# Path Integral in Quantum Field Theory Alexander Belyaev (Course Based on Lectures by Steven King) Contents

Total Page：16

File Type：pdf, Size：1020Kb

Path Integral in Quantum Field Theory Alexander Belyaev (course based on Lectures by Steven King) Contents 1 Preliminaries 5 1.1 Review of Classical Mechanics of Finite System . 5 1.2 Review of Non-Relativistic Quantum Mechanics . 7 1.3 Relativistic Quantum Mechanics . 14 1.3.1 Relativistic Conventions and Notation . 14 1.3.2 TheKlein-GordonEquation . 15 1.4 ProblemsSet1 ........................... 18 2 The Klein-Gordon Field 19 2.1 Introduction............................. 19 2.2 ClassicalScalarFieldTheory . 20 2.3 QuantumScalarFieldTheory . 28 2.4 ProblemsSet2 ........................... 35 3 Interacting Klein-Gordon Fields 37 3.1 Introduction............................. 37 3.2 PerturbationandScatteringTheory. 37 3.3 TheInteractionHamiltonian. 43 3.4 Example: K π+π− ....................... 45 S → 3.5 Wick’s Theorem, Feynman Propagator, Feynman Diagrams . .. 47 3.6 TheLSZReductionFormula. 52 3.7 ProblemsSet3 ........................... 58 4 Transition Rates and Cross-Sections 61 4.1 TransitionRates .......................... 61 4.2 TheNumberofFinalStates . 63 4.3 Lorentz Invariant Phase Space (LIPS) . 63 4.4 CrossSections............................ 64 4.5 Two-bodyScattering . 65 4.6 DecayRates............................. 66 4.7 OpticalTheorem .......................... 66 4.8 ProblemsSet4 ........................... 68 1 2 CONTENTS 5 Path Integrals in Quantum Mechanics 69 5.1 Introduction............................. 69 5.2 The Point to Point Transition Amplitude . 70 5.3 ImaginaryTime........................... 74 5.4 Transition Amplitudes With an External Driving Force . ... 77 5.5 Expectation Values of Heisenberg Position Operators . .... 81 5.6 Appendix .............................. 83 5.6.1 GaussianIntegration . 83 5.6.2 Functionals ......................... 85 5.7 ProblemsSet5 ........................... 87 6 Path Integral Quantisation of the Klein-Gordon Field 89 6.1 Introduction............................. 89 6.2 TheFeynmanPropagator(again) . 91 6.3 Green’s Functions in Free Field Theory . 94 6.4 Green’s Functions for λφ4 Theory ................. 97 6.5 The 2 2ScatteringAmplitudefromLSZ . 102 → 6.6 Appendix: ProofofEq.6.4.6 . 104 6.7 ProblemsSet6 ........................... 105 7 The Dirac Equation 107 7.1 Relativistic Wave Equations: Reprise . 107 7.2 TheDiracEquation. 108 7.3 Free Particle Solutions I: Interpretation . 109 7.4 FreeParticleSolutionsII:Spin. 112 7.5 Normalisation,GammaMatrices. 114 7.6 LorentzCovariance . 115 7.7 Parity ................................ 119 7.8 BilinearCovariants . 119 7.9 ChargeConjugation. 120 7.10Neutrinos .............................. 121 8 The Free Dirac Field 125 8.1 CanonicalQuantisation. 125 8.2 PathIntegralQuantisation. 127 8.3 TheFeynman PropagatorfortheDiracField. 129 8.4 Appendix: Grassmann Variables . 131 9 The Free Electromagnetic Field 133 9.1 The Classical Electromagnetic Field . 133 9.2 CanonicalQuantisation. 136 9.3 PathIntegralQuantisation. 138 CONTENTS 3 10 Quantum Electrodynamics 141 10.1 FeynmanRulesofQED . 141 10.2 Electron–MuonScattering . 143 10.3 Electron–ElectronScattering. 145 10.4 Electron–Positron Annihilation . 146 10.4.1 e+e− e+e− ........................ 146 → 10.4.2 e+e− µ+µ− and e+e− hadrons. 146 10.5 ComptonScattering→ . .→ . 