DOCSLIB.ORG

NP As Minimization of Degree 4 Polynomial, Integration Or Grassmann Number Problem, and New Graph Isomorphism Problem Approac

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
NP As Minimization of Degree 4 Polynomial, Integration Or Grassmann Number Problem, and New Graph Isomorphism Problem Approac 1 P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches Jarek Duda Jagiellonian University, Golebia 24, 31-007 Krakow, Poland, Email: [email protected] Abstract—While the P vs NP problem is mainly approached problem, vertex cover problem, independent set problem, form the point of view of discrete mathematics, this paper subset sum problem, dominating set problem and graph proposes reformulations into the field of abstract algebra, ge- coloring problem. All of them stay in widely understood ometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and field of discrete mathematics, like combinatorics, graph approaches for this question. The first one is equivalence of theory, logic. satisfaction of 3-SAT problem with the question of reaching The unsuccessfulness of a half century search for zero of a nonnegative degree 4 multivariate polynomial (sum the answer might suggest to try to look out of this of squares), what could be tested from the perspective of relatively homogeneous field - try to apply advances algebra by using discriminant. It could be also approached as a continuous global optimization problem inside [0; 1]n, of more distant fields of mathematics, like abstract for example in physical realizations like adiabatic quantum algebra fluent in working with the ring of polynomials, computers. However, the number of local minima usually use properties of multidimensional geometry, or other grows exponentially. Reducing to degree 2 polynomial plus continuous mathematics including numerical methods n constraints of being in f0; 1g , we get geometric formulations perfecting approaches for common problem of continuous as the question if plane or sphere intersects with f0; 1gn. There will be also presented some non-standard perspectives for the global optimization. Subset-Sum, like through convergence of a series, or zeroing R 2π Q of 0 i cos('ki)d' fourier-type integral for some natural While there are approaches for reformulation of 3-SAT ki. The last discussed approach is using anti-commuting into continuous constrained optimization [1] of a complex n Grassmann numbers θi, making (A·diag(θi)) nonzero only if formula which has to additionally fulfill some constraints, A has a Hamilton cycle. Hence, the P6=NP assumption implies exponential growth of matrix representation of Grassmann this article shows that 3-SAT can be reformulated as the numbers. There will be also discussed a looking promising question of just reaching zero of a multivariate real nonneg- algebraic/geometric approach to the graph isomorphism prob- ative degree 4 polynomial, and this degree cannot be further lem - tested to successfully distinguish strongly regular graphs reduced. with up to 29 vertices. This reformulation (reduction) allows to place this prob- Keywords: 3-SAT, Hamilton cycle, discriminant, fourier lem in both global continuous optimization (unconstrained) analysis, Grassmann numbers, adiabatic quantum comput- and abstract algebra. Hence, it allows to translate the com- ers, cryptography, graph isomorphism problem plexity of NP-complete problems to possibly exponential growth of the number of local minima of such multivariate polynomial, what suggests similar issue for recently popular I. INTRODUCTION arXiv:1703.04456v13 [cs.CC] 5 Nov 2020 adiabatic quantum computers also representing combinato- The P versus NP question is a major unsolved problem of rial tasks as (energy) optimization problems - might require computer science. It asks about existence of a polynomial exponential reduction of temperature to even distinguish time algorithm for so called NP-complete problems, for between exponentially growing number of local minima. which being a solution can be tested in a polynomial time, This degree of polynomial can be reduced further to 2 if however, there is not known efficient way to locate the we additionally enforce variables to obtain boolean values solution in exponentially large set of possibilities. This f0; 1gN . Zero of degree 2 polynomial is simple linear plane, class contains many problems which can be reformulated allowing to polynomially reduce 3-SAT to a geometric (reduced) one into another through a polynomial transfor- problem of plane or sphere intersecting with f0; 1gN . mation. Hence, existence of a polynomial time algorithm Alternatively, from the abstract algebra point of view, for one of them would imply polynomial time algorithm for polynomial formulation allows to shift the difficulty for ex- all of them. Additionally, such hypothetical efficient method ample into the problem of