DOCSLIB.ORG

16 Path Integrals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

16 Path Integrals 16.1 Path Integrals and Classical Physics Since Richard Feynman invented them over 60 years ago, path integrals have been used with increasing frequency in high-energy and condensed-matter physics, in finance, and in biophysics (Kleinert, 2009). Feynman used them to express matrix elements of the time-evolution operator exp( itH/~)in − terms of the classical action. Others have used them to compute matrix elements of the Boltzmann operator exp( H/kT ) which in the limit of zero − temperature projects out the ground state E of the system | 0i 1 (H E0)/kT (En E0)/kT lim e− − =lim En e− − En = E0 E0 (16.1) T 0 T 0 | i h | | ih | ! ! n=0 X a trick used in lattice gauge theory. Path integrals magically express the quantum-mechanical probability am- plitude for a process as a sum of exponentials exp(iS/~) of the classical action S of the various ways that process might occur. 16.2 Gaussian Integrals The path integrals we can do are gaussian integrals of infinite order. So we begin by recalling the basic integral formula (5.167) b 2 ⇡ 1 exp ia x dx = (16.2) − − 2a ia Z1 " ✓ ◆ # r 16.2 Gaussian Integrals 671 which holds for real a and b, and also the one (5.168) c 2 ⇡ 1 exp r x dx = (16.3) − − 2r r Z1 ⇣ ⌘ r which is true for positive r and complex c. Equivalent formulas for real a and b, positive r, and complex c are ⇡ b2 1 exp iax2 + ibx dx = exp i (16.4) − ia 4a Z1 r ✓ ◆ ⇡ c2 1 exp rx2 + cx dx = exp . (16.5) − r 4r Z r ✓ ◆ 1 This last formula will be useful with x = p, r = ✏/(2m), and c = i✏q˙ p2 2⇡m 1 1 exp ✏ + i✏ qp˙ dp = exp ✏ mq˙2 (16.6) − 2m ✏ − 2 Z1 ✓ ◆ r ✓ ◆ as will (16.4)withx = p, a = ✏/(2m), and b = ✏q˙ p2 2⇡m 1 1 exp i✏ + i✏ qp˙ dp = exp i✏ mq˙2 . (16.7) − 2m i✏ 2 Z1 ✓ ◆ r ✓ ◆ Doable path integrals are multiple gaussian integrals. One may show (ex- ercise 16.1) that for positive r1,...,rN and complex c1,...,cN , the integral (16.5) leads to N N 2 1 2 ⇡ 1 ci exp rixi + cixi dxi = exp . (16.8) − ! ri ! 4 ri ! Z1 Xi Yi=1 Yi=1 r Xi If R is the N N diagonal matrix with positive entries r ,r ,...,r , and ⇥ { 1 2 N } X and C are N-vectors with real x and complex c entries, then this { i} { i} formula (16.8) in matrix notation is N N 1 T T ⇡ 1 T 1 exp X RX + C X dx = exp C R− C . (16.9) − i det(R) 4 Z i=1 s ✓ ◆ 1 Y Now every positive symmetric matrix S is of the form S = OROT for some positive diagonal matrix R.SoinsertingR = OTSO into the previous equa- tion (16.9) and using the invariance of determinants under orthogonal trans- formations, we find N N 1 T T T ⇡ 1 T T 1 exp X O SOX + C X dxi = exp