147 4 CONTENTS Chapter 1 Preliminaries 1.1 Review of Classical Mechanics of Finite Sys- tem Consider a ﬁnite classical system of particles whose generalised coordinates are qi where i =1,...,N. For example N =3n for n particles in three dimensions. Suppose the Lagrangian given by L = T V , T and V being the kinetic and − potential energy, is: L = L(qi, q˙i) (1.1.1) whereq ˙i denotes the time derivative of qi. For each set of paths qi(t) connecting (qi1, t1) to (qi2, t2) the action is deﬁned by: t2 S[qi(t)] = Ldt (1.1.2) Zt1 Note that S is a functional, i.e. a number whose value depends on a set of functions qi(t). The classical paths are those which minimise (or in general extremise) the numerical value of S, subject to the boundary conditions δqi =0 at the initial and ﬁnal times t1 and t2. This is the well-known action principle (which can be derived from Feynman’s path integral approach to quantum mechanics – see later) and it leads to the N second order (in time) diﬀerential Lagrange equations of motion: d ∂L ∂L =0 (1.1.3) dt ∂q˙i − ∂qi In the Lagrange formalism the independent variables are the coordinates qi and the velocitiesq ˙i. The generalised momenta pi are derived quantities deﬁned by: ∂L pi (1.1.4) ≡ ∂q˙i In the Hamiltonian formalism the variablesq ˙i are traded in for pi, and instead of working with a Lagrangian L = L(qi, q˙i) one works with a Hamiltonian 5 6 CHAPTER 1. PRELIMINARIES H = H(qi,pi) deﬁned by the so-called Legendre transformation: H(q ,p ) p q˙ L(q , q˙ ) (1.1.5) i i ≡ i i − i i Xi where it is understood that theq ˙i are to be written as functions of qi and pi. From Eqs.1.1.3-1.1.5 one readily obtains the Hamilton equations of motion: ∂H ∂H p˙i = , q˙i = (1.1.6) − ∂qi ∂pi which are 2N ﬁrst order (in time) diﬀerential equations. Just as (for a conser- vative force) L = T V , so H = T + V , so the Hamiltonian is just the total − energy in this case. Example 1: The Harmonic Oscillator First consider a single oscillator in oscillating in one dimension (e.g. mass on a spring) The Lagrangian is: 1 1 L = T V = mq˙2 ω2q2 (1.1.7) − 2 − 2 The generalised momentum is: ∂L p = = mq˙ (1.1.8) ∂q˙ The Hamiltonian is: p2 ω2 H = pq˙ L = + q2 (1.1.9) − 2m 2 Example 2: Coupled Harmonic Oscillators Now consider a linear chain of coupled oscillators, say corresponding to a vibrating line of atoms. Let the coordinate qi denote the displacement of the i th atom in the line from its equilibrium position, where we assume that all displacements− correspond to longitudinal vibrations along the length of the line of atoms. To make the system ﬁnite, let us form a closed loop of N atoms such that qi+N = qi. The Lagrangian is: N 1 Ω2 Ω2 L = q˙ 2 (q q )2 0 (q )2 (1.1.10) 2 i − 2 i − i+1 − 2 i Xi=1 where we have taken the atoms to have unit mass, and have included a nearest neighbour coupling plus a second independent frequency Ω0. The Lagrange equations of motion are: q¨ = Ω2(2q q q ) Ω2q (1.1.11) i − i − i−1 − i+1 − 0 i 1.2. REVIEW OF NON-RELATIVISTIC QUANTUM MECHANICS 7 The generalised momenta are: ∂L pi = =q ˙i (1.1.12) ∂q˙i The Hamiltonian is: N 1 Ω2 Ω2 H = p 2 + (q q )2 + 0 (q )2 (1.1.13) 2 i 2 i − i+1 2 i Xi=1 Normal Modes The oscillators above can be uncoupled by taking a discretised Fourier transform of the form N/2 q˜ q = k eij2πk/N (1.1.14) j N 1/2 k=X−N/2 N/2 p˜ p = k e−ij2πk/N (1.1.15) j N 1/2 k=X−N/2 which satisﬁes the periodic boundary conditions. Note that i2 = 1 and we have labelled the oscillators by j to avoid confusion. − Since the qj are real theq ˜k andp ˜k are complex and satisfy ⋆ ⋆ q˜−k =q ˜k, p˜−k =p ˜k. (1.1.16) Using the result: N ij2π(k−k′)/N e = δkk′ N jX=1 the Hamiltonian becomes, 1 N/2 H = p˜ p˜⋆ + ω2q˜ q˜⋆ (1.1.17) 2 k k k k k k=X−N/2 which corresponds to a sum of uncoupled harmonic oscillators of frequencies given by 2 2 2 2 ωk = Ω 4 sin (πk/N) + Ω0 (1.1.18) 1.2 Review of Non-Relativistic Quantum Me- chanics In quantum mechanics, 1 physical states are represented by vectors ψ > in a vector space (Hilbert space). Physical observables are represented by| linear 1We shall set h/(2π)= c = 1 throughout. 