DOCSLIB.ORG

Feynman Diagram Method for Atomic Collisions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Feynman Diagram Method for Atomic Collisions Pramfi.na, Vol. 6, No. 5, 1976, pp. 312-322. © Printed in India. Feynman diagram method for atomic collisions T PRADHAN* and AVINASH KHARE Institute of Physics, Bhubaneswar 751007 MS received 1 December 1975; after revision 5 January 1976 Abstract. Feynmandiagram method of treating atomic collision problems in per- turbation theory is presented and matrix elements are calculated for a number of pro- cesses. The result for the resonant charge transfer in hydrogen is identical to the well known OBK value. However, in processes like collisional ionisation, the results are different from those obtained by conventional methods. Keywords. Electron exchange; ion exchange; vertex function; collisional ionisation; photo-effect charge transfer. 1. Introduction Feynman diagram method has proved to be very useful in a variety of problems in relativistic as well as non-relativistic physics. Apart from its indispensability in relativistic particle physics (Akhiezer and Beretetskii 1965), it has been found to be a very successful tool in handling many-body problems in nuclear, solid state (Fetter and Walecka 1971)and atomic physics (Kelley 1974). However, no attempt seems to have been made to apply this method to atomic collision problems. While exploring such possibilities the authors have found that in problems such as charge exchange processes the diagrammatic method not only gives an elegant physical picture of the phenomena but at the same time makes the calculation simple. The lowest order diagram consists of an electron exchange quite similar to the photon exchange in elastic scattering of charged particles. For the resonant charge transfer in hydrogen proton collision one obtains the familiar OBK amplitude (Brinkman and Kramers 1930); there is no diagram corresponding to the JS amplitude (Jackson and Schiff 1953a) in the lowest order. Another charge transfer mechanism operates in exchange scattering of electrons by hydrogen which can lze represented by a Feynman diagram with a proton exchange. Tb.e success of the diagrammatic method in these processes has prompted us to apply it to a host of other atomic collision processes. Principal among these are the photo-electric effect and its inverse (the radiative recombination), ionisation of atoms by electron and ionic impact. Detailed calculation of the amplitude of these processes are presented in this paper. The theory of charge exchange in positive ion-atom collision relevant to Feynman diagram method is presented in section 2 where amplitudes of other typical rearrange- ment processes of experimental interest have also been evaluated. Section 3 is *On leave of absence from Saha Institute of Nuclear Physics, Calcutta 700009. 312 Feynman diagram method for atomic collisions 313 devoted to the calculation of photo-electric effect and its inverse process of radiative recombination where relativistic and Coulomb effects have been ignored for the sake of simplicity. In section 4, calculation of matrix element for ionisation of atoms by electron and iortic impact is presented. Section 5 is devoted to discussions. 2. Charge transfer collisions We consider the process A+ + B--~ A + B+ where A+ is a positive singly ionized atom which picks up art electron from the atom B after collision. In other words, exchange of an electron between B and A+ characterises the collision. This exchange is, "literally" expressed by the Feynman diagram, of figure 1. The momenta of various particles indicated in tb.e diagram refer to the centre of mass system. The rules for writing down the matrix elements being quite standard (Bhasin et al 1965) need not be repeated here. Using these rules, we obtain M = ? (X,, K, -- X,, to)/ F (X, -- X,, to, m) × I"(K,, to,; Kt--K,, to) (1) where $I, = St, -- (2rr) 4 8(8) (P~ -- PI) ~ ( E, -- Et) M, or = KI~/2M. +, ~'~ = K*2/2Ma +, to = 12~- to1 = f2t- tot, I2~ = K,2/2Ma ~ ~B, 01 -- KI~ 2M a ~'~' a,~, an being bittding energy of the electron in atoms ' A' and ' B' respectively. r' (p~, to1; Pz, to~) is the vertex function representing the interaction of particles 1 and 2 and the bound state composed of these particles and AF (P, ~o, m) is the propagator for the electron. The vertex function/~(Pz, wx; P2, w2)is related to the Fourier transform ( : m pq gxs ~, mx + ms / of the Schr6dinger wave-function of the bound state formed by particles 1 and 2 as follows (see Appendix): (s) (~,, ~-,) (B+) (.k'f, wf) ,) i (A~) (~,~) t (A) (~'f,_n.f) lllgure I. Feyaraan diagram for charge transfer process A+ + B.