Insights The Need of Infinity in Physics - Comments

1. Jan 12, 2016

dextercioby

2. Jan 12, 2016

Staff: Mentor

Nice write up.

Indeed QM requires an infinite dimensional vector space. But I have to mention in modern times neither Martix Mechanics or Wave Mechanics are used. What's used is the more general transformation theory worked out by Dirac that incorporates both, plus his rather strange q numbers:

Thanks
Bill

3. Jan 12, 2016

Helios

The notion of infinity in cosmology was the proposition of Anaximander in the 6th century BC.

4. Jan 12, 2016

Isn't one of the reasons that time and space are continuous?
So we need the concept of infinity to calculate dynamics in such a environment?

5. Jan 12, 2016

davidbenari

Hmm. Don't they use the wave formulation a lot in chemistry and matter physics?

6. Jan 12, 2016

mathman

As a mathematician, it seems to me that the insight is making a mountain out of a molehill. Infinity is a useful item to have when carrying out calculations. Until you get to Cantor, it doesn't have any deep meaning that needs to be probed.

7. Jan 12, 2016

Greg Bernhardt

8. Jan 12, 2016

Dr. Courtney

Maybe not in your end, but I recall writing and running codes to diagonalize really big matrices (100k by 100k) in atomic physics.

9. Jan 12, 2016

10. Jan 12, 2016

Jano L.

This paragraph is really weird. How can the author possibly label person working in team in a research facility/institute as "modern Michael Faraday"? The work they do in modern laboratories is rarely even close to what Michael Faraday was doing in his research. He did moderate-cost research of basic EM phenomena with a small-size self-made equipment (he studied EM induction with magnets and coils). In CERN, they do immense-cost research of subtle and exotic EM phenomena with expensive machinery which takes years to build (they study what detectors say happens after microscopic particles collide).

And why does the author suggest that everybody else is doing "no more than applied mathematics"?

I can say that whenever "I was thinking of physics deeply", how rare soever it was, I have never had a feeling like I'm doing applied mathematics. I am not sure what the author thinks doing applied mathematics means, but I guess it means you're not doing physics at all and you're either calculating consequences of a mathematical model given to you or you're developing such a model based on some mathematically formulated requirements.

I think that when you think about theoretical physics deeply, you're thinking about how the claims from professor, peers, textbook or paper are inconsistent either with themselves or with other physics known. You're trying to discern which ideas are experimental facts, which are questionable interpretations of such facts, which are just a popular way to think of them but not really necessary. You're thinking whether they can possibly be consistent with that or that theory and facts. Or you think about statements of a person who claims he solves some physics problem and you're trying to find whether he's right by analyzing the arguments and validity of the assumptions made. In many ways, deep thinking in theoretical physics is much like deep thinking in philosophy (it really originated in there). Applied mathematics is not a good name for such endeavour, I would say.

It is true that the standard way to talk about Heisenberg matrices and Schroedinger operators is using the concept of infinity. However, neither matrices nor operators really are the core part of the theory that implies the predictions and explanations derived from it.

The core is the Schroedinger equation and the Born interpretation. The equation is a partial differential equation in coordinates and time.
This equation works with concepts of derivative and differentiable function, which are close to concept of infinity. But it can also be discretized and its solutions calculated in computer with no use of infinity. This can be done so it leads to predictions/explanations arbitrarily close to those you would get from the partial differential equation. The infinity has no more special significance for Schroedinger equation any more it has for the heat conduction equation or wave equation.

11. Jan 12, 2016

jambaugh

" It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics..." I dispute vehemently. The "core" of all mathematics is the rigor of deductive logic applied to axiomatics. Infinities manifest when it is improper or inconvenient to impose the actual finiteness we find in applications of mathematics.

The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary.
Physically we never actualize infinities except possibly in the measure of ignorance which is always infinite in contrast to our finite knowledge. The corollary to this is finite information encoded in an infinitude of possible ways which underlies the mysteries of quantum mechanics.

This is not to say that we should discard the (*mathematical*) concept. Many people measuring distances each using distinct minimal units of precision would not have their measurements readily comparable unless we mapped them all into the "infinite precision" ideal of a continuum of measurements. We, also, may extrapolate well beyond the effects of given conditions in a certain application model and when doing so it is convenient to speak of "behavior at infinity" but this is simply short hand for "behavior beyond the significant influence of the aforementioned effects."

In short Infinity = Ignorance (as to where the boundary lies in some application of the theory).

If some construct, (such as the continuum of space-time) is necessarily infinite then we should always second guess any attempt to treat such a construct as manifestly physical. (Hence, do NOT take too seriously the ontological reality of the "space-time" manifold and its geometry.)

Or so I would assert.

12. Jan 12, 2016

Staff: Mentor

The point was its not one or the other. They are both different aspects of an even more general theory.

Thanks
Bill

13. Jan 12, 2016

Staff: Mentor

We don't know one way or the other, but calculus is so powerful a tool you model it that way. In QM we don't know if an actual infinite dimensional space is needed, but powerful theorems from functional analysis such as Stones theorem cant be used if its not modelled that way.

Personally in QM I consider the physical realizable states to be finite dimensional, but perhaps of very large dimension, experimentally indistinguishable from an actual infinite one. One then, for mathematical convenience, and since we don't actually know the dimension, introduces states of actual infinite dimension so the powerful theorems of functional analysis can be used.

Thanks
Bill

14. Jan 13, 2016

mma

>Avogadro’s number is big enough to be considered the physicists’ true infinity.

I think that the notion of infinity in mathematics is quite different from "very big". If you add one mole of oxygen to one mole oxygen, the you get two mole oxygen. This is definitely more, than one mole. But if you add infinite number of elements to an infinite set, then the "size" of the set remains unchanged. This is the difference.

15. Jan 13, 2016

Samy_A

Actually, the "size" (or cardinality) of an infinite set can change when you add an infinite number of elements to an infinite set (example: add a set of the size (cardinality) of $\mathbb R$ to a countable infinite set).

16. Jan 13, 2016

mma

You are right. I should have told countably infinite.

Last edited: Jan 13, 2016
17. Jan 13, 2016

mma

This interesting fact was new to me. I found the details in Wikipedia:

18. Jan 13, 2016

ShayanJ

What you say is very hazy and strange to me. Could you give a reference?

19. Jan 13, 2016

Staff: Mentor

The dual of all the row vectors of finite dimension contains everthing used in QM - its in fact the maximal space of a Gelfland tripple:
https://en.wikipedia.org/wiki/Rigged_Hilbert_space

All the spaces used in QM are a subset of this space.

Thanks
Bill

20. Jan 13, 2016

ShayanJ

I think what I actually don't understand is what definition of dimension you are using. Because for me dimension is a property of the space not the states!