DOCSLIB.ORG

A Brief Introduction to Supersymmetric Quantum Mechanics ∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Brief Introduction to Supersymmetric Quantum Mechanics ∗ A Brief Introduction to Supersymmetric Quantum Mechanics ∗ November 28, 2007 In class, we've learned about the supersymmetric, or factorization, method for solving some quantum mechanical potentials. It's difficult to see how this method relates to everything else you might have heard about supersymmetry and high energy particle physics. In these notes, we will attempt to build a bridge between these two applications, first by giving an introduction to super- symmetric quantum mechanics, then by connecting it to the methods learned in class. Finally, we will mention the essential elements that generalize to quantum field theory and those that do not. Most of the exposition in these notes will follow closely the very excellent notes of [2]. 1 SUSY QM Let us postulate a Hilbert space of quantum mechanical states. In addition to the Hamiltonian, H, we will assume that their exist two other operators on the space, Q and Qy. We require the operators to satisfy the supersymmetry algebra: fQ; Qg = 0 fQy;Qyg = 0 fQ; Qyg = 2H; (1) where fA; Bg ≡ AB + BA is known as the anticommutator of A and B. Notice, in particular, that (1) implies that Q2 = (Qy)2 = 0. It is also a straightforward exercise to show that [Q; H] = [Qy;H] = 0. From the algebra alone, we can immediately derive one of the most important consequences of supersymmetry, the positivity of energy. In any state, the expectation value of the Hamiltonian is 1 hHi = hΩjfQy;QgjΩi 2 ∗These notes have been prepared as a supplement for 8.05 students by David Guarrera. 1 1 2 1 y 2 + jQjΩij + Q jΩi 2 2 ≥ 0: (2) In particular, for any energy eigenstate, the energy is always greater than or equal to zero. Also, a zero energy state is necessarily annihilated by both Q and Qy. We can diagonalize the Hamiltonian and work with a particular subspace spanned by eigenstates of a particular energy, En. On this subspace, if En > 0, we can define Q Qy a = p and ay = p : (3) 2En 2En The supersymmetry algebra (1) becomes fa; ag = fay; ayg = 0 fa; ayg = 1: (4) y y Since [a; H] = [a ;H] = 0, if jni has energy eigenvalue En, so will a jni (if it is nonzero). In fact, since (a)2 = 0, there will exist some |−i such that a|−i = 0. y y 2 j+i ≡ a |−i will then also have energy En.(a ) |−i is zero and the algebra (4) tells us that aay|−i = |−i. In this way, for a particular energy that is greater than zero, a and ay are raising and lowering operators. We see that, for any positive energy, states always come in pairs. In the j+i; |−i basis, the raising and lowering operators are 0 0 0 1 a = ; ay = : (5) 1 0 0 0 We can define another operator, which anticommutes with all others. It is called the fermion number operator (the name will be explained later) 1 1 0 (−)F ≡ 2aya − a = [Qy;Q] = : (6) 2E 0 −1 Note, in particular, its action on states (−)F j+i = j+i (−)F |−i = −|−i: (7) We must now deal with the possible E = 0 states. For these states fQ; Qyg = 0. We've also shown that for such a state, j0i, Qj0i = Qyj0i = 0 (2) . Therefore, we cannot form raising and lowering operators for this state and, in general, there is no degeneracy. We still have not addressed the problem of whether or not a zero energy state exists; this will be discussed in the next section. If such a state does exist, we'd still like to define how the fermion number operator acts on it. There is no way to do this in general, though in specific contexts, the correct action is usually clear (we will encounter this in the next 2 section). Since we'd still like (−)F = 1, (−)F j0i = ±|0i. 2 2 A One Dimensional Spinning Particle Let's be more concrete and look in particular at a particle with spin moving in one dimension. The wavefunction is (x) Ψ(x) = + : (8) −(x) The first component represents spin up in the z direction, while the second component is spin down. We define the operators 0 0 Q ≡ (p − iW 0(x)) 1 0 0 1 Qy ≡ (p + iW 0(x)); (9) 0 0 d with p = −i¯h dx and W (x) is a yet unspecified function known as the superpo- tential. We are using units where the mass of the particle m = 1. For any W , we can satisfy the supersymmetry algebra (1) by requiring 1 p2 (W 0)2 ¯h H = fQ; Qyg = + 11 − σ3W 00: (10) 2 2 2 2 The first term is the usual kinetic energy, the second a potential energy for a spinless particle, and the third is the potential energy for a spin 1/2 particle in a (possibly position dependent) magnetic field. We can now define the fermion number operator for all states (not just the positive energy ones) as 1 0 (−)F = σ3 = : (11) 0 −1 On positive energy states, it is an easy check that this reduces to the definition F 1 y (6), (−) = 2E [Q ;Q]. Now, though, we have a perfectly good definition for all states. As an example, let's take the particularly simple superpotential W (x) = 1 2 2 !x . The Hamiltonian (10) becomes p2 1 ¯h!