6. Systems of Identical Particles
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Finite Generation of Symmetric Ideals
FINITE GENERATION OF SYMMETRIC IDEALS MATTHIAS ASCHENBRENNER AND CHRISTOPHER J. HILLAR In memoriam Karin Gatermann (1965–2005 ). Abstract. Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R[SX ]. We prove that all ideals of R invariant under the action of SX are finitely generated as R[SX ]-modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gr¨obner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it. 1. Introduction A pervasive theme in invariant theory is that of finite generation. A fundamen- tal example is a theorem of Hilbert stating that the invariant subrings of finite- dimensional polynomial algebras over finite groups are finitely generated [6, Corol- lary 1.5]. In this article, we study invariant ideals of infinite-dimensional polynomial rings. Of course, when the number of indeterminates is finite, Hilbert’s basis theo- rem tells us that any ideal (invariant or not) is finitely generated. Our setup is as follows. Let X be an infinite collection of indeterminates, and let SX be the group of permutations of X. -
Identical Particles
8.06 Spring 2016 Lecture Notes 4. Identical particles Aram Harrow Last updated: May 19, 2016 Contents 1 Fermions and Bosons 1 1.1 Introduction and two-particle systems .......................... 1 1.2 N particles ......................................... 3 1.3 Non-interacting particles .................................. 5 1.4 Non-zero temperature ................................... 7 1.5 Composite particles .................................... 7 1.6 Emergence of distinguishability .............................. 9 2 Degenerate Fermi gas 10 2.1 Electrons in a box ..................................... 10 2.2 White dwarves ....................................... 12 2.3 Electrons in a periodic potential ............................. 16 3 Charged particles in a magnetic field 21 3.1 The Pauli Hamiltonian ................................... 21 3.2 Landau levels ........................................ 23 3.3 The de Haas-van Alphen effect .............................. 24 3.4 Integer Quantum Hall Effect ............................... 27 3.5 Aharonov-Bohm Effect ................................... 33 1 Fermions and Bosons 1.1 Introduction and two-particle systems Previously we have discussed multiple-particle systems using the tensor-product formalism (cf. Section 1.2 of Chapter 3 of these notes). But this applies only to distinguishable particles. In reality, all known particles are indistinguishable. In the coming lectures, we will explore the mathematical and physical consequences of this. First, consider classical many-particle systems. If a single particle has state described by position and momentum (~r; p~), then the state of N distinguishable particles can be written as (~r1; p~1; ~r2; p~2;:::; ~rN ; p~N ). The notation (·; ·;:::; ·) denotes an ordered list, in which different posi tions have different meanings; e.g. in general (~r1; p~1; ~r2; p~2)6 = (~r2; p~2; ~r1; p~1). 1 To describe indistinguishable particles, we can use set notation. -
Magnetic Materials I
5 Magnetic materials I Magnetic materials I ● Diamagnetism ● Paramagnetism Diamagnetism - susceptibility All materials can be classified in terms of their magnetic behavior falling into one of several categories depending on their bulk magnetic susceptibility χ. without spin M⃗ 1 χ= in general the susceptibility is a position dependent tensor M⃗ (⃗r )= ⃗r ×J⃗ (⃗r ) H⃗ 2 ] In some materials the magnetization is m / not a linear function of field strength. In A [ such cases the differential susceptibility M is introduced: d M⃗ χ = d d H⃗ We usually talk about isothermal χ susceptibility: ∂ M⃗ χ = T ( ∂ ⃗ ) H T Theoreticians define magnetization as: ∂ F⃗ H[A/m] M=− F=U−TS - Helmholtz free energy [4] ( ∂ H⃗ ) T N dU =T dS− p dV +∑ μi dni i=1 N N dF =T dS − p dV +∑ μi dni−T dS−S dT =−S dT − p dV +∑ μi dni i=1 i=1 Diamagnetism - susceptibility It is customary to define susceptibility in relation to volume, mass or mole: M⃗ (M⃗ /ρ) m3 (M⃗ /mol) m3 χ= [dimensionless] , χρ= , χ = H⃗ H⃗ [ kg ] mol H⃗ [ mol ] 1emu=1×10−3 A⋅m2 The general classification of materials according to their magnetic properties μ<1 <0 diamagnetic* χ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ B=μ H =μr μ0 H B=μo (H +M ) μ>1 >0 paramagnetic** ⃗ ⃗ ⃗ χ μr μ0 H =μo( H + M ) → μ r=1+χ μ≫1 χ≫1 ferromagnetic*** *dia /daɪə mæ ɡˈ n ɛ t ɪ k/ -Greek: “from, through, across” - repelled by magnets. We have from L2: 1 2 the force is directed antiparallel to the gradient of B2 F = V ∇ B i.e. -
Math 263A Notes: Algebraic Combinatorics and Symmetric Functions
MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS AARON LANDESMAN CONTENTS 1. Introduction 4 2. 10/26/16 5 2.1. Logistics 5 2.2. Overview 5 2.3. Down to Math 5 2.4. Partitions 6 2.5. Partial Orders 7 2.6. Monomial Symmetric Functions 7 2.7. Elementary symmetric functions 8 2.8. Course Outline 8 3. 9/28/16 9 3.1. Elementary symmetric functions eλ 9 3.2. Homogeneous symmetric functions, hλ 10 3.3. Power sums pλ 12 4. 9/30/16 14 5. 10/3/16 20 5.1. Expected Number of Fixed Points 20 5.2. Random Matrix Groups 22 5.3. Schur Functions 23 6. 10/5/16 24 6.1. Review 24 6.2. Schur Basis 24 6.3. Hall Inner product 27 7. 10/7/16 29 7.1. Basic properties of the Cauchy product 29 7.2. Discussion of the Cauchy product and related formulas 30 8. 10/10/16 32 8.1. Finishing up last class 32 8.2. Skew-Schur Functions 33 8.3. Jacobi-Trudi 36 9. 10/12/16 37 1 2 AARON LANDESMAN 9.1. Eigenvalues of unitary matrices 37 9.2. Application 39 9.3. Strong Szego limit theorem 40 10. 10/14/16 41 10.1. Background on Tableau 43 10.2. KOSKA Numbers 44 11. 10/17/16 45 11.1. Relations of skew-Schur functions to other fields 45 11.2. Characters of the symmetric group 46 12. 10/19/16 49 13. 10/21/16 55 13.1. -
5.1 Two-Particle Systems
5.1 Two-Particle Systems We encountered a two-particle system in dealing with the addition of angular momentum. Let's treat such systems in a more formal way. The w.f. for a two-particle system must depend on the spatial coordinates of both particles as @Ψ well as t: Ψ(r1; r2; t), satisfying i~ @t = HΨ, ~2 2 ~2 2 where H = + V (r1; r2; t), −2m1r1 − 2m2r2 and d3r d3r Ψ(r ; r ; t) 2 = 1. 1 2 j 1 2 j R Iff V is independent of time, then we can separate the time and spatial variables, obtaining Ψ(r1; r2; t) = (r1; r2) exp( iEt=~), − where E is the total energy of the system. Let us now make a very fundamental assumption: that each particle occupies a one-particle e.s. [Note that this is often a poor approximation for the true many-body w.f.] The joint e.f. can then be written as the product of two one-particle e.f.'s: (r1; r2) = a(r1) b(r2). Suppose furthermore that the two particles are indistinguishable. Then, the above w.f. is not really adequate since you can't actually tell whether it's particle 1 in state a or particle 2. This indeterminacy is correctly reflected if we replace the above w.f. by (r ; r ) = a(r ) (r ) (r ) a(r ). 1 2 1 b 2 b 1 2 The `plus-or-minus' sign reflects that there are two distinct ways to accomplish this. Thus we are naturally led to consider two kinds of identical particles, which we have come to call `bosons' (+) and `fermions' ( ). -
Categories with Negation 3
CATEGORIES WITH NEGATION JAIUNG JUN 1 AND LOUIS ROWEN 2 In honor of our friend and colleague, S.K. Jain. Abstract. We continue the theory of T -systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). Prime ground systems are introduced as a way of developing geometry. One basic result is that the polynomial system over a prime system is prime. For module systems, special attention also is paid to tensor products and Hom. Abelian categories are replaced by “semi-abelian” categories (where Hom(A, B) is not a group) with a negation morphism. The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Contents 1. Introduction 2 1.1. Objectives 3 1.2. Main results 3 2. Background 4 2.1. Basic structures 4 2.2. Symmetrization and the twist action 7 2.3. Ground systems and module systems 8 2.4. Morphisms of systems 10 2.5. Function triples 11 2.6. The role of universal algebra 13 3. The structure theory of ground triples via congruences 14 3.1. The role of congruences 14 3.2. Prime T -systems and prime T -congruences 15 3.3. Annihilators, maximal T -congruences, and simple T -systems 17 4. The geometry of prime systems 17 4.1. Primeness of the polynomial system A[λ] 17 4.2. -
Identical Particles in Classical Physics One Can Distinguish Between
Identical particles In classical physics one can distinguish between identical particles in such a way as to leave the dynamics unaltered. Therefore, the exchange of particles of identical particles leads to di®erent con¯gurations. In quantum mechanics, i.e., in nature at the microscopic levels identical particles are indistinguishable. A proton that arrives from a supernova explosion is the same as the proton in your glass of water.1 Messiah and Greenberg2 state the principle of indistinguishability as \states that di®er only by a permutation of identical particles cannot be distinguished by any obser- vation whatsoever." If we denote the operator which interchanges particles i and j by the permutation operator Pij we have ^ y ^ hªjOjªi = hªjPij OPijjÃi ^ which implies that