DOCSLIB.ORG

Introduction to SUSY

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to SUSY (...) an invariance of the theory under interchange of fermions and bosons[9] Introduction to Supersymmetry m {Qα, Qα˙ } = 2(σ )αα˙ Pm Contents 1 Spinors in four dimensional space time 2 1.1 Clifford Algebra . 2 1.2 Dirac Spinor Representation of SO(3, 1) . 2 1.3 Transformation of Weyl Spinors [4] . 3 2 (N =1) Supersymmetry Algebra: From Coleman-Mandula to Haag-Lopuszanski-Sohnius 3 3 Chiral Field Component Representation 5 4 Superspace 6 4.1 Representation of supersymmetry generators . 6 4.2 Superfields . 7 5 Chiral Superfields 8 5.1 K¨ahlerpotential . 8 5.2 Superpotential . 10 6 Guide to the Literature 11 D.Schmidt Talk given at the PhD/Diploma Seminar Electroweak Symmetry Breaking and Supersymmetry Heidelberg, March 29 - 31, 2010 1 1 Spinors in four dimensional space time • Group theory classifies all particles as members of irreducible represen- tations of the underlying symmetry group, i.e. particles are classified according to their transformation law under the symmetry group [1]. • symmetry group of four dimensional space time: Lorentz group SO(3, 1) (translations not involved; → Poincar´egroup) • In what representation of the Lorentz group live fermions? 1.1 Clifford Algebra µ ν µν µν •{γ , γ } = 2 g 1; we use the signature g = gµν = diag (−, +, +, +). • specialµ four dimensional¶ representation using (hermitian) Pauli σ matrices: 0 σµ γµ = σµ 0 µ ¶ −1 0 Attention: σ0 = in the given metric gµν . 0 −1 1.2 Dirac Spinor Representation of SO(3, 1) µ ¶ σµσν − σν σµ 0 • J µν ≡ i [γµ, γν ] = i satisfies Lie Algebra 4 4 0 σµσν − σν σµ of SO(3,1): [J µν ,J ρσ] = i(ηνρJ µσ − ηµρJ νσ − ηνσJ µν + ηµσJ νρ). Remark~: In all representations of the Lorentz group, the corresponding generators are Bose type symmetry generators, i.e. they act on particles of equal statistics [7]. µ • Transformation law of a Dirac Spinor (ΨD)a(x ) a = 1, 2, 3, 4 [2]: µ Λ −1 ν ν (ΨD)a(x ) −→ S(Λ)ab(ΨD)b((Λ )µ x ) i ω M µν with a general Lorentz transformation Λ = e 2 µν and corresponding i ω J µν spinor transformation S(Λ) = e 2 µν . Avoid confusion: A generic element Λ of the Lie group SO(3, 1) can always i ω M µν µν be written as Λ = e 2 µν with M being the generators of the cor- responding Lie algebra. S(Λ) is a special representation of Λ, the spinor representation, for which the generators are J µν . • Dirac spinor representation completely reducible because generators J µν are diagonal. Get irreducible representation with the known left- and right-handed pro- jection opertors PL and PR: µ ¶ µ ¶ PL (ΨD)a Ψα (ΨD)a = ≡ α˙ . PR (ΨD)a χ 2 1 • Ψα ; α = 1, 2 left-handed Weyl spinor (( 2 , 0) representation of Lorentz group) α˙ 1 χ ;α ˙ = 1, 2 right-handed Weyl spinor ((0, 2 ) representation of Lorentz group) 1.3 Transformation of Weyl Spinors [4] 4 m • v ∈ R → ˆv = vmσ hermitian • M ∈ SL(2, C) → ˆv0 = MˆvM† is hermitian: expansion in σ matrices → 0 0 m ˆv = vmσ → v02 = det ˆv0 = det ˆv = v2 , because det M = 1. Result: Any M ∈ SL(2, C) defines via ˆv → ˆv0 = MˆvM† a Lorentz 0 n transformation vm → vm = Λm vn . • Weyl spinors transform under SL(2, C) (Weyl spinor has complex components.): 0 β α˙ α˙ 0 ∗−1 α˙ β˙ ~~ Ψα → Ψα = Mα Ψβ and χ → χ = (M )β˙ χ • index zoo: αβ Lowering and raising indices with antisymmetric tensor ² = (² ) = iσ2: β α˙ α˙ β˙ Ψα = ²αβΨ and χ = ² χβ˙ . 