# Quantum effects caused by our mathematics?

1. Nov 30, 2015

### ande4jo

Forgive me if I sound ignorant, but is it possible that quantum affects (value of h, why physical dimensions appear to come in discreet chuncks, etc) all stem from our mathematics (which originated as counting numbers -quantum steps). I do understand that calculus was invented to help remedy this, but it still bothers me that at the heart of calculus we take limits that approach zero in a denominator but never can actually be zero. Thus an infinitely small nonzero number which is at the heart of calculus is also at the heart of quantum physics. Please don't beat me up too badly for asking ☺

2. Nov 30, 2015

Staff Emeritus
Effects. And no, since quantum behavior predates human mathematics by about 13.7 billion years.

3. Dec 1, 2015

### Demystifier

And we know that by the method of science, which is based on mathematics invented by humans.

More seriously, are you suggesting that mathematics is a human construct, and that physics is not? (I know mathematicians who would claim the opposite.)

In any case, this is an interesting variant of the chicken-or-egg dilemma. What is first, physics or mathematics?

Last edited: Dec 1, 2015
4. Dec 1, 2015

### mathman

Physics obviously came first. Math is used to describe what is already there.

5. Dec 1, 2015

### ande4jo

I am suggesting that mathematics and physics are both human constructs and that the limit we have with mathematics (infinities tat we get when we try and divide by zero), are showing up as a physical phenomenon in physics as we try and understand smaller and smaller time frames, energy levels, etc. Thus I ponder if there is a new means (similar to today's mathematics) that is still in need of discovering / inventing that could help us better understand the world we live in.

6. Dec 1, 2015

### trendal

I've been thinking about the nature of light, recently, and it ties in to what you are proposing.

What if light actually is a "wave", and its just that we can only detect it in certain energy levels? What if much of what we attribute to "quantum mechanics" is merely a result of our detection equipment (on a fundamental level)?

7. Dec 1, 2015

### Staff: Mentor

Mathematics is a human invention, but due to the interplay between pure and applied mathematics is not just pure thought - it too is influenced by applications to the world out there.

Physics on the other hand is entirely constrained by correspondence with experiment.

As to QM we have a much better idea these days what its basis is:
http://arxiv.org/pdf/quant-ph/0101012.pdf

It seems we have two choices - usual probability theory or QM - nature chose QM. Such is dictated by experiment - not mathematics.

Thanks
Bill

8. Dec 1, 2015

### Staff: Mentor

People have considered such things - none of them work.

We now know pretty well why QM is as it is - see my post above.

What it means - that is another matter - but the why of the QM formalism is much better understood these days. And even finding an alternative is severely constrained:
http://arxiv.org/pdf/quant-ph/0401062.pdf

Thanks
Bill

Last edited: Dec 1, 2015
9. Dec 1, 2015

### OCR

10. Dec 1, 2015

### ande4jo

Thanks for the feed back. Those 2 links will keep me busy.

11. Dec 1, 2015

### FactChecker

But also atoms only react to it at certain energy levels.

12. Dec 2, 2015

### Staff: Mentor

It is a wave.

It's harder to detect light at some energy levels than others, but the energy is related to the frequency and there are no frequencies at which we cannot detect light so there are no missing levels.

13. Dec 2, 2015

### Staff: Mentor

(There's nothing in quantum mechanics that says that physical dimensions come in discrete chunks)

Consider a beam of silver ions passing through a Stern-Gerlach device.
If we start with the postulates of classical physics and then apply math, we will conclude that a single diffuse beam will come out. If we start with the postulates of quantum physics and then apply math, we will conclude that two distinct beams will come out. Thus it can't be that the quantum mechanical effect is caused by some limitation of our mathematics; the math works just fine either way.

Instead, the quantum mechanical model is forced on on us by the results of the experiment. We try it and we find two distinct beams coming out.

14. Dec 2, 2015

### entropy1

I am not trained in maths very much. Nevertheless I find that mathematics is a means to describe phenomena or abstract notions. The kind of mathematics used may vary, and, in my eyes, that influences the power, accuracy range and descriptive range of it depending on the subjects it is applied to. So no, not an ignorant notion of you in my eyes (at all).

Last edited: Dec 2, 2015
15. Dec 2, 2015

### ande4jo

As I understand it (coming only from an engineering background), light (including all radio frequencies outside of our visible spectrum) come in discreet chunks called photos from E =hf equation. Does quantum mechanics dispute photon chunks of energy?

16. Dec 2, 2015

### Staff: Mentor

Of course not.

