Complex Questions: Exploring Physical Correlations and Interpretations

[danitaber]

On page 4 or so. . .

Finally there is the problem of complex numbers. Now don't get me wrong complex number theory is (in my humble opinion) the most beautiful piece of mathematics I have ever encountered. However that doesn't get over the "pie" problem. Integers are whole pies, rational numbers allow for slices of pies.. but what is a complex pie? The answer is that there is no such thing! Complex numbers have no physical correspondence. So here we have a theory where the main object - "the wave-function" has no physical meaning at all and that does separate qm from the other theories in physics.

I can't let this one go, sorry. In your "pie" problem, what about negative numbers? Pieces one already ate? What physical correspondence is there? I have yet to understand people's objections to complex numbers (including my husband's.) A complex number makes sense: you multiply it to another complex number to get a real number, you multiply a negative number to another negative number to get a positive number. You just can't add them to get a real number. I think it's just the names that get to people.

Mathematics comes from the world around us; that's how we got it. Complex numbers wouldn't have been formulated if we didn't need them.
(This is also why I'm dismayed when others are surprised about the applicability of an obscure branch of mathematics to some physical system. It has to apply to *something*. . .)

Anyway, I guess my point is, why would Psi need to be anything more than a "probability density"? Does the square root of any other physical quantity have to have a separate physical meaning? As far as I have read, Psi contains what we can speak about. That's a tall order in and of itself.

As for the rest of the discussion:

I agree it would be *nice* to have some clasically-compatible interpretations for these things, but does the universe require it? Must the universe behave according to our ability to create analogies? I don't think so.

I am pro "let's play with the math and see what we can find", but let the experiments and the math do the actual talking. :-X


To ZapperZ;
if this is too naive, let me know and I'll shut my mouth.

In fact, I think I'll save ZapperZ the time and shut it now.

 

[humanino]

I don't find your post naive danitaber, maybe I am naive too. I read somewhere that Zz can't get enough of people arguing complex numbers are not real. He wants a couple of them for breakfast every day

Following your lines i would add :

but what is a complex pie?

What is a negative pie ?
What is one third of car ?
What is pi humans ?
Where are matrix of pie ?
Who needs tensors of chicken ?
How do I get a twistor of this wine ?
Wonderful dress ! It will fit perfectly with your propagator !

 

[bd1976]

"A complex number makes sense: you multiply it to another complex number to get a real number, you multiply a negative number to another negative number to get a positive number. You just can't add them to get a real number. I think it's just the names that get to people."

Sorry about the pies thing, terribly simplistic I'm afraid. I agree that complex number make compete sense mathematically but they are a mathematical device. There is no natural physical correspodance to such a number. Yes you can define one - as in quantum mechanics - but yoiur still left with the fundamental problem of how the maths relates to reality.

 

[humanino]

Only natural integers are real. The rest is human invention. (Or something close to a quotation.)

I think we discover math.

 

[bd1976]

What is a negative pie ?
What is one third of car ?
What is pi humans ?
Where are matrix of pie ?
Who needs tensors of chicken ?
How do I get a twistor of this wine ?
Wonderful dress ! It will fit perfectly with your propagator !

Firstly money!

integers = how many pennies you have,
real numbers = how many bits of pennies you have,
negitive numbers = how much debt you have,
Complex numbers = ermmm... well maybe the square of the number is the probabilty
that your shares are about to collapse?

Secondly my lecturer told me that numbers actually arise from sets so there we are - maths exists without the real world anyway!

p.s He went on to say that sets can arise from numbers


*scratches head* Glad I gave up number theory

 

[Rothiemurchus]

When do complex numbers arise?
Is there a well established theory already for what type of equations produce complex numbers as solutions? And do different equations, with complex solutions, force us
to reach some similar conclusions?

 

[selfAdjoint]

Any polynomial equation above the linear could have complex roots. They occur in conjugate pairs. Quadratics, for example, have two complex conjugte roots if $$b^2 - 4ac < 0$$.

 

[selfAdjoint]

Any polynomial equation above the linear could have complex roots. They occur in conjugate pairs. Quadratics, for example, have two complex conjugate roots if $$b^2 - 4ac < 0$$.