0 † m † ~~ Observe from ˆv = MˆvM = vmMσ M and that the index structure m m αα˙ α˙ β˙ αβ m of the Pauli σ matrices is given as: (σ )αα˙ and (σ ) = ² ² (σ )ββ˙ with latin letters as space time indices and greek letters as spinor indices. † α˙ † Attention to dotted and undotted spinor indices: Ψα = Ψα˙ and χ = χα . Kinetic term for spinor m αβ˙ α m β˙ iΨd/Ψ = iΨα˙ (σ ) ∂mΨβ + iχ (σ )αβ˙ ∂mχ results in equation of motion for a massless left-handed Weyl spinor: m αβ˙ i(σ ) ∂mΨβ = 0. 2 (N =1) Supersymmetry Algebra: From Coleman-Mandula to Haag-Lopuszanski- Sohnius • Coleman-Mandula no-go theorem for (...) an invariance of the theory un- der interchange of fermions and bosons : Any symmetry group (for which the commutator is the bilinear operation in the Lie algebra) of a quantum field theory (QFT) is locally isomorphic to the direct product of an inter- nal symmetry group, i.e. a group for which the generators have no matrix elements between particles of different spin, and the Poincar´egroup [6]. −→~ Symmetry between fermions and bosons impossible? 3 • Bypass Coleman-Mandula theorem: no-go theorem holds for Lie alge- bras with commutator as bilinear operation. Consider graded Lie algebras whose generators obey commutator and anticommutator relations. • new generator (N =1): Qα and its hermitian conjugate Qα˙ • Consider the following graded Lie algebra1: – vanishing anticommutators (due to spinor character): {Qα,Qβ} = {Qα˙ , Qβ˙ } = 0 – transformation under Lorentz group: mn i m n n m β [M ,Qα] = (σ σ − σ σ )α Qβ |4 {z } transformation of left-handed Weyl spinor 1 1 Qα in ( 2 , 0) representation, Qα˙ in (0, 2 ) representation of Lorentz group – All operators should commute with space time translations for energy m momentum conversation: [P ,Qα] = 0. m – The new anticommutator: {Qα, Qα˙ } = 2(σ )αα˙ Pm 1 1 Motivation: QαQα˙ transforms in ( 2 , 2 ) representation of the Lorentz group what is the spinor description of a four vector that has to be proportional to P m for energy momentum conversation. The only objects which have space time and spinor indices in their structures are the Pauli σ matrices. m αα˙ m With (σ ) · (σ )αα˙ = −4 the new anticommutator takes on the form P m = − 1 (σm)αα˙ {Q , Q } and we find 8 ³ α α˙ ´ 0 1 † † † † H = P = + 8 Q1Q1 + Q1Q1 + Q2Q2 + Q2Q2 , what shows ↑ 0 σ =−12×2 that the expectation value of the Hamiltonian H is not negative in any state |ηi constructed by applying creation operators, which are proportional to Qα˙ , to the vaccum state |Ωi : hη|H|ηi = 0 ; |Ωi is defined as usual: Qα|Ωi = 0 with annihilation operators being proportional to Qα [4]. • Haag, Lopuszanski and Sohnius: The given graded Lie algebra is the unique graded Lie algebra of symmetries consistent with QFT. → (N =1) Supersymmetry Algebra [7] • (global) infinitesimal supersymmetry transformation: α α˙ α α˙ δξ = ξ Qα − ξ Qα˙ = ξ Qα + ξα˙ Q ↑ indices with spinor parameter ξ. The commutator of two supersymmetry transformations leads to space time translations (Pm = i∂m): 1 m Action S invariant under supersymmetry transformationR → get supercurrents Jα after 3 0 applying Noether’s theorem → supercharges Qα = d x Jα fulfill the given graded Lie algebra [4]. 