C O S− OC . − sdet(S) 4 Z1 i=1 Y (16.10) 672 Path Integrals The jacobian of the orthogonal transformations Y = OX and D = OC is unity, and so N N 1 T T ⇡ 1 T 1 exp Y SY + D Y dy = exp D S− D (16.11) − i det(S) 4 Z i=1 s ✓ ◆ 1 Y in which S is a positive symmetric matrix, and D is a complex vector. The other basic gaussian integral (16.4) leads for real S and D to (exer- cise 16.2) N N 1 T T ⇡ i T 1 exp iY SY + iD Y dyi = exp D S− D . − sdet(iS) 4 Z1 i=1 ✓ ◆ Y (16.12) The vector Y that makes the argument iY TSY + iDTY of the expo- − nential of this multiple gaussian integral (16.12) stationary is (exercise 16.3) 1 1 Y = S− D. (16.13) 2 The exponential of that integral evaluated at its stationary point Y is T T i T 1 exp iY SY + iD Y =exp D S− D . (16.14) − 4 ✓ ◆ Thus, the multiple gaussian integral (16.12) is equal to its exponential eval- uated at its stationary point Y , apart from a prefactor involving the deter- minant det iS. Similarly, the vector Y that makes the argument Y TSY + DTY of − the exponential of the multiple gaussian integral (16.11) stationary is Y = 1 S− D/2, and that exponential evaluated at Y is T T 1 T 1 exp Y SY + D Y =exp D S− D . (16.15) − 4 ✓ ◆ Once again, a multiple gaussian integral is simply its exponential evaluated at its stationary point Y , apart from a prefactor involving a determinant, det S. 16.3 Path Integrals in Imaginary Time At the imaginary time t = iβ~, the time-evolution operator exp( itH/~) − − is exp( βH) in which the inverse temperature β =1/kT is the reciprocal of − Boltzmann’s constant k =8.617 10 5 eV/K times the absolute temperature ⇥ − T . In the low-temperature limit, exp( βH) is a projection operator (16.1) − 16.3 Path Integrals in Imaginary Time 673 on the ground state of the system. These path integrals in imaginary time are called euclidian path integrals. Let us consider a quantum-mechanical system with hamiltonian p2 H = + V (q) (16.16) 2m in which the commutator of the position q and momentum p operators is [q, p]=i in units in which ~ = 1. For tiny ✏, the corrections to the approxi- mation p2 p2 exp ✏ + V (q) exp ✏ exp ( ✏V(q)) + (✏2) (16.17) − 2m ⇡ − 2m − O ✓ ◆ ✓ ◆ are of second order in ✏. To evaluate the matrix element q exp( ✏H) q ,weinserttheidentity h 1| − | 0i operator I in the form of an integral over the momentum eigenstates 1 I = p0 p0 dp0 (16.18) | ih | Z1 and use the inner product q p =exp(iq p )/p2⇡ so as to get as ✏ 0 h 1| 0i 1 0 ! 