-~ A + B+. ]P--7 314 T Pradhan and A vinash Khare I" (p 1 col; (m~pl- : m'P~ F ( p'2 -- ws -- w, Pz, w2)=gl~\ mi-}-ms /Lk2m,. ) c\2ml )3 f2) The non-relativistic propagator cart be written as 1 AF (P, % m) -- pS/2 m _ co -- b7 (3) Substituting eqs. (2) and (3) in eq. (1) and taking into account the fact that particles A+ and B+ are on the mass-shell i.e., pS/2m = oJ we get M = g,~+,(aK, -- Kt)g,~+,(bKt--K,)[(Kt~mm K')2 o~ ] , (4) where a = MB+/M~ b = M,~+/M,~, The general formula eq. (4) can now be applied to the particular cases of (a) H++ H(ls) -+ H(ls) 4- H -~, (b) e+ + (e+ e-)l. -+ (e+ e-)i, + e+, (c) H + + (tl+ e-)i, -+ H (ls) + t,+. For the process (a) M = g~n+. M K,- K t ) gall% (Mm + M 1(1 l- K? 1 X L(K -- K,)2/2m--Kill2 (M 4- m) 4- ~-M 4- e J, (5) where ~ is the binding energy of the hydrogen atom ground state and M is the mass of the proton. Using the fact that K~2 = Kt 2 -- (K s) which follows from over-all energy-momentum conservation and II 8 "/'/'1/2 S O/`' g,~+. (p) = ~<+-j~y, s = a: i , ao being Bokr radius for ground state, Eq. (5) simplifies to M = .--32rrs5 1 (6) m [#' + K 2 (/9 ~ 4- 4 sin s 0/2)] 3, with fi = m/M and cos 0 = K I • KdK 2 which is readily seen to be the OBK anapli- tude. It is worth pointing out that figure 1 being the only possible Feynman diagram for the charge transfer process in the lowest order, OBK amplitude (Brinkman and Kramers 1930) constitutes the entire matrix element. There being no other diagram we conclude that JS amplitude (Jackson and Schiff 1953 a) is not obtainable from diagrammatic formalism. It seems therefore that Wick's contention (Jackson and Schiff 1953 b) that proton-proton potential should not contribute to the capture amplitude is borne out by Feynman diagram formalism. In view of the absence of the JS term, one may say that diagram method suffers from the drawback that in the lowest order there ks no agreement with experiment. This is mainly due to ignoring Feynman diagram method for atomic collisions 315 the long-range Coulomb interaction between the bound electron and the colliding proton, which may be regarded as an irtitial state (or final state) interaction. The successful method of taking this effect into account is the Impulse Approximation initiated by one of us (TP) (Pradhan 1957) and refined by several others (Mcdowell and Coleman 1970). For the process (b) the matrix element is simpler on account of equality of masses of all the particles involved. We find IB im M = g.+,- (½ K, -- Kt) g°+,'- (½ K, -- K,) 1 × ~-~ Is, 2 + ¼K2 (1 + 8 sin s 0/2)], (7) where sp is the inverse Bohr radius of positronium. It is to be noted that we are using non-relativistic dynamics and as such ignoring spins of electron and positron. Using the formula x° 8rr112 s hlz g'+'-@) - + and the relation K, z = Kt z = K ~ in eq. (7) we obtain M = 32rrs~ 1 K 2 m [sp2 + ~-(1 + 8 sin e 0/2)] z (8) which is identical to the OBK amplitude for the process under consideration. For the process (C), the matrix element works out to 32~rss 1 M -- -- (9) m [ s" + K2(m 2~ + 4 sirt 2 0/2) )3j where K ~= K~2@ Kt z and cos0=K t . K~/ I Kt I K~ [ As in the two previous cases, this is also identical to tb.e corresponding OBK ampli. tude. The mechanism for all the charge transfer processes considered so far is electron exchange. It may be worthwhile to consider a process which occurs through ionic exchange. In electron scattering by atoms, apart from the direct scattering of the electron the process where the incident electron is captured and the atomic electron ejected must be considered.* This latter process e+A--X+e can be represented by the Feynman diagra m given in figure-2. It is clear from the diagram that the mechanism for this process is through the exchange of the iort A +. The matrix element for tb.is diagram is * This latter process is the only mechanism for inelastic electron scattering. e+He(l'S)~He(23P)+e by Helium. The direct and exchange amplitudes can be measured individually by spin analysis of the scattered beam in experiments using polarised electrons on atoms. 316 T Pradhan and Avinash Khare (A) (~i, .t~.,) (e')( K-'f, ~Of) < II ................. > ('e) ('ffi,~i) CA) CK~ ,~f) Figure 2. Feynman diagram for electron-atom scattering e + A ~ A + e through ion exchange. M = P (/(1, cot; Kt -- Ks, co) Av (KI -- K~) co, M,t÷) r (K, -- X,, ,o; K,, co,) fl0) For the particular case of electron scattering by hydrogen where the exchanged particle is a proton this works out to M = 32rrs5 1 -m- [s 2 -+- K 2 (1 + 4/3 sin z 0/2)] 3, ~ = m/M (I 1) which is somewhat different from the corresponding expression obtained by using the Born-Oppenheimer approximation (Mott and Massey 1965).