4 ³ ´ α m β˙ β m α˙ [δξ1 , δξ2 ] = 2i ξ1 (σ )αβ˙ ξ 2 − ξ2 (σ )βα˙ ξ 1 ∂m . Remark: ξ does not depend on space time, i.e. we consider only global supersymmetry. The theory of local supersymmetry transformations with ξ = ξ(xm), called Supergravity, is beyond the scope of this introduction. • Find a representation of this algebra on four dimensional space time to describe a QFT ! 3 Chiral Field Component Representation • theoryR of a complex scalar z and a left-handed Weyl spinor Ψα: 4 m m S = d x (−∂mz ∂ z − iΨ σ ∂mΨ) • supersymmetry transformations: For a supersymmetric theory the action ! S has to be invariant under a supersymmetry transformation δξ, δξS = 0. Therefore we have to define the following transformations: √ √ α α˙ δξz√≡ 2 ξ Ψα δξz ≡√ 2 ξ Ψα˙ m α˙ α m δξΨα ≡ 2 i(σ )αα˙ ξ ∂mz δξΨα˙ ≡ − 2 iξ (σ )αα˙ ∂mz Observe: In the first row bosons are transformed into fermions, in the second row fermions are transformed into bosons. • Consistency: The given definitions above must close the supersymmetry algebra. Problem for left-handed Weyl spinor: √ √ m α˙ m α˙ [δξ1 , δξ2 ]Ψα = δξ1 2 i(σ )αα˙ ξ 2 ∂mz − δξ2 2 i(σ )αα˙ ξ 1 ∂mz √ √ √ √ m α˙ β m α˙ β = 2 i(σ )αα˙ ξ ∂m 2 ξ Ψβ − 2 i(σ )αα˙ ξ ∂m 2 ξ Ψβ ³ 2 1 ´ 1 2 m α˙ β m α˙ β = 2i (σ )αα˙ ξ 2 ξ1 ∂m − (σ )αα˙ ξ 1 ξ2 ∂m Ψβ ³ ´ (?) α m β˙ β m β˙ = 2i −ξ1 (σ )ββ˙ ξ 2 ∂mΨβ + ξ1 (σ )ββ˙ ξ 2 ∂mΨα − (1 ↔ 2) ³ ´ (??) β˙ m ββ˙ α β m β˙ = 2i ξ 2 (σ ) ξ1 ∂mΨβ + ξ1 (σ )ββ˙ ξ 2 ∂mΨα − (1 ↔ 2) ³ ´ α β˙ m ββ˙ β m β˙ = 2i −ξ1 ξ 2 (σ ) ∂mΨβ + ξ1 (σ )ββ˙ ξ 2 ∂mΨα − (1 ↔ 2) ³ ´ β m β˙ β m β˙ = 2i ξ1 (σ )ββ˙ ξ 2 − ξ2 (σ )ββ˙ ξ 1 ∂mΨα ³ ´ α β˙ m ββ˙ α β˙ m ββ˙ − 2i ξ1 ξ 2 (σ ) ∂mΨβ − ξ2 ξ 1 (σ ) ∂mΨβ The algebra closes only for Ψβ fulfilling the equation of motion (eom): m αβ˙ (σ ) ∂mΨβ = 0. (?) Use Fierz rearrangement idendity: χα(ξς) = −ξα(ςχ) − ςα(χξ); α fixed, not summed m α˙ β with χ = (σ )αα˙ ξ 2 , ξ = ξ1 and ς = ∂mΨβ . (??) Use χσmξ† = −ξ†σ m. 5 • Counting the degrees of freedom (dof): Without use of the eom, there are two independent complex dof for the Weyl spinor, i.e. four independent real fermion dof. For the complex scalar there are two independent real boson dof.
Load more

Recommended publications
  • Arxiv:2009.05574V4 [Hep-Th] 9 Nov 2020 Predict a New Massless Spin One Boson [The ‘Lorentz’ Boson] Which Should Be Looked for in Experiments
    Trace dynamics and division algebras: towards quantum gravity and unification Tejinder P. Singh Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] Accepted for publication in Zeitschrift fur Naturforschung A on October 4, 2020 v4. Submitted to arXiv.org [hep-th] on November 9, 2020 ABSTRACT We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang-Mills fields, and fermions. Dynamical variables are described by odd- grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G2, replace space- time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1)em, describes the electro-colour symmetry, as shown earlier by Furey. We arXiv:2009.05574v4 [hep-th] 9 Nov 2020 predict a new massless spin one boson [the `Lorentz' boson] which should be looked for in experiments.
