P.S. This question also applies to momentum-position uncertainty.

The uncertainty principles relate the fluctuations of energy, momentum, etc. to those of their conjugate variables; not their individual values.

What Feynman diagrams compute are such fluctuations. The uncertainty principles then describe the fact that the correlation functions of certain quantities can"t all be delta functions. This is what's found. It suffices to do the calculation for any field theory, this isn't a property of quantum electrodynamics. Just take free fields.

Thanks Mr. Guynn and Dr. Nicolas for your answers.

First, i would think that proof of structure in the electron would require experimental evidence. Maybe there is, but i am not aware of it. Theories can never really prove anything, but they might direct experiment to the proof.

Dr. Nicolas' response got me to thinking. I don't believe i put my question clearly. My problem is not with the integrals involving real particles. These are represented as plane waves and are eigenfunctions of momentum and energy. The integration over all space and time holding momentum and energy constant is therefore not just OK but necessary as position and time for the particles are totally undefined. My problem is with the integrals over all 4-momentum space involving virtual particles. I wonder how you can take into account the fact the higher energies correspond to shorter virtual-particle lifetimes. Is it appropriate to add up the contributions of all energies from zero to infinity when the higher energies contribute over shorter times?

In one sense i see that the answer may be similar to that for the real particles. The virtual particle propagators can be devised from "the inverse of the appropriate wave operator as evaluated when acting on plane-wave solutions" -- Gauge Theories in Particle Physics by Aitchson and Hey, p. 81. Again, plane waves are eigenfunctions of 4-momentum and time and position are completely undefined for the particle. However, when this approach is employed, the plane wave exponential cancels out and there is no more reference to time or position. Maybe this is OK, but it still leaves me wondering whether or not too much weight is given to higher energies in the virtual particle 4-momentum integrals.

Francis,

Preston is correct. You are asking a question of mathematicians, that can only really be answered by engineers and practical people.

I have to deal with this routinely. Each group has only a partial model of any phenomena. The representations where "infinite" integrals occur, can better be simulated, allowing noise levels to determine when to truncate the summations. It is a practical solution because the mathematics just says "keep going forever". But the real world has limits. If you are trying to model something in the solar system and your equipment only capable of detecting femtometer per second squared fluctuations in the gravitational fields at 16 bits and 1 megasample per second, you can mostly ignore the rest of the universe and just keep watch on rare events or known sources.

I have to read everything on the Internet. To keep all the scientific, technical, financial, organizational, social, mathematical and other language separate and organized, I put on different hats. You are looking at the problem as a mathematician. What you need to do is look at it from an engineering standpoint. Ask yourself, "What can be measured that would allow me to calibrate my model against data from the real world?" And then ask, "Who is doing that kind of measurement, what tools do they use, what software and algorithms have they found that allow them to actually build things?" I do this every day, 12-18 hours a day. It is extremely hard to force myself to spend another few weeks learning this language, or another few weeks learning that language.

Also, if you would clarify your goals and tell people what you want to do, rather than asking about things that people have argued endlessly. This "infinite energy" has been argued much longer than when I first heard it more than 50 years ago. Frankly I am tired of hearing it.

I will say that I had to deal with this recently as I was trying for the hundredth time to set a workable mass for the graviton. You have to understand that such things serve a very human purpose to give a means of visualizing and remembering how to quickly calculate the correct scale and energies and othe properties. I would like to be able to simulate and display a model of the vacuum at Planck scale, but we simply do not have computers for that yet. We can, as a human species, handle something that is down in the picometer range, or rarely in the femtometer range. But then you are not modeling kilograms, but picograms.

If you would pick a problem in the real world, and not something in mathematics that has no basis in the world, I can tell you all the people in the world who are working on it, what they have found, and my best guess as to where it can be applied to help the human race. If you want to spend all your time on paper mathematics, I just cannot afford the time. I have gotten to the point where I am converting all the mathematical papers to symbolic math in the computer, and then "solving" mathematical papers that way. The paper notations have gotten so bizarre and arcane, it hurts my head to jump from even five or ten papers in one tiny area. But most papers can be transcribed to their equivalent computer representations and at least managed, audited and verified. It takes a human to check, but even that can be farmed out by having large groups of partially qualified people check something independently and then piece the results together. A variation on the "shot gun" method used for DNA.

I like the questions you ask. But if you are going to pick ones that people only talk about, and never actually do any work or create new things, or best help all the people in the world quit having to deal with sickness, hunger, pain, endless years of learning for simple things. Then I would be much happier to try to help.

I have layers of resolution that I have found by spending the time to learn what everyone is doing. It is not perfect, I can only estimate the sizes and importance of all the memes involved. But for everyday gravity a pico meter is fine and energies less than what we can easily produce with lasers and electronic devices. For gravitational imaging and low frequency electromagnetic imaging and communication, femtometers. For vacuum processes where matter is formed, attometers.

When I visualize all these things, I have to use human displays, even in my mind. So I visualize gravitons or particles or small glints and reflections of a light at every scale. That is because that is all I know. If I use dots, they are really particles with reflective and refractive properties - regardless of the scale. The real things might not work that way, but when I, a human, have to think about it, I use a display that any human can see.

So if you are asking "Why does it hurt when I keep banging my head on the wall?" I would suggest you stop, and find something better to do.

Have you done anything to help train a new generation to deal effectively with a simple global epidemic? What about helping the terribly young and inexperienced engineers trying to build spaceships how to model the devices and vehicles they are building, but do not really understand. More remind them that they need to take time to learn the fundamentals, besides meet a production schedule.

I like you. You are trying to ask the right questions. You are not giving up. But please pick ANY real world problem and try that. I think you have some fundamental questions to ask. But they are not how many Planck units can dance on the head of an axion, and should I continue my calculation for a billion terms or a trillion trillion terms? A wise person knows when to stop repeating things that don't work. There are priorities in the world and for the human species. If you simple ask "what is the required precision" and then truncate accordingly, you will not have to deal with infinite loops. It will always converge or termination. Every constrained optimization program or algorithm on the planet has practical ways to know when to stop.

Not sure if this helps. I want you to succeed and achieve your goals. Until you tell people what those are, they cannot work with you and help.

Richard Collins, Director, The Internet Foundation