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Haag–Łopuszański–Sohnius theorem

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In theoretical physics, the Haag–Łopuszański–Sohnius theorem states that if both commutating and anticommutating generators are considered, then the only way to nontrivially mix spacetime and internal symmetries is through supersymmetry. The anticommutating generators must be spin-1/2 spinors which can additionally admit their own internal symmetry known as R-symmetry. The theorem is a generalization of the Coleman–Mandula theorem to Lie superalgebras. It was proved in 1975 by Rudolf Haag, Jan Łopuszański, and Martin Sohnius[1] as a response to the development of the first supersymmetric field theories by Julius Wess and Bruno Zumino in 1974.

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Commutative property

Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it ; such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.

Anticommutative property

Anticommutative property

In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of a − b gives b − a = −(a − b); for example, 2 − 10 = −(10 − 2) = −8. Another prominent example of an anticommutative operation is the Lie bracket.

Generator (mathematics)

Generator (mathematics)

In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set.

Spacetime symmetries

Spacetime symmetries

Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact solutions of Einstein's field equations of general relativity. Spacetime symmetries are distinguished from internal symmetries.

Spin (physics)

Spin (physics)

Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nuclei.

Spinor

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 slight (infinitesimal) rotation. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360°. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors.

R-symmetry

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 its discrete subgroup (for the Z2 subgroup it is called R-parity). For extended supersymmetry, the R-symmetry group becomes a global U(N) non-abelian group.

Coleman–Mandula theorem

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 Lorentz scalars. Some notable exceptions to the no-go theorem are conformal symmetry and supersymmetry. It is named after Sidney Coleman and Jeffrey Mandula who proved it in 1967 as the culmination of a series of increasingly generalized no-go theorems investigating how internal symmetries can be combined with spacetime symmetries. The supersymmetric generalization is known as the Haag–Łopuszański–Sohnius theorem.

Rudolf Haag

Rudolf Haag

Rudolf Haag 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 identified the formal structure in terms of the principle of locality and local observables. He also made important advances in the foundations of quantum statistical mechanics.

Jan Łopuszański (physicist)

Jan Łopuszański (physicist)

Jan Łopuszański was a Polish theoretical physicist and author of several textbooks about classical, statistical and quantum physics. In the field of quantum field theory, he is most famous as co-author of the Haag-Lopuszanski-Sohnius theorem concerning the possibility of supersymmetry in renormalizable QFT's.

Julius Wess

Julius Wess

Julius Erich Wess 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 recipient of the Max Planck medal, the Wigner medal, the Gottfried Wilhelm Leibniz Prize, the Heineman Prize, and of several honorary doctorates.

Bruno Zumino

Bruno Zumino

Bruno Zumino 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.

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.[2] It showed that the Lie group symmetry of an interacting theory must necessarily be a direct product of the Poincaré group with some compact 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 from Wrocław shortly after Julius Wess and Bruno Zumino constructed the first supersymmetric quantum field theory, the Wess–Zumino model.[3] 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 extended it to the massless case. Later, after Łopuszański went back to Worclaw, Sohnius went to CERN to finish the paper with Haag, which was published in 1975.

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Lie group

Lie group

In mathematics, a Lie group is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be a group, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.

Symmetry (physics)

Symmetry (physics)

In physics, a symmetry of a physical system is a physical or mathematical feature of the system that is preserved or remains unchanged under some transformation.

Direct product

Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

Poincaré group

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 understanding of the most basic fundamentals of physics.

Compact group

Compact group

In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

Karlsruhe

Karlsruhe

Karlsruhe is the second-largest city of the German state (Land) of Baden-Württemberg after its capital of Stuttgart, and Mannheim, and the 21st-largest city in the nation, with 308,436 inhabitants. It is also a former capital of Baden, a historic region named after Hohenbaden Castle in the city of Baden-Baden. Located on the right bank of the Rhine near the French border, between the Mannheim/Ludwigshafen conurbation to the north and Strasbourg/Kehl to the south, Karlsruhe is Germany's legal center, being home to the Federal Constitutional Court (Bundesverfassungsgericht), the Federal Court of Justice (Bundesgerichtshof) and the Public Prosecutor General of the Federal Court of Justice.

Wrocław

Wrocław

Wrocław 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, roughly 350 kilometres (220 mi) from the Baltic Sea to the north and 40 kilometres (25 mi) from the Sudeten Mountains to the south. As of 2022, the official population of Wrocław is 672,929, with a total of 1.25 million residing in the metropolitan area, making it the third largest city in Poland.

Quantum field theory

Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.

Wess–Zumino model

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 terminology, dynamics of a single chiral superfield whose cubic superpotential leads to a renormalizable theory.

CERN

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 a northwestern suburb of Geneva, on the France–Switzerland border. It comprises 23 member states, and Israel is currently the only non-European country holding full membership. CERN is an official United Nations General Assembly observer.