    [Show full text]
  • String Theory. Volume 1, Introduction to the Bosonic String
    This page intentionally left blank String Theory, An Introduction to the Bosonic String The two volumes that comprise String Theory provide an up-to-date, comprehensive, and pedagogic introduction to string theory. Volume I, An Introduction to the Bosonic String, provides a thorough introduction to the bosonic string, based on the Polyakov path integral and conformal field theory. The first four chapters introduce the central ideas of string theory, the tools of conformal field theory and of the Polyakov path integral, and the covariant quantization of the string. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree-level and one loop amplitudes. Chapter eight covers toroidal compactification and many important aspects of string physics, such as T-duality and D-branes. Chapter nine treats higher-order amplitudes, including an analysis of the finiteness and unitarity, and various nonperturbative ideas. An appendix giving a short course on path integral methods is also included. Volume II, Superstring Theory and Beyond, begins with an introduction to supersym- metric string theories and goes on to a broad presentation of the important advances of recent years. The first three chapters introduce the type I, type II, and heterotic superstring theories and their interactions. The next two chapters present important recent discoveries about strongly coupled strings, beginning with a detailed treatment of D-branes and their dynamics, and covering string duality, M-theory, and black hole entropy. A following chapter collects many classic results in conformal field theory. The final four chapters are concerned with four-dimensional string theories, and have two goals: to show how some of the simplest string models connect with previous ideas for unifying the Standard Model; and to collect many important and beautiful general results on world-sheet and spacetime symmetries.
    [Show full text]
  • Introduction to Supersymmetry(1)
    Introduction to Supersymmetry(1) J.N. Tavares Dep. Matem¶aticaPura, Faculdade de Ci^encias,U. Porto, 4000 Porto TQFT Club 1Esta ¶euma vers~aoprovis¶oria,incompleta, para uso exclusivo nas sess~oesde trabalho do TQFT club CONTENTS 1 Contents 1 Supersymmetry in Quantum Mechanics 2 1.1 The Supersymmetric Oscillator . 2 1.2 Witten Index . 4 1.3 A fundamental example: The Laplacian on forms . 7 1.4 Witten's proof of Morse Inequalities . 8 2 Supergeometry and Supersymmetry 13 2.1 Field Theory. A quick review . 13 2.2 SuperEuclidean Space . 17 2.3 Reality Conditions . 18 2.4 Supersmooth functions . 18 2.5 Supermanifolds . 21 2.6 Lie Superalgebras . 21 2.7 Super Lie groups . 26 2.8 Rigid Superspace . 27 2.9 Covariant Derivatives . 30 3 APPENDIX. Cli®ord Algebras and Spin Groups 31 3.1 Cli®ord Algebras . 31 Motivation. Cli®ord maps . 31 Cli®ord Algebras . 33 Involutions in V .................................. 35 Representations . 36 3.2 Pin and Spin groups . 43 3.3 Spin Representations . 47 3.4 U(2), spinors and almost complex structures . 49 3.5 Spinc(4)...................................... 50 Chiral Operator. Self Duality . 51 2 1 Supersymmetry in Quantum Mechanics 1.1 The Supersymmetric Oscillator i As we will see later the \hermitian supercharges" Q®, in the N extended SuperPoincar¶eLie Algebra obey the anticommutation relations: i j m ij fQ®;Q¯g = 2(γ C)®¯± Pm (1.1) m where ®; ¯ are \spinor" indices, i; j 2 f1; ¢ ¢ ¢ ;Ng \internal" indices and (γ C)®¯ a bilinear form in the spinor indices ®; ¯. When specialized to 0-space dimensions ((1+0)-spacetime), then since P0 = H, relations (1.1) take the form (with a little change in notations): fQi;Qjg = 2±ij H (1.2) with N \Hermitian charges" Qi; i = 1; ¢ ¢ ¢ ;N.