σ2 H = + !2x2 − : (12) 2 2 2 This is the Hamiltonian for a spin 1/2 particle in a harmonic oscillator potential and a magnetic field ! (in m = 1 units). After 8.05, we can immediately write down the energies 1 1 En = ¯h!(n + ∓ ) (13) ± 2 2 and eigenvectors (x) 0 n and ; (14) 0 n(x) 3 Figure 1: The energy diagram for the supersymmetric harmonic oscillator. This diagram was shamelessly stolen from [2]. with the n's the usual Hermite polynomials. The first solutions have fermion number +1, and the second have −1. This simple system exhibits many of the important qualities of supersymmetry discussed in the previoius section: • All E ≥ 0. • All E > 0 states are paired. The paired states with degenerate energy have opposite fermion number. • There is at most one E = 0 state. It's fermion number is, in general, not determined. In this case, there is one state with fermion number (−)F = +1. Going back to the case of general W , it is very easy to determine the zero energy states. Equation (2) says that for a state to satisfy E = 0, the first order equations QjΨi = QyjΨi = 0 (15) are obeyed. For a state (8), this implies that 0 (p − iW ) + = 0 0 (p + iW ) − = 0: (16) Integrating, −W=h¯ + / e +W=h¯ − / e : (17) If W ! 1 as x ! ±∞, then + is normalizable while − is not. Similarly if W ! −∞ as x ! ±∞, then − is normalizable and + is not. In all other cases, neither is normalizable and there is no zero energy state. In this case, we say (for reasons that will be explained later) that supersymmetry is broken. 4 3 Connection to the Supersymmetric Method As you've read the previous sections, you may have noticed many parallels between some of the calculations in SUSY QM and the supersymmetric method. In this section, we try and make those connections more concrete. Recall that in the supersymmetric method H(1) = AyA and H(2) = AAy. Here, there exists a symmetry between eigenstates of H(1) and H(2), while in SUSY QM there exists a symmetry between eigenstates of the same Hamiltonian, though the eigenstates differ by fermion number. By now, you might have noticed that the symmetry of the two systems is basically the same with replacements Q $ A and Qy $ Ay. The proofs of all the useful properties of the supersymmetric method map directly onto proofs of the interesting properties of the SUSY QM spectrum. We remind the reader of some of those proofs now: 1. In SUSY QM, the proof that the spectrum is nonnegative (E ∼ hΩjfQy;QgjΩi ≥ 0) is exactly the same as the proof that the spectrum of H(1) and H(2) are nonnegative, e.g. E(1) ∼ hΩjAyAjΩi ≥ 0. 2. Both the supersymmetric method and SUSY QM have twofold energy de- generacies for positive energy states. In particular A (orAy) maps eigen- states of H(1) (H(2)) to eigenstates of H(2) (H(1)) with the same energy. In an exactly parallel manner, in SUSY QM, Q and Qy map eigenstates of the Hamiltonian to different eigenstates of the same Hamiltonian, but with different fermion numbers. This equivalence can be made more con- y y crete. On states with En > 0, QQ |−i = Enj+i and Q Qj+i = En|−i. F y Therefore on \bosonic states" (states with (−) = +1), H = Hb = Q Q, F y while on \fermionic states" ((−) = +1), H = Hf = QQ . Therefore, in SUSY QM we can actually think of Q and Qy as mapping eigenstates between different Hamiltonians that each act on states of opposite fermion number. This is the origin of equation (66) in reference [1]. 3. In both the supersymmetric method and SUSY QM, the method for solv- ing for zero energy solutions is exactly the same. In particular, one must solve Qj i = 0 (Aj i = 0) or Qyj i = 0 (Ayj i = 0). In both cases, at most one of these will have a solution. In SUSY QM, this one state can have either fermion number, while in the supersymmetric method the state can be a solution to either Hamiltonian. In both cases, if there is no zero energy solution, we say that SUSY is broken.