Load more

Recommended publications
  • Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany
    Generating Functionals Functional Integration Renormalization Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany 2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective field theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 1: Basics of Quantum Field Theory Thomas Mannel Theoretische Physik I, Universität Siegen f q f et Yerevan, October 2014 T. Mannel, Siegen University Effective Quantum
    [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]
  • 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]
  • Path Integral and Asset Pricing
    Path Integral and Asset Pricing Zura Kakushadze§†1 § Quantigicr Solutions LLC 1127 High Ridge Road #135, Stamford, CT 06905 2 † Free University of Tbilisi, Business School & School of Physics 240, David Agmashenebeli Alley, Tbilisi, 0159, Georgia (October 6, 2014; revised: February 20, 2015) Abstract We give a pragmatic/pedagogical discussion of using Euclidean path in- tegral in asset pricing. We then illustrate the path integral approach on short-rate models. By understanding the change of path integral measure in the Vasicek/Hull-White model, we can apply the same techniques to “less- tractable” models such as the Black-Karasinski model. We give explicit for- mulas for computing the bond pricing function in such models in the analog of quantum mechanical “semiclassical” approximation. We also outline how to apply perturbative quantum mechanical techniques beyond the “semiclas- sical” approximation, which are facilitated by Feynman diagrams. arXiv:1410.1611v4 [q-fin.MF] 10 Aug 2016 1 Zura Kakushadze, Ph.D., is the President of Quantigicr Solutions LLC, and a Full Professor at Free University of Tbilisi. Email: [email protected] 2 DISCLAIMER: This address is used by the corresponding author for no purpose other than to indicate his professional affiliation as is customary in publications. In particular, the contents of this paper are not intended as an investment, legal, tax or any other such advice, and in no way represent views of Quantigic Solutions LLC, the website www.quantigic.com or any of their other affiliates. 1 Introduction In his seminal paper on path integral formulation of quantum mechanics, Feynman (1948) humbly states: “The formulation is mathematically equivalent to the more usual formulations.
    [Show full text]
  • On Topological Vertex Formalism for 5-Brane Webs with O5-Plane
    DIAS-STP-20-10 More on topological vertex formalism for 5-brane webs with O5-plane Hirotaka Hayashi,a Rui-Dong Zhub;c aDepartment of Physics, School of Science, Tokai University, 4-1-1 Kitakaname, Hiratsuka-shi, Kanagawa 259-1292, Japan bInstitute for Advanced Study & School of Physical Science and Technology, Soochow University, Suzhou 215006, China cSchool of Theoretical Physics, Dublin Institute for Advanced Studies 10 Burlington Road, Dublin, Ireland E-mail: [email protected], [email protected] Abstract: We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of [1]. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO(N) gauge theories and the pure G2 gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level 9. At the end we rewrite the O-vertex in a form of a vertex operator. arXiv:2012.13303v2 [hep-th] 12 May 2021 Contents 1 Introduction1 2 O-vertex 3 2.1 Topological vertex formalism with an O5-plane3 2.2 Proposal for O-vertex6 2.3 Higgsing and O-planee 9 3 Examples 11 3.1 SO(2N) gauge theories 11 3.1.1 Pure SO(4) gauge theory 12 3.1.2 Pure SO(6) and SO(8) Theories 15 3.2 SO(2N
    [Show full text]
  • Quantum Electrodynamics in Strong Magnetic Fields. 111* Electron-Photon Interactions
    Aust. J. Phys., 1983, 36, 799-824 Quantum Electrodynamics in Strong Magnetic Fields. 111* Electron-Photon Interactions D. B. Melrose and A. J. Parle School of Physics, University of Sydney. Sydney, N.S.W. 2006. Abstract A version of QED is developed which allows one to treat electron-photon interactions in the magnetized vacuum exactly and which allows one to calculate the responses of a relativistic quantum electron gas and include these responses in QED. Gyromagnetic emission and related crossed processes, and Compton scattering and related processes are discussed in some detail. Existing results are corrected or generalized for nonrelativistic (quantum) gyroemission, one-photon pair creation, Compton scattering by electrons in the ground state and two-photon excitation to the first Landau level from the ground state. We also comment on maser action in one-photon pair annihilation. 1. Introduction A full synthesis of quantum electrodynamics (QED) and the classical theory of plasmas requires that the responses of the medium (plasma + vacuum) be included in the photon properties and interactions in QED and that QED be used to calculate the responses of the medium. It was shown by Melrose (1974) how this synthesis could be achieved in the unmagnetized case. In the present paper we extend the synthesized theory to include the magnetized case. Such a synthesized theory is desirable even when the effects of a material medium are negligible. The magnetized vacuum is birefringent with a full hierarchy of nonlinear response tensors, and for many purposes it is convenient to treat the magnetized vacuum as though it were a material medium.