    [Show full text]
  • Clifford Algebra and the Interpretation of Quantum
    In: J.S.R. Chisholm/A.K. Commons (Eds.), Cliord Algebras and their Applications in Mathematical Physics. Reidel, Dordrecht/Boston (1986), 321–346. CLIFFORD ALGEBRA AND THE INTERPRETATION OF QUANTUM MECHANICS David Hestenes ABSTRACT. The Dirac theory has a hidden geometric structure. This talk traces the concep- tual steps taken to uncover that structure and points out signicant implications for the interpre- tation of quantum mechanics. The unit imaginary in the Dirac equation is shown to represent the generator of rotations in a spacelike plane related to the spin. This implies a geometric interpreta- tion for the generator of electromagnetic gauge transformations as well as for the entire electroweak gauge group of the Weinberg-Salam model. The geometric structure also helps to reveal closer con- nections to classical theory than hitherto suspected, including exact classical solutions of the Dirac equation. 1. INTRODUCTION The interpretation of quantum mechanics has been vigorously and inconclusively debated since the inception of the theory. My purpose today is to call your attention to some crucial features of quantum mechanics which have been overlooked in the debate. I claim that the Pauli and Dirac algebras have a geometric interpretation which has been implicit in quantum mechanics all along. My aim will be to make that geometric interpretation explicit and show that it has nontrivial implications for the physical interpretation of quantum mechanics. Before getting started, I would like to apologize for what may appear to be excessive self-reference in this talk. I have been pursuing the theme of this talk for 25 years, but the road has been a lonely one where I have not met anyone travelling very far in the same direction.
    [Show full text]
  • An Introduction to Supersymmetry
    An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems.
    [Show full text]
  • A Clifford Dyadic Superfield from Bilateral Interactions of Geometric Multispin Dirac Theory
    A CLIFFORD DYADIC SUPERFIELD FROM BILATERAL INTERACTIONS OF GEOMETRIC MULTISPIN DIRAC THEORY WILLIAM M. PEZZAGLIA JR. Department of Physia, Santa Clam University Santa Clam, CA 95053, U.S.A., [email protected] and ALFRED W. DIFFER Department of Phyaia, American River College Sacramento, CA 958i1, U.S.A. (Received: November 5, 1993) Abstract. Multivector quantum mechanics utilizes wavefunctions which a.re Clifford ag­ gregates (e.g. sum of scalar, vector, bivector). This is equivalent to multispinors con­ structed of Dirac matrices, with the representation independent form of the generators geometrically interpreted as the basis vectors of spacetime. Multiple generations of par­ ticles appear as left ideals of the algebra, coupled only by now-allowed right-side applied (dextral) operations. A generalized bilateral (two-sided operation) coupling is propoeed which includes the above mentioned dextrad field, and the spin-gauge interaction as partic­ ular cases. This leads to a new principle of poly-dimensional covariance, in which physical laws are invariant under the reshuffling of coordinate geometry. Such a multigeometric su­ perfield equation is proposed, whi~h is sourced by a bilateral current. In order to express the superfield in representation and coordinate free form, we introduce Eddington E-F double-frame numbers. Symmetric tensors can now be represented as 4D "dyads", which actually are elements of a global SD Clifford algebra.. As a restricted example, the dyadic field created by the Greider-Ross multivector current (of a Dirac electron) describes both electromagnetic and Morris-Greider gravitational interactions. Key words: spin-gauge, multivector, clifford, dyadic 1. Introduction Multi vector physics is a grand scheme in which we attempt to describe all ba­ sic physical structure and phenomena by a single geometrically interpretable Algebra.
    [Show full text]
  • Clifford Algebras, Spinors and Supersymmetry. Francesco Toppan
    IV Escola do CBPF – Rio de Janeiro, 15-26 de julho de 2002 Algebraic Structures and the Search for the Theory Of Everything: Clifford algebras, spinors and supersymmetry. Francesco Toppan CCP - CBPF, Rua Dr. Xavier Sigaud 150, cep 22290-180, Rio de Janeiro (RJ), Brazil abstract These lectures notes are intended to cover a small part of the material discussed in the course “Estruturas algebricas na busca da Teoria do Todo”. The Clifford Algebras, necessary to introduce the Dirac’s equation for free spinors in any arbitrary signature space-time, are fully classified and explicitly constructed with the help of simple, but powerful, algorithms which are here presented. The notion of supersymmetry is introduced and discussed in the context of Clifford algebras. 1 Introduction The basic motivations of the course “Estruturas algebricas na busca da Teoria do Todo”consisted in familiarizing graduate students with some of the algebra- ic structures which are currently investigated by theoretical physicists in the attempt of finding a consistent and unified quantum theory of the four known interactions. Both from aesthetic and practical considerations, the classification of mathematical and algebraic structures is a preliminary and necessary require- ment. Indeed, a very ambitious, but conceivable hope for a unified theory, is that no free parameter (or, less ambitiously, just few) has to be fixed, as an external input, due to phenomenological requirement. Rather, all possible pa- rameters should be predicted by the stringent consistency requirements put on such a theory. An example of this can be immediately given. It concerns the dimensionality of the space-time.