2 1 p q exp( ✏H) q = q exp ✏ p0 p0 exp ( ✏V(q)) q dp0 h 1| − | 0i h 1| − 2m | ih | − | 0i Z1 ✓ ◆ 2 ✏V(q0) 1 p0 dp0 = e− exp ✏ + ip0 (q q ) . (16.19) − 2m 1 − 0 2⇡ Z1 If we adopt the suggestive notation q q 1 − 0 =˙q (16.20) ✏ 0 and use the integral formula (16.6), then we find 2 1 ✏V(q0) 1 p0 q exp( ✏H) q = e− exp ✏ + i✏p0 q˙ dp0 h 1| − | 0i 2⇡ − 2m 0 Z1 ✓ ◆ m 1/2 = exp ✏ 1 m q˙2 + V (q ) (16.21) 2⇡✏ − 2 0 0 ⇣ ⌘ ⇥ ⇤ in which q1 enters through the notation (16.20). The next step is to link two of these matrix elements together 2✏H 1 ✏H ✏H q e− q = q e− q q e− q dq (16.22) h 2| | 0i h 2| | 1ih 1| | 0i 1 Z1 m = 1exp ✏ 1 m q˙2 + V (q )+ 1 m q˙2 + V (q ) dq . 2⇡✏ − 2 1 1 2 0 0 1 Z 1 ⇥ ⇤ 674 Path Integrals Linking three of these matrix elements together and using subscripts instead of primes, we have 3✏H 1 ✏H ✏H ✏H q e− q = q e− q q e− q q e− q dq dq (16.23) h 3| | 0i h 3| | 2ih 2| | 1ih 1| | 0i 1 2 ZZ1 2 m 3/2 = 1exp ✏ 1 m q˙2 + V (q ) dq dq . 2⇡✏ 8 − 2 j j 9 2 1 ZZ j=0 ⇣ ⌘ 1 < X ⇥ ⇤= Boldly passing from 3 to n and suppressing: some integral signs,; we get n✏H 1 ✏H ✏H qn e− q0 = qn e− qn 1 ... q1 e− q0 dqn 1 ...dq1 (16.24) h | | i h | | − i h | | i − ZZZ 1 n 1 n/2 m 1 − 1 2 = exp ✏ m q˙j + V (qj) dqn 1 ...dq1. 2⇡✏ 8 − 2 9 − ZZZ j=0 ⇣ ⌘ 1 < X ⇥ ⇤= Writing dβ for ✏ and taking the: limits ✏ 0 and n β/;✏ ,wefind ! ⌘ !1 that the matrix element q e βH q is a path integral of the exponential h β| − | 0i of the average energy multiplied by β − β βH 1 2 q e− q = exp mq˙ (β0)+V (q(β0)) dβ0 Dq (16.25) h β| | 0i − 2 Z Z0 2 n/2 in which Dq (n m/2⇡~ ) dqn 1 ...dq2 dq1 andq ˙ is the β-derivative of ⌘ − the coordinate q(β). We sum over all paths q(t) that go from q(0) = q0 at inverse temperature β =0toq(β)=qβ at inverse temperature β. In the limit β , the operator exp( βH) becomes proportional to a !1 − projection operator (16.1) on the ground state of the theory. In three dimensions with q˙(β)=dq(β)/dβ, and ~ = 1, the analog of 6 equation (16.25) is (exercise 16.28) β βH m 2 q e− q = exp q˙ (β0)+V (q(β0)) dβ0 Dq (16.26) h β| | 0i − 2 2 Z Z0 ~ 2 3n/2 where Dq (n m/2⇡~ ) dqn 1 ...dq2 dq1. ⌘ − Path integrals in imaginary time are called euclidian mainly to distinguish them from Minkowski path integrals, which represent matrix elements of the time-evolution operator exp( itH) in real time. − 16.4 Path Integrals in Real Time 675 16.4 Path Integrals in Real Time Path integrals in real time represent the time-evolution operator exp( itH). − Using the integral formula (16.7), we find in the limit ✏ 0 ! 2 i✏H 1 p q e− q = q exp i✏ p0 p0 exp [ i✏V(q)] q dp0 h 1| | 0i h 1| − 2m | ih | − | 0i Z1 2 1 i✏V(q0) 1 p0 = e− exp i✏ + ip0 (q q ) dp0 2⇡ − 2m 1 − 0 Z1 2 1 i✏V(q0) 1 p0 = e− exp i✏ + i✏p0q˙ dp0 2⇡ − 2m 0 Z1 m 1/2 m q˙2 = exp i✏ 0 V (q ) .
Recommended publications
  • Lecture 3-7: General Formalism at Finite Temperature