calculating multivariate analogue would endanger most of currently used cryptography. of discriminant of polynomial. Some well known representants of this family are: 3- There will be also presented solution of Hamilton cycle SAT, Hamilton cycle problem, knapsack problem, travelling problem using anti-commuting Grassmann numbers, sug- salesman problem, subgraph isomorphism problem, clique gesting requirement of exponential growth of representation 2 for this algebra. Alternatively, derivatives can be also used The first one looks at four possibilities for x and y variables, for a related approach. enforcing the use of v = x _ y. Thanks of it, the second alternative becomes equivalent to x _ y _ z. II. 3-SAT AS GLOBAL MINIMIZATION OF POLYNOMIAL The left hand side alternative has 4 possibilities for being 3-SAT is the problem of determining if we can assign 0/1 satisfied, hence can be transformed into minimization of values to boolean variables, such that all alternatives from degree 8 nonnegative polynomial. Summing such 2m polynomials for all m c, we get n+m a chosen set of clauses (triples) are satisfied: 8i=1;:::;m Ci variable nonnegative polynomial of degree 8, which reaches . The alternatives may contain negation, for example C1 ^ zero if and only if the 3-SAT can be satisfied. C2 = (x _:y _ z) ^ (:x _ y _ u). Denote n as the number of variables, m as the number of clauses. We will now translate this conjugation of alternatives C. Approach for degree 6 into a nonnegative polynomial which zeros correspond to The main contribution of this paper is alternative approach satisfying variable assignments. The boolean variables will which directly obtains degree 6 polynomial and can be be transformed into real continuous variables, which are further reduced to degree 4 by adding m variables. enforced to finally choose 0 or 1 by the condition of zeroing Specifically, observe that C = x _ y _ z is satisfied when the polynomial. While the final values have to be discrete, the sum of representing 0/1 numbers is in f1; 2; 3g, leading their search might involve intermediately using real values to degree 6 polynomial: - especially from the [0; 1]n hypercube. _ 2 2 2 A. Degree 14 polynomial P3 (C) := (x+y+z−1) (x+y+z−2) (x+y+z−3) (3) The original author’s approach [2] from 2010 has used Reaching zero of this polynomial does not enforce vari- degree 14 polynomial, reduced to 8 by introducing one ables being in f0; 1g yet, but it can be done by additional additional variable per clause (triple). degree 4 polynomials: Specifically, the C = x _ y alternative is satisfied in 3 cases: 01, 10 and 11. It is equivalent to zeroing of degree 6 X _ X 2 2 nonnegative polynomial: p(x1; : : : ; xn) = P3 (Ci)+ xj (1−xj) (4) i=1;:::;m j=1;:::;n _ 2 2 p2 (C) := (x − 1) + y · This final polynomial of n variables is nonnegative, · (x − 1)2 + (y − 1)2 · x2 + (y − 1)2 (1) degree 6, and reaches zero if and only if the 3-SAT is satisfied. Alternative of three variables is satisfied in 7 cases: 001, 010, 100, 011, 101, 110, 111. Analogously we get degree _ D. The final reduction: degree 4 7 · 2 = 14 nonnegative polynomial p3 (C), which zeroes if and only if the alternative C is satisfied. To reduce to degree 4 polynomial, observe that the x+y+ _ We can now construct the final polynomial as sum of p3 z = 3 possibility can be avoided by adding a new variable _ for all m clauses: v - instead of P3 using polynomial: X _ p(x1; : : : ; xn) = p3 (Ci) (2) i=1;:::;m (x + y + v − 1)2(x + y + v − 2)2 + (z − v)2(z − v − 1)2 where for the negated variables we use 1 − x instead of x. For x = y = 0, the zeroing of the left hand side part This nonnegative polynomial is zero if and only if all p_ are 3 enforces v = 1, for which the right hand side part enforces zero, what is equivalent with all alternatives being satisfied. z = 1. In the remaining cases, the right hand side part allows We got degree 14 polynomial of n variables - there is a for z equal 0 or 1. natural question if this degree can be reduced at cost of at Summing the corresponding polynomials for all m most polynomial growth of the number of variables. 2 2 clauses with polynomials xi (1 − xi) for all original n variables and m additional ones, we get a nonnegative B. Reduction to degree 8 degree 4 polynomial of n+m variables, which reaches zero The original reduction has used additional variables (one if and only if the 3-SAT is satisfied. per clause) to reduce the number of possibilities satisfying Observe that if P6=NP, this degree 4 generally cannot alternative from 7 to 4, hence reducing the degree of be further reduced. Nonnegativity requires the degree to polynomial from 14 to 8. be even, so such hypothetical reduction would need to For this purpose, for each C = x _ y _ z clause introduce be to degree 2, which can be minimized in polynomial variable v and replace C with conjunction of the following time.