8 CHAPTER 1. PRELIMINARIES Hermitian operatorsω ˆ and the result of a measurement of some observableω ˆ is one of its (real) eigenvalues ω where ωˆ ω>= ω ω> (1.2.1) | | A very useful result is the completeness relation: ∞ I = ω ><ω Z−∞ | | which follows from the fact that the eigenvectors of an Hermitian operator form a complete orthonormal basis, where I is the unit operator. If the system is in some state ψ > and some observableω ˆ is measured then the system will be forced into one| of the eigenstates ψ > ω >. The result of the measurement | →| will be one of the eigenvalues ω which is not classically predictable, but which occurs with a probability P (ω) given by <ω ψ> 2 P (ω)= | | | (1.2.2) < ψ ψ > | The average or expectation value is therefore, < ψ ωˆ ψ > < ωˆ >= | | (1.2.3) < ψ ψ > | Now let us assume we are given a classical Hamiltonian in some particular basis of coordinates H = H(qi,pi). According to quantum mechanics we must replace the number-valued variables qi and pi by Hermitian operators which satisfy the commutation relations: [ˆqi, pˆj]= iIδij, [ˆqi, qˆj]=[ˆpi, pˆj]=0 (1.2.4) where I is the unit operator, 0 is the null operator, [A, B]= AB BA and δij is the Kronecker delta symbol deﬁned in the usual way − δij =1 (i = j) δ =0 (i = j) ij 6 Each coordinate has its own complete set of eigenvectors and real eigenvalues qˆ q >= q q >, pˆ p >= p p > i| i i| i i| i i| i The position and momentum space wavefunctions are thus ψ(q , q ,...)=< q < q . ψ > 1 2 1| 2| | and ψ(p ,p ,...)=<p <p . ψ > 1 2 1| 2| | 1.2. REVIEW OF NON-RELATIVISTIC QUANTUM MECHANICS 9 In one dimension we have simply, ψ(q)=< q ψ >, ψ(p)=<p ψ > | | The way the time dependence of a quantum system is described is a matter of taste and convenience.

150
Recommended publications
• Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation speciﬁes how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases.
[Show full text]
• A New Renormalization Scheme of Fermion Field in Standard Model
A New Renormalization Scheme of Fermion Field in Standard Model Yong Zhoua a Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China (Jan 10, 2003) In order to obtain proper wave-function renormalization constants for unstable fermion and consist with Breit-Wigner formula in the resonant region, We have assumed an extension of the LSZ reduction formula for unstable fermion and adopted on-shell mass renormalization scheme in the framework of without ﬁeld renormaization. The comparison of gauge dependence of physical amplitude between on-shell mass renormalization and complex-pole mass renormalization has been discussed. After obtaining the fermion wave-function renormalization constants, we extend them to two matrices in order to include the contributions of oﬀ-diagonal two-point diagrams at fermion external legs for the convenience of calculations of S-matrix elements. 11.10.Gh, 11.55.Ds, 12.15.Lk I. INTRODUCTION We are interested in the LSZ reduction formula [1] in the case of unstable particle, for the purpose of obtaining a proper wave function renormalization constant(wrc.) for unstable fermion. In this process we have demanded the form of the unstable particle’s propagator in the resonant region resembles the relativistic Breit-Wigner formula. Therefore the on-shell mass renormalization scheme is a suitable choice compared with the complex-pole mass renormalization scheme [2], as we will discuss in section 2. We also discard the ﬁeld renormalization constants which have been proven at question in the case of having fermions mixing [3]. The layout of this article is as follows.