Load more

Recommended publications
  • On Supergravity Domain Walls Derivable from Fake and True Superpotentials
    On supergravity domain walls derivable from fake and true superpotentials Rommel Guerrero •, R. Omar Rodriguez •• Unidad de lnvestigaci6n en Ciencias Matemtiticas, Universidad Centroccidental Lisandro Alvarado, 400 Barquisimeto, Venezuela ABSTRACT: We show the constraints that must satisfy the N = 1 V = 4 SUGRA and the Einstein-scalar field system in order to obtain a correspondence between the equations of motion of both theories. As a consequence, we present two asymmetric BPS domain walls in SUGRA theory a.ssociated to holomorphic fake superpotentials with features that have not been reported. PACS numbers: 04.20.-q, 11.27.+d, 04.50.+h RESUMEN: Mostramos las restricciones que debe satisfacer SUGRA N = 1 V = 4 y el sistema acoplado Einstein-campo esca.lar para obtener una correspondencia entre las ecuaciones de movimiento de ambas teorias. Como consecuencia, presentamos dos paredes dominic BPS asimetricas asociadas a falsos superpotenciales holomorfos con intersantes cualidades. I. INTRODUCTION It is well known in the brane world [1) context that the domain wa.lls play an important role bece.UIIe they allow the confinement of the zero mode of the spectra of gravitons and other matter fields (2], which is phenomenologically very attractive. The domain walls are solutions to the coupled Einstein-BCa.lar field equations (3-7), where the scalar field smoothly interpolatesbetween the minima of the potential with spontaneously broken of discrete symmetry. Recently, it has been reported several domain wall solutions, employing a first order formulation of the equations of motion of coupled Einstein-scalar field system in terms of an auxiliaryfunction or fake superpotential (8-10], which resembles to the true superpotentials that appear in the supersymmetry global theories (SUSY).
    [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]
  • One-Loop and D-Instanton Corrections to the Effective Action of Open String
    One-loop and D-instanton corrections to the effective action of open string models Maximilian Schmidt-Sommerfeld M¨unchen 2009 One-loop and D-instanton corrections to the effective action of open string models Maximilian Schmidt-Sommerfeld Dissertation der Fakult¨at f¨ur Physik der Ludwig–Maximilians–Universit¨at M¨unchen vorgelegt von Maximilian Schmidt-Sommerfeld aus M¨unchen M¨unchen, den 18. M¨arz 2009 Erstgutachter: Priv.-Doz. Dr. Ralph Blumenhagen Zweitgutachter: Prof. Dr. Dieter L¨ust M¨undliche Pr¨ufung: 2. Juli 2009 Zusammenfassung Methoden zur Berechnung bestimmter Korrekturen zu effektiven Wirkungen, die die Niederenergie-Physik von Stringkompaktifizierungen mit offenen Strings er- fassen, werden erklaert. Zunaechst wird die Form solcher Wirkungen beschrieben und es werden einige Beispiele fuer Kompaktifizierungen vorgestellt, insbeson- dere ein Typ I-Stringmodell zu dem ein duales Modell auf Basis des heterotischen Strings bekannt ist. Dann werden Korrekturen zur Eichkopplungskonstante und zur eichkinetischen Funktion diskutiert. Allgemeingueltige Verfahren zu ihrer Berechnung werden skizziert und auf einige Modelle angewandt. Die explizit bestimmten Korrek- turen haengen nicht-holomorph von den Moduli der Kompaktifizierungsmannig- faltigkeit ab. Es wird erklaert, dass dies nicht im Widerspruch zur Holomorphie der eichkinetischen Funktion steht und wie man letztere aus den errechneten Re- sultaten extrahieren kann. Als naechstes werden D-Instantonen und ihr Einfluss auf die Niederenergie- Wirkung detailliert analysiert, wobei die Nullmoden der Instantonen und globale abelsche Symmetrien eine zentrale Rolle spielen. Eine Formel zur Berechnung von Streumatrixelementen in Instanton-Sektoren wird angegeben. Es ist zu erwarten, dass die betrachteten Instantonen zum Superpotential der Niederenergie-Wirkung beitragen. Jedoch wird aus der Formel nicht sofort klar, inwiefern dies moeglich ist.
