Lagrangian Field Theory
|
|
- Joseph Daniels
- 5 years ago
- Views:
Transcription
1 Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part, we wll use the same notaton as Mandl and Shaw. The components of a contravarant four-vector x are denoted by x µ for µ = 0, 1,, 3 where x 0 = ct s the tme component and x j for j = 1,, 3 are the three spatal components. Unless otherwse specfed, any tme a Greek letter ndex s used, t wll range over 0, 1,, 3 and any tme a Latn letter ndex s used, t wll range over 1,, 3. We wll also use the bold face x to denote the three spatal coordnates of the four-vector x. The covarant metrc tensor we use s gven by g = (1.1) The notaton g µν refers to the entry n the µth row and νth column. A covarant vector s defned from the contravarant vector by the usual ndex-lowerng x µ = g µν x ν (1.) where the usual Ensten summaton conventons are used. The contravarant metrc tensor s defned by the equaton { g λµ g µν = δν λ 1 ν = λ = (1.3) 0 ν λ so t follows that g µν = g µν for every µ, ν,.e. the contravarant and covarant metrc tensors are the same. The metrc tensor s used to defne the generalzed scalar product of two four-vectors. For two four-vectors a and b, ther scalar product s defned as ab := a µ b µ = a µ g µν b ν = g µν a µ b ν = a 0 b 0 (a 1 b 1 + a b + a 3 b 3 ). (1.4) A Lorentz transformaton s a matrx Λ that preserves the scalar product xx for any four-vector x. Ths means that g µν (Λx) µ (Λx) ν = g µν Λ µ αx α Λ ν βx β = g µν x µ x ν. (1.5) We also nsst that each entry of the Lorentz transformaton s real. Lorentz transformatons also preserve the scalar product ab for any four-vectors a, b. We now defne some operators that we wll need later. µ := x µ (1.6) µ := x µ (1.7) := µ µ = 1 c t. (1.8) We also adopt the notaton that f an ndex s preceded by a comma, t means we are consderng the dervatve wth respect to that ndex. So for example, F,µ means F x µ. 1
2 Classcal pcture We start wth N felds φ r for 1 r N. We consder each feld to be a scalar feld on four-dmensonal spacetme. We assume that the system can be descrbed by a Lagrangan densty L (φ r, φ r,α ), (.1).e. the Lagrangan densty s only a functon of the felds and ther frst dervatves wth respect to tme and space. For an arbtrary regon Ω n spacetme, the acton ntegral s defned by S(Ω) := d 4 xl (φ r, φ r,α ). (.) Ω By mposng the usual prncple of least acton and nsstng that δs(ω) = 0, we can derve the usual Euler-Lagrange equatons L ( ) L φ r x α = 0 (.3) φ r,α for 1 r N and 0 α 3. We want to ntroduce the noton of a conjugate feld (analogous to the conjugate momentum to a generalzed coordnate n classc Lagrangan mechancs), but the problem s that system we are workng wth has uncountably many degrees of freedom. To get around ths, we approxmate t by a system wth only countably many degrees of freedom n the followng way. For a fxed tme t, decompose the three-dmensonal space nto small cells ndexed by, each of volume δx. If x denotes the center pont of the th cell, we approxmate each feld φ r by lettng φ r take the value φ r (x ) everywhere n the th cell. What ths accomplshes s that now we have a countable set of generalzed coordnates q r (t) := φ r (t, x ) =: φ r (t, ) (.4) that descrbe the system. We can also approxmate the spatal dervatves of φ r (t, ) n terms of the values of φ r n the adjacent cells. Thus the Lagrangan densty for the th cell takes the form L (φ r (t, ), φ r (t, ), φ r (t, )) (.5) where the dot represents the tme dervatve and denotes the ndex of any cell adjacent to the th. Then the total Lagrangan for the system s gven by L(t) = δx L (φ r (t, ), φ r (t, ), φ r (t, )) (.6) Now that we have dscrete varables descrbng the system, t s possble to defne the conjugate momenta n the usual way. We defne p r := L = L q r φ r (t, ) = δx π r (t, ) (.7) where π r (t, ) s defned to be π r (t, ) := We now can defne a Hamltonan densty and the usual Hamltonan L φ r (t, ). (.8) H := π r (t, ) φ r (t, ) L (.9) H = p r q r L = ( δx π r (t, ) φ ) r (t, ) L = δx H. (.10)