[Show full text]
• An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
[Show full text]
• External Fields and Measuring Radiative Corrections
Quantum Electrodynamics (QED), Winter 2015/16 Radiative corrections External ﬁelds and measuring radiative corrections We now brieﬂy describe how the radiative corrections Πc(k), Σc(p) and Λc(p1; p2), the ﬁnite parts of the loop diagrams which enter in renormalization, lead to observable effects. We will consider the two classic cases: the correction to the electron magnetic moment and the Lamb shift. Both are measured using a background electric or magnetic ﬁeld. Calculating amplitudes in this situation requires a modiﬁed perturbative expansion of QED from the one we have considered thus far. Feynman rules with external ﬁelds An important approximation in QED is where one has a large background electromagnetic ﬁeld. This external ﬁeld can effectively be treated as a classical background in which we quantise the fermions (in analogy to including an electromagnetic ﬁeld in the Schrodinger¨ equation). More formally we can always expand the ﬁeld operators around a non-zero value (e) Aµ = aµ + Aµ (e) where Aµ is a c-number giving the external ﬁeld and aµ is the ﬁeld operator describing the ﬂuctua- tions. The S-matrix is then given by R 4 ¯ (e) S = T eie d x : (a=+A= ) : For a static external ﬁeld (independent of time) we have the Fourier transform representation Z (e) 3 ik·x (e) Aµ (x) = d= k e Aµ (k) We can now consider scattering processes such as an electron interacting with the background ﬁeld. We ﬁnd a non-zero amplitude even at ﬁrst order in the perturbation expansion. (Without a background ﬁeld such terms are excluded by energy-momentum conservation.) We have Z (1) 0 0 4 (e) hfj S jii = ie p ; s d x : ¯A= : jp; si Z 4 (e) 0 0 ¯− µ + = ie d x Aµ (x) p ; s (x)γ (x) jp; si Z Z 4 3 (e) 0 µ ik·x+ip0·x−ip·x = ie d x d= k Aµ (k)u ¯s0 (p )γ us(p) e ≡ 2πδ(E0 − E)M where 0 (e) 0 M = ieu¯s0 (p )A= (p − p)us(p) We see that energy is conserved in the scattering (since the external ﬁeld is static) but in general the initial and ﬁnal 3-momenta of the electron can be different.
[Show full text]
• Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
[Show full text]
• Introductory Lectures on Quantum Field Theory
Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum ﬁeld theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difﬁcult to give a reasonable introduction to a subject as vast as quantum ﬁeld theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum ﬁeld theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties.
[Show full text]
• Lecture Notes Particle Physics II Quantum Chromo Dynamics 6
Lecture notes Particle Physics II Quantum Chromo Dynamics 6. Feynman rules and colour factors Michiel Botje Nikhef, Science Park, Amsterdam December 7, 2012 Colour space We have seen that quarks come in three colours i = (r; g; b) so • that the wave function can be written as (s) µ ci uf (p ) incoming quark 8 (s) µ ciy u¯f (p ) outgoing quark i = > > (s) µ > cy v¯ (p ) incoming antiquark <> i f c v(s)(pµ) outgoing antiquark > i f > Expressions for the> 4-component spinors u and v can be found in :> Griﬃths p.233{4. We have here explicitly indicated the Lorentz index µ = (0; 1; 2; 3), the spin index s = (1; 2) = (up; down) and the ﬂavour index f = (d; u; s; c; b; t). To not overburden the notation we will suppress these indices in the following. The colour index i is taken care of by deﬁning the following basis • vectors in colour space 1 0 0 c r = 0 ; c g = 1 ; c b = 0 ; 0 1 0 1 0 1 0 0 1 @ A @ A @ A for red, green and blue, respectively. The Hermitian conjugates ciy are just the corresponding row vectors. A colour transition like r g can now be described as an • ! SU(3) matrix operation in colour space. Recalling the SU(3) step operators (page 1{25 and Exercise 1.8d) we may write 0 0 0 0 1 cg = (λ1 iλ2) cr or, in full, 1 = 1 0 0 0 − 0 1 0 1 0 1 0 0 0 0 0 @ A @ A @ A 6{3 From the Lagrangian to Feynman graphs Here is QCD Lagrangian with all colour indices shown.29 • µ 1 µν a a µ a QCD = (iγ @µ m) i F F gs λ j γ A L i − − 4 a µν − i ij µ µν µ ν ν µ µ ν F = @ A @ A 2gs fabcA A a a − a − b c We have introduced here a second colour index a = (1;:::; 8) to label the gluon ﬁelds and the corresponding SU(3) generators.
