DOCSLIB.ORG

A List of All the Errata Known As of 19 December 2013

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A List of All the Errata Known As of 19 December 2013 14 Linear Algebra that is nonnegative when the matrices are the same N L N L † ∗ 2 (A, A)=TrA A = AijAij = |Aij| ≥ 0 (1.87) !i=1 !j=1 !i=1 !j=1 which is zero only when A = 0. So this inner product is positive definite. A vector space with a positive-definite inner product (1.73–1.76) is called an inner-product space,ametric space, or a pre-Hilbert space. A sequence of vectors fn is a Cauchy sequence if for every ϵ>0 there is an integer N(ϵ) such that ∥fn − fm∥ <ϵwhenever both n and m exceed N(ϵ). A sequence of vectors fn converges to a vector f if for every ϵ>0 there is an integer N(ϵ) such that ∥f −fn∥ <ϵwhenever n exceeds N(ϵ). An inner-product space with a norm defined as in (1.80) is complete if each of its Cauchy sequences converges to a vector in that space. A Hilbert space is a complete inner-product space. Every finite-dimensional inner-product space is complete and so is a Hilbert space. But the term Hilbert space more often is used to describe infinite-dimensional complete inner-product spaces, such as the space of all square-integrable functions (David Hilbert, 1862–1943). Example 1.17 (The Hilbert Space of Square-Integrable Functions) For the vector space of functions (1.55), a natural inner product is b (f,g)= dx f ∗(x)g(x). (1.88) "a The squared norm ∥ f ∥ of a function f(x)is b ∥ f ∥2= dx |f(x)|2. (1.89) "a A function is square integrable if its norm is finite. The space of all square-integrable functions is an inner-product space; it also is complete and so is a Hilbert space. Example 1.18 (Minkowski Inner Product) The Minkowski or Lorentz in- ner product (p, x) of two 4-vectors p =(E/c, p1,p2,p3) and x =(ct, x1,x2,x3) is p · x − Et . It is indefinite, nondegenerate (1.79), and invariant under Lorentz transformations, and often is written as p · x or as px.Ifp is the 4-momentum of a freely moving physical particle of mass m,then p · p = p · p − E2/c2 = − c2m2 ≤ 0. (1.90) The Minkowski inner product satisfies the rules (1.73, 1.74, and 1.79), but 22 Linear Algebra and the formula (1.118) for the kth orthonormal linear combination of the vectors |Vℓ⟩ is |uk⟩ |Uk⟩ = . (1.139) ⟨u |u ⟩ ! k k The vectors |Uk⟩ are not unique; they vary with the order of the |Vk⟩. Vectors and linear operators are abstract. The numbers we compute with are inner products like ⟨g|f⟩ and ⟨g|A|f⟩. In terms of N orthonormal basis ∗ vectors |n⟩ with fn = ⟨n|f⟩ and gn = ⟨g|n⟩, we can use the expansion (1.131) to write these inner products as N N ⟨g|f⟩ = ⟨g|I|f⟩ = ⟨g|n⟩⟨n|f⟩ = g∗f " " n n n=1 n=1 N N (1.140) ⟨g|A|f⟩ = ⟨g|IAI|f⟩ = ⟨g|n⟩⟨n|A|ℓ⟩⟨ℓ|f⟩ = g∗ A f " " n nℓ ℓ n,ℓ=1 n,ℓ=1 in which Anℓ = ⟨n|A|ℓ⟩. We often gather the inner products fℓ = ⟨ℓ|f⟩ into a column vector f with components fℓ = ⟨ℓ|f⟩ ⟨1|f⟩ f1 ⎛ ⎞ ⎛ ⎞ ⟨2|f⟩ f2 f = ⎜ . ⎟ = ⎜ . ⎟ (1.141) ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⟨N|f⟩ ⎠ ⎝ f3 ⎠ and the ⟨n|A|ℓ⟩ into a matrix A with matrix elements Anℓ = ⟨n|A|ℓ⟩.Ifwe also line up the inner products ⟨g|n⟩ = ⟨n|g⟩∗ in a row vector that is the transpose of the complex conjugate of the column vector g † ∗ ∗ ∗ ∗ ∗ ∗ g =(⟨1|g⟩ , ⟨2|g⟩ ,...,⟨N|g⟩ )=(g1,g2,...,gN ) (1.142) then we can write inner products in matrix notation as ⟨g|f⟩ = g†f and as ⟨g|A|f⟩ = g†Af. If we switch to a