    [Show full text]
  • Dr. Joseph Conlon Supersymmetry: Example
    DR. JOSEPH CONLON Hilary Term 2013 SUPERSYMMETRY: EXAMPLE SHEET 3 1. Consider the Wess-Zumino model consisting of one chiral superfield Φ with standard K¨ahler potential K = Φ†Φ and tree-level superpotential. 1 1 W = mΦ2 + gΦ3. (1) tree 2 3 Consider the couplings g and m as spurion fields, and define two U(1) symmetries of the tree-level superpotential, where both U(1) factors act on Φ, m and g, and one of them also acts on θ (making it an R-symmetry). Using these symmetries, infer the most general form that any loop corrected su- perpotential can take in terms of an arbitrary function F (Φ, m, g). Consider the weak coupling limit g → 0 and m → 0 to fix the form of this function and thereby show that the superpotential is not renormalised. 2. Consider a renormalisable N = 1 susy Lagrangian for chiral superfields with F-term supersymmetry breaking. By analysing the scalar and fermion mass matrix, show that 2 X 2j+1 2 STr(M ) ≡ (−1) (2j + 1)mj = 0, (2) j where j is particle spin. Verfiy that this relation holds for the O’Raifertaigh model, and explain why this forbids single-sector susy breaking models where the MSSM superfields are the dominant source of susy breaking. 3. This problem studies D-term susy breaking. Consider a chiral superfield Φ of charge q coupled to an Abelian vector superfield V . Show that a nonvanishing vev for D, the auxiliary field of V , cam break supersymmetry, and determine the goldstino in this case.
    [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]
  • 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]
  • TASI 2008 Lectures: Introduction to Supersymmetry And
    TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg
    [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]
  • SUSY and XD Notes
    Supersymmetry and Extra Dimensions Flip Tanedo LEPP, Cornell University Ithaca, New York A pedagogical set of notes based on lectures by Fernando Quevedo, Adrian Signer, and Csaba Cs´aki,as well as various books and reviews. Last updated: January 9, 2009 ii iii Abstract This is a set of combined lecture notes on supersymmetry and extra dimensions based on various lectures, textbooks, and review articles. The core of these notes come from Professor Fernando Quevedo's 2006- 2007 Lent Part III lecture course of the same name [1]. iv v Acknowledgements Inspiration to write up these notes in LATEX came from Steffen Gielen's excel- lent notes from the Part III Advanced Quantum Field Theory course and the 2008 ICTP Introductory School on the Gauge Gravity Correspondence. Notes from Profes- sor Quevedo's 2005-2006 Part III Supersymmetry and Extra Dimensions course exist in TEXform due to Oliver Schlotterer. The present set of notes were written up indepen- dently, but simliarities are unavoidable. It is my hope that these notes will provide a broader pedagogical introduction supersymmetry and extra dimensions. vi vii Preface These are lecture notes. Version 1 of these notes are based on Fernando Quevedo's lecture notes and structure. I've also incorporated some relevant topics from my research that I think are important to round-out the course. Version 2 of these notes will also incorporate Csaba Cs´aki'sAdvanced Particle Physics notes. Framed text. Throughout these notes framed text will include parenthetical dis- cussions that may be omitted on a first reading. They are meant to provide a broader picture or highlight particular applications that are not central to the main purpose of the chapter.
    [Show full text]
  • Introduction to Supersymmetry
    Introduction to Supersymmetry Pre-SUSY Summer School Corpus Christi, Texas May 15-18, 2019 Stephen P. Martin Northern Illinois University [email protected] 1 Topics: Why: Motivation for supersymmetry (SUSY) • What: SUSY Lagrangians, SUSY breaking and the Minimal • Supersymmetric Standard Model, superpartner decays Who: Sorry, not covered. • For some more details and a slightly better attempt at proper referencing: A supersymmetry primer, hep-ph/9709356, version 7, January 2016 • TASI 2011 lectures notes: two-component fermion notation and • supersymmetry, arXiv:1205.4076. If you find corrections, please do let me know! 2 Lecture 1: Motivation and Introduction to Supersymmetry Motivation: The Hierarchy Problem • Supermultiplets • Particle content of the Minimal Supersymmetric Standard Model • (MSSM) Need for “soft” breaking of supersymmetry • The Wess-Zumino Model • 3 People have cited many reasons why extensions of the Standard Model might involve supersymmetry (SUSY). Some of them are: A possible cold dark matter particle • A light Higgs boson, M = 125 GeV • h Unification of gauge couplings • Mathematical elegance, beauty • ⋆ “What does that even mean? No such thing!” – Some modern pundits ⋆ “We beg to differ.” – Einstein, Dirac, . However, for me, the single compelling reason is: The Hierarchy Problem • 4 An analogy: Coulomb self-energy correction to the electron’s mass A point-like electron would have an infinite classical electrostatic energy. Instead, suppose the electron is a solid sphere of uniform charge density and radius R. An undergraduate problem gives: 3e2 ∆ECoulomb = 20πǫ0R 2 Interpreting this as a correction ∆me = ∆ECoulomb/c to the electron mass: 15 0.86 10− meters m = m + (1 MeV/c2) × .
