426 views

Skip to first unread message

Nov 6, 2001, 3:41:04 AM11/6/01

to

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.

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"???

Nov 7, 2001, 10:00:53 PM11/7/01

to

John Baez <ba...@galaxy.ucr.edu> wrote in message

news:9s87n0$i...@gap.cco.caltech.edu...

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]

Nov 7, 2001, 10:01:18 PM11/7/01

to

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"???

<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.

-------------------------------------------------------------------------------

Write to me at:

[Note: the fourth letter of the English alphabet is used in the following

exclusively as anti-spam]

dSdqudarkd_...@excite.com

Nov 7, 2001, 10:01:05 PM11/7/01

to ba...@galaxy.ucr.edu

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"???

...

> 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

Nov 11, 2001, 5:11:48 PM11/11/01

to

Ralph E. Frost <ref...@dcwi.com> wrote in message

news:tufslt1...@corp.supernews.com...Doh!

news:tufslt1...@corp.supernews.com...

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

Nov 12, 2001, 11:50:02 PM11/12/01

to

In article <3BE80919...@aic.nrl.navy.mil>,

Ralph Hartley <har...@aic.nrl.navy.mil> wrote:

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.

Reply all

Reply to author

Forward

0 new messages

Search

Clear search

Close search

Google apps

Main menu