Haag–Łopuszański–Sohnius Theorem
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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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, 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, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
and internal symmetries is through
supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
. The anticommutating generators must be
spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...
-1/2
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
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 or ...
. 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 Lore ...
to
Lie superalgebras In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a \Z/2\Zgraded algebra, grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. The notio ...
. 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 identifie ...
, 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 pro ...
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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
of an interacting theory must necessarily be a
direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
of the
Poincaré group The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our unde ...
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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
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 German, South Franconian: ''Kallsruh'') is the List of cities in Baden-Württemberg by population, third-largest city of the States of Germany, German state of Baden-Württemberg, after its capital Stuttgart a ...
from
Wrocław Wrocław is a city in southwestern Poland, and the capital of the Lower Silesian Voivodeship. It is the largest city and historical capital of the region of Silesia. It lies on the banks of the Oder River in the Silesian Lowlands of Central Eu ...
shortly after Julius Wess and Bruno Zumino constructed the first supersymmetric
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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 to
CERN The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in Meyrin, western suburb of Gene ...
where he worked with Rudolf Haag to significantly refine the argument and also extend it to the
massless In particle physics, a massless particle is an elementary particle whose invariant mass is zero. At present the only confirmed massless particle is the photon. Other particles and quasiparticles Standard Model gauge bosons The photon (carrier of ...
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 is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
with
analytic Analytic or analytical may refer to: Chemistry * Analytical chemistry, the analysis of material samples to learn their chemical composition and structure * Analytical technique, a method that is used to determine the concentration of a chemical ...
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. Formulation Scattering in quantum mechanics begins with a p ...
s such that any two-
particle In the physical sciences, a particle (or corpuscle 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 s ...
state must undergo some reaction at almost all
energies Energy () 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 light. Energy is a conserved quantity—the law of conservation of energy sta ...
and scattering angles. Furthermore, there must only be a finite number of particle types below any
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
, disqualifying massless particles. The theorem then restricts the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of the theory to be a direct sum of the Poincare algebra with some internal symmetry
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. 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 \mathcal N pairs of
supercharge In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge (physics), charge in physics. Supercharge, denoted by the symbol Q, is an operator which transforms bosons in ...
s Q^L_\alpha and \bar Q^R_, indexed by \alpha, which commute with the momentum generator and transform as left-handed and right-handed
Weyl spinors Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
. 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 (\bar Q^A_) = (Q^A_\alpha)^\dagger, these supercharges satisfy : \ = \epsilon_ Z^, \ \ \ \ \ \ \ \ \ \ \ = \delta^\sigma^\mu_P_\mu, where Z^ 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 element ...
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 sym ...
. 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 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 \ \mathr ...
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 B_i satisfy : ^A_\alpha, B_i=\sum_B s^_i Q^B_\alpha, where s_i^ 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 me ...
representation matrices of the generators in the \mathcal N-dimensional representation of the R-symmetry group. For \mathcal N=1 the central charge must vanish and the R-symmetry is given by a \text(1) group, while for
extended supersymmetry In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal where \mathcal is integer (such as 2 or 4). Extended ...
\mathcal N>1 the central charges need not vanish, while the R-symmetry is a \text(\mathcal N) group. If massless particles are allowed, then the algebra can additionally be extended using conformal generators: the
dilaton In particle physics, the hypothetical dilaton 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 theory's compa ...
generator D 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 K_\mu. For \mathcal N supercharges, there must also be the same number of superconformal generators S_\alpha which satisfy : \ = \delta^\sigma^\mu_K_\mu, with both the supercharges and the superconformal generators being charged under a \text(\mathcal N) 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, superc ...
, which in this four dimensional case is denoted by \mathfrak(2,2, \mathcal N). Unlike for non-conformal supersymmetric algebras, 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 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, R-symmetry groups differ according to the dimensionality and the number of supercharges. This superalgebra also only applies in Minkowski spacetime, being modified in other spacetimes. For example, there exists an extension to
anti-de Sitter space In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a symmetric_space, maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are na ...
for one or more supercharges, while an extension to
de Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often denoted dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canonical Rie ...
only works if multiple supercharges are present. 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 In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square ...
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 is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...


References

{{DEFAULTSORT:Haag-Łopuszański-Sohnius theorem Supersymmetry Quantum field theory Theorems in quantum mechanics No-go theorems