    Lecture 3-7: General Formalism at Finite Temperature

    Lecture 3-7: General formalism at finite temperature Reference: Negele & Orland (N&O) Chapter 2 Lecture 3 Introduction Quantum statistical mechanics (Home reading) Three ensembles: microcanonical, canonical, grand canonical essemble Partition function and thermodynamics Physical response functions and Green’s function Consider a system subjected to a time-dependent external field, where the operators and corresponding states are in the Schrödinger picture. We shall study the system through the evolution operator. Time-ordered operator product where Time-ordered exponential t t where b a and t t n. It may be expanded in a Taylor series as follows, M n a The evolution operator Using the time-ordered exponential, the evolution operator may be written It is easy to verify that it satisfies the equation of motion and the boundary condition The response to an infinitesimal perturbation in the external field The Schrödinger picture and the Heisenberg picture. (Home reading) The response of a wavefunction to an infinitesimal perturbation by an external field is given by the functional derivative where the operator and the state in the Heisenberg picture is related to the operator in the Schrödinger picture by and Now, consider the response of the expectation value of an operator to an infinitesimal perturbation in the external field. ˆ ˆ The response of a measurement of O2 (t2 ) to a perturbation couple to O1 is specified by the response function, The above is one of century results in this chapter. The n-body real-time Green’s function The n-body imaginary-time Green’s function where Approximation strategies (Home reading) Asymptotic expansions Weak coupling and strong coupling Functional integral formulation A powerful tool for the study of many-body systems The Feynman path integral for a single particle system A different formulation to the canonical formalism, the propagator (or the matrix element of the evolution operator) plays the basic role.
  • 10 Field Theory on a Lattice

    10 Field Theory on a Lattice

    10 Field Theory on a Lattice 10.1 Scalar Fields To represent a hermitian field φ(x), we put a real number φ(i, j, k, `) at each vertex of the lattice and set x = a(i, j, k, `) (10.1) where a is the lattice spacing. The derivative @iφ(x) is approximated as φ(x + ˆi) φ(x) @ φ(x) − (10.2) i ⇡ a in which x is the discrete position 4-vector and ˆi is a unit 4-vector pointing in the i direction. So the euclidian action is the sum over all lattice sites of 1 1 S = (@ φ(x))2 a4 + m2 φ2(x) a4 e 2 i 2 x X 1 2 1 = φ(x + ˆi) φ(x) a2 + m2 φ2(x) a4 2 − 2 0 (10.3) x ⇣ ⌘ X1 1 = φ(x + ˆi)φ(x) a2 + 8+m2 φ2(x) a4 − 2 2 0 xi X if the self interaction happens to be quartic. 1 1 λ S = (@ φ(x))2 a4 + m2 φ2(x) a4 + φ4(x) a4 (10.4) e 2 i 2 0 4 if the self interaction happens to be quartic. 10.2 Pure Gauge Theory 249 10.2 Pure Gauge Theory The gauge-covariant derivative is defined in terms of the generators ta of a compact Lie algebra [ta,tb]=ifabctc (10.5) b and a gauge-field matrix Ai = igAi tb as D = @ A = @ igAb t (10.6) i i − i i − i b summed over all the generators, and g is a coupling constant. Since the group is compact, we may raise and lower group indexes without worrying about factors or minus signs.
  • Derivation: the Volume of a D-Dimensional Hypershell

    Derivation: the Volume of a D-Dimensional Hypershell

    Derivation: the volume of a D‐dimensional hypershell Phys 427 Yann Chemla Sept. 1, 2011 In class we worked out the multiplicity of an ideal gas by summing over volume elements in k‐space (each of which contains one quantum state) that have energy E. We showed that the surface of states that have energy E is a hypershell of radius k in D‐dimensions where . Integrating in D‐dimensions in Cartesian coordinates is easy, but how do you integrate over a hypershell in D‐ dimensions? We used dimensional arguments in class to relate a volume of k‐space in Cartesian vs. hyperspherical coordinates: Ω where Ω is the solid angle subtended by the hypershell. If we integrate over all D‐dimensional space in Cartesian vs. hyperspherical coordinates, we should get the same answer: where we integrated over the solid angle to obtain the factor gD. We would like to determine gD. The above formula is not very useful in that respect because both sides are infinite. What we want is a function that can be integrated over all of D‐dimensional space in both Cartesian and hyperspherical coordinates and give a finite answer. A Gaussian function over k1 ... kD will do the trick. We can write: where we wrote the Gaussian in terms of k on the right side of the equation. Let’s look at the left side of the equation first. The integral can be written in terms of the product of D identical 1‐dimensional integrals: / where we evaluated the Gaussian integral in the last step (look up Kittel & Kroemer Appendix A for help on evaluating Gaussian integrals).
  • Notas De Física CBPF-NF-011/96 March 1996