    [Show full text]
  • Super-Higgs in Superspace
    Article Super-Higgs in Superspace Gianni Tallarita 1,* and Moritz McGarrie 2 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago 7941169, Chile 2 Deutsches Elektronen-Synchrotron, DESY, Notkestrasse 85, 22607 Hamburg, Germany; [email protected] * Correspondence: [email protected] or [email protected] Received: 1 April 2019; Accepted: 10 June 2019; Published: 14 June 2019 Abstract: We determine the effective gravitational couplings in superspace whose components reproduce the supergravity Higgs effect for the constrained Goldstino multiplet. It reproduces the known Gravitino sector while constraining the off-shell completion. We show that these couplings arise by computing them as quantum corrections. This may be useful for phenomenological studies and model-building. We give an example of its application to multiple Goldstini. Keywords: supersymmetry; Goldstino; superspace 1. Introduction The spontaneous breakdown of global supersymmetry generates a massless Goldstino [1,2], which is well described by the Akulov-Volkov (A-V) effective action [3]. When supersymmetry is made local, the Gravitino “eats” the Goldstino of the A-V action to become massive: The super-Higgs mechanism [4,5]. In terms of superfields, the constrained Goldstino multiplet FNL [6–12] is equivalent to the A-V formulation (see also [13–17]). It is, therefore, natural to extend the description of supergravity with this multiplet, in superspace, to one that can reproduce the super-Higgs mechanism. In this paper we address two issues—first we demonstrate how the Gravitino, Goldstino, and multiple Goldstini obtain a mass. Secondly, by using the Spurion analysis, we write down the most minimal set of new terms in superspace that incorporate both supergravity and the Goldstino multiplet in order to reproduce the super-Higgs mechanism of [5,18] at lowest order in M¯ Pl.
    [Show full text]
  • Lecture 1 – Symmetries & Conservation
    LECTURE 1 – SYMMETRIES & CONSERVATION Contents • Symmetries & Transformations • Transformations in Quantum Mechanics • Generators • Symmetry in Quantum Mechanics • Conservations Laws in Classical Mechanics • Parity Messages • Symmetries give rise to conserved quantities . Symmetries & Conservation Laws Lecture 1, page1 Symmetry & Transformations Systems contain Symmetry if they are unchanged by a Transformation . This symmetry is often due to an absence of an absolute reference and corresponds to the concept of indistinguishability . It will turn out that symmetries are often associated with conserved quantities . Transformations may be: Active: Active • Move object • More physical Passive: • Change “description” Eg. Change Coordinate Frame • More mathematical Passive Symmetries & Conservation Laws Lecture 1, page2 We will consider two classes of Transformation: Space-time : • Translations in (x,t) } Poincaré Transformations • Rotations and Lorentz Boosts } • Parity in (x,t) (Reflections) Internal : associated with quantum numbers Translations: x → 'x = x − ∆ x t → 't = t − ∆ t Rotations (e.g. about z-axis): x → 'x = x cos θz + y sin θz & y → 'y = −x sin θz + y cos θz Lorentz (e.g. along x-axis): x → x' = γ(x − βt) & t → t' = γ(t − βx) Parity: x → x' = −x t → t' = −t For physical laws to be useful, they should exhibit a certain generality, especially under symmetry transformations. In particular, we should expect invariance of the laws to change of the status of the observer – all observers should have the same laws, even if the evaluation of measurables is different. Put differently, the laws of physics applied by different observers should lead to the same observations. It is this principle which led to the formulation of Special Relativity.
