In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the Haag–Łopuszański–Sohnius theorem states that if both
commutating and
anticommutating generators are considered, then the only way to nontrivially mix
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
and internal symmetries is through
supersymmetry. The anticommutating generators must be
spin-1/2
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s which can additionally admit their own internal symmetry known as
R-symmetry
In theoretical physics, the R-symmetry is the symmetry transforming different supercharges in a theory with supersymmetry into each other. In the simplest case of the ''N''=1 supersymmetry, such an R-symmetry is isomorphic to a global U(1) group o ...
. The theorem is a generalization of the
Coleman–Mandula theorem
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 Lor ...
to
Lie superalgebras. It was proved in 1975 by
Rudolf Haag
Rudolf Haag (17 August 1922 – 5 January 2016) was a German theoretical physicist, who mainly dealt with fundamental questions of quantum field theory. He was one of the founders of the modern formulation of quantum field theory and he identifi ...
,
Jan Łopuszański, and Martin Sohnius as a response to the development of the first supersymmetric field theories by
Julius Wess
Julius Erich Wess (5 December 19348 August 2007) was an Austrian theoretical physicist noted as the co-inventor of the Wess–Zumino model and Wess–Zumino–Witten model in the field of supersymmetry and conformal field theory. He was also a ...
and
Bruno Zumino
Bruno Zumino (28 April 1923 − 21 June 2014) was an Italian theoretical physicist and faculty member at the University of California, Berkeley. He obtained his DSc degree from the University of Rome in 1945.
He was renowned for his rigorous p ...
in 1974.
History
During the 1960s, a set of theorems investigating how internal symmetries can be combined with spacetime symmetries were proved, with the most general being the Coleman–Mandula theorem. It showed that the
Lie group symmetry of an interacting theory must necessarily be a
direct product of the
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
with some
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
internal group. Unaware of this theorem, during the early 1970s a number of authors independently came up with supersymmetry, seemingly in contradiction to the theorem since there some generators do transform non-trivially under spacetime transformations.
In 1974 Jan Łopuszański visited
Karlsruhe
Karlsruhe ( , , ; South Franconian: ''Kallsruh'') is the third-largest city of the German state (''Land'') of Baden-Württemberg after its capital of Stuttgart and Mannheim, and the 22nd-largest city in the nation, with 308,436 inhabitants. ...
from
Wrocław
Wrocław (; german: Breslau, or . ; Silesian German: ''Brassel'') is a city in southwestern Poland and the largest city in the historical region of Silesia. It lies on the banks of the River Oder in the Silesian Lowlands of Central Europe, rou ...
shortly after Julius Wess and Bruno Zumino constructed the first supersymmetric
quantum field theory, the
Wess–Zumino model
In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with linearly realised supersymmetry. In 1974, Julius Wess and Bruno Zumino studied, using modern termino ...
. Speaking to Wess, Łopuszański was interested in figuring out how these new theories managed to overcome the Coleman–Mandula theorem. While Wess was too busy to work with Łopuszański, his doctoral student Martin Sohnius was available. Over the next few weeks they devised a proof of their theorem after which Łopuszański went on to
CERN where he worked with Rudolf Haag to significantly refine the argument and also extend it to the
massless case. Later, after Łopuszański went back to Wrocław, Sohnius went to CERN to finish the paper with Haag, which was published in 1975.
Theorem
The main assumptions of the Coleman–Mandula theorem are that the theory includes an
S-matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
with
analytic scattering amplitude
In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
state must undergo some reaction at almost all
energies
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
and scattering angles. Furthermore, there must only be a finite number of particle types below any
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
, disqualifying massless particles. The theorem then restricts the
Lie algebra of the theory to be a direct sum of the Poincare algebra with some internal symmetry
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
.
The Haag–Łopuszański–Sohnius theorem is based on the same assumptions, except for allowing additional anticommutating generators, elevating the Lie algebra to a Lie superalgebra. In four dimensions the theorem states that the only nontrivial anticommutating generators that can be added are a set of
pairs of
supercharges
and
which commute with the
momentum generator and transform as
left-handed
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subject ...
and right-handed
Weyl spinors. The undotted and dotted index notation, known as
Van der Waerden notation, distinguishes left-handed and right-handed Weyl spinors from each other. Generators of other spin, such spin-3/2 or higher, are disallowed by the theorem. In a basis where
, these supercharges satisfy
:
where
are known as
central charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other elemen ...
s, which commute with all generators of the superalgebra. Together with the Poincaré algebra, this Lie superalgebra is known as the
super-Poincaré 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 (without central charges or internal symme ...
. Since four dimensional Minkowski spacetime also admits
Majorana spinors as fundamental spinor representations, the algebra can equivalently be written in terms of four-component Majorana spinor supercharges, with the algebra generally expressed in terms of
gamma matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
and the
charge conjugation operator rather than Pauli matrices used for the two-component Weyl spinors.
The supercharges can also admit an additional Lie algebra symmetry known as R-symmetry, whose generators
satisfy
:
where
are
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
representation matrices of the generators in the
-dimensional representation of the R-symmetry group. For
the central charge must vanish and the R-symmetry is given by a
group, while for
extended supersymmetry
In theoretical physics, extended supersymmetry is supersymmetry whose Lie group#The Lie algebra associated to a Lie group, infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal w ...
, central charges need not be vanish, while the R-symmetry is a
group.
If massless particles are allowed, then the algebra can additionally be extended using
conformal generators: the
dilaton
In particle physics, the hypothetical dilaton particle is a particle of a scalar field \varphi that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theor ...
generator
and the
special conformal transformation
In projective geometry, a special conformal transformation is a linear fractional transformation that is ''not'' an affine transformation. Thus the generation of a special conformal transformation involves use of multiplicative inversion, which ...
s generator
. For
supercharges, there must also be the same number of superconformal transformations
which satisfy
:
with both the supercharges and the superconformal generators being charged under a
R-symmetry. This algebra is an example of a
superconformal algebra
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superco ...
, in this case denoted by
. Unlike for non-conformal supersymmetric algebras, the R-symmetry is always present in superconformal algebras.
Limitations
The Haag–Łopuszański–Sohnius theorem was originally derived in
four dimensions, however the result that supersymmetry is the only nontrivial extension to the spacetime symmetries holds in all dimensions greater than two. The form of the supersymmetry algebra however changes. Depending on the dimension, the supercharges can be Weyl, Majorana, Weyl–Majorana, or symplectic Weyl–Majorana spinors. Furthermore, the R-symmetry groups also differ according to the dimensionality and the number of supercharges.
In two or fewer dimensions the theorem breaks down. The reason for this is that analyticity of the scattering amplitudes can no longer hold since for example in two dimensions the only scattering is forward and backward scattering. The theorem also does not apply to
discrete symmetries or to
spontaneously broken symmetries since these are not symmetries at the level of the S-matrix.
See also
*
Supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
*
S-matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
References
{{DEFAULTSORT:Haag-Łopuszański-Sohnius theorem
Theoretical physics
Supersymmetry
Quantum field theory
Theorems in quantum mechanics
No-go theorems