different basis, say from |n⟩’s to |αn⟩’s, then the com- ′ ponents of the column vectors change from fn = ⟨n|f⟩ to fn = ⟨αn|f⟩, and similarly those of the row vectors g† and of the matrix A change, but the bras, the kets, the linear operators, and the inner products ⟨g|f⟩ and ⟨g|A|f⟩ 28 Linear Algebra Remarkably, this translation operator is an exponential of the momentum operator U(a)=exp(−ipa/!)inwhich! = h/2π =1.054 × 10−34 Js is Planck’s constant divided by 2π. In two-dimensions, with basis states |x, y⟩ that are orthonormal in Dirac’s sense, ⟨x, y|x′,y′⟩ = δ(x − x′)δ(y − y′), the unitary operator U(θ)= |x cos θ − y sin θ,xsin θ + y cos θ⟩⟨x, y| dxdy (1.171) ! rotates a system in space by the angle θ. This rotation operator is the exponential U(θ)=exp(−i θ Lz/!)inwhichthez component of the angular momentum is Lz = xpy − ypx. We may carry most of our intuition about matrices over to these unitary transformations that change from one infinite basis to another. But we must use common sense and keep in mind that infinite sums and integrals do not always converge. 1.18 Antiunitary, Antilinear Operators Certain maps on states |ψ⟩→|ψ′⟩, such as those involving time reversal, are implemented by operators K that are antilinear ∗ ∗ ∗ ∗ K (zψ + wφ)=K (z|ψ⟩ + w|φ⟩)=z K|ψ⟩ + w K|φ⟩ = z Kψ + w Kφ (1.172) and antiunitary ∗ ∗ (Kφ,Kψ)=⟨Kφ|Kψ⟩ =(φ, ψ) = ⟨φ|ψ⟩ = ⟨ψ|φ⟩ =(ψ, φ) . (1.173) In Dirac notation, these rules are K(z|ψ⟩)=z∗⟨ψ| and K(w⟨φ|)=w∗|φ⟩. 1.19 Symmetry in Quantum Mechanics In quantum mechanics, a symmetry is a one-to-one map of states |ψ⟩↔|ψ′⟩ and |φ⟩↔|φ′⟩ that preserves probabilities ′ ′ |⟨φ |ψ ⟩|2 = |⟨φ|ψ⟩|2. (1.174) Eugene Wigner (1902–1995) showed that every symmetry in quantum me- chanics can be represented either by an operator U that is linear and uni- tary or by an operator K that is antilinear and antiunitary. The antilinear, antiunitary case seems to occur only when the symmetry involves time re- versal. Most symmetries are represented by operators that are linear and unitary. Unitary operators are of great importance in quantum mechanics. 34 Linear Algebra shows that the operations (1.189) on columns that don’t change the value of the determinant can be written as matrix multiplication from the rightbya matrix that has unity on its main diagonal and zeros below. Now consider the matrix product A 0 IB A AB = (1.202) !−IB"!0 I " !−I 0 " in which A and B are N × N matrices, I is the N × N identity matrix, and 0istheN × N matrix of all zeros. The second matrix on the left-hand side has unity on its main diagonal and zeros below, and so it does not change the value of the determinant of the matrix to its left, which then must equal that of the matrix on the right-hand side: A 0 A AB det =det . (1.203) !−IB" !−I 0 " By using Laplace’s expansion (1.183) along the first column to evaluate the determinant on the left-hand side and his expansion along the last row to compute the determinant on the right-hand side, one finds that the de- terminant of the product of two matrices is the product of the determinants det A det B =detAB. (1.204) Example 1.27 (Two 2 × 2 Matrices) When the matrices A and B are both 2 × 2, the two sides of (1.203) are a11 a12 00 A 0 ⎛a21 a22 00⎞ det =det (1.205) !−IB" −10b b ⎜ 11 12⎟ ⎜ 0 −1 b b ⎟ ⎝ 21 22⎠ = a11a22 det B − a21a12 det B =detA det B and a11 a12 (ab)11 (ab)12 A AB ⎛a21 a22 (ab)21 (ab)22⎞ det =det (1.206) !