    Notas De Física CBPF-NF-011/96 March 1996

    CBPF CENTRO BRASILEIRO DE PESQUISAS FÍSICAS Rio de Janeiro Notas de Física CBPF-NF-011/96 March 1996 Non-Abelian BF Theories With Sources and 2-d Gravity J. P. Lupi, A. Restuccia and J. Stephany CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico VOL. 2 ( í:1 " CBPF-NF-011/96 Non-Abelian BF Theories With Sources and 2-d Gravity by J. P. Lupi1, A. Restuccia1 and J. Stephany1'* "Centro Brasileiro de Pesquisas Físicas - CBPF "Departamento de Campos e Partículas Rua Dr. Xavier Sigaud, 150 22290-180 - Rio de Janeiro, RJ - Brazil ^niversidad Simón Bolívar, Departamento de Física, Apartado Postal 89000, Caracas 1080-A, Venezuela e-mail: [email protected], [email protected], [email protected] ABSTRACT r (We study the interaction of non-Abelian topological BF theories defined on two dimensional manifolds with point sources carrying non-Abelian charges. We identify the most general solution for the field equations on simply and multiply connected two- manifolds. Taking the particular choice of the so-called extended Poincaré group as the gauge group we discuss how recent discussions of two dimensional gravity models do fit in this formalism. J Key-words: BF theories; 2-d gravity. - 1 - CBPF-NF-011/96 1 Introduction The adequate description of classical and first quantized relativistic objects, like particles and strings, is an essential point of discussion in any attempt to set a unified model of physical interactions, Our understanding of this issue for relativistic objects has been continually improved in recent years mainly by the application of BRST techniques, In particular, a satisfactory relativistic covariant treatment of isolated scalar [1] and spin- ning particles [2], and more recently, superparticles [3] which allows to discuss most of the kinematical aspects of such systems have been developed.
  • The Amazing Poisson Calculation: a Method Or a Trick?

    The Amazing Poisson Calculation: a Method Or a Trick?

    The amazing Poisson calculation: a method or a trick? Denis Bell April 27, 2017 The Gaussian integral and Poisson's calculation The Gaussian function e−x2 plays a central role in probability and statistics. For this reason, it is important to know the value of the integral 1 Z 2 I = e−x dx: 0 The function e−x2 does not have an elementary antiderivative, so I cannot be evaluated by the FTC. There is the following argument, attributed to Poisson, for calculating I . Consider 1 1 Z 2 Z 2 I 2 = e−x dx · e−y dy: 0 0 Interpret this product as a double integral in the plane and transform to polar coordinates x = r cos θ; y = r sin θ to get 1 1 Z Z 2 2 I 2 = e−(x +y )dxdy 0 0 π=2 1 Z Z 2 = e−r r drdθ 0 0 1 π Z d 2 π = − e−r dr = : 4 0 dr 4 p π Thus I = 2 . Can this argument be used to evaluate other seemingly intractable improper integrals? Consider the integral Z 1 J = f (x)dx: 0 Proceeding as above, we obtain Z 1 Z 1 J2 = f (x)f (y) dxdy: 0 0 In order to take the argument further, there will need to exist functions g and h such that f (x)f (y) = g(x2 + y 2)h(y=x): Transformation to polar coordinates and the substitution u = r 2 will then yield 1 Z 1 Z π=2 J2 = g(u)du h(tan θ)dθ: (1) 2 0 0 Which functions f satisfy f (x)f (y) = g(x2 + y 2)h(y=x) (2) Theorem Suppose f : (0; 1) 7! R satisfies (2) and assume f is non-zero on a set of positive Lebesgue measure, and the discontinuity set of f is not dense in (0; 1).
  • Contents 1 Gaussian Integrals ...1 1.1