 

[danitaber]

Sorry about the pies thing, terribly simplistic I'm afraid. I agree that complex number make compete sense mathematically but they are a mathematical device. There is no natural physical correspodance to such a number. Yes you can define one - as in quantum mechanics - but yoiur still left with the fundamental problem of how the maths relates to reality.

What I'm trying to say is that all math, whether we like it or not, comes directly from the world around us. All of the laws, all of the postulates, all of the calculi, everything. So there is *always* some kind of correspondence to the physical world. If you want to try it, try thinking of complex numbers simply as another dimension, another axis. But we can only see the "real" shadows. There's a correspondence with math you see every day. Or take a Moebus strip. You go around it once, you are on the *opposite* edge of the strip. Go around it again, and you're back where you started. You have to go around the strip twice to map the circumference once. Or logic. You take the inverse of the inverse, and you're back where you started.

But this is way off topic, and I didn't really mean to start a new discussion; I'm still trying to wrap my brain around and keep up with the main one

 

[vanesch]

If you want to try it, try thinking of complex numbers simply as another dimension, another axis. But we can only see the "real" shadows.

This comes up regularly: complex numbers are not "physical". My usual reaction is to say that *real* numbers are the true mystery! The Pythagoreans who discovered them (or at least the fact that rational numbers aren't the last word) thought of this as such a terrible discovery that they kept it secret.
MOST real numbers cannot be specified. You cannot write down a formula, an equation, that has as a solution most of the real numbers.
Yet nobody complains about the fact that what we call Euclidean space is specified by real numbers. Nevertheless, *couples* of real numbers suddenly give difficulties ??

cheers,
Patrick.

 

[bd1976]

One should study things in order
You talk about quantum gravity and then you ask the above question ; I don't want to sound offending, but it is a bit as if you subscribed to Formula 1 contests, and ask the technician on the starting line, what do people mean by "changing gears" ?

Ah but you forget complex number theory is not required reading for a physics degree! (Crackers I think but that's the truth!) Many physicsts I have talked to don't have a firm grounding in these ideas. It just shows how far modern education has fallen!

I think possibly the real mystery is why maths is a tool for describing physics at all!

Yet nobody complains about the fact that what we call Euclidean space is specified by real numbers. Nevertheless, *couples* of real numbers suddenly give difficulties ??

Erm.. This is just confusing the issue. Complex numbers are pairs of real numbers which obey specific mathematical rules which define their behaviour. These rules don't derive from the real world they are a mathematical invention. That's what people have a problem with.

 

[bd1976]

p.s

What I'm trying to say is that all math, whether we like it or not, comes directly from the world around us. All of the laws, all of the postulates, all of the calculi, everything.

I hope the pure maths department at Bristol never has to read this.. you just wiped out the meaning of their lives!

 

[Kane O'Donnell]

There are, in fact, quite a lot of motivations for the existence of complex numbers. I keep reading people saying 'they're just a mathematical invention'. They're no more an invention than the reals, or even the rationals. They're a perfectly natural extension of the reals. You can, for example, define the complex numbers as field isomorphic to a particular quotient group of a group of polynomials with particular symbology. Or you can define them by extending the reals such that the square root function is defined everywhere (this is the usual way people meet them). Or...blah blah blah.

The reals, similarly, can be thought to arise as the 'completion' (in the Cauchy sequence sense) of the rationals, or you can define them using other methods that arise from the rationals. Note that it is impossible (well, infinitely tiring, at least!) to write out a complete decimal expansion for *any* real number (if we remember that 1 for example is really 1.00000000... in the reals) so no 'real' numbers exist in the 'real' world either!

Maths is about (or *more* about) finding out what you *can* do given a particular structure, not about relating it to the real world. The fact is, mathematics is an extremely precise and efficient language for representing the universe. Why this may be is a matter for philosophy, and hopefully one day it will come within the reach of Physics. Until then, stress less. Complex numbers exist. Move on :)

Kane O'Donnell

PS - complex number theory isn't taught and quantum gravity is? What absolute rubbish. If you're advanced enough to be seeing quantum gravity (which is obviously a *very* leading edge field, being the last of the forces yet to be quantised successfully) you should be beyond undergraduate, at *least* in Honours and probably post-grad. If you haven't had a thorough grounding in complex analysis, there is something seriously wrong - even the engineers have to take it at our uni.