[Show full text]
• Quantum Vacuum Energy Density and Unifying Perspectives Between Gravity and Quantum Behaviour of Matter
Annales de la Fondation Louis de Broglie, Volume 42, numéro 2, 2017 251 Quantum vacuum energy density and unifying perspectives between gravity and quantum behaviour of matter Davide Fiscalettia, Amrit Sorlib aSpaceLife Institute, S. Lorenzo in Campo (PU), Italy corresponding author, email: [email protected] bSpaceLife Institute, S. Lorenzo in Campo (PU), Italy Foundations of Physics Institute, Idrija, Slovenia email: [email protected] ABSTRACT. A model of a three-dimensional quantum vacuum based on Planck energy density as a universal property of a granular space is suggested. This model introduces the possibility to interpret gravity and the quantum behaviour of matter as two diﬀerent aspects of the same origin. The change of the quantum vacuum energy density can be considered as the fundamental medium which determines a bridge between gravity and the quantum behaviour, leading to new interest- ing perspectives about the problem of unifying gravity with quantum theory. PACS numbers: 04. ; 04.20-q ; 04.50.Kd ; 04.60.-m. Key words: general relativity, three-dimensional space, quantum vac- uum energy density, quantum mechanics, generalized Klein-Gordon equation for the quantum vacuum energy density, generalized Dirac equation for the quantum vacuum energy density. 1 Introduction The standard interpretation of phenomena in gravitational ﬁelds is in terms of a fundamentally curved space-time. However, this approach leads to well known problems if one aims to ﬁnd a unifying picture which takes into account some basic aspects of the quantum theory. For this reason, several authors advocated diﬀerent ways in order to treat gravitational interaction, in which the space-time manifold can be considered as an emergence of the deepest processes situated at the fundamental level of quantum gravity.
[Show full text]
• Solving the Quantum Scattering Problem for Systems of Two and Three Charged Particles
Solving the quantum scattering problem for systems of two and three charged particles Solving the quantum scattering problem for systems of two and three charged particles Mikhail Volkov c Mikhail Volkov, Stockholm 2011 ISBN 978-91-7447-213-4 Printed in Sweden by Universitetsservice US-AB, Stockholm 2011 Distributor: Department of Physics, Stockholm University In memory of Professor Valentin Ostrovsky Abstract A rigorous formalism for solving the Coulomb scattering problem is presented in this thesis. The approach is based on splitting the interaction potential into a ﬁnite-range part and a long-range tail part. In this representation the scattering problem can be reformulated to one which is suitable for applying exterior complex scaling. The scaled problem has zero boundary conditions at inﬁnity and can be implemented numerically for ﬁnding scattering amplitudes. The systems under consideration may consist of two or three charged particles. The technique presented in this thesis is ﬁrst developed for the case of a two body single channel Coulomb scattering problem. The method is mathe- matically validated for the partial wave formulation of the scattering problem. Integral and local representations for the partial wave scattering amplitudes have been derived. The partial wave results are summed up to obtain the scat- tering amplitude for the three dimensional scattering problem. The approach is generalized to allow the two body multichannel scattering problem to be solved. The theoretical results are illustrated with numerical calculations for a number of models. Finally, the potential splitting technique is further developed and validated for the three body Coulomb scattering problem. It is shown that only a part of the total interaction potential should be split to obtain the inhomogeneous equation required such that the method of exterior complex scaling can be applied.
