical Physics. 142 (1): 99–138. doi:10.1007/BF02099173.
  14. ^ Fotopoulos, A.; Tsulaia, M. (2010). "On the Tensionless Limit of String theory, Off - Shell Higher Spin Interaction Vertices and BCFW Recursion Relations". JHEP. 11: 086. arXiv:1009.0727. doi:10.1007/JHEP11(2010)086.
  15. ^ Fabrizio, N.; Percacci, R. (2008). "Graviweak Unification". J. Phys. A. 41: 075405. arXiv:0706.3307. doi:10.1088/1751-8113/41/7/075405.
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