Massless particle

Massless particle

In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon and the gluon. However, gluons are never observed as free particles, since they are confined within hadrons. In addition the Weyl semimetal or Weyl fermion discovered in 2015 is also massless.

Theorem

The main assumptions of the Coleman–Mandula theorem are that the theory includes an S-matrix with analytic scattering amplitudes such that any two-particle state must undergo some reaction at almost all energies and scattering angles.[4] Furthermore, there must only be a finite number of particle types below any mass, 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.

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 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.[5] In a basis where , these supercharges satisfy

where are known as central charges, which commute with all generators of the superalgebra. Together with the Poincaré algebra, this Lie superalgebra is known as the super-Poincaré algebra.

The supercharges can also admit an additional Lie algebra symmetry known as R-symmetry, whose generators satisfy

where are Hermitian representation matrices of the generators in the -dimensional representation of the R-symmetry group.[6] For the central charge must vanish and the R-symmetry is given by a group, while for extended supersymmetry , 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 generator and the special conformal transformations 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 this case denoted by .[7] Unlike for non-conformal supersymmetric algebras, the R-symmetry is always present in superconformal algebras.[8]

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Analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain.

Energy

Energy

In physics, 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 states that energy can be converted in form, but not created or destroyed. The unit of measurement for energy in the International System of Units (SI) is the joule (J).

Mass

Mass

Mass is the quantity of matter in a physical body. It is also a measure of the body's inertia, the resistance to acceleration when a net force is applied. An object's mass also determines the strength of its gravitational attraction to other bodies.

Lie algebra

Lie algebra

In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

Algebra over a field

Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

Chirality (physics)

Chirality (physics)

A chiral phenomenon is one that is not identical to its mirror image. The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity transformation. Invariance under parity transformation by a Dirac fermion is called chiral symmetry.

Central charge

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 elements of the original group—often embedded within a Lie algebra. In some cases, such as two-dimensional conformal field theory, a central charge may also commute with all of the other operators, including operators that are not symmetry generators.

Hermitian matrix

Hermitian matrix

In mathematics, a Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:

Group representation

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself ; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication.

Extended supersymmetry

Extended supersymmetry

In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators carry not only a spinor index , but also an additional index where is integer.

Conformal symmetry

Conformal symmetry

In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation.

Homothety

Homothety

In mathematics, a homothety is a transformation of an affine space determined by a point S called its center and a nonzero number k called its ratio, which sends point to a point by the rule for a fixed number .

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.[9]

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.

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Four-dimensional space

Four-dimensional space

A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height.

Majorana equation

Majorana equation

In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this equation are termed Majorana particles, although that term now has a more expansive meaning, referring to any fermionic particle that is its own anti-particle.

Discrete symmetry

Discrete symmetry

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's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges. In mathematics and theoretical physics, a discrete symmetry is a symmetry under the transformations of a discrete group—e.g. a topological group with a discrete topology whose elements form a finite or a countable set.

Spontaneous symmetry breaking

Spontaneous symmetry breaking

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

Source: "Haag–Łopuszański–Sohnius theorem", Wikipedia, Wikimedia Foundation, (2022, November 30th), https://en.wikipedia.org/wiki/Haag–Łopuszański–Sohnius_theorem.

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Further Reading

References
  1. ^ Haag, R.; Łopuszański, J.T.; Sohnius, M. (1975). "All possible generators of supersymmetries of the S-matrix". Nuclear Physics B. 88 (2): 257–274. doi:10.1016/0550-3213(75)90279-5.
  2. ^ Weinberg, S. (2005). "24". The Quantum Theory of Fields: Supersymmetry. Vol. 3. Cambridge University Press. p. 1-2. ISBN 978-0521670555.
  3. ^ Sohnius, M.F. (2001). "Recollections of a young contributor". Nucl. Phys. B Proc. Suppl. 101: 129–132. doi:10.1016/S0920-5632(01)01499-2.
  4. ^ Coleman, S.R.; Mandula, J. (1967). "All Possible Symmetries of the S Matrix". Phys. Rev. 159: 1251–1256. doi:10.1103/PhysRev.159.125.
  5. ^ Duplij, S. (2003). Concise Encyclopedia of Supersymmetry. Springer. p. 181–182. ISBN 978-1402013386.
  6. ^ Wess, J.; Bagger, B. (1992). "1". Supersymmetry and Supergravity. Princeton University Press. p. 3–9. ISBN 978-0691025308.
  7. ^ Freund, P.G.O. (1988). "4". Introduction to Supersymmetry. Cambridge University Press. p. 26. ISBN 978-0521356756.
  8. ^ Akhond, M.; et al. (2021). "The Hitchhiker's Guide to 4d N=2 Superconformal Field Theories". SciPost Phys. Lect. Notes. arXiv:2112.14764. doi:10.21468/SciPostPhysLectNotes.64.
  9. ^ Dall'Agata, G.; Zagermann, M. (2021). "10". Supergravity: From First Principles to Modern Applications. Springer. p. 264. ISBN 978-3662639788.

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