Week 8 Lecture: Concepts of Quantum Field Theory (QFT) Klein-Gordon Green s Functions and Raising/Lowering Operators

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1 Week 8 Lecture: Concepts of Quantum Fiel Theory (QFT) Anrew Forrester February 29, 2008 Klein-Goron Green s Functions an aising/lowering Operators This Week s Questions How o the Green s functions of the classical Klein-Goron fiel relate to the raising operators of the quantum Klein-Goron fiel? (Maybe they on t really relate.) Also, how oes the propagator relate to the Green s function an raising/lowering operators? What are the general solutions to the Klein-Goron equation? Clue: A Klein-Goron Moel Particle, by J. L. Synge Proceeings of the oyal Society of Lonon. Series A, Mathematical an Physical Sciences, Copyright 965 The oyal Society Motivation / Inspiration In Morse an Feshbach ], pages 22-24, the connection between a concentrate force on a string an a Green s function is iscusse. The force raises (or isplaces) the string, of course, an this puts energy into the string system. This iea coul work in the same way for fiel theory, so that external forces are what cause the creation of particles an raise the fiel.

2 Klein-Goron Green s Function an Propagator The Klein-Goron operator D K-G is an the Klein-Goron equation is D K-G v 2 t ν 2 D K-G f 0. However, in general we may have a source term s(x), an this relates to the Green s functions G(x, x ) an g(x, x ): v 2 t ν 2 f(x) s(x) () λ v 2 t ν 2 G(x, x ) δ(x x ) v 2 t ν 2 g(x, x ) 2 + ν 2] g(x, x ) δ(x x ) The general solution is given by f(x) v 2 t ν 2 f(x) 4 x λ s(x ) G(x, x ) v 2 t ν 2 4 x λ s(x ) G(x, x ) 4 x λ s(x ) v 2 t ν 2 G(x, x ) λ s(x) 4 x λ s(x ) δ(x x ) Time-inepenent, stationary solutions are given by f stationary (x) 3 x λ s stationary(x ) g(x, x ) v 2 t ν 2 f stationary (x) v 2 t ν 2 3 x λ s stationary(x ) g(x, x ) 3 x λ s stationary(x ) v 2 t ν 2 g(x, x ) 3 x λ s stationary(x ) δ(x x ) λ s stationary(x) Let s try to solve explicitly for what G(x, x ) is for the Klein-Goron equation. If G(x, x ) G(x x ), (x x ) (t t, x x ) (τ, ) () (vτ, ), (q q ) (ω ω, q q ) (Ω, Q) (Q) (Ω/v, Q), Q Ωτ Q, 2

3 2 v 2 τ 2 2 Q 2 Q Q Ω 2 /v 2 Q 2 G(x, x ) G() G(τ, ) then the Fourier transform of G is an noting this, G(Q) G() δ() v 2 t ν 2 e iq 4 G() e iq 4 4 Q G(Q) e iq 4 4 Q e iq 4 ] Ω2 v 2 + Q2 + ν 2 e iq Q 2 + ν 2] e iq we have v 2 t ν 2 4 Q G(Q) 4 4 v 2 t ν 2 G() δ() 4 Q G(Q) e iq 4 v 2 t ν 2 ] e iq 4 Q G(Q) Q 2 + ν 2] e iq G(Q) Q 2 + ν 2] 4 4 Q e iq 4 Q e iq 4 4 Q e iq 4 G(Q) Q 2 ν 2 This is the Fourier transform of the Klein-Goron Green s function. It is a propagator that we see in quantum fiel theory. 3

4 Aitional Notes an Scratch Work pg? Morse an Feshbach G(x ξ) g(, r) G(q) G (+) (q) { 2ν eν(x ξ) ; 2ν eν(ξ x) ; x < ξ x > ξ δτ (/c)] κ τ 2 (/c) J κc τ 2 (/c) 2 ] Θτ (/c)] 2 4 q G(x) e iq x q 2 m 2 q 2 m 2 + iɛ pg 856 Morse an Feshbach (here, g means the Green s function over all space, with no bounary) pg Aitchison an Hey (x) (t, x) (t, x, x 2, x 3 ) (q) (ω, q) (ω, q, q 2, q 3 ) v 2 t ν 2 f(x) λ s(x) v 2 t ν 2 G(x, x ) δ(x x ) v 2 t ν 2 g(x, x ) δ(x x ) Shoul the measure be Lorentz invariant? ] G(q, q ) G 0 (q, 0) G(x, 0) δ(x x ) 4 x 4 x G(x, x ) e iq x e iq x 8 4 x G(x, 0) e iq x 4 4 q G 0 (q, 0) e iq x 4 4 q e iq (x x ) 4 4

