Infinitely maddening

Originally posted by DeepThought
The classic example of that kind of game going well is the Dirac delta function, which mathematicians initially dismissed as invalid. Happily Dirac wasn't to be put off and as he continued to get good results with it they were forced to take it seriously and the field of distributions was born.

In a sense the Millennium prize for showing that Yang-Mi ...[text shortened]... theories we can make up to describe it. When things go well new fields of mathematics open up.

Necessity is the mother of invention.

I've created my own typeset notes on differential equations which I use when I teach a course on the subject. Almost everything is proved and made as rigorous as feasible at the undergraduate level -- much more so that in the textbook. But there is the problem of what to do with the Dirac delta "doodad." The book says, in essence, it's "zero everywhere, except at zero where it equals infinity." Which is ridiculous. In this case I simply define it in the manner that we use it for:

∫δ(t)dt := 1

over any interval of integration in ℝ that includes 0 (otherwise the integral is simply 0). Because in an introductory differential equations course it's not practical to develop the theory of distributions.

Originally posted by Soothfast
Necessity is the mother of invention.

I've created my own typeset notes on differential equations which I use when I teach a course on the subject. Almost everything is proved and made as rigorous as feasible at the undergraduate level -- much more so that in the textbook. But there is the problem of what to do with the Dirac delta "doodad." The bo ...[text shortened]... ductory differential equations course it's not practical to develop the theory of distributions.

Yes, that's roughly what they did in my undergraduate course, and then happily used it outside integrals...

Originally posted by Soothfast
Necessity is the mother of invention.

I've created my own typeset notes on differential equations which I use when I teach a course on the subject. Almost everything is proved and made as rigorous as feasible at the undergraduate level -- much more so that in the textbook. But there is the problem of what to do with the Dirac delta "doodad." The bo ...[text shortened]... ductory differential equations course it's not practical to develop the theory of distributions.

If I remember correctly, when the Dirac delta function was introduced to me, it was shown to be a limiting case of a series of functions (e.g. tent functions with ever smaller bases and ever greater peaks). They then went on to say it wasn't a real function, but a distribution, without explaining what that is, and showing the definition that you give here.

Originally posted by KazetNagorra
If I remember correctly, when the Dirac delta function was introduced to me, it was shown to be a limiting case of a series of functions (e.g. tent functions with ever smaller bases and ever greater peaks). They then went on to say it wasn't a real function, but a distribution, without explaining what that is, and showing the definition that you give here.

Yes, I have the same thing in my notes: "impulse functions" inside an integral that get narrower and taller, yet with area under the curve remaining at 1. Then presto: define the Dirac delta. Yet, let's admit, the Dirac delta is not only a dubious mathematical device (at least as presented in introductory differential equations courses), it is also dubious physics. No "impulse" is truly instantaneous and infinite! But we all just go along with the joke because of convenience and the empirical fact that it works.

You take your Dirac delta. I'll keep the Kronecker delta. 😉

Originally posted by DeepThought
Yes, that's roughly what they did in my undergraduate course, and then happily used it outside integrals...

Correction: sadly used it outside integrals, at the uttermost end of need.

Originally posted by Soothfast
Yes, I have the same thing in my notes: "impulse functions" inside an integral that get narrower and taller, yet with area under the curve remaining at 1. Then presto: define the Dirac delta. Yet, let's admit, the Dirac delta is not only a dubious mathematical device (at least as presented in introductory differential equations courses), it is also dub ...[text shortened]... e empirical fact that it works.

You take your Dirac delta. I'll keep the Kronecker delta. 😉

The same applies to plane waves, which people normally accept without blinking:

W(x) = exp(ikx)/N

But for the wavefunction to be normalized over the whole of space N has to be infinite...

Originally posted by DeepThought
The same applies to plane waves, which people normally accept without blinking:

W(x) = exp(ikx)/N

But for the wavefunction to be normalized over the whole of space N has to be infinite...

Or we can apply our procedures in a finite box and then generalize to an infinite space (waving things over by applying periodic boundary conditions). 😵😵😵

Originally posted by Soothfast
Yes, I have the same thing in my notes: "impulse functions" inside an integral that get narrower and taller, yet with area under the curve remaining at 1. Then presto: define the Dirac delta. Yet, let's admit, the Dirac delta is not only a dubious mathematical device (at least as presented in introductory differential equations courses), it is also dub ...[text shortened]... e empirical fact that it works.