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]
  • Gaussian Operator Bases for Correlated Fermions
    Gaussian operator bases for correlated fermions J. F. Corney and P. D. Drummond ARC Centre of Excellence for Quantum-Atom Optics, University of Queensland, Brisbane 4072, Queensland, Australia. (Dated: 19th October 2018) We formulate a general multi-mode Gaussian operator basis for fermions, to enable a positive phase-space representation of correlated Fermi states. The Gaussian basis extends existing bosonic phase-space methods to Fermi systems and thus enables first-principles dynamical or equilibrium calculations in quantum many-body Fermi systems. We prove the completeness and positivity of the basis, and derive differential forms for products with one- and two-body operators. Because the basis satisfies fermionic superselection rules, the resulting phase space involves only c-numbers, without requiring anti-commuting Grassmann variables. I. INTRODUCTION integrals, except that here the Gaussian basis is used to expand the fermionic states themselves, rather than a In this paper we address the issue of how to represent path integral, which has advantages in terms of giving a highly correlated fermionic states, for the purposes of ef- greater physical understanding and fewer restrictions in ficient calculations in fermionic many-body physics. To the resulting applications. this end, we introduce a normally ordered Gaussian op- To begin, we establish in Sec. II the definition of a erator basis for fermionic density operators. With this Gaussian operator in unnormalised form, and follow this basis, earlier phase-space techniques used to represent in Sec. III with elementary examples of one- and two- atomic transitions[1, 2] can be extended to general Fermi mode Gaussians in order to illustrate the basic structure systems.
    [Show full text]
  • Classical and Quantum Mechanics Via Supermetrics in Time
    Noname manuscript No. (will be inserted by the editor) E. GOZZI Classical and Quantum Mechanics via Supermetrics in Time the date of receipt and acceptance should be inserted later Abstract Koopman-von Neumann in the 30’s gave an operatorial formul- ulation of Classical Mechancs. It was shown later on that this formulation could also be written in a path-integral form. We will label this functional ap- proach as CPI (for classical path-integral) to distinguish it from the quantum mechanical one, which we will indicate with QPI. In the CPI two Grassman- nian partners of time make their natural appearance and in this manner time becomes something like a three dimensional supermanifold. Next we intro- duce a metric in this supermanifold and show that a particular choice of the supermetric reproduces the CPI while a different one gives the QPI. Keywords quantum mechanics, classical mechanics, supermetric,path- integral 1 INTRODUCTION. The topic of this conference has been Spin-Statistics. One of the things which is most difficult to accept to our common sense is the anticommuting nature of Fermions. Actually anticommuting variables have a long history that goes back to Grassmann [1] and are not strictly related to spin. Grassmann in- vented them for abstract reasons but he discovered that they are usefull in the description of ruled surfaces. Later on the exterior product introduced arXiv:0910.1812v1 [quant-ph] 9 Oct 2009 by Grassmann was used in the field of differential forms by Cartan. Grass- mannian variables have made their appearance in
    [Show full text]
  • Geometric Approaches to Quantum Field Theory
    GEOMETRIC APPROACHES TO QUANTUM FIELD THEORY A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2020 Kieran T. O. Finn School of Physics and Astronomy Supervised by Professor Apostolos Pilaftsis BLANK PAGE 2 Contents Abstract 7 Declaration 9 Copyright 11 Acknowledgements 13 Publications by the Author 15 1 Introduction 19 1.1 Unit Independence . 20 1.2 Reparametrisation Invariance in Quantum Field Theories . 24 1.3 Example: Complex Scalar Field . 25 1.4 Outline . 31 1.5 Conventions . 34 2 Field Space Covariance 35 2.1 Riemannian Geometry . 35 2.1.1 Manifolds . 35 2.1.2 Tensors . 36 2.1.3 Connections and the Covariant Derivative . 37 2.1.4 Distances on the Manifold . 38 2.1.5 Curvature of a Manifold . 39 2.1.6 Local Normal Coordinates and the Vielbein Formalism 41 2.1.7 Submanifolds and Induced Metrics . 42 2.1.8 The Geodesic Equation . 42 2.1.9 Isometries . 43 2.2 The Field Space . 44 2.2.1 Interpretation of the Field Space . 48 3 2.3 The Configuration Space . 50 2.4 Parametrisation Dependence of Standard Approaches to Quan- tum Field Theory . 52 2.4.1 Feynman Diagrams . 53 2.4.2 The Effective Action . 56 2.5 Covariant Approaches to Quantum Field Theory . 59 2.5.1 Covariant Feynman Diagrams . 59 2.5.2 The Vilkovisky–DeWitt Effective Action . 62 2.6 Example: Complex Scalar Field . 66 3 Frame Covariance in Quantum Gravity 69 3.1 The Cosmological Frame Problem .