    Contents 1 Gaussian Integrals ...1 1.1

    Contents 1 Gaussian integrals . 1 1.1 Generating function . 1 1.2 Gaussian expectation values. Wick's theorem . 2 1.3 Perturbed gaussian measure. Connected contributions . 6 1.4 Expectation values. Generating function. Cumulants . 9 1.5 Steepest descent method . 12 1.6 Steepest descent method: Several variables, generating functions . 18 1.7 Gaussian integrals: Complex matrices . 20 Exercises . 23 2 Path integrals in quantum mechanics . 27 2.1 Local markovian processes . 28 2.2 Solution of the evolution equation for short times . 31 2.3 Path integral representation . 34 2.4 Explicit calculation: gaussian path integrals . 38 2.5 Correlation functions: generating functional . 41 2.6 General gaussian path integral and correlation functions . 44 2.7 Harmonic oscillator: the partition function . 48 2.8 Perturbed harmonic oscillator . 52 2.9 Perturbative expansion in powers of ~ . 54 2.10 Semi-classical expansion . 55 Exercises . 59 3 Partition function and spectrum . 63 3.1 Perturbative calculation . 63 3.2 Semi-classical or WKB expansion . 66 3.3 Path integral and variational principle . 73 3.4 O(N) symmetric quartic potential for N . 75 ! 1 3.5 Operator determinants . 84 3.6 Hamiltonian: structure of the ground state . 85 Exercises . 87 4 Classical and quantum statistical physics . 91 4.1 Classical partition function. Transfer matrix . 91 4.2 Correlation functions . 94 xii Contents 4.3 Classical model at low temperature: an example . 97 4.4 Continuum limit and path integral . 98 4.5 The two-point function: perturbative expansion, spectral representation 102 4.6 Operator formalism. Time-ordered products . 105 Exercises . 107 5 Path integrals and quantization .
  • Arxiv:1412.2393V4 [Gr-Qc] 27 Feb 2019 2.6 Geodesics and Normal Coordinates

    Arxiv:1412.2393V4 [Gr-Qc] 27 Feb 2019 2.6 Geodesics and Normal Coordinates

    Riemannian Geometry: Definitions, Pictures, and Results Adam Marsh February 27, 2019 Abstract A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. As much material as possible is developed for manifolds with connection (omitting a metric) to make clear which aspects can be readily generalized to gauge theories. The presentation in most cases does not assume a coordinate frame or zero torsion, and the coordinate-free, tensor, and Cartan formalisms are developed in parallel. Contents 1 Introduction 2 2 Parallel transport 3 2.1 The parallel transporter . 3 2.2 The covariant derivative . 4 2.3 The connection . 5 2.4 The covariant derivative in terms of the connection . 6 2.5 The parallel transporter in terms of the connection . 9 arXiv:1412.2393v4 [gr-qc] 27 Feb 2019 2.6 Geodesics and normal coordinates . 9 2.7 Summary . 10 3 Manifolds with connection 11 3.1 The covariant derivative on the tensor algebra . 12 3.2 The exterior covariant derivative of vector-valued forms . 13 3.3 The exterior covariant derivative of algebra-valued forms . 15 3.4 Torsion . 16 3.5 Curvature .
  • Time-Ordered Exponential for Unbounded Operators with Applications to Quantum Field Theory

    Time-Ordered Exponential for Unbounded Operators with Applications to Quantum Field Theory

