Supersymmetric Dualities and Quantum Entanglement in Conformal Field Theory
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Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases. -
An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ........................... -
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). -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
Introductory Lectures on Quantum Field Theory
Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties. -
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. -
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. -
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. -
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. -
Andrew Thomson Dissertation.Pdf
PHENOMENOLOGY OF COMPACTIFICATIONS OF M-THEORY ON MANIFOLDS OF G2 HOLONOMY by Andrew James Thomson Supervisor: Kelly Stelle A dissertation submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London 24 September 2010 Contents I. Introduction 3 II. Holonomy and G2 manifolds 7 III. Chirality and singular manifolds 11 IV. Moduli stabilization and the hierarchy problem 14 V. Conclusion 17 References 19 2 I. Introduction The Standard Model is a phenomenally successful theory of particle physics, well tested at quantum mechanical level up to the TeV energy scale, and renormalizable (notwithstanding that the Higgs particle has not been observed yet) [1]. However there are several issues (or at least perceived ones) with the model. First of all, it only describes three of the four fundamental forces (the strong, electromagnetic and weak forces) in a quantum manner, leaving gravity still described only in a classical fashion; however we will not be concerned with this any further during this dissertation. The primary issue with which we will be concerned is the so-called hierarchy problem (also known as the weak-scale instability) – why there are two (electroweak and Planck) mass scales and why they are so far apart, why the electroweak scale is not modified at loop level by the Planck scale and whether there are any other scales between them. [1,2] One wonders why we should be concerned with as yet purely theoretical imperfections which have not yet manifested themselves in any experimental observations. One only has to look to the history of the standard model itself for a precedent for looking for physics beyond it – the earliest theories of the weak interaction, which were based on the interaction of four fermions at a point, broke down when calculated to higher order at the then unimaginably high energies of 300 GeV (Heisenberg 1939) before they had been violated by any observation. -
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. -
(Aka Second Quantization) 1 Quantum Field Theory
221B Lecture Notes Quantum Field Theory (a.k.a. Second Quantization) 1 Quantum Field Theory Why quantum field theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum field theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content. 1.1 Limitations of Multi-body Schr¨odinger Wave Func- tion We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) difference. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The final state contains yet another particle, photon in this case. But the Schr¨odinger equation is a differential equation acting on the arguments of the Schr¨odingerwave function, and can never change the number of arguments.