    [Show full text]
  • Interpreting Supersymmetry
    Interpreting Supersymmetry David John Baker Department of Philosophy, University of Michigan [email protected] October 7, 2018 Abstract Supersymmetry in quantum physics is a mathematically simple phenomenon that raises deep foundational questions. To motivate these questions, I present a toy model, the supersymmetric harmonic oscillator, and its superspace representation, which adds extra anticommuting dimensions to spacetime. I then explain and comment on three foundational questions about this superspace formalism: whether superspace is a sub- stance, whether it should count as spatiotemporal, and whether it is a necessary pos- tulate if one wants to use the theory to unify bosons and fermions. 1 Introduction Supersymmetry{the hypothesis that the laws of physics exhibit a symmetry that transforms bosons into fermions and vice versa{is a long-standing staple of many popular (but uncon- firmed) theories in particle physics. This includes several attempts to extend the standard model as well as many research programs in quantum gravity, such as the failed supergravity program and the still-ascendant string theory program. Its popularity aside, supersymmetry (SUSY for short) is also a foundationally interesting hypothesis on face. The fundamental equivalence it posits between bosons and fermions is prima facie puzzling, given the very different physical behavior of these two types of particle. And supersymmetry is most naturally represented in a formalism (called superspace) that modifies ordinary spacetime by adding Grassmann-valued anticommuting coordinates. It 1 isn't obvious how literally we should interpret these extra \spatial" dimensions.1 So super- symmetry presents us with at least two highly novel interpretive puzzles. Only two philosophers of science have taken up these questions thus far.
    [Show full text]
  • Observable Algebra
    Observable Algebra Merab Gogberashvili Andronikashvili Institute of Physics 6 Tamarashvili Str.,Tbilisi 380077, Georgia E-mail: [email protected] (October 24, 2018) A physical applicability of normed split-algebras, such as hyperbolic numbers, split-quaternions and split-octonions is considered. We argue that the observable geometry can be described by the algebra of split-octonions. In such a picture physical phenomena are described by the ordinary elements of chosen algebra, while zero divisors (the elements of split-algebras corresponding to zero norms) give raise the coordinatization of space-time. It turns to be possible that two fundamental constants (velocity of light and Planck constant) and uncertainty principle have geometrical meaning and appears from the condition of positive definiteness of norms. The property of non-associativity of octonions could correspond to the appearance of fundamental probabilities in four dimensions. Grassmann elements and a non-commutativity of space coordinates, which are widely used in various physical theories, appear naturally in our approach. PACS numbers: 04.50.+h; 11.25.Hf; 01.55.+b I. INTRODUCTION nals), one must be able to introduce a distance between two objects. Introduction of a distance always means Real physical phenomena always expose themselves ex- some comparison of two objects using one of them as an perimentally as a set of measured numbers. In a general etalon. Thus we need an algebra with a unit element. way one can say that the measurement process (even In the algebraic language, these requirements mean simple counting) consists of matching the physical phe- that to describe the geometry of the real world we need a nomenon to be measured with some familiar set of ob- normed composition algebra with the unit element over jects, or the number words.
    [Show full text]
  • Lecture Notes on Supersymmetry
    Lecture Notes on Supersymmetry Kevin Zhou [email protected] These notes cover supersymmetry, closely following the Part III and MMathPhys Supersymmetry courses as lectured in 2017/2018 and 2018/2019, respectively. The primary sources were: • Fernando Quevedo's Supersymmetry lecture notes. A short, clear introduction to supersym- metry covering the topics required to formulate the MSSM. Also see Ben Allanach's revised version, which places slightly more emphasis on MSSM phenomenology. • Cyril Closset's Supersymmetry lecture notes. A comprehensive set of supersymmetry lecture notes with more emphasis on theoretical applications. Contains coverage of higher SUSY, and spinors in various dimensions, the dynamics of 4D N = 1 gauge theories, and even a brief overview of supergravity. • Aitchison, Supersymmetry in Particle Physics. A friendly introductory book that covers the basics with a minimum of formalism; for instance, the Wess{Zumino model is introduced without using superfields. Also gives an extensive treatment of subtleties in two-component spinor notation. The last half covers the phenomenology of the MSSM. • Wess and Bagger, Supersymmetry and Supergravity. An incredibly terse book that serves as a useful reference. Most modern sources follow the conventions set here. Many pages consist entirely of equations, with no words. We will use the conventions of Quevedo's lecture notes. As such, the metric is mostly negative, and a few other signs are flipped with respect to Wess and Bagger's conventions. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Motivation.........................................3 1.2 The Poincare Group....................................6 1.3 Spinors in Four Dimensions................................9 1.4 Supersymmetric Quantum Mechanics..........................