Pij commutes with an arbitrary observable O. In particular, it commutes with the Hamiltonian. All operators are permutation invariant. Law of nature: The wave function of a collection of identical half-odd-integer spin particles is completely antisymmetric while that of integer spin particles or bosons is completely symmetric3. De¯ning the permutation operator4 by Pij ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) = ª(1; 2; ¢ ¢ ¢ ; j; ¢ ¢ ¢ ; i; ¢ ¢ ¢ N) we have Pij ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) = § ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) where the upper sign refers to bosons and the lower to fermions. Emphasize that the symmetry requirements only apply to identical particles. For the hydrogen molecule if I and II represent the coordinates and spins of the two protons and 1 and 2 of the two electrons we have ª(I;II; 1; 2) = ¡ ª(II;I; 1; 2) = ¡ ª(I;II; 2; 1) = ª(II;I; 2; 1) 1On the other hand, the elements on earth came from some star/supernova in the distant past. -
The Conventionality of Parastatistics
The Conventionality of Parastatistics David John Baker Hans Halvorson Noel Swanson∗ March 6, 2014 Abstract Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. This has been seen as a deep mystery: paraparticles make perfect physical sense, so why don't we see them in nature? We consider one potential answer: every paraparticle theory is physically equivalent to some theory of bosons or fermions, making the absence of paraparticles in our theories a matter of convention rather than a mysterious empirical discovery. We argue that this equivalence thesis holds in all physically admissible quantum field theories falling under the domain of the rigorous Doplicher-Haag-Roberts approach to superselection rules. Inadmissible parastatistical theories are ruled out by a locality- inspired principle we call Charge Recombination. Contents 1 Introduction 2 2 Paraparticles in Quantum Theory 6 ∗This work is fully collaborative. Authors are listed in alphabetical order. 1 3 Theoretical Equivalence 11 3.1 Field systems in AQFT . 13 3.2 Equivalence of field systems . 17 4 A Brief History of the Equivalence Thesis 20 4.1 The Green Decomposition . 20 4.2 Klein Transformations . 21 4.3 The Argument of Dr¨uhl,Haag, and Roberts . 24 4.4 The Doplicher-Roberts Reconstruction Theorem . 26 5 Sharpening the Thesis 29 6 Discussion 36 6.1 Interpretations of QM . 44 6.2 Structuralism and Haecceities . 46 6.3 Paraquark Theories . 48 1 Introduction Our most fundamental theories of matter provide a highly accurate description of subatomic particles and their behavior. -
Talk M.Alouani
From atoms to solids to nanostructures Mebarek Alouani IPCMS, 23 rue du Loess, UMR 7504, Strasbourg, France European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 1/107 Course content I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 2/107 Course content I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 3/107 Magnetic moments In classical electrodynamics, the vector potential, in the magnetic dipole approximation, is given by: μ μ × rˆ Ar()= 0 4π r2 where the magnetic moment μ is defined as: 1 μ =×Id('rl ) 2 ∫ It is shown that the magnetic moment is proportional to the angular momentum μ = γ L (γ the gyromagnetic factor) The projection along the quantification axis z gives (/2Bohrmaμ = eh m gneton) μzB= −gmμ Be European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 4/107 Magnetization and Field The magnetization M is the magnetic moment per unit volume In free space B = μ0 H because there is no net magnetization. -
BRST Inner Product Spaces and the Gribov Obstruction
PITHA – 98/27 BRST Inner Product Spaces and the Gribov Obstruction Norbert DUCHTING¨ 1∗, Sergei V. SHABANOV2†‡, Thomas STROBL1§ 1 Institut f¨ur Theoretische Physik, RWTH Aachen D-52056 Aachen, Germany 2 Department of Mathematics, University of Florida, Gainesville, FL 32611–8105, USA Abstract A global extension of the Batalin–Marnelius proposal for a BRST inner product to gauge theories with topologically nontrivial gauge orbits is discussed. It is shown that their (appropriately adapted) method is applicable to a large class of mechanical models with a semisimple gauge group in the adjoint and fundamental representa- tion. This includes cases where the Faddeev–Popov method fails. arXiv:hep-th/9808151v1 25 Aug 1998 Simple models are found also, however, which do not allow for a well– defined global extension of the Batalin–Marnelius inner product due to a Gribov obstruction. Reasons for the partial success and failure are worked out and possible ways to circumvent the problem are briefly discussed. ∗email: [email protected] †email: [email protected]fl.edu ‡On leave from the Laboratory of Theoretical Physics, JINR, Dubna, Russia §email: [email protected] 1 Introduction and Overview Canonical quantization of gauge theories leads, in general, to an ill–defined scalar product for physical states. In the Dirac approach [1] physical states are selected by the quantum constraints. Assuming that the theory under consideration is of the Yang–Mills type with its gauge group acting on some configuration space, the total Hilbert space may be realized by square in- tegrable functions on the configuration space and the quantum constraints imply gauge invariance of these wave functions. -
