DOCSLIB.ORG

Hyperdiamond Feynman Checkerboard in 4-Dimensional Spacetime

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hyperdiamond Feynman Checkerboard in 4-Dimensional Spacetime View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server quant-ph/950301 5 THEP-95-3 March 1995 Hyp erDiamond Feynman Checkerb oard in 4-dimensional Spacetime Frank D. (Tony) Smith, Jr. e-mail: [email protected] and [email protected] P.O.Box for snail-mail: P.O.Box 430, Cartersville, Georgia 30120 USA WWW URL http://www.gatech.edu/tsmith/home.html Scho ol of Physics Georgia Institute of Technology Atlanta, Georgia 30332 Abstract A generalized Feynman Checkerb oard mo del is constructed using a processed by the SLAC/DESY Libraries on 22 Mar 1995. 〉 4-dimensional Hyp erDiamond lattice. The resulting phenomenological mo del is the D D E mo del describ ed in hep-ph/9501252 and 4 5 6 quant-ph/9503009. PostScript c 1995 Frank D. (Tony) Smith, Jr., Atlanta, Georgia USA QUANT-PH-9503015 Contents 1 Intro duction. 2 2 Feynman Checkerb oards. 3 3 Hyp erDiamond Lattices. 10 3.1 8-dimensional Hyp erDiamond Lattice. :: :: ::: :: :: :: 12 3.2 4-dimensional Hyp erDiamond Lattice. :: :: ::: :: :: :: 14 3.3 From 8 to 4 Dimensions. : ::: :: :: :: :: ::: :: :: :: 19 4 Internal Symmetry Space. 21 5 Protons, Pions, and Physical Gravitons. 25 5.1 3-Quark Protons. : :: :: ::: :: :: :: :: ::: :: :: :: 29 5.2 Quark-AntiQuark Pions. : ::: :: :: :: :: ::: :: :: :: 32 5.3 Spin-2 Physical Gravitons. ::: :: :: :: :: ::: :: :: :: 35 1 1 Intro duction. The 1994 Georgia Tech Ph. D. thesis of Michael Gibbs under David Finkel- stein [11] constructed a discrete 4-dimensional spacetime Hyp erDiamond lat- tice, here denoted a 4HD lattice, in the course of building a physics mo del. The 1992 MIT Ph. D. thesis of Hrvo je Hrgovcic under Tommasso To oli [12] constructed a discrete 4-dimensional spacetime Minkowski lattice in the course of building a simulation of solutions to the Dirac equation and other equations. Hrgovcic's lattice is closely related to the 4HD lattice. Both the Gibbs mo del and the Hrgovcic simulation di er in signi cant ways from the D D E mo del constructed from 3 3 o ctonion matrices 4 5 6 in hep-ph/9501252 and quant-ph/9503009 . However, b oth theses have b een very helpful with resp ect to this pap er, which generalizes the 2-dimensional Feynman checkerb oard [8, 9] to a phys- ically realistic 4-dimensional spacetime. In this 4-dimensional 4HD lattice generalized Feynman Hyp erDiamond checkerb oard, the prop erties of the particles that move around on the checkerb oard are determined by the physically realistic D D E mo del constructed from 3 3 o ctonion matrices 4 5 6 in hep-ph/9501252 and quant-ph/9503009 . The 4-dimensional Hyp erDiamond 4HD checkerb oard is related to the 8-dimensional Hyp erDiamond 8HD = E in the same way 8 the 4-dimensional physical asso ciative spacetime is related to the 8-dimensional o ctonionic spacetime in the D D E mo del, 4 5 6 so the Hyp erDiamond 4HD Feynman checkerb oard is fundamentally the discrete version of the D D E mo del. 4 5 6 For more ab out o ctonions and lattices, see the works of Go e rey Dixon [3, 4, 5, 6]. 2 2 Feynman Checkerb oards. The 2-dimensional Feynman checkerb oard [8, 9] is a notably successful and useful representation of the Dirac equation in 2-dimensional spacetime. To build a Feynman 2-dimensional checkerb oard, start with a 2-dimensional Diamond checkerb oard with two future lightcone links and two past lightcone links at eachvertex. The future lightcone then lo oks like @I @ If the 2-dimensional Feynman checkerb oard is co ordinatized by the complex plane C : the real axis 1 is identi ed with the time axis t; the imaginary axis i is identi ed with the space axis x; and p p the two future lightcone links are (1= 2 )(1 + i) and (1= 2)(1 i). 