The Coleman-Mandula Theorem

Prerequisites: Poincare Group, Canonical Quantum Field Theory. Full path (23 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Particle physics uses two kinds of symmetry. Spacetime symmetries (translations, rotations, boosts) act on positions and momenta. Internal symmetries, like the isospin symmetry relating protons and neutrons or the colour symmetry of quarks, mix particle types but do not touch momentum or spin. In the 1960s physicists looked for a larger symmetry that would combine the two, so that particles of different spin would sit in one multiplet, in the same way that isospin puts the proton and neutron together.

Sidney Coleman and Jeffrey Mandula showed in 1967 that this is impossible in an interacting relativistic theory with a few reasonable properties. Any extra conserved quantity that transforms nontrivially under rotations or boosts restricts collisions so much that particles could only scatter straight forward or straight backward. Real particles scatter at all angles, so such conserved quantities cannot exist. The only possible symmetries of the scattering of particles are the Poincaré symmetry of spacetime together with an internal symmetry that commutes with it.

The theorem has loopholes, and they turned out to be important. It assumes that symmetry generators obey commutation relations. If anticommuting generators are allowed, spacetime and internal symmetries can be combined after all, which leads to supersymmetry. The theorem also does not apply to theories without massive particles, to two-dimensional theories, or to symmetries that are spontaneously broken.

Concrete

Statement. Consider a quantum field theory in four spacetime dimensions with an S-matrix that satisfies:

• Poincaré invariance.
• For any mass $M$ there are only finitely many particle types with mass below $M$, and there is a mass gap.
• Two-particle scattering is nontrivial: elastic amplitudes are nonzero at almost all energies and angles.
• The two-particle scattering amplitudes are analytic functions of the energy and scattering angle (plus a technical assumption on their behaviour).

Then every Lie algebra of symmetry generators that commute with the S-matrix is a direct sum of the Poincaré algebra, with generators $P_\mu$ and $M_{\mu\nu}$, and an internal algebra with generators $B_a$: $$[B_a, P_\mu] = 0, \qquad [B_a, M_{\mu\nu}] = 0, \qquad [B_a, B_b] = i f_{ab}{}^{c} B_c .$$ The internal algebra is the direct sum of a compact semisimple Lie algebra and abelian $\mathfrak{u}(1)$ factors. In particular, the $B_a$ act on one-particle states without changing momentum or spin, so all members of an internal multiplet have the same mass and spin.

Why tensor charges kill scattering. The core of the argument can be seen in an example. Suppose there is a conserved charge $Q_{\mu\nu}$ that transforms as a symmetric tensor. On a one-particle state of a spinless particle with momentum $p$, Lorentz covariance restricts its action to $$Q_{\mu\nu} |p\rangle = \left( \alpha\, p_\mu p_\nu + \beta\, \eta_{\mu\nu} \right) |p\rangle .$$ A conserved charge built from local currents acts additively on well-separated particles in the incoming and outgoing states. For elastic scattering $p_1 + p_2 \to p_3 + p_4$, conservation of $Q_{\mu\nu}$ and of $P_\mu$ then requires $$p_{1\mu}p_{1\nu} + p_{2\mu}p_{2\nu} = p_{3\mu}p_{3\nu} + p_{4\mu}p_{4\nu}, \qquad p_1 + p_2 = p_3 + p_4 .$$ Take equal masses and go to the centre-of-mass frame, $\vec p_1 = -\vec p_2 = \vec p$ and $\vec p_3 = -\vec p_4 = \vec{p}^{\,\prime}$ with $|\vec{p}^{\,\prime}| = |\vec p|$. The spatial components of the first condition give $2 p_i p_j = 2 p'_i p'_j$, so $\vec{p}^{\,\prime} = \pm \vec p$. Only forward and backward scattering are allowed. Combined with analyticity of the amplitude in the scattering angle, this forces the amplitude to vanish, which contradicts the assumption of nontrivial scattering. The same reasoning excludes any conserved charge carrying Lorentz indices beyond those of $P_\mu$ and $M_{\mu\nu}$.

Loopholes. Each assumption marks a way out:

• Generators that anticommute, forming a Lie superalgebra, are not covered. The Haag-Łopuszański-Sohnius theorem extends the analysis to this case and shows that the supersymmetry algebras (with possible central charges, and superconformal algebras in the massless case) are the only allowed extensions, see supersymmetry.
• Without a mass gap (massless particles only), the conformal algebra is allowed.
• In two spacetime dimensions, scattering is only forward or backward anyway, and integrable models have infinitely many higher-spin conserved charges.

• Spontaneously broken symmetries and discrete symmetries do not act as Lie algebra symmetries of the S-matrix in this sense.

• The Coleman-Mandula Theorem by Sascha Leonhardt

Abstract

Loopholes

• For a nice discussion, see Section 5.2 "Living without Supersymmetry" in New Symmetry Concepts for Spacetime and Unification by Sebastian Ohmer.

There are also symmetries that are not captured by the theorem at all. These symmetries are for example discrete symmetries and spontaneously broken symmetries. The latter being symmetries that do not act on the S-matrix level and therefore evading the requirement to commute with S. Rather, these symmetries are symmetries of the action of the theory and are spontaneously broken before any calculation.The Coleman-Mandula Theorem by Sascha Leonhardt

Why is it interesting?

The Coleman-Mandula theorem is often invoked as a motivation for Supersymmetry.

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Group Theory
6. Hilbert Space
7. Legendre Transformation
8. Newton's Second Law
9. Vector Calculus
10. Newtonian Mechanics
11. Variational Calculus
12. Configuration Space
13. Euler-Lagrange Equations
14. Special Relativity
15. Lagrangian Formalism
16. Lorentz Group
17. Phase Space
18. Classical Field Theory
19. Hamiltonian Formalism
20. Poincare Group
21. Schrödinger Equation
22. Canonical Quantum Mechanics
23. Canonical Quantum Field Theory