## Archive for the ‘Relativity’ Category

### Summary Outline of Richard Feynmans Thesis – Framework for learning QED and Quantum Mechanics in general

Friday, August 21st, 2009

A reasonable strategy for learning QED would be to try to recreate Richard Feynman's path of discovery that lead him to his thesis paper.   If you are an electrical or computer engineer you should probably set about the task of learning the subject for sooner or later quantum computing and nanotechnology are going to go mainstream.  If I am successful in learning this subject it will mark the first time I have not been at least 10 years behind the technology curve when it finally hits.

— The following summary is from his Nobel prize address in 1965 and yields some insight as to how he went about the problem of reformulating QED into a more usable form.   It includes some information about the wrong turns he took which of course is very instructive in and of itself. —

A = ~i mi J (Xi,uXi,u )1 d(Xi + t ~ ejej J Jc5(Iij2) Xi,u ((Xi) xj,u ((Xj) d(Xi d(Xj (I)
.1· I-r-J
where
Iij2 = [Xi,u ((Xi) – Xi, (exj)] [Xi,u ((Xi) – Xj,u(exj)]
whereXi,u ((Xi) is the four-vector position of the i Ihparticle as a function of

s = S L(x, x) dt (2)

then you can start with the Lagrangian and then create a Hamiltonian and work out the quantum mechanics, more or lessuniquely. But this thing (I) involves the key variables, positions, at two different times and therefore, it was not obvious what to do to make the quantum-mechanical analogue. I tried – I would struggle in various ways. One of them was this; if I had harmonic oscillators interacting with a delay in time, I could work out what the normal modes were and guess that the quantum theory of the normal modes was the same as for simple oscillators and kind of work my way back in terms of the original variables. I succeeded in doing that, but I hoped then to generalize to other than a harmonic oscillator, but I learned to my regret something, which many people have learned. The harmonic oscillator is too imple; very often you can work out what it should do in quantum theory without getting much of a clue as to how to generalize your results to other systems. So that didn't help me very much, but when I was struggling with this problem, I went to a beer party in the Nassau Tavern in Princeton. There was a gentleman, newly arrived from Europe (Herbert Jehle) who came and sat next to me. Europeans are much more serious than we are in America because they think that a good place to discuss intellectual matters is a beer party. So, he sat by me and asked, «what are you doing» and so on, and I said, « I'm drinking beer. » Then I realized that he wanted to know what work I was doing and I told him I was struggling with this problem, and I simply turned to him and said, ((listen, do you know any way of doing quantum mechanics, starting withaction – where the action integral comes into the quantum mechanics? » « No », he said, « but Dirac has a paper in which the Lagrangian, at least, comes into quantum mechanics. I will show it to you tomorrow. » Next day we went to the Princeton Library, they have little rooms on the side to discuss things, and he showed me this paper. What Dirac said was the following : There is in quantum mechanics a very important quantity which carries the wave function from one time to another, besides the differential quation but equivalent to it, a kind of a kernal, which we might call K(x', x), which carries the wave function \j1 (x) known at time t, to the wave function 'If (x ') at time, t +e. Dirac points out that this function K was analogous to the quantity in classical mechanics that you would calculate if you took the exponential of ie, multiplied by the Lagrangian L(:ie, x) imagining that these two positions x,x' corresponded t and t +e. In other words,
x' x
K(x ' ,x) .IS anaI ogous to ei e L(-e–'x )/rr

Professor Jehle showed me this, I read it, he explained it to me, and I said, « what does he mean, they are analogous; what does that mean, analogous?  What is the use of that? » He said, « you Americans ! You al ways want to find a use for everything! » I said, that I thought that Dirac must mean that they were equal. « No », he explained, « he doesn't mean they are equal.» «Well », I said, « let's see what happens if we make them equal. » So I simply put them equal, taking the simplest example where the Lagrangian is 112 Mx2 – V( x) but soon found I had to put a constant of proportionality A in, suitably adjusted. When I substituted AeiEL/Ii for K to get

1p(x',t+s) = J A exp[~L(x'~x,x)J 1p(x,t) dx (3)

So what happened to the old theory that I fell in love with as a youth? Well, I would say it's become an old lady, that has very little attractive left in her and the young today will not have their hearts pound when they look at her anymore. But, we can say the best we can for any old woman, that she has been a very good mother and she has given birth to some very good children. And, I thank the Swedish Academy of Sciences for complimenting one of them. Thank you.

R.P. Feynman

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Feynman's Thesis: A New Approach to Quantum Theory
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Richard Feynman's never previously published doctoral thesis formed the heart of much of his brilliant and profound work in theoretical physics. Entitled "The Principle of Least Action in Quantum Mechanics," its original motive was to quantize the classical action-at-a-distance electrodynamics. Because that theory adopted an overall space-time viewpoint, the classical Hamiltonian approach used in the conventional formulations of quantum theory could not be used, so Feynman turned to the Lagrangian function and the principle of least action as his points of departure." The result was the path integral approach, which satisfied – and transcended – its original motivation, and has enjoyed great success in renormalized quantum field theory, including the derivation of the ubiquitous Feynman diagrams for elementary particles. Path integrals have many other applications, including atomic, molecular, and nuclear scattering, statistical mechanics, quantum liquids and solids, Brownian motion, and noise theory. It also sheds new light on fundamental issues like the interpretation of quantum theory because of its new overall space-time viewpoint.

Preface vii
The Principle of Least Action in Quantum Mechanics
R. P. Feynman
I. Introduction 1
II. Least Action in Classical Mechanics 6
1. The Concept of Functional 6
2. The Principle of Least Action 9
3. Conservation of Energy. Constants of the Motion 10
4. Particles Interacting through an Intermediate Oscillator 16
III. Least Action in Quantum Mechanics 24
1. The Lagrangian in Quantum Mechanics 26
2. The Calculation of Matrix Elements in the
Language of a Lagrangian 32
3. The Equations of Motion in Lagrangian Form 34
4. Translation to the Ordinary Notation of Quantum
Mechanics 39
5. The Generalization to Any Action Function 41
6. Conservation of Energy. Constants of the Motion 42
7. The Role of the Wave Function 44
8. Transition Probabilities 46
9. Expectation Values for Observables 49
10. Application to the Forced Harmonic Oscillator 55
11. Particles Interacting through an Intermediate Oscillator 61
12. Conclusion 68
Space-time Approach to Non-Relativistic Quantum Mechanics 71
The Lagrangian in Quantum Mechanics 111

Product Details

* ISBN: 9812563660
* ISBN-13: 9789812563668
* Format: Hardcover, 119pp
* Publisher: World Scientific Publishing Company, Incorporated
* Pub. Date: August 2005

### Optical Effects of Special Relativity

Thursday, July 2nd, 2009

Race down a road at near the speed of light.  What do you see?

### Physics of Relativity

Friday, February 27th, 2009

Professor Richard A. Muller does a good job of brushing through Einstein’s theory of relativity.