Quantum Mechanics over Finite Fields?

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madaco
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Quantum Mechanics over Finite Fields?

Postby madaco » Tue Nov 22, 2016 8:46 am UTC

Sorry if this isn't a good place to put this.

This thread is about me wondering how much it would make sense to try to do quantum mechanics type calculations except over finite fields instead of the field C.

I am under the impression that the Bekenstein Bound (and related things) suggest that the number of ways some stuff in a region can be arranged is finite, because of there being a maximum entropy and such. (However, I don't know if that maybe is more that the measure of the set of possible states is finite, instead of the cardinality of possible states? I was assuming it was the cardinality, but I'm not sure. Maybe this is a question where there isn't much consensus yet? If you know what the consensus is, I'd appreciate being told what it is.)

So, if the number of possible states is finite, it seems to me like it would be fitting for the numbers involved to be from finite fields, seeing as only finitely many values would be possible.

And, a nice analogy of the complex numbers seems to be fields of order p^2 where p is a prime and p is congruent to 3 mod 4 (so that -1 is not a quadratic residue mod p. I don't know why these are equivalent but wikipedia says they are.). (So, you take the field of order p, and then add in i, the square root of p-1=-1). What I saw on wikipedia calls this field GI(p). It also defines complex conjugates in the way that makes sense. Also it defines absolute value based on quadratic residues (a number in F_p is treated as "negative" if it is not a quadratic residue, and positive if it is. negating flips between quadratic residues and non quadratic residues, I'm guessing because of -1 being a non quadratic residue)

So, unitary matrices over GI(p) work like normal unitary matrices as far as I can tell.

One thing I am unsure about is how to define the inner product on vector spaces over GI(p) (the norm being needed for QM), and how to define the derivative of functions over GI(p) (needed for schrodinger equation), because the normal definition doesn't seem to work. It seems that inner products cannot exist over finite fields, but that bilinear forms can? So the normal dot product (with complex conjugation) works. And this is preserved by the unitary matrices.

But, if the vector space is GI(p)->GI(p) , then the constant 1 function would have a norm of 0, which is probably not the most desirable thing in the world.

The wikipedia page I saw that defined GI(p) also defined sine and cosine, using powers of elements of GI(p).
The elements of GI(p) with norm 1 form a group under multiplication of order p+1 , so can be used to define things like sine and cosine.
But if one picks a generating element of this group, then the cosine+i*sine defined this way has a period of p+1 rather than p.

But this seems (to me) to work nicely enough, because while the vector of p 1s would have a norm of 0, the vector of p+1 1s would have a norm of 1.

And it turns out that the f_n(x)=cos(n*x)+i*sine(n*x) (for x from 0 to p) (from n from 0 to p) functions form a orthonormal basis for GI(p)^(p+1)

So I thought that defining the derivative of these as just being a linear function that sends each of these basis elements to a good result could make sense. So, for example, with p=7

d/dx[f_2(x)]=2*i*f_2(x)

d/dx[f_6(x)]=2*i*f_6(x)


(aside: f_0(x) is the constant 1 function. f_((p+1)/2)(x)=(-1)^x . All the other p-1 of them come in pairs which make the derivatives make sense, but I don't know what derivatives would make sense for these two.)

However this does not work with the product rule, at least as far as I've seen, so that seems like a thing that wouldn't work so well. I've seen at least 2 definitions for derivatives on functions over finite fields (for different notions of derivative.) and this might be connected to one of them, but I didn't understand it well enough to tell how similar or different it is. (I think it was called cyclomatic? something like that.)

So, I thought maybe this would work, but using f_n(x) as the basis vectors of GI(p)^(p+1) takes up all of them, which doesn't seem to leave room for having solutions like x*sine(x) as being the solution to some differential equations. It would only allow solutions that are sums of sines and cosines, with no powers of x.

So maybe if some of the basis vectors with derivatives defined directly were of the form (x^m)*(cos(n*x)+i*sine(n*x)) instead of all of them being the special case with m=0. So, this would mean the values of n would not cover all the possible values, but it would maybe allow for greater variety in the types of solutions you could have to differential equations. And this seems important if one wants to do stuff with the schrodinger equation with this.

(if this did relate to reality at all, p would, I assume, be very very large, so not having all the values of n seems like maybe not a terrible problem.)
_______________


alright uh, so to be clear I'm not trying to claim anything substantial. I assume that if there are any ideas in this post that are new that they are not particularly useful, and if there are any that are true and useful that they are not new.

this is just a thing I've been thinking about and I want to know, uh, which parts of it are wrong and which parts of it are obvious or already known, I guess.

I'd appreciate any feedback on this post. If part of this post is unclear, please ask me to clarify that part.

Thank you for your time.

-madaco
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Re: Quantum Mechanics over Finite Fields?

Postby doogly » Tue Nov 22, 2016 12:57 pm UTC

I know there are some books on this. Or at least, I know there are one book on this. And this is a thing:
https://en.wikipedia.org/wiki/P-adic_quantum_mechanics
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Re: Quantum Mechanics over Finite Fields?

Postby madaco » Tue Nov 22, 2016 10:45 pm UTC

doogly wrote:I know there are some books on this. Or at least, I know there are one book on this. And this is a thing:
https://en.wikipedia.org/wiki/P-adic_quantum_mechanics


Oh! Thank you!

I had briefly looked at that page, but didn't read it in depth, and didn't notice it had a section on finite fields.

I've now looked at one of the papers that that section cited on arxiv (https://arxiv.org/abs/hep-th/0605294) which I have not understood all of, but which was interesting. I note that all 3 of the papers it cites about specifically finite fields are by Felix Lev.

It seemed to me like the first half of the one that I read seemed to me to almost be philosophy, but I guess it was important to give a motivation for their work.

I am going to read more about this. Thank you!

Also, do you happen to remember the title of the book in question?

(Also, I was pleased to find that some of my blind searching seemed to be generally in the direction that Felix Lev found stuff in, though I of course, didn't know enough to find anything.)

Again, thank you for your response.

edit: oh, I wonder if I just found the book you mean.

Is it by any chance, "Quantum Theory on a Galois Field" by Felix Lev ?
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Re: Quantum Mechanics over Finite Fields?

Postby doogly » Wed Nov 23, 2016 12:40 am UTC

The one I was specifically thinking of was Adler's on quaternions. Which are not finite fields. But, still, quantum mechanics over a field that's not C, so, maybe of interest?
https://www.amazon.com/Quaternionic-Qua ... 019506643X
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