    Title Time-ordered Exponential for Unbounded Operators with Applications to Quantum Field Theory Author(s) 二口, 伸一郎 Citation 北海道大学. 博士(理学) 甲第11798号 Issue Date 2015-03-25 DOI 10.14943/doctoral.k11798 Doc URL http://hdl.handle.net/2115/58756 Type theses (doctoral) File Information Shinichiro_Futakuchi.pdf Instructions for use Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP Time-ordered Exponential for Unbounded Operators with Applications to Quantum Field Theory (非有界作用素に対する time-ordered exponential と 場の量子論への応用) A dissertation submitted to Hokkaido University for the degree of Doctor of Sciences presented by Shinichiro Futakuchi Department of Mathematics Graduate School of Science Hokkaido University advised by Asao Arai March 2015 Abstract Time-ordered exponential is a fundamental tool in theoretical physics and mathematical physics, and often used in quantum theory to give the perturbative expansion of significant objects, such as the time evolution, the n-point correlation functions, and the scattering amplitudes. The time-ordered exponential generated by a bounded operator-valued function has already been well researched, but the one generated by an unbounded operator-valued function has not fully investigated so far; the mathematical theory which is applicable to the analysis of concrete models of quantum field theory has been inadequately studied. The first main purpose of this work is to provide a general mathematical theory on time-ordered exponential. The second main purpose is to construct concrete quantum field models and to analyze them rigorously. In this paper, we study the following: (I) construction of dynamics for non- symmetric Hamiltonians, (II) Gupta-Bleuler formalism, (III) Gell-Mann{Low formula, (IV) criteria for essential self-adjointness.
  • Geometric Flows and (Some Of) Their Physical Applications

    Geometric Flows and (Some Of) Their Physical Applications

    hep–th/0511057 CERN-PH-TH/2005-211 October 2005 Geometric flows and (some of) their physical applications∗ Ioannis Bakas† Theory Division, Department of Physics, CERN CH-1211 Geneva 23, Switzerland [email protected] Abstract The geometric evolution equations provide new ways to address a variety of non-linear problems in Riemannian geometry, and, at the same time, they enjoy numerous physical applications, most notably within the renormalization group analysis of non-linear sigma models and in general relativity. They are divided into classes of intrinsic and extrinsic curvature flows. Here, we review the main aspects of intrinsic geometric flows driven by the Ricci curvature, in various forms, and explain the intimate relation between Ricci and Calabi flows on K¨ahler manifolds using the notion of super-evolution. The integration of these flows on two-dimensional surfaces relies on the introduction of a novel class of arXiv:hep-th/0511057 v1 4 Nov 2005 infinite dimensional algebras with infinite growth. It is also explained in this context 2 how Kac’s K2 simple Lie algebra can be used to construct metrics on S with prescribed scalar curvature equal to the sum of any holomorphic function and its complex conjugate; applications of this special problem to general relativity and to a model of interfaces in statistical mechanics are also briefly discussed. ∗Based on an invited lecture at the Alexander von Humboldt foundation international conference on Advances in Physics and Astrophysics of the 21st Century, held from 6 to 11 September 2005 in Varna, Bulgaria; to appear in a Supplement to the Bulgarian Journal of Physics †Present (permanent) address: Department of Physics, University of Patras, 26500 Patras, Greece; e-mail: [email protected] The geometric evolution equations are parabolic systems that describe the deformation of metrics on Riemannian manifolds driven by their curvature in various forms.
  • Gauge Theory and Gravity in the Loop Formulation

    Gauge Theory and Gravity in the Loop Formulation

    6 Cb 3 Li I } ` VO Center for Gravitational Physics and Geometry The Pennsylvania State University University Pant, PA 16802-6300 ,. ,_ 3 (F V9 ;m8eio uyemusrpiroo etmrm CGPG-94/ 1-2 Gauge Theory and Gravity in the Loop Formulation Renate Lcll cam Lieeemi-;s. cassava Center fg; Ggavitatignal and Geometry, Pennsylvania State University, F‘@B@El 125 University Psi-k, PA 16802-62:00, USA 1 Introduction This article introduces the basic notions and mathematical structures of the so-called loop approach to gauge theory and gravity. The two theories will be treated in parallel as far as this is possible. Emphasis will be put on those aspects of the loop formulation that make it conceptually different from other, local formulations. This contribution is meant to be complementary to other articles in this volume, but some overlap has been unavoidable to ensure a coherent presentation. Many more mathematical details and references for those who want to learn more about loops are contained in a related review (Loll 1993d). Another useful reference focussing on the gravitational application is the review article by Pullin (1993). The- following Section 2 summarizes the classical and quantum description of both gauge theory and gravity as theories on connection space, to set the stage for Sections 3-5 which deal with specific aspects of the loop formulation. In Section 3, I define paths, loops and holonomies, and briefly review past applications in physics. Section 4 is devoted to the iden tities satisfied by the holonomies and their traces, the latter also known as the Mandelstam constraints.
  • Reduced Density Matrix Cumulants: the Combinatorics of Size-Consistency and Generalized Normal Ordering