    [Show full text]
  • 1 Drawing Feynman Diagrams
    1 Drawing Feynman Diagrams 1. A fermion (quark, lepton, neutrino) is drawn by a straight line with an arrow pointing to the left: f f 2. An antifermion is drawn by a straight line with an arrow pointing to the right: f f 3. A photon or W ±, Z0 boson is drawn by a wavy line: γ W ±;Z0 4. A gluon is drawn by a curled line: g 5. The emission of a photon from a lepton or a quark doesn’t change the fermion: γ l; q l; q But a photon cannot be emitted from a neutrino: γ ν ν 6. The emission of a W ± from a fermion changes the flavour of the fermion in the following way: − − − 2 Q = −1 e µ τ u c t Q = + 3 1 Q = 0 νe νµ ντ d s b Q = − 3 But for quarks, we have an additional mixing between families: u c t d s b This means that when emitting a W ±, an u quark for example will mostly change into a d quark, but it has a small chance to change into a s quark instead, and an even smaller chance to change into a b quark. Similarly, a c will mostly change into a s quark, but has small chances of changing into an u or b. Note that there is no horizontal mixing, i.e. an u never changes into a c quark! In practice, we will limit ourselves to the light quarks (u; d; s): 1 DRAWING FEYNMAN DIAGRAMS 2 u d s Some examples for diagrams emitting a W ±: W − W + e− νe u d And using quark mixing: W + u s To know the sign of the W -boson, we use charge conservation: the sum of the charges at the left hand side must equal the sum of the charges at the right hand side.
    [Show full text]
  • Old-Fashioned Perturbation Theory
    2012 Matthew Schwartz I-4: Old-fashioned perturbation theory 1 Perturbative quantum field theory The slickest way to perform a perturbation expansion in quantum field theory is with Feynman diagrams. These diagrams will be the main tool we’ll use in this course, and we’ll derive the dia- grammatic expansion very soon. Feynman diagrams, while having advantages like producing manifestly Lorentz invariant results, give a very unintuitive picture of what is going on, with particles that can’t exist appearing from nowhere. Technically, Feynman diagrams introduce the idea that a particle can be off-shell, meaning not satisfying its classical equations of motion, for 2 2 example, with p m . They trade on-shellness for exact 4-momentum conservation. This con- ceptual shift was critical in allowing the efficient calculation of amplitudes in quantum field theory. In this lecture, we explain where off-shellness comes from, why you don’t need it, but why you want it anyway. To motivate the introduction of the concept of off-shellness, we begin by using our second quantized formalism to compute amplitudes in perturbation theory, just like in quantum mechanics. Since we have seen that quantum field theory is just quantum mechanics with an infinite number of harmonic oscillators, the tools of quantum mechanics such as perturbation theory (time-dependent or time-independent) should not have changed. We’ll just have to be careful about using integrals instead of sums and the Dirac δ-function instead of the Kronecker δ. So, we will begin by reviewing these tools and applying them to our second quantized photon.