−I 0 " −10 0 0 ⎜ ⎟ ⎜ 0 −10 0⎟ ⎝ ⎠ =(−1)C42 =(−1)(−1) det AB =detAB and so they give the product rule det A det B =detAB. 44 Linear Algebra remains true when the matrix A replaces the variable λ N k P (A, A)= pk A =0. (1.258) !k=0 To see why, we use the formula (1.196) N δkℓ det A = AikCiℓ (1.259) !i=1 to write the determinant |A − λI| = P (λ,A) as the product of the matrix A − λI and the transpose of its matrix of cofactors (A − λI) C(λ,A)T = |A − λI| I = P (λ,A) I. (1.260) The transpose of the matrix of cofactors of the matrix A−λI is a polynomial in λ with matrix coefficients T N−1 C(λ,A) = C0 + C1λ + ···+ CN−1λ . (1.261) The left-hand side of equation (1.260) is then T 2 (A − λI)C(λ,A) = AC0 +(AC1 − C0)λ +(AC2 − C1)λ + ... N−1 N +(ACN−1 − CN−2)λ − CN−1λ . (1.262) Equating equal powers of λ on both sides of (1.260), we have using (1.257) and (1.262) AC0 = p0I AC1 − C0 = p1I AC2 − C1 = p2I ...= ... (1.263) ACN−1 − CN−2 = pN−1I −CN−1 = pN I. We now multiply from the left the first of these equations by I, the second by A,thethirdbyA2, . , and the last by AN and then add the resulting equations. All the terms on the left-hand sides cancel, while the sum of those on the right gives P (A, A). Thus a square matrix A obeys its characteristic equation 0 = P (A, A) or N k N−1 N−1 N N 0= pk A = |A| I+p1A+···+(−1) (TrA) A +(−1) A (1.264) !k=0 1.27 Functions of Matrices 45 a result known as the Cayley-Hamilton theorem (Arthur Cayley, 1821– 1895, and William Hamilton, 1805–1865).
Load more

Recommended publications
  • Clifford Algebras, Spinors and Supersymmetry. Francesco Toppan
    IV Escola do CBPF – Rio de Janeiro, 15-26 de julho de 2002 Algebraic Structures and the Search for the Theory Of Everything: Clifford algebras, spinors and supersymmetry. Francesco Toppan CCP - CBPF, Rua Dr. Xavier Sigaud 150, cep 22290-180, Rio de Janeiro (RJ), Brazil abstract These lectures notes are intended to cover a small part of the material discussed in the course “Estruturas algebricas na busca da Teoria do Todo”. The Clifford Algebras, necessary to introduce the Dirac’s equation for free spinors in any arbitrary signature space-time, are fully classified and explicitly constructed with the help of simple, but powerful, algorithms which are here presented. The notion of supersymmetry is introduced and discussed in the context of Clifford algebras. 1 Introduction The basic motivations of the course “Estruturas algebricas na busca da Teoria do Todo”consisted in familiarizing graduate students with some of the algebra- ic structures which are currently investigated by theoretical physicists in the attempt of finding a consistent and unified quantum theory of the four known interactions. Both from aesthetic and practical considerations, the classification of mathematical and algebraic structures is a preliminary and necessary require- ment. Indeed, a very ambitious, but conceivable hope for a unified theory, is that no free parameter (or, less ambitiously, just few) has to be fixed, as an external input, due to phenomenological requirement. Rather, all possible pa- rameters should be predicted by the stringent consistency requirements put on such a theory. An example of this can be immediately given. It concerns the dimensionality of the space-time.
    [Show full text]
  • 5 the Dirac Equation and Spinors
    5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory.
