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Coleman Mandula theorem

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David A Roberts edited this page Dec 2, 2015 · 1 revision

__AUTOLINKER{1|memory} This article is dedicated to the memory of Sidney Coleman.

The Coleman-Mandula theorem states what Lie group symmetries are possible in a relativistic theory of interacting particles.

Table of Contents

Statement of the Coleman-Mandula theorem

The Coleman-Mandula Theorem concerns what symmetries are possible in a relativistic theory of interacting particles (Coleman and Mandula, 1967). It states that the only possible such Lie group symmetries are direct products of the Poincaré group and an internal symmetry group. If we denote the state of a single particle by $\left| i\lambda p \right\rangle$, where $i$ denotes the particle type, $\lambda$ its intrinsic spin, and $p$ its momentum, an internal symmetry transformation is represented by a matrix acting only on the particle type index while a Poincaré transformation leaves the particle type unchanged and only affects its spin and momentum. The sorts of transformations excluded by the Coleman-Mandula theorem are those whose action on the $i\lambda p$ labels cannot be factored into the product of two terms, one acting only on the particle type index and the other acting only on the space-time indices. We would call such transformations hybrid symmetries.

Two important properties of a symmetry group of a relativistic particle theory are that single particle states form representations of the group, and that multiparticle states transform like direct products of single-particle representations. We will assume that the elements of a symmetry group are unitary operators acting on the space of states.

Corresponding to the unitarity of the group elements, the generators of the symmetry group are assumed Hermitean. The eigenvalues of the generators are the quantum numbers carried by the particles of the theory. The statement that multiparticle states transform like direct products of single particle representations is equivalent to the additive conservation of these quantum numbers. All of the generators of an internal symmetry group commute with each of the generators of the Poincaré group.

The theorem is phrased purely in terms of observable properties of particles and their interactions. Neither the statement of the theorem nor its proof use the full apparatus of quantum field theory. This pedagogical orientation is an artifact of the state of particle theory when the questions that led to its discovery arose. This will be reviewed in the next section of this article.

Context of the theorem

In the 1950's, experiments at particle accelerators had discovered many new particles and scattering resonances. The masses and spins of these particles were measured, and the patterns of allowed and forbidden decays were observed. These discoveries were new information about the strong, nuclear force, at much higher energies than in previous experiments. They invigorated the attempt to build a theory of the strong interactions.

At that time the only successful theory of elementary particle interactions known was quantum electrodynamics, which seemed to describe all electromagnetic phenomena. This was the natural model from which to start, but it was immediately realized that trying to describe the strong interactions as a quantum field theory was not straightforward. Among the host of obstacles was the fact that there were no reliable methods of calculation in strongly coupled field theory, and that there was no method for choosing a set of fundamental fields.

Without a clear path forward, progress came from attempts to exploit general properties of relativistic quantum field theories. The results that came from such work could be true properties of the strong interactions. One approach was the use of dispersion relations, which express the principle of causality in terms of the analytic properties of scattering amplitudes as functions of complex momenta. Dispersion relations, supplemented by quite reasonable simplifying assumptions, gave good descriptions of many aspects of high energy scattering, such as the electromagnetic form factors of nucleons. Attempts to build a complete theory of the strong interactions were not immediately successful, but the ideas developed in this context have had a powerful and continuing effect on elementary particle theory.

A particularly successful approach was to look for symmetry principles to organize the data. Of course, symmetries alone could not give a complete description of strong interactions, but it was plausible that one could discover the symmetries of the theory underlying the resonances and stable particles before having found the true theory. Furthermore, discovering the symmetries of the theory could be a major step in finding the theory itself.

Symmetry groups had been successfully applied in nuclear physics. The paradigm of an internal symmetry was isotopic spin, which related the nuclei of isotopes of different elements, but with the same atomic number. Isotopic spin was seen to be a symmetry of the strong interactions as well. Resonances with different charges but (almost) identical mass could be grouped into multiplets that formed representations of the isospin group, SU(2), and their allowed decays followed the patterns expected from group representation theory.

Besides isospin, a new internal quantum number was discovered. It was observed that many possible decay modes that were allowed by conservation of energy, charge, and isotopic spin, did not occur at anything like the expected natural strong interaction rate. This puzzle was solved by assigning to each particle a new quantum number, called "strangeness", and positing that it had to be conserved in strong interaction decays (Nakano and Nishijima, 1953; Gell-Mann, 1956). Particles in the same isotopic spin multiplet had the same value of this new quantum number, that is, strangeness commuted with isotopic spin.