 

[TenaliRaman]

*Dennis defines a new system the "sqrt(three)" number system
where each number is of the form (a+sqrt(3)b)*

Peewee : does this have physical intepretation ?
Dennis : ... errr ummm
Peewee : is this useful?
Dennis : ... err ummm
Peewee : well?
Dennis : hey that doesn't stop me from defining this
Peewee : right maybe some one will find use for it someday
Dennis : yeah i hope the same too

Peewee : hey if i replace ur 3 with -1 then i see that is highly useful in many problems
Dennis : huh?
Peewee : i will call it the complex number system
Dennis : hey does this have a physical interpretation?
Peewee : did urs have?
Dennis : right u are! cheers!

-- AI
Moral of the story : It was actually peewee who found the complex number system
Ofcourse Gauss then simply developed it into a beautiful complex number theory with gaussian integers and gaussian primes and what not?

 

[robert Ihnot]

Complex numbers were called "imaginary," because since the days of the Babylonians up to Cardan there was a very strong desire to reject such an entity, which came about originally from observing quadratic equations.

This is interesting since I heard that the neutrino "just fell out of equations," and was a thing that nobody had ever seen or measured, or even thought about previously--just like imaginary numbers!

This gives us a strange observation: while some think that all math comes from the physical world, could we turn the argument around and suggest, "Our underlying understanding of the real world comes from math?" Lord Calvin said something about, "when you can tell it to me in numbers, then I think you understand it."

At the time of Newton’s theory of gravitation, there were certainly other theories making the rounds, yet it was almost instantly recognized that Newton’s theory, grounded in math, was much superior, and for example could be used to study the orbit of the moon and the tides here on earth. Other theories based on the difference between heavenly matter and earthly matter could not see any relationship between the tides and the moon.

Now if philosophers along with the general public decided that mathematical relationship were not important and misleading, then Newton’s theory might have been rejected and forgotten, for all I know. This strongly suggests that it is the highly positive acceptance of a mathematical framework that gives us our accepted physical world belief system.

 

[Kane O'Donnell]

There is yet another aspect to complex numbers in that they are the basically the most 'complicated' field you can construct using 'numbers' as the objects - if you go up to quarternions, for example, you lose commutativity.

So complex numbers, numbers that look like $$Ae^{i\theta}$$, seem to be the most 'evolved' version of the abelian counting numbers you can have.

By the way, the previous post about numbers of the form $$a+\sqrt{3}b$$ was a little strange - these numbers (if a and b are rational) do actually form a field, the only problem is once you start doing analysis, the fields $$\mathbb{R}$$ and $$\mathbb{C}$$ sort of stand out, so most analysis is done using these fields. There are also apparently some subtleties involving linear algebra when you aren't using the real or complex numbers as the base fields.

Cheerio!

Kane

 

[TenaliRaman]

...By the way, the previous post about numbers of the form $$a+\sqrt{3}b$$ was a little strange - these numbers (if a and b are rational) do actually form a field, ...

It was intended to mean a "field" ofcourse.. just tried to put it in a more subtle form since Q[sqrt(3)] being a field wasn't the question here ...

-- AI

 

[matt grime]

So complex numbers, numbers that look like $$Ae^{i\theta}$$, seem to be the most 'evolved' version of the abelian counting numbers you can have.

p-adics? p-local?

 

[matt grime]

What I'm trying to say is that all math, whether we like it or not, comes directly from the world around us. All of the laws, all of the postulates, all of the calculi, everything.

That is a matter of opinion. At what point do we stop saying something came from something else? What earthly thing motivated the Third Isomorphism Theorem or the definition of an Exact Category? (And, yes, I am picking something as obtuse as I can reasonably think of.)

 

[matt grime]

p.s



I hope the pure maths department at Bristol never has to read this.. you just wiped out the meaning of their lives!

Was that aimed directly at me or is it just a coincidence?

 

[Kane O'Donnell]

p-adics? p-local?

Yuck!

Ok you have me on that one, but I guess I was more focussed on fields with similar structure to those below them but with extra features.

Don't even talk to me about p-adic numbers. It brings back memories of having to show whether functions from a p-adic topology were continuous in certain circumstances. Yuck.

Cheerio!

Kane