    [Show full text]
  • About the Inversion of the Dirac Operator
    About the inversion of the Dirac operator Version 1.3 Abstract 5 In this note we would like to specify a bit what is done when we invert the Dirac operator in lattice QCD. Contents 1 Introductive remark 2 2 Notations 3 2.1 Thegaugefieldsandgaugeconfigurations . 4 5 2.2 General properties of the Dirac operator matrix . ... 5 2.3 Wilson-twistedDiracoperator. 6 3 Even-odd preconditionning 9 3.1 implementation of even-odd preconditionning . ... 10 3.1.1 ConjugateGradient . 11 10 4 Inexact deflation 15 4.1 Deflationmethod........................... 15 4.2 DeflationappliedtotheDiracOperator . 16 4.3 DeflationFAPP............................ 16 4.4 Deflation:theory........................... 18 15 4.4.1 Some properties of PR and PR ............... 18 4.4.2 TheLittleDiracOperator. 19 5 Mathematic tools 20 6 Samples from ETMC codes 22 1 Chapter 1 Introductive remark The inversion of the Dirac operator is an important step during the building of a statistical sample of gauge configurations: indeed in the HMC algorithm 5 it appears in the expression of what is called ”the fermionic force” used to update the momenta associated with the gauge fields along a trajectory. It is the most expensive part in overall computation time of a lattice simulation with dynamical fermions because it is done many times. Actually the computation of the quark propagator, i.e. the inverse of the Dirac 10 operator, is already necessary to compute a simple 2pts correlation function from which one extracts a hadron mass or its decay constant: indeed that correlation function is roughly the trace (in the matricial language) of the product of 2 such propagators.
    [Show full text]
  • Chapter 2: Linear Algebra User's Manual
    Preprint typeset in JHEP style - HYPER VERSION Chapter 2: Linear Algebra User's Manual Gregory W. Moore Abstract: An overview of some of the finer points of linear algebra usually omitted in physics courses. May 3, 2021 -TOC- Contents 1. Introduction 5 2. Basic Definitions Of Algebraic Structures: Rings, Fields, Modules, Vec- tor Spaces, And Algebras 6 2.1 Rings 6 2.2 Fields 7 2.2.1 Finite Fields 8 2.3 Modules 8 2.4 Vector Spaces 9 2.5 Algebras 10 3. Linear Transformations 14 4. Basis And Dimension 16 4.1 Linear Independence 16 4.2 Free Modules 16 4.3 Vector Spaces 17 4.4 Linear Operators And Matrices 20 4.5 Determinant And Trace 23 5. New Vector Spaces from Old Ones 24 5.1 Direct sum 24 5.2 Quotient Space 28 5.3 Tensor Product 30 5.4 Dual Space 34 6. Tensor spaces 38 6.1 Totally Symmetric And Antisymmetric Tensors 39 6.2 Algebraic structures associated with tensors 44 6.2.1 An Approach To Noncommutative Geometry 47 7. Kernel, Image, and Cokernel 47 7.1 The index of a linear operator 50 8. A Taste of Homological Algebra 51 8.1 The Euler-Poincar´eprinciple 54 8.2 Chain maps and chain homotopies 55 8.3 Exact sequences of complexes 56 8.4 Left- and right-exactness 56 { 1 { 9. Relations Between Real, Complex, And Quaternionic Vector Spaces 59 9.1 Complex structure on a real vector space 59 9.2 Real Structure On A Complex Vector Space 64 9.2.1 Complex Conjugate Of A Complex Vector Space 66 9.2.2 Complexification 67 9.3 The Quaternions 69 9.4 Quaternionic Structure On A Real Vector Space 79 9.5 Quaternionic Structure On Complex Vector Space 79 9.5.1 Complex Structure On Quaternionic Vector Space 81 9.5.2 Summary 81 9.6 Spaces Of Real, Complex, Quaternionic Structures 81 10.
    [Show full text]
  • Exact Solutions of a Dirac Equation with a Varying CP-Violating Mass Profile and Coherent Quasiparticle Approximation
    Exact solutions of a Dirac equation with a varying CP-violating mass profile and coherent quasiparticle approximation Olli Koskivaara Master’s thesis Supervisor: Kimmo Kainulainen November 30, 2015 Preface This thesis was written during the year 2015 at the Department of Physics at the University of Jyväskylä. I would like to express my utmost grat- itude to my supervisor Professor Kimmo Kainulainen, who guided me through the subtle path laid out by this intriguing subject, and provided me with my first glimpses of research in theoretical physics. I would also like to thank my fellow students for the countless number of versatile conversations on physics and beyond. Finally, I wish to thank my family and Ida-Maria for their enduring and invaluable support during the entire time of my studies in physics. i Abstract In this master’s thesis the phase space structure of a Wightman function is studied for a temporally varying background. First we introduce coherent quasiparticle approximation (cQPA), an approximation scheme in finite temperature field theory that enables studying non-equilibrium phenom- ena. Using cQPA we show that in non-translation invariant systems the phase space of the Wightman function has in general structure beyond the traditional mass-shell particle- and antiparticle-solutions. This novel structure, describing nonlocal quantum coherence between particles, is found to be living on a zero-momentum k0 = 0 -shell. With the knowledge of these coherence solutions in hand, we take on a problem in which such quantum coherence effects should be promi- nent. The problem, a Dirac equation with a complex and time-dependent mass profile, is manifestly non-translation invariant due to the temporal variance of the mass, and is in addition motivated by electroweak baryo- genesis.