You take your Dirac delta. I'll keep the Kronecker delta. 😉

It's not always dubious physics, in plenty of cases it is a pretty good approximation, and for e.g. a short pulse the physics related to the finiteness of the pulse is not always relevant and/or interesting.

Originally posted by KazetNagorra
It's not always dubious physics, in plenty of cases it is a pretty good approximation, and for e.g. a short pulse the physics related to the finiteness of the pulse is not always relevant and/or interesting.

That does depend on the level of description. At a fundamental level, if something's infinite in physics it either means there's a phase transition and it's a derived quantity like heat capacity or there's an event horizon neatly hiding it.

Originally posted by DeepThought
That does depend on the level of description. At a fundamental level, if something's infinite in physics it either means there's a phase transition and it's a derived quantity like heat capacity or there's an event horizon neatly hiding it.

There doesn't have to be anything fancy. Take for instance the classical ideal gas of infinitely small particles interacting infinitely strongly when two particles meet (bouncing elastically), which is still a decent description of a dilute gas.

Originally posted by KazetNagorra
There doesn't have to be anything fancy. Take for instance the classical ideal gas of infinitely small particles interacting infinitely strongly when two particles meet (bouncing elastically), which is still a decent description of a dilute gas.

In the ideal gas model the particles don't interact with anything other than the walls of the container. The particles can't scatter off each other as they're infinitely small so the cross-section for contact scattering is zero. You could develop a classical ideal gas model of a molecular cloud where the density is on the order of a few molecules per cubic metre and have them only interacting gravitationally. Since they basically aren't going to get close enough to each other for their size to matter the model is fine. So in a practical sense I agree with you that one can build workable models with these things.

At a fundamental level the point like nature of the particles is a problem as it implies an infinite field strength near the particle. Quantum field theory half deals with the problem by having a cloud of virtual particles shielding the bare electron and String Theory goes further by avoiding point-like particles altogether. In QFT the propagators are all spherical waves which have much better normalisation properties than plane ones, although there's still a pole at k² = m².

Originally posted by DeepThought
In the ideal gas model the particles don't interact with anything other than the walls of the container. The particles can't scatter off each other as they're infinitely small so the cross-section for contact scattering is zero. You could develop a classical ideal gas model of a molecular cloud where the density is on the order of a few molecules per c ...[text shortened]... much better normalisation properties than plane ones, although there's still a pole at k² = m².

You are right, my language was sloppy. The ideal gas is non-interacting and thermalization happens through the container wall. Still, if one models the finiteness of particles using a weak delta peak repulsion the behaviour of a dilute gas is pretty much the same as an ideal gas.

Originally posted by KazetNagorra
You are right, my language was sloppy. The ideal gas is non-interacting and thermalization happens through the container wall. Still, if one models the finiteness of particles using a weak delta peak repulsion the behaviour of a dilute gas is pretty much the same as an ideal gas.

I think we're saying the same thing, but with different emphasis. Your dilute gas model needs to be a quantum model to work though. A delta peak will scatter an incoming quantum particle, as if it didn't then a finite, but extended, potential wouldn't scatter either and nothing would ever interact. A classical particle on the other hand is just going to miss.

Originally posted by DeepThought
I think we're saying the same thing, but with different emphasis. Your dilute gas model needs to be a quantum model to work though. A delta peak will scatter an incoming quantum particle, as if it didn't then a finite, but extended, potential wouldn't scatter either and nothing would ever interact. A classical particle on the other hand is just going to miss.

Yes, you are right. I guess I am too used to quantum thinking to make sense of the classical world. The classical analog would require finite-sized "billiard balls."

Originally posted by KazetNagorra
Yes, you are right. I guess I am too used to quantum thinking to make sense of the classical world. The classical analog would require finite-sized "billiard balls."

You're the quantum physicist who determinedly watches his beer to use the quantum Zeno Effect to prevent the beer tunnelling out of the glass while he's not looking!