Coleman-Mandula theorem
Encyclopedia
The Coleman–Mandula theorem, named after Sidney Coleman
and Jeffrey Mandula
, is a no-go theorem
in theoretical physics
. It states that "space-time and internal symmetries cannot be combined in any but a trivial way". The only conserved quantities in a "realistic" theory with a mass gap
, apart from the generators of the Poincaré group
, must be Lorentz scalar
s.
satisfying certain technical assumptions about its S-matrix that has non-trivial interactions can only have a symmetry Lie algebra
which is always a direct product
of the Poincaré group
and an internal group if there is a mass gap
: no mixing between these two is possible. As the authors say in the introduction to the 1967 publication, "We prove a new theorem on the impossibility of combining space-time and internal symmetries in any but a trivial way."
Note that this theorem only constrains the symmetries of the S-matrix itself. As such, it places no constraints on spontaneously broken symmetries which do not show up directly on the S-matrix level. In fact, it is easy to construct spontaneously broken symmetries (in interacting theories) which unify spatial and internal symmetries.
This theorem also only applies to Lie algebras and not Lie groups. As such, it does not apply to discrete symmetries or globally for Lie groups. As an example of the latter, we might have a model where a rotation by 2π (a spacetime symmetry) is identified with an involutive internal symmetry which commutes with all the other internal symmetries.
If there is no mass gap, it could be a tensor product of the conformal algebra with an internal Lie algebra. But in the absence of a mass gap, there are also other possibilities. For example, quantum electrodynamics
has vector and tensor conserved charges. See infraparticle
for more details.
Supersymmetry
may be considered a possible "loophole" of the theorem because it contains additional generators (supercharge
s) that are not scalars but rather spinor
s. This loophole is possible because supersymmetry is a Lie superalgebra
, not a Lie algebra
. The corresponding theorem for supersymmetric theories with a mass gap is the Haag-Lopuszanski-Sohnius theorem
.
Quantum group
symmetry, present in some two-dimensional integrable quantum field theories like the sine-Gordon model, exploits a similar loophole.
Sidney Coleman
Sidney Richard Coleman was an American theoretical physicist who studied under Murray Gell-Mann.- Life and work :Sidney Coleman grew up on the Far North Side of Chicago...
and Jeffrey Mandula
Jeffrey Mandula
Jeffrey Ellis Mandula is a physicist well-known for the Coleman–Mandula theorem from 1967. He got his Ph.D. 1966 under Sidney Coleman at Harvard University. Today, he is responsible for the funding of science in the U.S. Department of Energy.-References:* at University of California, Davis* at...
, is a no-go theorem
No-go theorem
In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible.-Examples of no-go theorems:* Bell's theorem* Coleman–Mandula theorem* Haag-Lopuszanski-Sohnius theorem* Earnshaw's theorem...
in theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
. It states that "space-time and internal symmetries cannot be combined in any but a trivial way". The only conserved quantities in a "realistic" theory with a mass gap
Mass gap
In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest...
, apart from the generators of the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
, must be Lorentz scalar
Lorentz scalar
In physics, a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar may be generated from multiplication of vectors or tensors...
s.
Description
Every quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
satisfying certain technical assumptions about its S-matrix that has non-trivial interactions can only have a symmetry Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
which is always a direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
of the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
and an internal group if there is a mass gap
Mass gap
In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest...
: no mixing between these two is possible. As the authors say in the introduction to the 1967 publication, "We prove a new theorem on the impossibility of combining space-time and internal symmetries in any but a trivial way."
Note that this theorem only constrains the symmetries of the S-matrix itself. As such, it places no constraints on spontaneously broken symmetries which do not show up directly on the S-matrix level. In fact, it is easy to construct spontaneously broken symmetries (in interacting theories) which unify spatial and internal symmetries.
This theorem also only applies to Lie algebras and not Lie groups. As such, it does not apply to discrete symmetries or globally for Lie groups. As an example of the latter, we might have a model where a rotation by 2π (a spacetime symmetry) is identified with an involutive internal symmetry which commutes with all the other internal symmetries.
If there is no mass gap, it could be a tensor product of the conformal algebra with an internal Lie algebra. But in the absence of a mass gap, there are also other possibilities. For example, quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
has vector and tensor conserved charges. See infraparticle
Infraparticle
An infraparticle is an electrically charged particle and its surrounding cloud of soft photons—of which there are infinite number, by virtue of the infrared divergence of quantum electrodynamics. That is, it is a dressed particle rather than a bare particle...
for more details.
Supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
may be considered a possible "loophole" of the theorem because it contains additional generators (supercharge
Supercharge
In theoretical physics, a supercharge is a generator of supersymmetry transformations.Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and vice versa...
s) that are not scalars but rather spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s. This loophole is possible because supersymmetry is a Lie superalgebra
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
, not a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
. The corresponding theorem for supersymmetric theories with a mass gap is the Haag-Lopuszanski-Sohnius theorem
Haag-Lopuszanski-Sohnius theorem
In theoretical physics, the Haag–Lopuszanski–Sohnius theorem shows that the possible symmetries of a consistent 4-dimensional quantum field theory do not only consist of internal symmetries and Poincaré symmetry, but can also include supersymmetry as a nontrivial extension of the Poincaré algebra...
.
Quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
symmetry, present in some two-dimensional integrable quantum field theories like the sine-Gordon model, exploits a similar loophole.