    [Show full text]
  • Maxima by Example: Ch. 12, Dirac Algebra and Quantum Electrodynamics ∗
    Maxima by Example: Ch. 12, Dirac Algebra and Quantum Electrodynamics ∗ Edwin L. Woollett June 8, 2019 Contents 1 Introduction 5 2 Kinematics and Mandelstam Variables 5 2.1 1 + 2 3 + 4 Scattering for Non-Equal Masses . ....... 6 2.2 Mandelstam→ Variable Tools at Work . .................. 7 2.3 CM Frame Scattering Description . .................. 9 2.4 CM Frame Differential Scattering Cross Section for (AB) (CD) .................... 13 → 2.5 Lab Frame Differential Scattering Cross Section for Elastic Scattering (AB) (AB) .......... 13 2.6 DecayRateforTwoBodyDecay. .→............... 14 3 SpinSums 14 1 4 Spin 2 Examples 15 4.1 e e+ µ µ+,ee-mumu-AE.wxm .................................... 15 − → − 4.2 e µ e µ ,e-mu-scatt.wxm .................................... 16 − − → − − 4.3 e− e− e− e− (Moller Scattering), em-em-scatt-AE.wxm . .......... 16 + → + 4.4 e e e e (Bhabha Scattering), em-ep-scatt.AE.wxm . .......... 17 − → − 5 Theorems for Physical External Photons 17 5.1 Photon Real Linear Polarization 3-Vector Theorems: photon1.wxm .................... 17 5.2 Photon Circular Polarization 4-Vectors, photon2.wxm . ........................... 18 6 Examples for Spin 1/2 and External Photons 18 6.1 e e+ γ γ,ee-gaga-AE.wxm .................................... 18 − → 6.2 γ e− γ e−, compton1.wxm, compton-CMS-HE.wxm . ........ 19 6.3 Covariant→ Physical External Photon Polarization 4-Vectors-AReview................... 20 7 Examples Which Involve Scalar Particles 20 7.1 π+K+ π+K+ .............................................. 21 → 7.2 π+π+ π+π+ .............................................. 21 → 7.3 e π+ e π+ ............................................... 22 − → − 7.4 γ π+ γ π+ ................................................ 23 → 8 Electroweak Unification Examples 23 8.1 W-bosonDecay...................................
    [Show full text]
  • 1 the Gamma-Matrices A.) the Gamma-Matrices Satisfy the Clifford Algebra
    Prof Dr. Ulrich Uwer/ Dr. T. Weigand 27. April 2010 The Standard Model of Particle Physics - SoSe 2010 Assignment 3 (Due: May 6, 2010 ) 1 The gamma-matrices a.) The gamma-matrices satisfy the Clifford algebra fγµ; γνg = 2gµν: (1) The significance of the Clifford algebra is that it induces a representation of the Lorentz algebra as follows: Consider the set of matrices i σµν = [γµ; γν]: (2) 2 These satisfy the relation [σµν; σαβ] = 2i gµβσνα + gνασµβ − gµασνβ − gνβσµα (3) as a consequence of the Clifford algebra and thus form a representation of the Lorentz algebra, as promised (cf. Assignment 1, Exercise 1). Give the four-dimensional representation of the gamma-matrices introduced in the lecture and check explicitly that they satisfy (1) as well as γ0 = (γ0)y; γi = −(γi)y: (4) 0 1 2 3 b.) The matrix γ5 is defined as γ5 = iγ γ γ γ . Without using a concrete represen- tation of the gamma-matrices, prove that γ5 = (γ5)y; (γ5)2 = 1; fγ5; γµg = 0: (5) 2 The Dirac spinor a.) A Dirac spinor is defined by its properties under Lorentz transformations. Give these properties in terms of the matrix i S(Λ) = exp(− σµν! ): (6) 4 µν What's the difference between the transformation of a vector and a Dirac spinor? Optional: Show that i [σµν; γρ] ! = !ρ γν: (7) 4 µν ν Hint: Rewrite the commutators in terms of anti-commutators. Argue that this is the infinitesimal version of the more general relation −1 µ −1 µ ν S (Λ)γ S (Λ) = Λ νγ : (8) b.) The Dirac equation for a spinor field of mass m is µ (iγ @µ − m) (x) = 0: (9) Transform this equation into a new frame x0 = Λx and show that if (x) satisfies the Dirac equation in the original frame, then 0(x0) does so in the new frame.