But what's going on is far from simple. Its described by Quantum Field Theory where even the number of particles is not fixed:
https://en.wikipedia.org/wiki/Fock_space

That however has nothing to do with physical dimensions - a Quantum Field is continuous.

How particles emerge is complicated - but its similar to a quantum harmonic oscillator with its creation and annihilation operators:
https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

Thanks
Bill

17. Dec 2, 2015

### Staff: Mentor

You are right that when light interacts with matter, it always deposits its energy and momentum in discrete chunks whose size is given by E=hf.

Usually when someone says that physical "dimensions" come in discrete chunks, they're using the word to refer to time and space. If that's not what you meant, we're good.

18. Dec 3, 2015

### ddd123

Photons may deposit energy in discrete chunks, but the energy free photons can take (equivalently, frequency) is not discretized as far as we know. In quantum mechanics there are discrete and continuous aspects in a mix, both discrete and continuous mathematics is used. Both kinds of mathematics are mathematics.

If you contend that mathematics originated in counting, and only later continuous models were devised which you find "suspicious" (like infinity never actually being reached) consider this. Maths originated also as geometry, which was initially separate from algebra. But geometry has already aspects of continuity of the 1600's calculus, namely irrational numbers, which expose the "existence" of real numbers, an uncountably infinite set. So continuous maths is necessarily there as much as triangles, it's not "more fictitious" than discrete algebra.

And then there's the fact that continuous maths has effectively solved Zeno's paradox, infinity is actually reached, though you may be skeptical about it if you haven't seen the proof.

19. Dec 4, 2015

### ande4jo

I guess I struggle with an infinity being reached as that seems like a contradiction. By definition i expect that an infinity cannot be reached. Also I am intrigued by that fact that calculus has at its roots a "quantum" nature (since it is a limit of a ratio's denominator that is infinitesimal but not zero in size). I understand that this point of view is possibly just my lack of adequate training in much higher mathematics but I was interested in hearing what others thought on this.

20. Dec 4, 2015

### spino

I approve your reasoning regardless of what number base or type of math that's used "as a best fit model to describe said phenomenom" the phenomenon had to be present to be modeled after in the first place.

21. Dec 7, 2015

### DuckAmuck

Are you asking if our mathematics is limited in explaining physical phenomenon? Like perhaps nature only seems to obey math up to a point? It's possible, but that's getting into some philosophy.

22. Dec 7, 2015

### votingmachine

Slightly aside from the physics ... it should not bother you that calculus formalizes the approach to zero in the denominator. It is absolutely clean math. Remember that to have an answer, it is generally zero in the numerator as well. So an expression such as x/x is going to be absolutely equal to 1 at every number where x does not equal zero. And indeterminate at 0/0. Calculus does not recommend we divide by zero, but that we consider the limit value of a process that has no limit ... considering the smallest non-zero number. And x/x will have a different limit than 2x/x.

Generally, these limits are different from "quantum". There is no fundamental "quantum" difference that the limit of that smallest possible non-zero number is. If anything, it is the opposite of quantum.

23. Dec 7, 2015

### ande4jo

Still, at the heart of calulus, if I understand it correctly, is the fact that we take a number infinitely small but not zero and assume that if the limit exists as we approach the limit from both the high side and low side, then the limit must exist at all points (including zero). Because we cannot get a real mathematic answer at x/0 we are left with the infinitely small non - zero number assumption used to describe real phenomenon and I am proposing that this is where we may be introducing quantum. Thus, we use calculus to describe energy and then end up with plank constant (or any other very small non - zero number ). And to be clear, I'm not saying this is a definite result, I am simply saying this is where I am stuck in my understanding of quantum with respect to calculus and I am intrigued by the thought that we need the next new mathematical function (similar to when physics introduced the calculus).

24. Dec 7, 2015

### Staff: Mentor

That's not the modern view of calculus.

It was all sorted out in the 19th century by great mathematicians like Wiestrass and Cauchy.

Its part of an area of math called analysis:
https://en.wikipedia.org/wiki/Mathematical_analysis

And to make things even more interesting we now have what's called non standard analysis where things like dividing by zero exist:
https://en.wikipedia.org/wiki/Non-standard_analysis

Here is a modern view of what QM is about:
http://arxiv.org/abs/quant-ph/0101012

Thanks
Bill

25. Dec 8, 2015

### zmth

"an infinitely small nonzero number which is at the heart of calculus is also at the heart of quantum physics " In quantum physics it is not an infinitely small number. Plancks constant for example is finite. These two "infinitely small limits" you speak of in mathematics and quantum physics(which is not infinitely small) are not really related - in mathematics one might consider it as sort of a topological idea for example as in a closed or open interval