    [Show full text]
  • A Brief Look at Gaussian Integrals
    A Brief Look at Gaussian Integrals William O. Straub, PhD Pasadena, California January 11, 2009 Gaussian integrals appear frequently in mathematics and physics. Perhaps the oddest thing about these integrals is that they cannot be evaluated in closed form over finite limits but are exactly integrable over [−∞, 1]. A typical Gaussian expression has the familiar “bell curve” shape and is described by the equation 1 y = exp(− ax2) 2 where a is a constant. In the path integral approach to quantum field theory, these integrals typically involve exponential terms with quadratic and linear terms in the variables. They are always multi-dimensional with limits at [−∞; 1], and are therefore difficult to solve. For example, the probability amplitude for a scalar field '(x) to transition from one spacetime point to another can be expressed by 1 Z = D' exp[i= d4x[− '(@2 + m2)' + J'] ˆ ~ ˆ 2 where the first integral is of dimension n and D'= d'1(x)d'2(x):::d'n(x), where n goes to infinity. Needless to say, it helps to have a few formulas handy to calculate such quantities. In quantum field theory, Gaussian integrals come in two types. The first involves ordinary real or complex variables, and the other involves Grassmann variables. The second type is used in the path integral description of fermions, which are particles having half-integral spin. Grassmann variables are highly non-intuitive, but calculating Gaussian integrals with them is very easy. Here, we’ll briefly consider both types. Gaussian Integrals with Ordinary Variables The simplest Gaussian integral is 1 I = exp(−x2) dx (1) ˆ−∞ which can solved solved quite easily.
    [Show full text]
  • Supersymmetry, Nontrivial Fermionic Shell and Nilpotent Mechanics S
    УДК 539.12 SUPERSYMMETRY, NONTRIVIAL FERMIONIC SHELL AND NILPOTENT MECHANICS S. A. Duplij 1), A. Frydryszak 2) 1) Department of Physics and Technology, Kharkov National University, Kharkov 61077, Ukraine E-mail: [email protected]. Internet: http://gluon.physik.uni-kl.de/˜duplij 2)Institute of Theoretical Physics, University of Wrocław, Pl. Maxa Borna 9, 50-204 Wrocław, Poland E-mail: [email protected] Received January 20, 2001 A need of the introduction of nilpotent even variables to the consistent classical supersymmetric theory is justyfied on the example of simple supersymmetric system. It is claimed that any classical supersymmetric theory on fermionic shell is equivalent to the corresponding nilpotent mechanics. KEYWORDS : supermechanics, dual number, nilpotence, fermionic shell, Legendre transformation, Poisson bracket Natural formulation of supersymmetric models is based on the formalism involving so called Grassmannian coordinates [1, 2], i.e. functions taking values in an appropriate Grassmann algebra [3]. Usually nilpotence is associated to the fermions due to the anticommutativity of objects describing them. However even in simple supersymmetric mechanical models (for review see e.g. [4]) it turns out that we must consider nilpotent even coordinates to describe consistently the model [5, 6] and to have nontrivial description on the fermionic shell [7]. These nilpotent even elements appear as an extension of the field of numbers used in the formalism. In our approach such an algebra of the so called Study numbers [8] or dual numbers [9, 10] replaces the basic field of numbers (complex or real) and resulting configuration and phase spaces have a geometric structure which we shall call a nilfold.