Spontaneous Symmetry Breaking and Mass Generation As Built-In Phenomena in Logarithmic Nonlinear Quantum Theory
Vol. 42 (2011) ACTA PHYSICA POLONICA B No 2 SPONTANEOUS SYMMETRY BREAKING AND MASS GENERATION AS BUILT-IN PHENOMENA IN LOGARITHMIC NONLINEAR QUANTUM THEORY Konstantin G. Zloshchastiev Department of Physics and Center for Theoretical Physics University of the Witwatersrand Johannesburg, 2050, South Africa (Received September 29, 2010; revised version received November 3, 2010; final version received December 7, 2010) Our primary task is to demonstrate that the logarithmic nonlinearity in the quantum wave equation can cause the spontaneous symmetry break- ing and mass generation phenomena on its own, at least in principle. To achieve this goal, we view the physical vacuum as a kind of the funda- mental Bose–Einstein condensate embedded into the fictitious Euclidean space. The relation of such description to that of the physical (relativis- tic) observer is established via the fluid/gravity correspondence map, the related issues, such as the induced gravity and scalar field, relativistic pos- tulates, Mach’s principle and cosmology, are discussed. For estimate the values of the generated masses of the otherwise massless particles such as the photon, we propose few simple models which take into account small vacuum fluctuations. It turns out that the photon’s mass can be naturally expressed in terms of the elementary electrical charge and the extensive length parameter of the nonlinearity. Finally, we outline the topological properties of the logarithmic theory and corresponding solitonic solutions. DOI:10.5506/APhysPolB.42.261 PACS numbers: 11.15.Ex, 11.30.Qc, 04.60.Bc, 03.65.Pm 1. Introduction Current observational data in astrophysics are probing a regime of de- partures from classical relativity with sensitivities that are relevant for the study of the quantum-gravity problem [1,2]. -
Extending Robustness & Randomization from Consensus To
Extending robustness & randomization from Consensus to Symmetrization Algorithms Luca Mazzarella∗ Francesco Ticozzi† Alain Sarlette‡ June 16, 2015 Abstract This work interprets and generalizes consensus-type algorithms as switching dynamics lead- ing to symmetrization of some vector variables with respect to the actions of a finite group. We show how the symmetrization framework we develop covers applications as diverse as consensus on probability distributions (either classical or quantum), uniform random state generation, and open-loop disturbance rejection by quantum dynamical decoupling. Robust convergence results are explicitly provided in a group-theoretic formulation, both for deterministic and for randomized dynamics. This indicates a way to directly extend the robustness and randomization properties of consensus-type algorithms to more fields of application. 1 Introduction The investigation of randomized and robust algorithmic procedures has been a prominent development of applied mathematics and dynamical systems theory in the last decades [?, ?]. Among these, one of the most studied class of algorithms in the automatic control literature are those designed to reach average consensus [?, ?, ?, ?]: the most basic form entails a linear algorithm based on local iterations that reach agreement on a mean value among network nodes. Linear consensus, despite its simplicity, is used as a subtask for several distributed tasks like distributed estimation [?, ?], motion coordination [?], clock synchronization [?], optimization [?], and of course load balancing; an example is presented later in the paper, while more applications are covered in [?]. A universally appreciated feature of linear consensus is its robustness to parameter values and perfect behavior under time-varying network structure. In the present paper, inspired by linear consensus, we present an abstract framework that pro- duces linear procedures to solve a variety of (a priori apparently unrelated) symmetrization problems with respect to the action of finite groups.