2 2 In cylindrical co ordinates t; r with r = x , 2 2 2 2 the Euclidian metric is t + r = t + x and the Wick-Rotated Minkowski metric with sp eed of light c is 2 2 2 2 (ct) r =(ct) x . For the future lightcone links to lie on the 2-dimensional Minkowski lightcone, c =1. Either link is taken into the other link by complex multiplication by i. Now, consider a path in the Feynman checkerb oard. At a given vertex in the path, denote the future lightcone link in the same direction as the past path link by1,and the future lightcone link in the (only p ossible) changed direction by i. 3 1 i @I @ The Feynman checkerb oard rule is that if the future step at a vertex p oint of a given path is in a di erent direction from the immediately preceding step from the past, then the path at the p ointofchange gets a weightof im, where m is the mass (only massive particles can change directions),and is the length of a path segment. Here I have used the Gersch [10] convention of weighting each turn by im rather than the Feynman [8, 9] convention of weighting by+im, b ecause Gersch's convention gives a b etter nonrela- tivistic limit in the isomorphic 2-dimensional Ising mo del [10]. HOW SHOULD THIS BE GENERALIZED TO HIGHER DIMENSIONS? 22 0 The 2-dim future light-cone is the 0-sphere S = S = fi; 1g , with 1 representing a path step to the future in the same direction as the path step from the past, and i representing a path step to the future in a (only 1 in the 2-dimensional Feynman checkerb oard lattice) di erent direction from the path step from the past. The 2-dimensional Feynman checkerb oard lattice spacetime can b e represented by the complex numb ers C , with 1;i representing the two future lightcone directions and 1; i representing the two past lightcone directions. Consider a given path in the Feynman checkerb oard lattice 2-dimensional spacetime. 4 Atany given vertex on the path in the lattice 2-dimensional spacetime, the future lightcone direction representing the continuation of the path in the same direction can b e represented by1,and the future lightcone direction representing the (only 1 p ossible) change of direction can b e represented by i since either of the 2 future lightcone directions can b e taken into the other bymultiplication by i, + for a left turn and for a right turn. If the path do es change direction at the vertex, then the path at the p ointofchange gets a weightof im, where i is the complex imaginary, m is the mass (only massive particles can change directions), and is the timelike length of a path segment, where the 2-dimensional sp eed of light is taken to b e 1. Here I have used the Gersch [10] convention of weighting each turn by im rather than the Feynman [8, 9] convention of weighting by+im, b ecause Gersch's convention gives a b etter nonrela- tivistic limit in the isomorphic 2-dimensional Ising mo del [10]. For a given path, let C b e the total numb er of direction changes, and c b e the cth change of direction, and i b e the complex imaginary representing the cth change of direction. C can b e no greater than the timelikecheckerb oard distance D between the initial and nal p oints. The total weight for the given path is then Y Y C C im =(m) ( i)= (im) (1) 0cC 0cC The pro duct is a vector in the direction 1ori. 5 Let N (C ) b e the numb er of paths with C changes in direction. The propagator amplitude for the particle to go from the initial vertex to the nal vertex is the sum over all paths of the weights, that is the path integral sum over all weighted paths: X C N (C )(im) (2) 0C D The propagator phase is the angle b etween the amplitude vector in the complex plane and the complex real axis. Conventional attempts to generalize the Feynman checkerb oard from 2-dimensional spacetime to k -dimensional spacetime are based on the fact that the 2-dimensional future light-cone directions are 22 0 the 0-sphere S = S = fi; 1g. The k -dimensional continuous spacetime lightcone directions are k 2 the (k 2)-sphere S . 2 In 4-dimensional continuous spacetime, the lightcone directions are S . Instead of lo oking for a 4-dimensional lattice spacetime, Feynman 0 2 and other generalizers went from discrete S to continuous S for lightcone directions, and then tried to construct a weighting 2 using changes of directions as rotations in the continuous S , and never (as far as I know) got any generalization that worked. 6 THE D D E MODEL 4HD HYPERDIAMOND 4 5 6 GENERALIZATION HAS DISCRETE LIGHTCONE DIRECTIONS. If the 4-dimensional Feynman checkerb oard is co ordinatized by the quaternions Q: the real axis 1 is identi ed with the time axis t; the imaginary axes i; j; k are identi ed with the space axes x; y ; z ; and the four future lightcone links are (1=2)(1 + i + j + k ), (1=2)(1 + i j k ), (1=2)(1 i + j k ), and (1=2)(1 i j + k ). In cylindrical co ordinates t; r 2 2 2 2 with r = x + y + z , 2 2 2 2 2 2 the Euclidian metric is t + r = t + x + y + z and the Wick-Rotated Minkowski metric with sp eed of light c is 2 2 2 2 2 2 (ct) r =(ct) x y z .
