question for physicists

[Kuciwalker]

Is there any evidence that real physical processes are/aren't limited to computable numbers? For example, could the probability associated with some quantum state be non-computable?

[MRT144]

question for kuciwalker : is virginia awesome?

[Kuciwalker]

Yes.

[Perfection]

I seem to recall some things involving strong interaction possibly fitting your ideas, but I can't remember where I got it from and don't have any sigificant education in QCD so I couldn't tell you anything more then just a place to look.

[VetLegion]

That's a strange question Kuciwalker. Do you mean uncomputable as in an equation describing something having irrational (non-computable) numbers in it like sqrt(2)? Or do you mean non-determinable by observers?

[civman2000]

To clarify (I certainly don't know the answer to the question): are you essentially asking whether all of the mathematical formalism of physics could be done over some natural small (countable) subfield of the real numbers?

[Perfection]

No, I think he means an uncomputable number like Chaitin's Constants

[KrazyHorse]

Re: question for physicists

Originally posted by Kuciwalker
Is there any evidence that real physical processes are/aren't limited to computable numbers? For example, could the probability associated with some quantum state be non-computable?

It's possible. In QFT processes where the coupling constant is of order one (for instance, in QCD interactions in some range of temperatures) the normal perturbation expansion cannot give an estimate of increasing accuracy.

However, it is possible that other methods of computation can give the desired answer in finite time. Lattice QCD is but one of the "non-perturbative" calculation methods. It is possible (but not certain) that all quantities not currently computable may be determined through clever methodologies.

[KrazyHorse]

By the way, reading the Wikipedia entry on computable numbers, it seems as though somebody is slightly confused. Technically speaking, all real numbers are computable by the existence of the decimal expansion.

[Kuciwalker]

Eh. We learned the definition as: there exists a program that prints out the digits of the number in sequence, and for a given digit (by a finite index) it prints out that digit in a finite amount of time.

[C0ckney]

Re: Re: question for physicists

Originally posted by KrazyHorse
In QFT processes...

QFT now has processes? awesome.

[Kuciwalker]

Re: Re: question for physicists

Originally posted by KrazyHorse
It's possible. In QFT processes where the coupling constant is of order one (for instance, in QCD interactions in some range of temperatures) the normal perturbation expansion cannot give an estimate of increasing accuracy.

However, it is possible that other methods of computation can give the desired answer in finite time. Lattice QCD is but one of the "non-perturbative" calculation methods. It is possible (but not certain) that all quantities not currently computable may be determined through clever methodologies.

The issue is that if physics includes non-computable numbers, that violates the Church-Turing thesis (the computational power of the universe is equivalent to the computation power of a Turing machine, which is a theoretical model of a computer). If that's false, we can harness these non-computable numbers to pretty dramatically increase the computational power of real computers.

[Kuciwalker]

Originally posted by civman2000
To clarify (I certainly don't know the answer to the question): are you essentially asking whether all of the mathematical formalism of physics could be done over some natural small (countable) subfield of the real numbers?

I'm wondering if there exists a complete mathematical formalism of physics that doesn't use non-computable numbers (a countable subset of the reals).

[KrazyHorse]

Originally posted by Kuciwalker
Eh. We learned the definition as: there exists a program that prints out the digits of the number in sequence, and for a given digit (by a finite index) it prints out that digit in a finite amount of time.

The problem is that the definition should not be in terms of "numbers" (since there exists a definition which prints out the nth digit of ANY number in finite time). The definition should be in terms of DEFINITIONS. The space of definitions is far larger than the space of numbers, since there exist an uncountable number of definitions for every number, some of which may be uncomputable and some of which may be computable.

[Kuciwalker]

Originally posted by VetLegion
That's a strange question Kuciwalker. Do you mean uncomputable as in an equation describing something having irrational (non-computable) numbers in it like sqrt(2)? Or do you mean non-determinable by observers?

Many irrational numbers are computable, though all non-computable numbers are irrational. Computable numbers are those that can be generated by a computer program (which is a finite sequence of symbols). Thus in some sense they are the numbers which can be "compressed" to some finite representation, and whose value can be taken from that represenation.

There are actually infinitely more non-computable numbers than computable numbers.