quantum mechanics over a commutative rig

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John Baez

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Nov 6, 2001, 3:41:04 AM11/6/01
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A "commutative rig" is like a commutative ring but perhaps
lacking additive inverses. A good example is the natural
numbers. If you're interested you can dream up the precise
definition yourself.

Jim Dolan pointed out to me that there are just two commutative
rigs with two elements. One is Z/2 with the usual addition and
multiplication mod two. This is actually a commutative ring -
even a field. The other is {F,T} with "or" as addition and
"and" as multiplication. This doesn't have additive inverses.
The first example looks more like the second if we describe it
like this: it's {F,T} with "exclusive or" as addition and "and"
as multiplication. In both of these, F plays the role of 0 and
T plays the role of 1.

I was telling my pal Bruce Smith about matrix algebra over
various commutative rigs, starting with the case of the 2-element
commutative rig {F,T} with "or" as plus and "and" as times.
In this particular case Jim pointed out to me that matrices
are just what people call "relations", and matrix multiplication
is "composition of relations". For example, if we have a set
of n people and R is the Boolean n x n matrix where R_{ij} = T
iff j is i's father, and S is the Boolean n x n matrix where
S_{jk} = T if k is j's mother, then RS is the Boolean n x n
matrix where (RS)_{ik} = T if k is i's paternal grandmother.

It's also easy to grok these matrices by thinking of them as
linear operators. Say we have the operator R: V -> V and we
apply it to a vector v in V. Here a "vector" is just a list
of "numbers", by which I really mean a list of elements in
our commutative rig {T,F}. So really, a vector is just a list
of truth values. Similarly, R is a square array of truth values.

If v is the answer to the question "which cities on the list can
we start at?" - giving "true" or "false" for each city - and the
matrix for R is the answer to the question "which pairs of cities
are connected by a one day's drive?" then the vector Rv is the answer
to "which cities on the list can we end up at after a day's drive?"

Similarly R^2(v) tells us the answer for two days driving, etc..

But the cool part, to me, is that the same idea works for lots
of other choices of commutative rig.

It also works for the natural numbers, where instead of matrices
saying "can we get from this city to that one?" we get matrices
saying "how many ways can we get from this city to that one?"

And it works for nonnegative real numbers, where they say
"what's the probability to get from this city to that one?"

And it works for complex numbers, where they say "what's the
amplitude to get from this city to that one?"

(In the last two examples we need certain things to sum to 1
to get things to work perfectly. We need "normalized" vectors
and "Markov" or "unitary" matrices.)

Alas, I don't know quite as obvious a way of interpreting
the case Bruce likes best, where our rig is Z/2. It seems
that in this case, since addition is exclusive or, we're
living in a world where having two roads from Houston to
Boston is just like having none! "Complete destructive
interference"???


Ralph E. Frost

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Nov 7, 2001, 10:00:53 PM11/7/01
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John Baez <ba...@galaxy.ucr.edu> wrote in message
news:9s87n0$i...@gap.cco.caltech.edu...

> Alas, I don't know quite as obvious a way of interpreting
> the case Bruce likes best, where our rig is Z/2. It seems
> that in this case, since addition is exclusive or, we're
> living in a world where having two roads from Houston to
> Boston is just like having none! "Complete destructive
> interference"???

Might this Z/2 also be interpreted more like having one road between Boston
and Houston ends up being two roads -- one in each direction , or else the
short great circle and the long great circle?


[Moderator's note: Not really, since neither situation corresponds
to having no roads. -MM]

Squark

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Nov 7, 2001, 10:01:18 PM11/7/01
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On 6 Nov 2001 08:41:04 GMT, John Baez wrote (in
<9s87n0$i...@gap.cco.caltech.edu>):
>...

>Alas, I don't know quite as obvious a way of interpreting
>the case Bruce likes best, where our rig is Z/2. It seems
>that in this case, since addition is exclusive or, we're
>living in a world where having two roads from Houston to
>Boston is just like having none! "Complete destructive
>interference"???

Another interpretation would be that we're dealing with the number of roads mod
2 here, all the other properties of the roads being "unobservable".

Best regards,
Squark.

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Ralph Hartley

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Nov 7, 2001, 10:01:05 PM11/7/01
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John Baez wrote:
...

> Alas, I don't know quite as obvious a way of interpreting
> the case Bruce likes best, where our rig is Z/2. It seems
> that in this case, since addition is exclusive or, we're
> living in a world where having two roads from Houston to
> Boston is just like having none! "Complete destructive
> interference"???

Maybe you are thinking too hard. How about "Is there an odd number of
routes from Houston to Boston?"

That reminds me of something I have wondered about. What does quantum
"mechanics" look like if you use a finite field instead of C? I know
people have worked it out, at least to some extent, for R, H and even
O, so someone must have done the finite fields as well. Right?

I put "mechanics" in quotes above, because I would be surprised if the
resulting theory had any interpretations that resembled mechanics very
much. Certainly there would be no probabilistic interpretation, but
never mind that. If there is any interesting math there, it ought to
be good for something!

As a computer scientist, I am most interested in the computational
aspects of things (e.g. what can simulate what, etc.). But I am very
interested in physics as well (math from physics has computational
applications surprising often). Does this lead to anything interesting
(or at least non trivial)? If so, does anyone have any good
references? If not, is there a clear explanation of why not?

Ralph Hartley

Ralph E. Frost

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Nov 11, 2001, 5:11:48 PM11/11/01
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Ralph E. Frost <ref...@dcwi.com> wrote in message
news:tufslt1...@corp.supernews.com...
Doh!

So sort of like quantum tunneling, or teleportation then, huh?

John Baez

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Nov 12, 2001, 11:50:02 PM11/12/01
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In article <3BE80919...@aic.nrl.navy.mil>,
Ralph Hartley <har...@aic.nrl.navy.mil> wrote:

>John Baez wrote:

>> Alas, I don't know quite as obvious a way of interpreting
>> the case Bruce likes best, where our rig is Z/2. It seems
>> that in this case, since addition is exclusive or, we're
>> living in a world where having two roads from Houston to
>> Boston is just like having none! "Complete destructive
>> interference"???

>Maybe you are thinking too hard.

Either that, or not hard enough.

>How about "Is there an odd number of
>routes from Houston to Boston?"

Okay, fine! It's not thrilling, but maybe it's the best we should
expect from doing quantum mechanics over such a silly field as Z/2.

>That reminds me of something I have wondered about. What does quantum
>"mechanics" look like if you use a finite field instead of C?

I don't know anything other than what we've just worked out, and
its obvious generalization to Z/p for other primes p.

>I know people have worked it out, at least to some extent, for R, H and
>even O, so someone must have done the finite fields as well. Right?

Maybe, but I don't know anything about it. The closest thing
I've actually seen papers on is quantum mechanics over the p-adics,
which became popular after Witten wrote a paper on adelic string
theory. By some bizarre coincidence, right before reading this
article of yours I posted an article about a recent paper on
hep-th about this subject!

>I put "mechanics" in quotes above, because I would be surprised if the
>resulting theory had any interpretations that resembled mechanics very
>much. Certainly there would be no probabilistic interpretation, but
>never mind that. If there is any interesting math there, it ought to
>be good for something!

The idea of "adeles" is to cleverly do computations in the complex
numbers by doing computations in the p-adics for all p... in some
weird situations this is actually a good idea! Apart from that,
I bet the main application of p-adic quantum mechanics would be
to get results in number theory using tricks from physics.


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