Load more

Recommended publications
  • The Sphere Packing Problem in Dimension 8 Arxiv:1603.04246V2
    The sphere packing problem in dimension 8 Maryna S. Viazovska April 5, 2017 8 In this paper we prove that no packing of unit balls in Euclidean space R has density greater than that of the E8-lattice packing. Keywords: Sphere packing, Modular forms, Fourier analysis AMS subject classification: 52C17, 11F03, 11F30 1 Introduction The sphere packing constant measures which portion of d-dimensional Euclidean space d can be covered by non-overlapping unit balls. More precisely, let R be the Euclidean d vector space equipped with distance k · k and Lebesgue measure Vol(·). For x 2 R and d r 2 R>0 we denote by Bd(x; r) the open ball in R with center x and radius r. Let d X ⊂ R be a discrete set of points such that kx − yk ≥ 2 for any distinct x; y 2 X. Then the union [ P = Bd(x; 1) x2X d is a sphere packing. If X is a lattice in R then we say that P is a lattice sphere packing. The finite density of a packing P is defined as Vol(P\ Bd(0; r)) ∆P (r) := ; r > 0: Vol(Bd(0; r)) We define the density of a packing P as the limit superior ∆P := lim sup ∆P (r): r!1 arXiv:1603.04246v2 [math.NT] 4 Apr 2017 The number be want to know is the supremum over all possible packing densities ∆d := sup ∆P ; d P⊂R sphere packing 1 called the sphere packing constant. For which dimensions do we know the exact value of ∆d? Trivially, in dimension 1 we have ∆1 = 1.
    [Show full text]
  • Sphere Packing, Lattice Packing, and Related Problems
    Sphere packing, lattice packing, and related problems Abhinav Kumar Stony Brook April 25, 2018 Sphere packings Definition n A sphere packing in R is a collection of spheres/balls of equal size which do not overlap (except for touching). The density of a sphere packing is the volume fraction of space occupied by the balls. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 1, we can achieve density 1 by laying intervals end to end. Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem II In dimension 3, the best possible way is to stack layers of the solution in 2 dimensions. This is Kepler's conjecture, now a theorem of Hales and collaborators. mmm m mmm m There are infinitely (in fact, uncountably) many ways of doing this! These are the Barlow packings. Face centered cubic packing Image: Greg A L (Wikipedia), CC BY-SA 3.0 license But (until very recently!) no proofs. In very high dimensions (say ≥ 1000) densest packings are likely to be close to disordered.
    [Show full text]
  • Physical Interpretation of the 30 8-Simplexes in the E8 240-Polytope
    Physical Interpretation of the 30 8-simplexes in the E8 240-Polytope: Frank Dodd (Tony) Smith, Jr. 2017 - viXra 1702.0058 248-dim Lie Group E8 has 240 Root Vectors arranged on a 7-sphere S7 in 8-dim space. The 12 vertices of a cuboctahedron live on a 2-sphere S2 in 3-dim space. They are also the 4x3 = 12 outer vertices of 4 tetrahedra (3-simplexes) that share one inner vertex at the center of the cuboctahedron. This paper explores how the 240 vertices of the E8 Polytope in 8-dim space are related to the 30x8 = 240 outer vertices (red in figure below) of 30 8-simplexes whose 9th vertex is a shared inner vertex (yellow in figure below) at the center of the E8 Polytope. The 8-simplex has 9 vertices, 36 edges, 84 triangles, 126 tetrahedron cells, 126 4-simplex faces, 84 5-simplex faces, 36 6-simplex faces, 9 7-simplex faces, and 1 8-dim volume The real 4_21 Witting polytope of the E8 lattice in R8 has 240 vertices; 6,720 edges; 60,480 triangular faces; 241,920 tetrahedra; 483,840 4-simplexes; 483,840 5-simplexes 4_00; 138,240 + 69,120 6-simplexes 4_10 and 4_01; and 17,280 = 2,160x8 7-simplexes 4_20 and 2,160 7-cross-polytopes 4_11. The cuboctahedron corresponds by Jitterbug Transformation to the icosahedron. The 20 2-dim faces of an icosahedon in 3-dim space (image from spacesymmetrystructure.wordpress.com) are also the 20 outer faces of 20 not-exactly-regular-in-3-dim tetrahedra (3-simplexes) that share one inner vertex at the center of the icosahedron, but that correspondence does not extend to the case of 8-simplexes in an E8 polytope, whose faces are both 7-simplexes and 7-cross-polytopes, similar to the cubocahedron, but not its Jitterbug-transform icosahedron with only triangle = 2-simplex faces.