    [Show full text]
  • 'Diagramology' Types of Feynman Diagram
    `Diagramology' Types of Feynman Diagram Tim Evans (2nd January 2018) 1. Pieces of Diagrams Feynman diagrams1 have four types of element:- • Internal Vertices represented by a dot with some legs coming out. Each type of vertex represents one term in Hint. • Internal Edges (or legs) represented by lines between two internal vertices. Each line represents a non-zero contraction of two fields, a Feynman propagator. • External Vertices. An external vertex represents a coordinate/momentum variable which is not integrated out. Whatever the diagram represents (matrix element M, Green function G or sometimes some other mathematical object) the diagram is a function of the values linked to external vertices. Sometimes external vertices are represented by a dot or a cross. However often no explicit notation is used for an external vertex so an edge ending in space and not at a vertex with one or more other legs will normally represent an external vertex. • External Legs are ones which have at least one end at an external vertex. Note that external vertices may not have an explicit symbol so these are then legs which just end in the middle of no where. Still represents a Feynman propagator. 2. Subdiagrams A subdiagram is a subset of vertices V and all the edges between them. You may also want to include the edges connected at one end to a vertex in our chosen subset, V, but at the other end connected to another vertex or an external leg. Sometimes these edges connecting the subdiagram to the rest of the diagram are not included, in which case we have \amputated the legs" and we are working with a \truncated diagram".
    [Show full text]
  • Path Integral in Quantum Field Theory Alexander Belyaev (Course Based on Lectures by Steven King) Contents
    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 .
    [Show full text]
  • Feynman Diagrams Particle and Nuclear Physics
    5. Feynman Diagrams Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 5. Feynman Diagrams 1 In this section... Introduction to Feynman diagrams. Anatomy of Feynman diagrams. Allowed vertices. General rules Dr. Tina Potter 5. Feynman Diagrams 2 Feynman Diagrams The results of calculations based on a single process in Time-Ordered Perturbation Theory (sometimes called old-fashioned, OFPT) depend on the reference frame. Richard Feynman 1965 Nobel Prize The sum of all time orderings is frame independent and provides the basis for our relativistic theory of Quantum Mechanics. A Feynman diagram represents the sum of all time orderings + = −−!time −−!time −−!time Dr. Tina Potter 5. Feynman Diagrams 3 Feynman Diagrams Each Feynman diagram represents a term in the perturbation theory expansion of the matrix element for an interaction. Normally, a full matrix element contains an infinite number of Feynman diagrams. Total amplitude Mfi = M1 + M2 + M3 + ::: 2 Total rateΓ fi = 2πjM1 + M2 + M3 + :::j ρ(E) Fermi's Golden Rule But each vertex gives a factor of g, so if g is small (i.e. the perturbation is small) only need the first few. (Lowest order = fewest vertices possible) 2 4 g g g 6 p e2 1 Example: QED g = e = 4πα ∼ 0:30, α = 4π ∼ 137 Dr. Tina Potter 5. Feynman Diagrams 4 Feynman Diagrams Perturbation Theory Calculating Matrix Elements from Perturbation Theory from first principles is cumbersome { so we dont usually use it. Need to do time-ordered sums of (on mass shell) particles whose production and decay does not conserve energy and momentum. Feynman Diagrams Represent the maths of Perturbation Theory with Feynman Diagrams in a very simple way (to arbitrary order, if couplings are small enough).
    [Show full text]
  • The Link Between a Vacuum Diagram and the Renormalization Group Equation
    Radboud University Nijmegen Department of Theoretical High Energy Physics Master's thesis The link between a vacuum diagram and the renormalization group equation Author: Supervisor: Daphne van den Elzen Prof. dr. Wim Beenakker September 1, 2016 Contents 1 Introduction3 2 Theoretical context4 2.1 The cosmological constant problem.................4 2.2 Previous research on the cosmological constant problem.....5 2.3 The Higgs Lagrangian........................6 2.4 Scalar Yukawa theory........................8 2.5 Renormalization...........................9 3 Research 11 3.1 The 1-loop self-energy........................ 11 3.2 The K¨all´en-Lehmannspectral representation........... 16 3.3 The vacuum bubble......................... 21 3.4 Comparison with the renormalization group equation....... 25 4 Conclusion and outlook 32 A 'Altered' renormalization group equation 34 2 1 Introduction What exactly is the vacuum? It seems a rather simple question, but when zoom- ing in on this subject it actually turns out to be a pretty complex one. The most obvious definition of the vacuum is empty space; the absence of matter and radiation. When looking at it from the angle of quantum field theory however, the notion of empty space is replaced with that of a vacuum state. It is defined to be the lowest energy state, also called the ground state, of all fundamental fields. This energy isn't zero, as expected from classical mechanics, but it has a value due to zero-point fluctuations of the fields made possible by quantum mechanics. Summing all contributions of zero-point energies leads to an infinite vacuum energy. To make sense of a quantum field theory we need to remove this infinity.
    [Show full text]