Re: Space-Time, mathematics and physics

On Jul 29, 9:52 pm, Jean Paul <jcorriv...@xxxxxxx> wrote:
Hello.

In my post "are the space-time continuous or discrete", some members
have misunderstood my request for clarifications, in particular the
person who commented "he dont even grasp tha notion of continuous and
discrete". My MSc in Mathematics is 20 years old, so I forgot a lot
but I certainly did not forget what these two concepts mean. I did not
join this forum to be insulted, but rather to seek clarifications
about what the Space Dimension is *physically*, and what the Time
dimension is *physically*. I wish physical explanations, not
mathematical formulas (unless you are sure that they can lead me to
answers).

One member referred to the formula:
r = ct
(in the case of radiation which travels at the speed of light, if I
infer correctly from the formula.)
to show me that if "t", in Time dimension, is continuous, then so is
"r", in Space dimension. Similarly, if "t", in Time dimension, is
discrete, then so is "r", in Space dimension.

That is clear. I have no objection. I understand that.

But the set of coordinates (r, t) represent a *trajectory* of some
radiation (or whatever it is)

Appropriate caution. :-) Propagating electromagnetic
radiation is not well modeled as a particle with a
"trajectory". It is best (quqntitativly) model as a
particle that explores all classical paths with a
clock related to Fermat's principle of least time.

<< Snell's law may be derived from Fermat's principle,
which states that the light travels the path which takes
the least time. By taking the derivative of the optical path
length, the stationary point is found giving the path taken
by the light (though it should be noted that the result does
not show light taking the least time path, but rather one
that is stationary with respect to small variations as
there are cases where light actually takes the greatest
time path, as in a spherical mirror). In a classic analogy
by Richard Feynman, the area of lower refractive index
is replaced by a beach, the area of higher refractive
index by the sea, and the fastest way for a rescuer
on the beach to get to a drowning person in the sea
is to run along a path that follows Snell's law. >>
http://en.wikipedia.org/wiki/Snell%27s_law


http://en.wikipedia.org/wiki/Snell%27s_law#Derivations

http://en.wikipedia.org/wiki/Functional_integral
http://en.wikipedia.org/wiki/Feynman_path_integral

So... the classical paths can't be ignored.

Time-independent Maxwell equations
Time-dependent Maxwell's equations
Relativity and electromagnetism
http://farside.ph.utexas.edu/teaching/em/lectures/lectures.html

There are two reasons to favor a particle model of
light.
--A love of statistics.
--A hatered of Maxwell's equations.

The latter is the more common motivation. ;-)


in the Space-Time dimensions. In the
case of particules, I deducted by myself that their trajectory is
discrete. So then this formula reflects a discrete function,
suggesting that Space and Time dimensions are discrete. Again that
formula expresses the trajectory of a particle, or radiation, or
whatever. The formula does *not* imply anything about the *nature* of
the Time and Space dimensions. If the trajectory is discrete, does
that imply that the dimensions are discrete too? That is my big
question to you all, and the answer does not seem so immediate.

A particle model on this level is not a choice of tool
but rather a choice of career.
http://en.wikipedia.org/wiki/Sum_over_histories#Ward-Takahashi_identities

Can you devote a few years to study what is being modeled?


I have to admit that a continuous dimension appears to be an
impossibility in the physical world, if we assume the definition of
continuity from mathematics. So this suggests that all dimensions are
discrete. But philosophically, I find this extremely difficult to
grasp.

If the coordinate system you choose is continuous.
then you will interpret the results as continouos.

Atoms are discontinuites as are lenses, reflectors
gasses, etc.


Maybe we should not view particles travelling in Space-Time, but
rather view something less tangible like energy, waves... travelling
in space? If we refer to particles jumping around in discrete steps,
then it appears impossible to avoid the question of what is in between
those steps? This is my dilemma.

The wave model demands so much attention, it must be
imbedded for a particle model to be valid.


If anyone can help me, I would be grateful. I know that it would help
if I knew more about physics. All I learned in college is one course,
the classical Newton's mechanics. Is it at all possible to make me
understand without me having to take a BSc in Physics? I'm too busy to
take courses toward a BSc.

Learn Maxwell's equations. There is no getting around them.

Time-independent Maxwell equations
Time-dependent Maxwell's equations
Relativity and electromagnetism
http://farside.ph.utexas.edu/teaching/em/lectures/lectures.html

Maxwell's equations in classic electrodynamics
(classic field theory)_
a) Maxwell equations (no movement),
b) Maxwell equations (with moved bodies)
http://www.wolfram-stanek.de/maxwell_equations.htm#maxwell_classic_extended

http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/light/index.htm
http://www.ee.surrey.ac.uk/Personal/D.Jefferies/antennas.html

Sue...



Thank you.


.

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