    [Show full text]
  • Arxiv:2010.10200V3 [Math.GT] 1 Mar 2021 We Deduce in Particular the Following
    HYPERBOLIC MANIFOLDS THAT FIBER ALGEBRAICALLY UP TO DIMENSION 8 GIOVANNI ITALIANO, BRUNO MARTELLI, AND MATTEO MIGLIORINI Abstract. We construct some cusped finite-volume hyperbolic n-manifolds Mn that fiber algebraically in all the dimensions 5 ≤ n ≤ 8. That is, there is a surjective homomorphism π1(Mn) ! Z with finitely generated kernel. The kernel is also finitely presented in the dimensions n = 7; 8, and this leads to the first examples of hyperbolic n-manifolds Mfn whose fundamental group is finitely presented but not of finite type. These n-manifolds Mfn have infinitely many cusps of maximal rank and hence infinite Betti number bn−1. They cover the finite-volume manifold Mn. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes P5;:::;P8, and then applying some arguments of Jankiewicz { Norin { Wise [15] and Bestvina { Brady [6]. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to Pn, and the algebra of integral octonions for the crucial dimensions n = 7; 8. Introduction We prove here the following theorem. Every hyperbolic manifold in this paper is tacitly assumed to be connected, complete, and orientable. Theorem 1. In every dimension 5 ≤ n ≤ 8 there are a finite volume hyperbolic 1 n-manifold Mn and a map f : Mn ! S such that f∗ : π1(Mn) ! Z is surjective n with finitely generated kernel. The cover Mfn = H =ker f∗ has infinitely many cusps of maximal rank. When n = 7; 8 the kernel is also finitely presented. arXiv:2010.10200v3 [math.GT] 1 Mar 2021 We deduce in particular the following.
    [Show full text]
  • Optimality and Uniqueness of the Leech Lattice Among Lattices
    ANNALS OF MATHEMATICS Optimality and uniqueness of the Leech lattice among lattices By Henry Cohn and Abhinav Kumar SECOND SERIES, VOL. 170, NO. 3 November, 2009 anmaah Annals of Mathematics, 170 (2009), 1003–1050 Optimality and uniqueness of the Leech lattice among lattices By HENRY COHN and ABHINAV KUMAR Dedicated to Oded Schramm (10 December 1961 – 1 September 2008) Abstract We prove that the Leech lattice is the unique densest lattice in R24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R24 can 30 exceed the Leech lattice’s density by a factor of more than 1 1:65 10 , and we 8 C give a new proof that E8 is the unique densest lattice in R . 1. Introduction It is a long-standing open problem in geometry and number theory to find the densest lattice in Rn. Recall that a lattice ƒ Rn is a discrete subgroup of rank n; a minimal vector in ƒ is a nonzero vector of minimal length. Let ƒ vol.Rn=ƒ/ j j D denote the covolume of ƒ, i.e., the volume of a fundamental parallelotope or the absolute value of the determinant of a basis of ƒ. If r is the minimal vector length of ƒ, then spheres of radius r=2 centered at the points of ƒ do not overlap except tangentially. This construction yields a sphere packing of density n=2 r Án 1 ; .n=2/Š 2 ƒ j j since the volume of a unit ball in Rn is n=2=.n=2/Š, where for odd n we define .n=2/Š .n=2 1/.
