Re: Infinite precision floating-point



Mikko Heikelä wrote:
> On Thu, 26 May 2005, Raffael Cavallaro wrote:
>
> > On 2005-05-26 14:15:47 -0400, tar@xxxxxxxxxxxxx (Thomas A. Russ) said:
> >
> > Similarly, square roots are a mathematical construct. Basically this
> > is an argument between quantum mechanics and mathematical platonism.
> > Pysicists hold that the platonic view of the world is not borne out
> > by physical experimentation. If all physical quantities are
> > discrete, not continuous, then there are no real world irrational
> > quantities - every real world quantity can be reduced to an integer
> > number of the fundamental unit for that quantity (charge, length,
> > time, etc.).
>
> Could you (or someone else expressing similar views in this
> discussion) explain how do we know that length and time are quantized?
> Or, if this is an argument by authority, point us to the authority
> i.e. an article with the aforementioned explanation...
>
> -Mikko

This is by no means part of experimental physics. However, the general
way this kind of idea arises is in discussion of quantum gravity.
(setting aside electric charge, which seems to come in discrete
amounts, with no experimentally verified explanation, and only
speculations about magnetic monopoles, or strange topological
arguments, that have led pretty much nowhere.)

Now, I will preface this by saying it is not necessary to have a full
quantum theory of gravity to understand how ordinary Earth-strength
gravitational fields affect a laboratory experiment which explores
quantum mechanics. You just add a term in your Hamiltonian representing
the gravitational potential and move on, if you like. However, taken to
extremes (such as in the neighborhood of black holes, or when the
universe was born out of something *much* more compact), the
(continuous) geometry of Einstein's theory of gravity seriously mucks
with the variables of space and time which are used in quantum field
theories. So there is lots of fun to be had by people like Hawking and
Penrose and string theorists, and so forth, trying to replace this
oil-and-water combination with something new.

In any case, the key physical "landmarks" which indicate you are
approaching this extreme, are given by taking the physical contants
which determine relativity (the speed of light, c, and the
gravitational constant G), and combining them with the physical
constant which governs quantum mechanics: Planck's h.

You can combine these constants alone to calculate a length:

http://scienceworld.wolfram.com/physics/PlanckLength.html
sqrt(G h/c^3) = 4 x 10^-35 meter.

or a time

http://scienceworld.wolfram.com/physics/PlanckTime.html
sqrt(G h / c^5) = 1.35 x 10^-43 seconds,

or a mass
http://scienceworld.wolfram.com/physics/PlanckMass.html

sqrt(h c/G) = 5.45 x 10-8 kg.

Planck was actually quite proud that his constant led to these new
fundamental constants.

Now, these are not absolute limits. The Planck mass, especially, is a
perfectly "ordinary" mass, about 10^19 amu, or 10^18 carbon atoms.
Bacteria are much smaller than this, about 10^-15 kg, but don't even
require quantum mechanics to describe their motion.

Instead, we need some context in order to make sense of when they are
extreme. A fundamental subatomic particle of the Planck mass would be
far beyond our experimental experience. The most massive subatomic
particles we have direct evidence of from particle colliders are about
the mass of atoms.

For the Planck mass, we are actually concerned about the opposite
direction: black holes are the main hallmark of gravity; if a black
hole were as *small* as the Planck mass, it would definitely need
quantum mechanics to explain it, while a black hole as massive as a
star works pretty much classically. Such a Planck-mass black hole would
also have a size roughly the Planck length, if it were still classical.

The Planck length *is* extreme, because it is much smaller than we can
measure: much, much smaller than an atomic nucleus or even a proton
(10^-15 meter). We think an electron is point-like, but only because
(crudely) throwing electrons together as hard as we can, the closest we
can make them approach one another
is roughly 10^-18 meters.

This is really what makes the most sense to think about: it might be
that "space" and "time" appear smooth and continuous only because we
are used to probing regions much larger than the Planck length, on time
scales much longer than the Planck time.

Crudely speaking, space and time could actually "look like" some kind
of foam or wooly stuff, with bubbles in in on the Planck length, or
like a chain-link fence with links the size of the planck length, or
whatever. It could even look like graph paper, where God decided to
doodle on the edges of the squares, or a rasterized kind of cellular
automata, to get even more cartoony.

To "giant" things like protons & neutrons, and ordinary atoms, and us,
we would ride over all this chaos, like an ocean liner riding over the
ocean, without worrying about the bubbly foam on the ocean surface, or
even the fact that the water is made of tiny little molecules. We just
see that space time is curved enough overall in our neighborhood that
when we drop things, they fall down.

But all this is really the most extreme sort of speculation. To claim
on this basis that space & time are "discrete" on this scale is quite
glib.

.

Relevant Pages

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  • Re: values of the 26+ fundamental dimensionless constants?
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