3 At ths pont, we want to brng our approxmaton closer to the actual system by takng a lmt as δx 0. Ths gves us the followng defntons and relatons. π r (x) := L φ (feld conjugate to φ r ) (.11) r L = d 3 xl (φ r (x), φ r,α (x)) (Lagrangan) (.1) H (x) := π r (x) φ r (x) L (φ r (x), φ r,α (x)) (Hamltonan densty) (.13) H = d 3 xh (x) (Hamltonan). (.14) Note that n analogy to classcal mechancs, f the Lagrangan densty does not depend explctly on tme, then the Hamltonan s constant n tme. 3 Quantum pcture Recall the dscrete approxmatons to the system that we started wth. We now want to quantze the model by nterpretng the generalzed coordnates and conjugate momenta as operators and mposng commutaton relatons on them. The commutaton relatons are chosen n analogy to the usual quantum-mechancal commutaton relatons. Now agan we take a lmt as δx 0 and we get 4 Example [φ r (t, ), π s (t, j)] := δ rsδ j δx (3.1) [φ r (t, ), φ s (t, j)] := [π r (t, ), π s (t, j)] := 0. (3.) [φ r (t, x), π s (t, x )] := δ rs δ(x x ) (3.3) [φ r (t, x), φ s (t, x )] := [π r (t, x), π s (t, x )] := 0. (3.4) Ths secton wll be dedcated to workng out an example of the above theory for a specfc system. Consder a system wth one real-valued feld φ and Lagrangan densty L = 1 ( φ,α φ α, µ φ ) (4.1) where µ s a constant. It turns out that ths Lagrangan densty corresponds to a spnless neutral boson wth mass µ/c. Usng the equaton of moton (.3), we have L φ = ( ) L x α (4.) µ φ = ( ) 1 x α φ, α (4.3) = 1 α α φ (4.4) = 1 φ (4.5) so the equaton of moton s ( ) 1 + µ φ = 0. (4.6) 3
4 Ths s the Klen-Gordon equaton. The feld conjugate to φ defned by (.11) s The Hamltonan densty s The commutaton relatons become 5 Conservaton laws π(x) = L φ = 1 c φ(x). (4.7) H = π(x) φ(x) L (4.8) = 1 c φ(x) 1 ( φ,α φ α, µ φ ) (4.9) = 1 ( c π(x) + ( φ) + µ φ ). (4.10) [φ(t, x), φ(t, x )] = δ(x x ) (4.11) [φ(t, x), φ(t, x )] = 0 (4.1) [π(t, x), π(t, x )] = 1 c 4 [ φ(t, x), φ(t, x )] = 0. (4.13) It s a general prncple of physcs that any mathematcal symmetres n the Lagrangan of the system correspond to some conserved quantty n the physcal system. For example, n classcal mechancs, a translatonnvarant Lagrangan corresponds to the conservaton of energy and a rotaton-nvarant Lagrangan corresponds to the conservaton of angular momentum. We can also apply ths dea to the quantum case. Consder a transformaton of a feld φ of the form Ths wll cause the Lagrangan densty to change lke δl φ(x) φ(x) + δφ(x). (5.1) = L L δφ + δφ,α φ = α If the orgnal transformaton s a symmetry, then we wll have δl = 0, so where f α s defned as ( ) L δφ. (5.) α f α = 0 (5.3) f α := L δφ. (5.4) Now we want to nvestgate whch quantty wll be conserved as a result of ths symmetry. Defne F α (t) := d 3 xf α (t, x). (5.5) From equaton (5.3), we have It follows that the quantty s conserved. 1 df 0 (t) c dt = d 3 x j f j (t, x) = 0. (5.6) F 0 = d 3 xf 0 (t, x) (5.7) R 3 = d 3 x L δφ (5.8) φ = d 3 xπ(t, x)δφ (5.9) 4
5 6 Example In ths secton we wll consder an example of the theory developed n the prevous secton. If φ s a complexvalued feld, then we treat φ and φ as ndependent felds, where φ denotes the complex conjugate of φ. We wll suppose that the Lagrangan densty L s nvarant under nfntesmal rotatons of the form so that n the notaton of the above secton, we have φ exp(ɛ)φ (1 + ɛ)φ (6.1) φ exp( ɛ)φ (1 ɛ)φ (6.) δφ = ɛφ (6.3) δφ = ɛφ. (6.4) Now the conserved quantty from equaton (5.9) becomes F 0 = ɛc d 3 x ( π(x)φ(x) π(x)φ(x) ). (6.5) We can scale by any constant factor we want, so we rename Q := q d 3 x ( π(x)φ(x) π(x)φ(x) ) (6.6) where q s an undetermned constant at the moment. We want to see how the operator Q acts on our orgnal feld φ, so we compute the commutator [Q, φ(x)] = q d 3 x [π(x ), φ(x)]φ(x) (6.7) = qφ(x). (6.8) Ths result ndcates that when the operator Q acts on the feld φ, t scales t by a factor of q. Smlarly, f t were to act of φ, t would scale t by a factor of q. Because of ths, the operators φ and φ can be nterpreted as creaton and absorpton operators for electrc charge. References [1] F. Mandl and G. Shaw. Quantum Feld Theory. Wley,
Lecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 7 Specal Relatvty (Chapter 7) What We Dd Last Tme Worked on relatvstc knematcs Essental tool for epermental physcs Basc technques are easy: Defne all 4 vectors Calculate c-o-m
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationLAGRANGIAN MECHANICS
LAGRANGIAN MECHANICS Generalzed Coordnates State of system of N partcles (Newtonan vew): PE, KE, Momentum, L calculated from m, r, ṙ Subscrpt covers: 1) partcles N 2) dmensons 2, 3, etc. PE U r = U x 1,
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More informationQuantum Mechanics I Problem set No.1
Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationAnalytical classical dynamics
Analytcal classcal ynamcs by Youun Hu Insttute of plasma physcs, Chnese Acaemy of Scences Emal: yhu@pp.cas.cn Abstract These notes were ntally wrtten when I rea tzpatrck s book[] an were later revse to
More informationClassical Field Theory
Classcal Feld Theory Before we embark on quantzng an nteractng theory, we wll take a dverson nto classcal feld theory and classcal perturbaton theory and see how far we can get. The reader s expected to
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationHomework & Solution. Contributors. Prof. Lee, Hyun Min. Particle Physics Winter School. Park, Ye
Homework & Soluton Prof. Lee, Hyun Mn Contrbutors Park, Ye J(yej.park@yonse.ac.kr) Lee, Sung Mook(smlngsm0919@gmal.com) Cheong, Dhong Yeon(dhongyeoncheong@gmal.com) Ban, Ka Young(ban94gy@yonse.ac.kr) Ro,
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More informationLorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3
Lorentz Group Lng Fong L ontents Lorentz group. Generators............................................. Smple representatons..................................... 3 Lorentz group In the dervaton of Drac
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationDynamics of a Superconducting Qubit Coupled to an LC Resonator
Dynamcs of a Superconductng Qubt Coupled to an LC Resonator Y Yang Abstract: We nvestgate the dynamcs of a current-based Josephson juncton quantum bt or qubt coupled to an LC resonator. The Hamltonan of
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationwhere the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt
Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationON MECHANICS WITH VARIABLE NONCOMMUTATIVITY
ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY CIPRIAN ACATRINEI Natonal Insttute of Nuclear Physcs and Engneerng P.O. Box MG-6, 07725-Bucharest, Romana E-mal: acatrne@theory.npne.ro. Receved March 6, 2008
More informationHW #6, due Oct Toy Dirac Model, Wick s theorem, LSZ reduction formula. Consider the following quantum mechanics Lagrangian,
HW #6, due Oct 5. Toy Drac Model, Wck s theorem, LSZ reducton formula. Consder the followng quantum mechancs Lagrangan, L ψ(σ 3 t m)ψ, () where σ 3 s a Paul matrx, and ψ s defned by ψ ψ σ 3. ψ s a twocomponent
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More informationThree views of mechanics
Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationThe Noether theorem. Elisabet Edvardsson. Analytical mechanics - FYGB08 January, 2016
The Noether theorem Elsabet Evarsson Analytcal mechancs - FYGB08 January, 2016 1 1 Introucton The Noether theorem concerns the connecton between a certan kn of symmetres an conservaton laws n physcs. It
More informationCHAPTER 5: Lie Differentiation and Angular Momentum
CHAPTER 5: Le Dfferentaton and Angular Momentum Jose G. Vargas 1 Le dfferentaton Kähler s theory of angular momentum s a specalzaton of hs approach to Le dfferentaton. We could deal wth the former drectly,