    [Show full text]
  • Dirac Fermions on Rotating Space-Times
    DIRAC FERMIONS ON ROTATING SPACE-TIMES VICTOR EUGEN AMBRUS, PhD thesis Supervisor: Prof. Elizabeth Winstanley School of Mathematics and Statistics University of Sheffield December 2014 Abstract Quantum states of Dirac fermions at zero or finite temperature are investigated using the point-splitting method in Minkowski and anti-de Sitter space-times undergoing rotation about a fixed axis. In the Minkowski case, analytic expressions presented for the thermal expectation values (t.e.v.s) of the fermion condensate, parity violating neutrino current and stress-energy tensor show that thermal states diverge as the speed of light surface (SOL) is approached. The divergence is cured by enclosing the rotating system inside a cylinder located on or inside the SOL, on which spectral and MIT bag boundary conditions are considered. For anti-de Sitter space-time, renormalised vacuum expectation values are calcu- lated using the Hadamard and Schwinger-de Witt methods. An analytic expression for the bi-spinor of parallel transport is presented, with which some analytic expres- sions for the t.e.v.s of the fermion condensate and stress-energy tensor are obtained. Rotating states are investigated and it is found that for small angular velocities Ω of the rotation, there is no SOL and the thermal states are regular everywhere on the space-time. However, if Ω is larger than the inverse radius of curvature of adS, an SOL forms and t.e.v.s diverge as inverse powers of the distance to it. ii Contents Abstract ii Table of contents iii List of figures viii List of tables xvi Preface xvii 1 Introduction 1 2 General concepts 4 2.1 The quantised scalar field .
    [Show full text]
  • As Usual, These Notes Are Intended for Use by Class Participants Only, and Are Not for Circulation
    As usual, these notes are intended for use by class participants only, and are not for circulation. Week 6: Lectures 11, 12 March 5, 2012 The Dirac equation and algebra The Lagrange density for the Dirac equation is • = ψ¯(i∂ γµ m) ψ(x) , (2) L µ − where we define ψ¯ ψ† γ , (3) ≡ 0 so that 4 ψψ¯ ψ¯ ψ ≡ α α α!=1 2 2 a b ∗ = (ξa˙ )∗ η + η ξb˙ a!=1 b!=1 " # where we’ve inserted the sums for (hopefully) clarity. Notice that unlike spinor indices, Dirac indices are not raised or lowered. The fundamental relation for Dirac (or “gamma”) matrices, • γµγν + γν,γµ γµ,γν =2gµν (4) ≡{ } where the first line defines the anticommutator. These rules imply γµ γν = γνγµ ,µ= ν µ 2 −µν # (γ ) = g I4 4 . × Thus, Dirac matrices anticommute if their indices are different, while 0 the square of γ is +I4 4 (the four-by-four identity matrix), while the × 6 square of any of the “spatial” gammas is I4 4. (Note, it is not con- × ventional to exhibit the identity matrix, and− the right hand side of the second relation above is normally just written gµν.) Further, useful relations are 0 µ 0 µ γ γ γ = γ (2δµ,0 1) µ − = γ gµµ = γµ µ =(γ )† (5) where the first line follows from the anticommutation relations, the sec- ond from the difference between covariant and contravariant indices, and the third follows by inspection of the explicit γµ in Weyl represen- tation. By using the anticommutation relations (5), we can reduce the power • of any monomial involving more than four γ’s, since in this case, at least two must be the same, and any pair can be “anticommuted” past other gammas to give a square, which results in 1 times the monomial without the pair.