    [Show full text]
  • Functional Quantization of Spinor Fields
    PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 37 Wednesday, March 25, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topic: Functional Quantization of Spinor Fields. Functional Quantization of Spinor Fields In order to generalize the functional integral methods to spinor fields, which obey anticommutation relations, we must represent even the classical fields by anticommuting numbers. Grassmann Numbers Let us define anticommuting numbers (also called Grassmann numbers) by giving algebraic rules for manipulating them. They lead to the familiar quantum theory of the Dirac equation. A Grassmann algebra is a set of objects G that are generated by a basis fθig. These θi are Grassmann numbers, which anticommute with each other θiθj = −θjθi; (1) add commutatively θi + θj = θj + θj; (2) and can be multiplied by complex numbers, aθ 2 G for θ 2 G and a 2 C. The algebra must also have an element 0 so that θi + 0 = θi. For one θ, the most general element of the algebra is g = a + bθ; a; b 2 C; (3) since θ2 = 0. For two θ’s, the most general element is g = A + Bθ1 + Cθ2 + F θ1θ2; (4) PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 and so on. Elements of the algebra that have an even number of θi commute with all elements of the algebra, so they compose the even-graded or bosonic subalgebra. Similarly, the odd-graded or fermionic subalgebra has an odd number of θi and anticommutes within itself (but commutes with the bosonic subalgebra).
    [Show full text]
  • Gauge and Gravity Amplitudes from Trees to Loops
    FACULTY OF SCIENCE UNIVERSITY OF COPENHAGEN PhD thesis Rijun Huang Gauge and Gravity Amplitudes from trees to loops Principal Supervisor: Poul Henrik Damgaard Submitted: November 7, 2013 University of Copenhagen Doctoral Thesis Gauge and Gravity Amplitudes from trees to loops Author: Principal Supervisor: Rijun Huang Poul Henrik Damgaard A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Niels Bohr Institute, Faculty of Science University of Copenhagen December 2013 Abstract This thesis describes two subjects that I mainly work on during my PhD study. They are both about scattering amplitudes, covering gravity and gauge theories, tree and loop level, with or without supersymmetry. The first subject is Kawai-Lewellen-Tye(KLT) relation in field theory, which mysteriously relates Yang-Mills amplitudes to gravity am- plitudes. Based on many known works about KLT and super-KLT relations, we provide a complete map between super-gravity amplitudes and super-Yang-Mills amplitudes for any number of supersymmetry that allowed in 4-dimensional theory. We also provide an explanation for vanishing identities of Yang-Mills amplitudes as violation of linear symmetry groups based on KLT relation argument. The second subject is integrand re- duction of multi-loop amplitude. The recent methods based on computational algebraic geometry make it possible to systematically study multi-loop amplitude with general- ized unitarity cut. Using Gr¨obnerbasis and primary decomposition, we thoroughly study integrand basis and solution space of equations from maximal unitarity cut for all 4-dimensional two-loop topologies. Algorithm and examples of this computation are illustrated in this thesis.
    [Show full text]
  • Entanglement of Grassmannian Coherent States for Multi-Partite N-Level Systems
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 011, 11 pages Entanglement of Grassmannian Coherent States for Multi-Partite n-Level Systems Ghader NAJARBASHI and Yusef MALEKI Department of Physics, University of Mohaghegh Ardabili, Ardabil, 179, Iran E-mail: [email protected], [email protected] Received September 05, 2010, in final form January 19, 2011; Published online January 24, 2011 doi:10.3842/SIGMA.2011.011 Abstract. In this paper, we investigate the entanglement of multi-partite Grassmannian coherent states (GCSs) described by Grassmann numbers for n > 2 degree of nilpotency. Choosing an appropriate weight function, we show that it is possible to construct some well- known entangled pure states, consisting of GHZ, W, Bell, cluster type and bi-separable states, which are obtained by integrating over tensor product of GCSs. It is shown that for three level systems, the Grassmann creation and annihilation operators b and by together 2πi with bz form a closed deformed algebra, i.e., SUq(2) with q = e 3 , which is useful to construct entangled qutrit-states. The same argument holds for three level squeezed states. Moreover combining the Grassmann and bosonic coherent states we construct maximal entangled super coherent states. Key words: entanglement; Grassmannian variables; coherent states 2010 Mathematics Subject Classification: 81R30; 15A75; 81P40 1 Introduction Quantum entanglement has been recognized as the vital resource for the applications of quan- tum information and quantum computation. The emergence of entanglement is a fundamental difference between classical and quantum composite systems. Consequently The question of entanglement of composite systems has been intensively studied in last years [1,2].
    [Show full text]