5 v 2 t ν 2 v 2 t ν 2 G(x, 0) δ(x) 4 q G(q, 0) 2 4 q G(q, 0) e iq x 2 q 2 + ν 2] e iq x () G(q, 2 0) q 2 + ν 2] 2 4 q e iq x 4 q e iq x 2 G(q, 0) q 2 ν 2 Shoul the measure be Lorentz invariant? ] G(x x ) G(x x, t t ) G(, τ) g(x x ) g() Ĝ(, ω) t G(, τ) e iωτ G(, τ) δ(τ) ( v 2 t ν 2 ω ω Ĝ(, ω) e iωτ ω e iωτ v 2 t ν 2 G(x, x ) δ(x x ) ω Ĝ(, ω) e iωτ ) ( δ() ] ω2 v ν 2 Ĝ(, ω) e iωτ δ() 2 + (ν 2 ω 2 /v )] 2 Ĝ(, ω) δ() ω e iωτ ) ω e iωτ 5

6 Green s Functions an the eal-value Classical Klein-Goron Fiel We a another assumption to our physical moel for the real-value classical Klein-Goron fiel: (9) external forces S (both attractive an repulsive) may be exerte on the sheet from above, but only in the z-irection (the force area-ensity is s s(x, y, t). The origin of these forces is not containe in this moel let s just say someone s sticky fingers coul be involve.) The equation of motion is with the forces given by So we have F z (µ δx δy) t 2 f, F z F s z + F tx z + F ty z Fz s (κ δx δy)f Fz tx (λ δy) δ( x f) Fz ty (λ δx) δ( y f) S z δx δy s. + S z F z (κ δx δy)f + (λ δy) δ( x f) + (λ δx) δ( y f) + δx δy s (µ δx δy) t 2 f κ λ f + δ( xf) δx + δ( yf) δy + λ s µ λ t 2 f, after iviing by λ δx δy, an if we take the limit as δx 0 an δy 0, we get Thus, letting κ λ f + x 2 f + y 2 f + λ s µ λ t 2 f. v λ/µ ν κ/λ v 2 t 2 f 2 f + ν 2 f λ s we have an inhomogeneous Klein-Goron-type equation with a source function s/λ. Let s compare these equations: v 2 t 2 f 2 f 0 (Wave Equation; sheet without springy slab) v 2 t 2 f 2 f + ν 2 f 0 (Klein-Goron-type equation; sheet with springy slab) c 2 t 2 ϕ 2 ϕ + ν 2 ϕ 0 (Klein-Goron Equation) where ν mc/ for the Klein-Goron equation, since it is escribing the fiel ϕ of a particle of mass m. c 2 t 2 ϕ : the generalize momentum-ensity rate of change 2 ϕ : the generalize spring-force-ensity (perpenicular to spacetime) ν 2 ϕ : the generalize surface-tension-force-ensity (perpenicular to spacetime) 6

7 eferences ] Morse, Feshbach: Methos of Theoretical Physics, Part, McGraw-Hill Book Company, Inc. (953) This book is very goo. It takes a physically-base (as oppose to purely mathematical) approach to unerstaning the mathematics of physics an helps to create intuition. 2] I. J.. Aitchison, A. J. G. Hey: Gauge Theories in Particle Physics, A Practical Introuction, Thir Eition. Volume I: From elativistic Quantum Mechanics to QED, Taylor & Francis Group, LLC (2003) Appenix G is a goo, quick escription of a few Green s function examples, incluing that for the Klein-Goron equation. Appenix F, on contour integration, may be helpful in unerstaning the complex versions of the Green s functions / propagators. 3] Economou, L. N.: Green s Functions in Quantum Physics, Secon Correcte an Upate Eition, (Springer Series in Soli-State Sciences 7), Springer-Verlag (983) The first chapter seems to give a goo explanation of the general formalism for Green s functions, although I in t have time to go through it carefully. The titles of each part are: Part I: Green s Functions in Mathematical Physics Part II: Green s Functions in One-Boy Quantum Problems Part III: Green s Functions in Many-Boy Systems 7