    [Show full text]
  • 1 the Superalgebra of the Supersymmetric Quantum Me- Chanics
    CBPF-NF-019/07 1 Representations of the 1DN-Extended Supersymmetry Algebra∗ Francesco Toppan CBPF, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro (RJ), Brazil. E-mail: [email protected] Abstract I review the present status of the classification of the irreducible representations of the alge- bra of the one-dimensional N− Extended Supersymmetry (the superalgebra of the Supersym- metric Quantum Mechanics) realized by linear derivative operators acting on a finite number of bosonic and fermionic fields. 1 The Superalgebra of the Supersymmetric Quantum Me- chanics The superalgebra of the Supersymmetric Quantum Mechanics (1DN-Extended Supersymme- try Algebra) is given by N odd generators Qi (i =1,...,N) and a single even generator H (the hamiltonian). It is defined by the (anti)-commutation relations {Qi,Qj} =2δijH, [Qi,H]=0. (1) The knowledge of its representation theory is essential for the construction of off-shell invariant actions which can arise as a dimensional reduction of higher dimensional supersymmetric theo- ries and/or can be given by 1D supersymmetric sigma-models associated to some d-dimensional target manifold (see [1] and [2]). Two main classes of (1) representations are considered in the literature: i) the non-linear realizations and ii) the linear representations. Non-linear realizations of (1) are only limited and partially understood (see [3] for recent results and a discussion). Linear representations, on the other hand, have been recently clarified and the program of their classification can be considered largely completed. In this work I will review the main results of the classification of the linear representations and point out which are the open problems.
    [Show full text]
  • Differential Graded Lie Algebras and Deformation Theory
    Differential graded Lie algebras and Deformation theory Marco Manetti 1 Differential graded Lie algebras Let L be a lie algebra over a fixed field K of characteristic 0. Definition 1. A K-linear map d : L ! L is called a derivation if it satisfies the Leibniz rule d[a; b] = [da; b]+[a; db]. n d P dn Lemma. If d is nilpotent (i.e. d = 0 for n sufficiently large) then e = n≥0 n! : L ! L is an isomorphism of Lie algebras. Proof. Exercise. Definition 2. L is called nilpotent if the descending central series L ⊃ [L; L] ⊃ [L; [L; L]] ⊃ · · · stabilizes at 0. (I.e., if we write [L]2 = [L; L], [L]3 = [L; [L; L]], etc., then [L]n = 0 for n sufficiently large.) Note that in the finite-dimensional case, this is equivalent to the other common definition that for every x 2 L, adx is nilpotent. We have the Baker-Campbell-Hausdorff formula (BCH). Theorem. For every nilpotent Lie algebra L there is an associative product • : L × L ! L satisfying 1. functoriality in L; i.e. if f : L ! M is a morphism of nilpotent Lie algebras then f(a • b) = f(a) • f(b). a P an 2. If I ⊂ R is a nilpotent ideal of the associative unitary K-algebra R and for a 2 I we define e = n≥0 n! 2 R, then ea•b = ea • eb. Heuristically, we can write a • b = log(ea • eb). Proof. Exercise (use the computation of www.mat.uniroma1.it/people/manetti/dispense/BCHfjords.pdf and the existence of free Lie algebras).
    [Show full text]
  • SUPERSYMMETRY 1. Introduction the Purpose
    SUPERSYMMETRY JOSH KANTOR 1. Introduction The purpose of these notes is to give a short and (overly?)simple description of supersymmetry for Mathematicians. Our description is far from complete and should be thought of as a first pass at the ideas that arise from supersymmetry. Fundamental to supersymmetry is the mathematics of Clifford algebras and spin groups. We will describe the mathematical results we are using but we refer the reader to the references for proofs. In particular [4], [1], and [5] all cover spinors nicely. 2. Spin and Clifford Algebras We will first review the definition of spin, spinors, and Clifford algebras. Let V be a vector space over R or C with some nondegenerate quadratic form. The clifford algebra of V , l(V ), is the algebra generated by V and 1, subject to the relations v v = v, vC 1, or equivalently v w + w v = 2 v, w . Note that elements of· l(V ) !can"b·e written as polynomials· in V · and this! giv"es a splitting l(V ) = l(VC )0 l(V )1. Here l(V )0 is the set of elements of l(V ) which can bCe writtenC as a linear⊕ C combinationC of products of even numbers ofCvectors from V , and l(V )1 is the set of elements which can be written as a linear combination of productsC of odd numbers of vectors from V . Note that more succinctly l(V ) is just the quotient of the tensor algebra of V by the ideal generated by v vC v, v 1.
    [Show full text]