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationLecture Notes 7: The Unruh Effect
Quantum Feld Theory for Leg Spnners 17/1/11 Lecture Notes 7: The Unruh Effect Lecturer: Prakash Panangaden Scrbe: Shane Mansfeld 1 Defnng the Vacuum Recall from the last lecture that choosng a complex
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More information4. Laws of Dynamics: Hamilton s Principle and Noether's Theorem
4. Laws of Dynamcs: Hamlton s Prncple and Noether's Theorem Mchael Fowler Introducton: Galleo and Newton In the dscusson of calculus of varatons, we antcpated some basc dynamcs, usng the potental energy
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More informationQuantum Particle Motion in Physical Space
Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationIntegrals and Invariants of
Lecture 16 Integrals and Invarants of Euler Lagrange Equatons NPTEL Course Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng, Indan Insttute of Scence, Banagalore
More informationDifferentiating Gaussian Processes
Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationReview of Newtonian Mechanics
hapter 1 Revew of Newtonan Mechancs 1.1 Why Study lasscal Mechancs? Quantum lmt Relatvstc lmt General relatvty Mathematcal technques Frst approxmaton Intuton 1.2 Revew of Newtonan Mechancs Basc defntons
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More informationClassical Mechanics Virtual Work & d Alembert s Principle
Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UM-DAE Centre for Excellence n Basc Scences Kalna, Mumba 400098 August 15, 2016 1 Constrants Moton of a system of partcles s often
More information14 The Statement of AdS/CFT
14 The Statement of AdS/CFT 14.1 The Dctonary Choose coordnates ds 2 = `2 z 2 (dz2 + dx 2 ) (14.1) on Eucldean AdS d+1,wherex s a coordnate on R d.theboundarysatz =0. We showed above that scatterng problems
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationNATURAL 2-π STRUCTURES IN LAGRANGE SPACES
AALELE ŞTIIŢIFICE ALE UIVERSITĂŢII AL.I. CUZA DI IAŞI (S.. MATEMATICĂ, Tomul LIII, 2007, Suplment ATURAL 2-π STRUCTURES I LAGRAGE SPACES Y VICTOR LĂUŢA AD VALER IMIEŢ Dedcated to Academcan Radu Mron at
More informationAPPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More informationChapter 8. Potential Energy and Conservation of Energy
Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal
More information8.323: QFT1 Lecture Notes
8.33: QFT1 Lecture Notes Joseph A. Mnahan c MIT, Sprng 11 Preface Ths volume s a complaton of eght nstallments of notes that I provded for the students who took Relatvstc Quantum Feld Theory 1 8.33 at
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationOne Dimension Again. Chapter Fourteen
hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons
More informationPerfect Fluid Cosmological Model in the Frame Work Lyra s Manifold
Prespacetme Journal December 06 Volume 7 Issue 6 pp. 095-099 Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold Perfect Flud Cosmologcal Model n the Frame Work Lyra
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationCausal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
More informationSolutions to Problem Set 6
Solutons to Problem Set 6 Problem 6. (Resdue theory) a) Problem 4.7.7 Boas. n ths problem we wll solve ths ntegral: x sn x x + 4x + 5 dx: To solve ths usng the resdue theorem, we study ths complex ntegral:
More informationYukawa Potential and the Propagator Term
PHY304 Partcle Physcs 4 Dr C N Booth Yukawa Potental an the Propagator Term Conser the electrostatc potental about a charge pont partcle Ths s gven by φ = 0, e whch has the soluton φ = Ths escrbes the
More informationIterative General Dynamic Model for Serial-Link Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
More informationFor now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.
Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson
More informationFinslerian Nonholonomic Frame For Matsumoto (α,β)-metric
Internatonal Journal of Mathematcs and Statstcs Inventon (IJMSI) E-ISSN: 2321 4767 P-ISSN: 2321-4759 ǁ Volume 2 ǁ Issue 3 ǁ March 2014 ǁ PP-73-77 Fnsleran Nonholonomc Frame For Matsumoto (α,)-metrc Mallkarjuna
More informationPhysics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints
Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or
More informationA Quantum Gauss-Bonnet Theorem
A Quantum Gauss-Bonnet Theorem Tyler Fresen November 13, 2014 Curvature n the plane Let Γ be a smooth curve wth orentaton n R 2, parametrzed by arc length. The curvature k of Γ s ± Γ, where the sgn s postve
More informationScattering of two identical particles in the center-of. of-mass frame. (b)
Lecture # November 5 Scatterng of two dentcal partcle Relatvtc Quantum Mechanc: The Klen-Gordon equaton Interpretaton of the Klen-Gordon equaton The Drac equaton Drac repreentaton for the matrce α and
More informationPOINCARE ALGEBRA AND SPACE-TIME CRITICAL DIMENSIONS FOR PARASPINNING STRINGS
POINCARE ALGEBRA AND SPACE-TIME CRITICAL DIMENSIONS FOR PARASPINNING STRINGS arxv:hep-th/0055v 7 Oct 00 N.BELALOUI, AND H.BENNACER LPMPS, Département de Physque, Faculté des Scences, Unversté Mentour Constantne,
More informationQuantum Field Theory III
Quantum Feld Theory III Prof. Erck Wenberg February 16, 011 1 Lecture 9 Last tme we showed that f we just look at weak nteractons and currents, strong nteracton has very good SU() SU() chral symmetry,
More informationSpecial Relativity and Riemannian Geometry. Department of Mathematical Sciences
Tutoral Letter 06//018 Specal Relatvty and Reannan Geoetry APM3713 Seester Departent of Matheatcal Scences IMPORTANT INFORMATION: Ths tutoral letter contans the solutons to Assgnent 06. BAR CODE Learn
More informationLecture 4: Universal Hash Functions/Streaming Cont d
CSE 5: Desgn and Analyss of Algorthms I Sprng 06 Lecture 4: Unversal Hash Functons/Streamng Cont d Lecturer: Shayan Oves Gharan Aprl 6th Scrbe: Jacob Schreber Dsclamer: These notes have not been subjected
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationSUPPLEMENTARY INFORMATION
do: 0.08/nature09 I. Resonant absorpton of XUV pulses n Kr + usng the reduced densty matrx approach The quantum beats nvestgated n ths paper are the result of nterference between two exctaton paths of
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationRobert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations
Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton
More informationClassical Mechanics ( Particles and Biparticles )
Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles
More informationGroup Analysis of Ordinary Differential Equations of the Order n>2
Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationComparative Studies of Law of Conservation of Energy. and Law Clusters of Conservation of Generalized Energy
Comparatve Studes of Law of Conservaton of Energy and Law Clusters of Conservaton of Generalzed Energy No.3 of Comparatve Physcs Seres Papers Fu Yuhua (CNOOC Research Insttute, E-mal:fuyh1945@sna.com)
More informationNormal Coordinates Describing Coupled Oscillations in the Gravitational Field
Normal Coordnates Descrbng Coupled Oscllatons n the Gravtatonal Feld Walter James Chrstensen Jr. Department of Physcs Cal State Unversty, Fullerton 8 N. State College Blvd. Fullerton, CA 983 And Department
More information6. Hamilton s Equations
6. Hamlton s Equatons Mchael Fowler A Dynamcal System s Path n Confguraton Sace and n State Sace The story so far: For a mechancal system wth n degrees of freedom, the satal confguraton at some nstant
More information