    [Show full text]
  • U-Dual Fluxes and Generalized Geometry
    Published for SISSA by Springer Received: September 3, 2010 Accepted: October 26, 2010 Published: November 18, 2010 U-dual fluxes and Generalized Geometry JHEP11(2010)083 G. Aldazabal,a,b E. Andr´es,b P.G. C´amarac and M. Gra˜nad aCentro At´omico Bariloche, 8400 S.C. de Bariloche, Argentina bInstituto Balseiro (CNEA-UNC) and CONICET, 8400 S.C. de Bariloche, Argentina cCERN, PH-TH Division, CH-1211 Gen`eve 23, Switzerland dInstitut de Physique Th´eorique, CEA/ Saclay, 91191 Gif-sur-Yvette Cedex, France E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We perform a systematic analysis of generic string flux compactifications, making use of Exceptional Generalized Geometry (EGG) as an organizing principle. In particular, we establish the precise map between fluxes, gaugings of maximal 4d supergrav- ity and EGG, identifying the complete set of gaugings that admit an uplift to 10d heterotic or type IIB supegravity backgrounds. Our results reveal a rich structure, involving new deformations of 10d supergravity backgrounds, such as the RR counterparts of the β- deformation. These new deformations are expected to provide the natural extension of the β-deformation to full-fledged F-theory backgrounds. Our analysis also provides some clues on the 10d origin of some of the particularly less well understood gaugings of 4d supergrav- ity. Finally, we derive the explicit expression for the effective superpotential in arbitrary = 1 heterotic or type IIB orientifold compactifications, for all the allowed
    [Show full text]
  • Jordan Wigner Transformations and Quantum Spin Systems on Graphs
    LMU M T U M . E B TMP eoretical and Mathematical Physics Master’s Thesis Jordan Wigner Transformations and antum Spin Systems on Graphs Tobias Ried Thesis Adviser: Prof. Dr. Simone Warzel . Submission Date: 31st August 2013 First Reviewer: Prof. Dr. Simone Warzel Second Reviewer: Prof. Dr. Peter Müller Thesis Defence: 9th September 2013 A e Jordan Wigner transformation is an important tool in the study of one- dimensional quantum spin systems. Aer reviewing the one-dimensional Jordan Wigner transformation, generalisations to graphs are discussed. As a first res- ult it is proved that the occurrence of statistical gauge fields in the transformed Hamiltonian is related to the structure of the graph. To avoid the difficulties caused by statistical gauge fields, an indirect transformation due to W and S is introduced and applied to a particular quantum spin system on a square laice – the constrained Gamma matrix model. In a random exterior magnetic field, a locality estimate on the Heisenberg evolution – a zero-velocity Lieb Robinson type bound in disorder average – is proved for the constrained Gamma matrix model, using localisation results for the two-dimensional Ander- son model. A Firstly, I would like to thank Professor Simone Warzel for supervising this thesis. roughout my studies of physics and mathematical physics, her enthu- siasm for those subjects has been a constant source of inspiration. I am deeply grateful for her guidance and support. I am also indebted to Professor Peter Müller who agreed to second review my thesis. For leing me use one of his group’s offices, which enormously facilitated the writing process, I am thankful to Professor Jan von Del.
    [Show full text]
  • Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory
    Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory Alain Connes1 and Matilde Marcolli2 1 Coll`egedeFrance 3, rue Ulm F-75005 Paris and I.H.E.S. 35 route de Chartres F-91440 Bures-sur-Yvette France [email protected] 2 Max–Planck Institut f¨ur Mathematik Vivatsgasse 7 D-53111 Bonn Germany [email protected] 1 Introduction ..............................................618 2 Renormalization in Quantum Field Theory ................624 2.1 BasicformulasofQFT.......................................625 2.2 Feynman diagrams . 627 2.3 Divergences and subdivergences . 629 3 Affine group schemes......................................633 3.1 Tannakiancategories ........................................634 3.2 The Lie algebra and the Milnor-Moore theorem . 635 4 The Hopf algebra of Feynman graphs and diffeographisms 636 5 The Lie algebra of graphs .................................640 6 Birkhoff decomposition and renormalization...............642 7UnitofMass..............................................648 8 Expansional ...............................................651 9 Renormalization group ....................................654 10 Diffeographisms and diffeomorphisms .....................663 11 Riemann–Hilbert problem ................................665 11.1 Regular-singular case . 666 11.2 Local Riemann–Hilbert problem and Birkhoff decomposition . 668 618 Alain Connes and Matilde Marcolli 11.3 Geometric formulation . 669 11.4 Irregular case . 670 12 Local equivalence of meromorphic connections ............673 13 Classification
    [Show full text]