    [Show full text]
  • Octonions and the Leech Lattice
    Octonions and the Leech lattice Robert A. Wilson School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS 18th December 2008 Abstract We give a new, elementary, description of the Leech lattice in terms of octonions, thereby providing the first real explanation of the fact that the number of minimal vectors, 196560, can be expressed in the form 3 × 240 × (1 + 16 + 16 × 16). We also give an easy proof that it is an even self-dual lattice. 1 Introduction The Leech lattice occupies a special place in mathematics. It is the unique 24- dimensional even self-dual lattice with no vectors of norm 2, and defines the unique densest lattice packing of spheres in 24 dimensions. Its automorphism group is very large, and is the double cover of Conway’s group Co1 [2], one of the most important of the 26 sporadic simple groups. This group plays a crucial role in the construction of the Monster [13, 4], which is the largest of the sporadic simple groups, and has connections with modular forms (so- called ‘Monstrous Moonshine’) and many other areas, including theoretical physics. The book by Conway and Sloane [5] is a good introduction to this lattice and its many applications. It is not surprising therefore that there is a huge literature on the Leech lattice, not just within mathematics but in the physics literature too. Many attempts have been made in particular to find simplified constructions (see for example the 23 constructions described in [3] and the four constructions 1 described in [15]).
    [Show full text]
  • Symmetries and Exotic Smooth Structures on a K3 Surface WM Chen University of Massachusetts - Amherst, [email protected]
    University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Mathematics and Statistics Publication Series 2008 Symmetries and exotic smooth structures on a K3 surface WM Chen University of Massachusetts - Amherst, [email protected] S Kwasik Follow this and additional works at: https://scholarworks.umass.edu/math_faculty_pubs Part of the Physical Sciences and Mathematics Commons Recommended Citation Chen, WM and Kwasik, S, "Symmetries and exotic smooth structures on a K3 surface" (2008). JOURNAL OF TOPOLOGY. 268. Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/268 This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. SYMMETRIES AND EXOTIC SMOOTH STRUCTURES ON A K3 SURFACE WEIMIN CHEN AND SLAWOMIR KWASIK Abstract. Smooth and symplectic symmetries of an infinite family of distinct ex- otic K3 surfaces are studied, and comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order ≥ 7 is proved and nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 which has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of a prime order ≥ 5 is explicitly determined, provided that the action is homologically nontrivial.
    [Show full text]
  • Root System - Wikipedia
    Root system - Wikipedia https://en.wikipedia.org/wiki/Root_system Root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representations theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory.[1] Contents Definitions and examples Definition Weyl group Rank two examples Root systems arising from semisimple Lie algebras History Elementary consequences of the root system axioms Positive roots and simple roots Dual root system and coroots Classification of root systems by Dynkin diagrams Constructing the Dynkin diagram Classifying root systems Weyl chambers and the Weyl group Root systems and Lie theory Properties of the irreducible root systems Explicit construction of the irreducible root systems An Bn Cn Dn E6, E 7, E 8 F4 G2 The root poset See also Notes References 1 of 14 2/22/2018, 8:26 PM Root system - Wikipedia https://en.wikipedia.org/wiki/Root_system Further reading External links Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots .
    [Show full text]
  • Notes on Fourier Series and Modular Forms
    Notes on Fourier Series and Modular Forms Rich Schwartz August 29, 2017 This is a meandering set of notes about modular forms, theta series, elliptic curves, and Fourier expansions. I learned most of the statements of the results from wikipedia, but many of the proofs I worked out myself. I learned the proofs connected to theta series from some online notes of Ben Brubaker. Nothing in these notes is actually original, except maybe the mistakes. My purpose was just to understand this stuff for myself and say it in terms that I like. In the first chapter we prove some preliminary results, most having to do with single variable analysis. These assorted results will be used in the remaining chapters. In the second chapter, we deal with Fourier series and lattices. One main goal is to establish that the theta series associated to an even unimodular lattice is a modular form. Our results about Fourier series are proved for functions that have somewhat artificial restrictions placed on them. We do this so as to avoid any nontrivial analysis. Fortunately, all the functions we encounter satisfy the restrictions. In the third chapter, we discuss a bunch of beautiful material from the classical theory of modular forms: Eisenstein series, elliptic curves, Fourier expansions, the Discriminant form, the j function, Riemann-Roch. In the fourth chapter, we will define the E8 lattice and the Leech lattice, and compute their theta series. 1 1 Preliminaries 1.1 The Discrete Fourier Transform Let ω = exp(2πi/n) (1) be the usual nth root of unity.