    Reduced Density Matrix Cumulants: the Combinatorics of Size-Consistency and Generalized Normal Ordering

    Reduced Density Matrix Cumulants: The Combinatorics of Size-Consistency and Generalized Normal Ordering Jonathon P. Misiewicz, Justin M. Turney, and Henry F. Schaefer III∗ Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia, 30602 E-mail: [email protected] Abstract Reduced density matrix cumulants play key roles in the theory of both reduced den- sity matrices and multiconfigurational normal ordering, but the underlying formalism has remained mysterious. We present a new, simpler generating function for reduced density matrix cumulants that is formally identical to equating the coupled cluster and configuration interaction ans¨atze.This is shown to be a general mechanism to convert between a multiplicatively separable quantity and an additively separable quantity, as defined by a set of axioms. It is shown that both the cumulants of probability theory and reduced density matrices are entirely combinatorial constructions, where the differences can be associated to changes in the notion of \multiplicative separabil- ity" for expectation values of random variables compared to reduced density matrices. We compare our generating function to that of previous works and criticize previous claims of probabilistic significance of the reduced density matrix cumulants. Finally, we present the simplest proof to date of the Generalized Normal Ordering formalism to explore the role of reduced density matrix cumulants therein. 1 1 Introduction Reduced density matrix cumulants are fundamental in both reduced density matrix (RDM) theories and multireference theories that use the generalized normal ordering formalism (GNO) of Kutzelnigg and Mukherjee.1{5 To RDM theories, RDM cumulants are the size- consistent parts of the RDMs. This is one of the primary reasons why cumulants are either parameterized or varied directly in many RDM-based theories.6{11 In GNO, second quan- tized operators are decomposed into linear combinations of operators \normal ordered" with respect to an arbitrary reference, Ψ, via the Generalized Wick's Theorem.
  • Lecture Notes on General Relativity Sean M

    Lecture Notes on General Relativity Sean M

    Lecture Notes on General Relativity Sean M. Carroll Institute for Theoretical Physics University of California Santa Barbara, CA 93106 [email protected] December 1997 Abstract These notes represent approximately one semester’s worth of lectures on intro- ductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein’s equations, and three applications: grav- itational radiation, black holes, and cosmology. Individual chapters, and potentially updated versions, can be found at http://itp.ucsb.edu/~carroll/notes/. arXiv:gr-qc/9712019v1 3 Dec 1997 NSF-ITP/97-147 gr-qc/9712019 i Table of Contents 0. Introduction table of contents — preface — bibliography 1. Special Relativity and Flat Spacetime the spacetime interval — the metric — Lorentz transformations — spacetime diagrams — vectors — the tangent space — dual vectors — tensors — tensor products — the Levi-Civita tensor — index manipulation — electromagnetism — differential forms — Hodge duality — worldlines — proper time — energy-momentum vector — energy- momentum tensor — perfect fluids — energy-momentum conservation 2. Manifolds examples — non-examples — maps — continuity — the chain rule — open sets — charts and atlases — manifolds — examples of charts — differentiation — vectors as derivatives — coordinate bases — the tensor transformation law — partial derivatives are not tensors — the metric again — canonical form of the metric — Riemann normal coordinates — tensor densities — volume forms and integration 3. Curvature