The 1960's saw a great expansion of the use of symmetry groups in particle physics. A spectacular development was the observation that resonances with similar masses but different charge, isospin, and strangeness could be collected into multiplets that formed representations of a larger group, SU(3) (Gell-Mann, 1961; Ne'eman, 1961). This was called the "unitary symmetry" group at the time, and is called the "flavor" SU(3) group today. It was curious that, although the simplest representations of SU(3) have dimensions 3 and 6, only higher representations seemed to occur in nature. The lightest bosonic particles and resonances formed two 8-dimensional representations of SU(3), one consisting of pseudoscalar mesons and the other of vector mesons. The lightest spin $1/2$ baryons also formed an 8-dimensional representation while the lightest spin $3/2$ baryons formed a 10-dimensional representation. The assumption that the interactions that violated unitary symmetry transformed in a specific simple way under the symmetry group led to many relations among particle masses that were accurately obeyed (Coleman and Glashow, 1961; Gell-Mann, 1961; Okubo 1962).

Naturally, the success and utility of unitary symmetry led to searches for further symmetries of the strong interactions, and for models that would give the new abstract symmetries a natural dynamical setting.

An especially appealing generalization of the internal symmetry group SU(3) was the hybrid of internal SU(3) symmetry and intrinsic spin based on the group SU(6) (Sakita, 1964; Gürsey and Radicati, 1964; Pais, 1964; Gürsey ''et al''., 1964). As a method of grouping particles into multiplets it was strikingly successful. The ensemble of pseudoscalar and vector mesons formed a single 35-dimensional representation of SU(6) and the spin $1/2$ and spin $3/2$ baryons formed a single 56-dimensional representation. A similar hybrid symmetry had been successfully used in nuclear physics, but there was an conceptual difference between particle physics and nuclear physics. The theory of nuclei was framed in a non-relativistic context, and nuclear reactions could conserve angular momentum and intrinsic spin separately. However, relativity is inherent in particle physics, and intrinsic spin and orbital angular momentum are not separately conserved. The only conserved rotational quantum number in relativistic quantum mechanics is the total angular momentum.

Soon after SU(6) was proposed, several papers explored the problems associated with formulating SU(6) symmetry, and other hybrid symmetries, in a relativistic context (Coleman, 1964; Michel, 1964; Michel and Sakita, 1964; Bég and Pais, 1964; McGlinn, 1964; Weinberg, 1965; Jordan, 1965). The Coleman-Mandula theorem expressed clearly the reasons that hybrid symmetries could not be invariances of particle physics, and that the only possible Lie groups that can be symmetries of a relativistic particle theory are (locally) isomorphic to the direct product of the Poincaré group and an internal symmetry group.

Proof of the theorem

As we remarked in the introduction, the Coleman-Mandula theorem rests on the incompatibility of Poincaré invariance and the conservation of hybrid quantum numbers that involve spin. Because the result involves the interplay of relativistic scattering theory and group representation theory, the proof is quite convoluted. The logical structure of the argument is to begin with an arbitrary symmetry group generator and whittle its structure down to the sum of a translation, a pure Lorentz transformation, and an internal symmetry generator.

The basic assumptions about the theory are:

  1. The theory is Poincaré invariant, so its symmetry group includes the Poincaré group as a subgroup. In particular, any sum of translations or homogeneous Lorentz transformations of a generator is also in the generator algebra.
  2. The number of particle types whose masses are less than any given value is finite. The states of each particle type transform according to some positive mass representation of the Poincaré subgroup of the full symmetry group of the theory. There is no generalization of the Coleman-Mandula theorem to theories with massless particles.
  3. The $S$ matrix of the theory has the canonical analyticity properties. For a physical elastic scattering process $p + q \to p' + q'$, the momenta are on shell, $=m_{q}^{2}$, and the $T$ matrix, defined as $S - 1$ with the energy-momentum conservation $\delta$ function factored out, is an analytic function of the center-of-mass energy and momentum transfer variables \[\begin{align}\label{eq1}
]
#]The theory admits non-trivial scattering.  The $T$ matrix does not vanish, except possibly at a discrete set of exceptional values of the center-of mass energy and momentum transfer.

# An "ugly technical assumption", that the matrix elements of the group generators are distributions in momentum space.



To express these assumptions mathematically requires some notation.  We denote the generators of translations (the 4-momenta) by $$ and the generators of pure Lorentz transformations by ${M_ {\mu \nu} }$.  We will call a general element of the algebra of the generators of the symmetry group $A$, and work with representations in which the 4-momentum is diagonal.  On single particle states, the elements of the representation matrices are the matrix elements of $A$ between states of fixed momentum.


              


              

                
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