Haag-Lopuszanski-Sohnius theorem
Encyclopedia
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. This significantly generalized the Coleman–Mandula theorem.
One of the important results is that the fermionic part of the Lie superalgebra has to have spin-1/2 (spin 3/2 or higher are ruled out).
s, stating that the symmetry group of a consistent 4-dimensional quantum field theory
is the direct product
of the internal symmetry group and the Poincaré group
.
In 1975, Rudolf Haag
, Jan Łopuszański, and Martin Sohnius published their proof that weakening the assumptions of the Coleman–Mandula theorem by allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the Poincaré algebra, namely the supersymmetry algebra
.
ic particles into fermion
ic ones and vice versa, but the commutator
of two such transformations yields a translation
in spacetime
. Precisely such an interplay seemed excluded by the Coleman–Mandula theorem, which stated that (bosonic) internal symmetries cannot interact non-trivially with spacetime symmetry.
This theorem was also an important justification of the previously found Wess–Zumino model, an interacting four-dimensional quantum field theory
with supersymmetry
, leading to a renormalizable theory.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, the Haag–Lopuszanski–Sohnius theorem shows that the possible symmetries of a consistent 4-dimensional quantum field theory
Quantum 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...
do not only consist of internal symmetries and Poincaré symmetry, but can also include 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...
as a nontrivial extension of the Poincaré algebra. This significantly generalized the Coleman–Mandula theorem.
One of the important results is that the fermionic part of the Lie superalgebra has to have spin-1/2 (spin 3/2 or higher are ruled out).
History
Prior to the Haag–Lopuszanski–Sohnius theorem, the Coleman–Mandula theorem was the strongest of a series of no-go theoremNo-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...
s, stating that the symmetry group of a consistent 4-dimensional quantum field theory
Quantum 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...
is the direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of the internal symmetry group and 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:...
.
In 1975, Rudolf Haag
Rudolf Haag
Rudolf Haag is a German physicist. He is best known for his contributions to the algebraic formulation of axiomatic quantum field theory, namely the Haag-Kastler axioms...
, Jan Łopuszański, and Martin Sohnius published their proof that weakening the assumptions of the Coleman–Mandula theorem by allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the Poincaré algebra, namely the supersymmetry algebra
Supersymmetry algebra
In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras, and hence are Lie superalgebras...
.
Importance
What is most fundamental in this result (and thus in supersymmetry), is that there can be an interplay of spacetime symmetry with internal symmetry (in the sense of "mixing particles"): the supersymmetry generators transform bosonBoson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
ic particles into fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
ic ones and vice versa, but the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
of two such transformations yields a translation
Translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...
in spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. Precisely such an interplay seemed excluded by the Coleman–Mandula theorem, which stated that (bosonic) internal symmetries cannot interact non-trivially with spacetime symmetry.
This theorem was also an important justification of the previously found Wess–Zumino model, an interacting four-dimensional quantum field theory
Quantum 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...
with 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...
, leading to a renormalizable theory.