    [Show full text]
  • Lectures on Supersymmetry and Superstrings 1 General Superalgebra
    Lectures on supersymmetry and superstrings Thomas Mohaupt Abstract: These are informal notes on some mathematical aspects of su- persymmetry, based on two ‘Mathematics and Theoretical Physics Meetings’ in the Autumn Term 2009.1 First super vector spaces, Lie superalgebras and su- percommutative associative superalgebras are briefly introduced together with superfunctions and superspaces. After a review of the Poincar´eLie algebra we discuss Poincar´eLie superalgebras and introduce Minkowski superspaces. As a minimal example of a supersymmetric model we discuss the free scalar superfield in two-dimensional N = (1, 1) supersymmetry (this is more or less copied from [1]). We briefly discuss the concept of chiral supersymmetry in two dimensions. None of the material is original. We give references for further reading. Some ‘bonus material’ on (super-)conformal field theories, (super-)string theories and symmetric spaces is included. These are topics which will play a role in future meetings, or which came up briefly during discussions. 1 General superalgebra Definition: A super vector space V is a vector space with a decomposition V = V V 0 ⊕ 1 and a map (parity map) ˜ : (V V ) 0 Z · 0 ∪ 1 −{ } → 2 which assigns even or odd parity to every homogeneous element. Notation: a V a˜ =0 , ‘even’ , b V ˜b = 1 ‘odd’ . ∈ 0 → ∈ 1 → Definition: A Lie superalgebra g (also called super Lie algebra) is a super vector space g = g g 0 ⊕ 1 together with a bracket respecting the grading: [gi , gj] gi j ⊂ + (addition of index mod 2), which is required to be Z2 graded anti-symmetric: ˜ [a,b]= ( 1)a˜b[b,a] − − (multiplication of parity mod 2), and to satisfy a Z2 graded Jacobi identity: ˜ [a, [b,c]] = [[a,b],c] + ( 1)a˜b[b, [a,c]] .
    [Show full text]
  • Adinkras for Mathematicians
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 6, June 2014, Pages 3325–3355 S 0002-9947(2014)06031-5 Article electronically published on February 6, 2014 ADINKRAS FOR MATHEMATICIANS YAN X. ZHANG Abstract. Adinkras are graphical tools created to study representations of supersymmetry algebras. Besides having inherent interest for physicists, the study of adinkras has already shown non-trivial connections with coding the- ory and Clifford algebras. Furthermore, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computa- tional nature. We survey these topics for a mathematical audience, make new connections to other areas (homological algebra and poset theory), and solve some of these said problems, including the enumeration of all hyper- cube adinkras up through dimension 5 and the enumeration of odd dashings of adinkras for any dimension. 1. Introduction In a series of papers starting with [11], different subsets of the “DFGHILM collaboration” (Doran, Faux, Gates, H¨ubsch, Iga, Landweber, Miller) have built and extended the machinery of adinkras. Following the ubiquitous spirit of visual diagrams in physics, adinkras are combinatorial objects that encode information about the representation theory of supersymmetry algebras. Adinkras have many intricate links with other fields such as graph theory, Clifford theory, and coding theory. Each of these connections provide many problems that can be compactly communicated to a (non-specialist) mathematician. This paper is a humble attempt to bridge the language gap and generate communication. In short, adinkras are chromotopologies (a class of edge-colored bipartite graphs) with two additional structures, a dashing condition on the edges and a ranking condition on the vertices.
    [Show full text]