[Show full text]
• What Is the Dirac Equation?
What is the Dirac equation? M. Burak Erdo˘gan ∗ William R. Green y Ebru Toprak z July 15, 2021 at all times, and hence the model needs to be ﬁrst or- der in time, [Tha92]. In addition, it should conserve In the early part of the 20th century huge advances the L2 norm of solutions. Dirac combined the quan- were made in theoretical physics that have led to tum mechanical notions of energy and momentum vast mathematical developments and exciting open operators E = i~@t, p = −i~rx with the relativis- 2 2 2 problems. Einstein's development of relativistic the- tic Pythagorean energy relation E = (cp) + (E0) 2 ory in the ﬁrst decade was followed by Schr¨odinger's where E0 = mc is the rest energy. quantum mechanical theory in 1925. Einstein's the- Inserting the energy and momentum operators into ory could be used to describe bodies moving at great the energy relation leads to a Klein{Gordon equation speeds, while Schr¨odinger'stheory described the evo- 2 2 2 4 lution of very small particles. Both models break −~ tt = (−~ ∆x + m c ) : down when attempting to describe the evolution of The Klein{Gordon equation is second order, and does small particles moving at great speeds. In 1927, Paul not have an L2-conservation law. To remedy these Dirac sought to reconcile these theories and intro- shortcomings, Dirac sought to develop an operator1 duced the Dirac equation to describe relativitistic quantum mechanics. 2 Dm = −ic~α1@x1 − ic~α2@x2 − ic~α3@x3 + mc β Dirac's formulation of a hyperbolic system of par- tial diﬀerential equations has provided fundamental which could formally act as a square root of the Klein- 2 2 2 models and insights in a variety of ﬁelds from parti- Gordon operator, that is, satisfy Dm = −c ~ ∆ + 2 4 cle physics and quantum ﬁeld theory to more recent m c .
[Show full text]
• Dirac Equation - Wikipedia
Dirac equation - Wikipedia https://en.wikipedia.org/wiki/Dirac_equation Dirac equation From Wikipedia, the free encyclopedia In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it 1 describes all spin-2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity,[1] and was the ﬁrst theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the ﬁne details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally conﬁrmed several years later. It also provided a theoretical justiﬁcation for the introduction of several component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. Although Dirac did not at ﬁrst fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics.
[Show full text]
• Appendix: the Interaction Picture
Appendix: The Interaction Picture The technique of expressing operators and quantum states in the so-called interaction picture is of great importance in treating quantum-mechanical problems in a perturbative fashion. It plays an important role, for example, in the derivation of the Born–Markov master equation for decoherence detailed in Sect. 4.2.2. We shall review the basics of the interaction-picture approach in the following. We begin by considering a Hamiltonian of the form Hˆ = Hˆ0 + V.ˆ (A.1) Here Hˆ0 denotes the part of the Hamiltonian that describes the free (un- perturbed) evolution of a system S, whereas Vˆ is some added external per- turbation. In applications of the interaction-picture formalism, Vˆ is typically assumed to be weak in comparison with Hˆ0, and the idea is then to deter- mine the approximate dynamics of the system given this assumption. In the following, however, we shall proceed with exact calculations only and make no assumption about the relative strengths of the two components Hˆ0 and Vˆ . From standard quantum theory, we know that the expectation value of an operator observable Aˆ(t) is given by the trace rule [see (2.17)], & ' & ' ˆ ˆ Aˆ(t) =Tr Aˆ(t)ˆρ(t) =Tr Aˆ(t)e−iHtρˆ(0)eiHt . (A.2) For reasons that will immediately become obvious, let us rewrite this expres- sion as & ' ˆ ˆ ˆ ˆ ˆ ˆ Aˆ(t) =Tr eiH0tAˆ(t)e−iH0t eiH0te−iHtρˆ(0)eiHte−iH0t . (A.3) ˆ In inserting the time-evolution operators e±iH0t at the beginning and the end of the expression in square brackets, we have made use of the fact that the trace is cyclic under permutations of the arguments, i.e., that Tr AˆBˆCˆ ··· =Tr BˆCˆ ···Aˆ , etc.
[Show full text]