Quantum take on LGA
daniel B miller
Computing:
http://citeseer.ist.psu.edu/meyer96from.html
But of course it is radically different from what most of us here want
to achieve, which is _total_ discretization of all of physics. The
representation of the superposed quantum states is necessarily done with
(floating point) complex numbers.
What confuses me is this: even if we assume the extravagance of a
MWI-type approach, ISTM there should be a way to avoid floating point
(ie we-wish-they-was-reals) numbers, and do everything on an integer
basis. Rather than propagate a probability amplitude, as Meyer does in
this paper, the whole set of possible Universes should be computed in
parallel. Since all decisions are discrete and countable, the whole
shebang has to remain finite. The definition of the 'actual
probability' of making some observation must then be a rational number:
specifically, a ratio of the number of ways we can travel from all the
states consistent with our initial conditions to all the states
consistent with the outcome in question (ie, the path integral through
spacetime!)
So if we wanted to work backwards from this to a statistical theory, the
probabilities should still all be rational numbers. Even in the realm
of Quantum phenomena as concieved from the MWI perspective, I see no
need for the continuum -- just lots & lots of (normal, non-quantum) bits.
-dbm
ps I am aware some of these ideas have been proposed before in Yahoo.
I"m not claiming to have invented anything original; I'm just noticing
the subtle difference between a truly discrete/finite approach and the
traditional way thngs are done.
daniel B miller
daniel B miller wrote:
>...
> specifically, a ratio of the number of ways we can travel from all the
> states consistent with our initial conditions to all the states
> consistent with the outcome in question (ie, the path integral through
> spacetime!)
should be:
the ratio of the number of spacetime paths consistent with our initial
conditions and outcome in question, to the total number of spacetime
paths consistent with just our initial conditions.
dbm
Gerard Westendorp
[..]
> What confuses me is this: even if we assume the extravagance of a
> MWI-type approach, ISTM there should be a way to avoid floating point
> (ie we-wish-they-was-reals) numbers, and do everything on an integer
> basis.
Maybe the use of reals is just a kind of pragmatism. Even if you
could in principle use rationals, it is simply often a faster
route to solving a problem to use floats.
For example, it is quite laborious to add say 10 different fractions,
you would have to find the common denominator etc. But with floating
point arithmetic, you can quickly get an approximate answer.
The advantage becomes even greater in more complex cases as for
example in a set of N linear equations. You can get an exact
solution with Cramers rule, but that takes up a huge amount
of computation (~N!) for large N. Gauss elimination is
much faster (~N^2), but only if you use floats.
But I guess it is a valid question to investigate if nature can be
ultimately expressed in some nice way in terms of integers.
Would you accept other fields, e.g. Gaussian integers?
Gerard
dan
> daniel B miller wrote:
>
> [..]
>
> Maybe the use of reals is just a kind of pragmatism. Even if you
> could in principle use rationals, it is simply often a faster
> route to solving a problem to use floats.
>
>
> But I guess it is a valid question to investigate if nature can be
> ultimately expressed in some nice way in terms of integers.
> Would you accept other fields, e.g. Gaussian integers?
>
using a finite number of integers. sqrt(2) fits that bill, as do many
other types of information. The key is to avoid any representation
*or* computation that requires any kind of infinite process to be
exact. For instance, sqrt(2) could be used in the sense that we could
represent a quantity as sqrt(2), and if we needed to multiply it by
itself, the result would be 2. However, the general idea of using
PDE's to model physical behavior would be disallowed, because there is
no perfectly accurate solution to a complex PDE.
Thinking about it, it's not whether we are using reals or not that is
at question. It's simply whether we are dealing with a finite
representation of all quantities involved, *and* finite computation
time to get from one finite representation to another at a different
time.
Tim Tyler
> Maybe the use of reals is just a kind of pragmatism. Even if you
> could in principle use rationals, it is simply often a faster
> route to solving a problem to use floats.
Few computer programming languages provide syntactical support
for fixed point arithmetic - so using floats is often necessary.
However floats are just as finite and discrete as integers are ;-)
--
__________
|im |yler http://timtyler.org/ t...@tt1lock.org Remove lock to reply.
Ed Fredkin
>
> [..]
>
> > What confuses me is this: even if we assume the extravagance of a
> > MWI-type approach, ISTM there should be a way to avoid floating point
> > (ie we-wish-they-was-reals) numbers, and do everything on an integer
> > basis.
>
>
>
> Maybe the use of reals is just a kind of pragmatism. Even if you
> could in principle use rationals, it is simply often a faster
> route to solving a problem to use floats.
>
point processes in a computer. At the bottom (in the CPU chip) are
logic gates such as NAND or NOR which are both universal gates. The
only components needed to construct a computer are a collection
universal gates (such as NAND gates) and wires. It is easy to
construct the processor, memory, clock circuits, etc. out of just
these 2 elements. With a few million (or billion) such elements,
properly connected, we get a computer.
The analogy to physics is a bit strained but nevertheless useful.
There is no doubt that discrete (digital) processes at the bottom can
easily produce phenomena best described by differential equations
involving apperently continuous variables. Of course, the
differential equations are not mathematically an exact description.
We now know that there are discrete models of physics, basically made
up out of bits, that can function in ways that exactly conserve
quantities such as momentum or spin. The fact that quantities are
conserved exactly leads to the applicability of differential equations
that are excellent models.
What all this implies is only that the success of the calculus in
modeling physics is not a rational argument against the idea that at
the bottom - physics is a totally discrete process.
Ed F
ueb
> Gerard Westendorp wrote in message news:<4058D756...@xs4all.nl>...
The conserved quantities are integration constants of these differential
equations.
> What all this implies is only that the success of the calculus in
> modeling physics is not a rational argument against the idea that at
> the bottom - physics is a totally discrete process.
If you say "improper calculus", I can unconditionally agree.
However, the question keeps open with assumed proper calculus.
It is hardly known, that the known Einstein-Maxwell equations indeed
predict particle numbers, i.e. the particles are discrete solutions
of them, see http://home.t-online.de/home/Ulrich.Bruchholz/ .
As well, these solutions emerge from an algorithm that goes from
discrete time & length coordinates. Any convergences to very small
differences do *never* match "conventional" solutions. The latter
do not let see discrete values of the integration constants, unless
one introduces any strained margin.
Thus, I'd suppose that the widely known conventional solutions
come from an improper calculus. These are only good for the initial
conditions.
Ulrich Bruchholz
info at bruchholz minus acoustics dot de