    [Show full text]
  • Which Integer Is the Most Mysterious?
    Bridges 2018 Conference Proceedings Which Integer Is the Most Mysterious? Osmo Pekonen University of Jyväskylä, Finland; osmo.pekonen@jyu.fi Abstract The author argues that the number 248, together with some of its multiples, is the Most Mysterious (large) Integer that is encountered in Mathematics, Physics, and Mysticism. A Preliminary Discussion – and a Word of Warning "All things are number," Pythagoras may have said. The Pythagoreans assigned special meanings to small integers: For example, 1 was the number of reason and unity; 2 represented the female principle; 3, the male principle; 4, the four elements; 5 = 2 + 3, marriage; etc. Other philosophical traditions and religions have their sacred numbers, as well. Surely, the number 3 is important in Christian theology as it represents the Holy Trinity, whereas 5 is sacred in Islam as it corresponds to its Five Pillars. So there would be many candidates for the Most Mysterious Number in the Universe that we are looking for in this paper. Let us exclude from discussion, however, the smallest integers because far too many different meanings can be assigned to them. Among the bigger ones, 666, or the Number of the Beast, is famous enough but, in spite of two millennia of speculation, no conclusive mathematical interpretation has been found for it. So let us also exclude numbers whose only special meaning is religious or cultural. Some physicists are obsessed [5] with the number 137 because the (dimensionless) fine-structure con- stant is approximately 1/137. However, this is but an approximation, and no justification for a precise integer value has ever been found.
    [Show full text]
  • Sphere Packings, Density Fluctuations, Coverings and Quantizers
    2254-3 Workshop on Sphere Packing and Amorphous Materials 25 - 29 July 2011 Sphere Packings, Density Fluctuations, Coverings and Quantizers Salvatore TORQUATO Princeton University, Dept.of Chemistry & Physics Washington Road Princeton, NJ 08540 U.S.A. Sphere Packings, Density Fluctuations, Coverings, and Quantizers Salvatore Torquato Department of Chemistry, Department of Physics, Princeton Center for Theoretical Science, Program in Applied & Computational Mathematics, & Princeton Institute for the Science and Technology of Materials Princeton University http://cherrypit.princeton.edu .–p.1/35 Four Different Problems: Interplay Between Geometry and Physics 1. Sphere Packing Problem Applications: low-temperature states of matter (liquids, crystals and glasses), granular media, biological media, communications, string theory, etc. 2. Number Variance Problem Applications: equilibrium and nonequilibrium systems; critical-point phenomena, number theory, hyperuniformity, etc. 3. Covering Problem Applications: wireless communication network layouts, search of high-dimensional data parameter spaces, stereotactic radiation therapy, etc. 4. Quantizer Problem Applications: computer science (e.g., data compression), digital communications, coding and cryptography, optimal meshing of space for numerical applications, etc. .–p.2/35 Interaction Energies of Many-Particle Systems N Total potential energy ΦN (r ) of N identical particles with positions N r ≡ r1, r2,...,rN in some large volume in d-dimensional Euclidean space d R can be resolved into one-body, two-body, ..., N -body contributions: N N N N N ΦN (r )= u1(ri)+ u2(ri, rj)+ u3(ri, rj, rk)+···+uN (r ), i=1 i<j i<j<k To make the statistical-mechanical problem more tractable, this exact many-body potential is often replaced by a mathematically simpler form, e.g., pairwise interactions: N N ΦN (r )= u2(ri, rj).
    [Show full text]
  • An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices by Robert L
    Advanced Lectures in Mathematics Volume XV An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices by Robert L. Griess, Jr. International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Advanced Lectures in Mathematics, Volume XV An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices by Robert L. Griess, Jr. 2010 Mathematics Subject Classification. 11H56, 20C10, 20C34, 20D08. Copyright © 2011 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. Excluded from these provisions is material in articles to which the author holds the copyright. (If the author holds copyright, notice of this will be given with article.) In such cases, requests for permission to use or reprint should be addressed directly to the author. ISBN: 978-1-57146-206-0 Printed in the United States of America. 15 14 13 12 2 3 4 5 6 7 8 9 ADVANCED LECTURES
    [Show full text]