Comments on: The other meaning of “dimension” and its use in physics

By: yoron

yoron — Mon, 08 Feb 2010 01:42:22 +0000

Rereading myself, I mean that as I see it mathematics today seems to treat ‘dimensions’ as singular objects. I haven’t seen any real proof for that view though. If one considered them emergences they would ‘appear’ as objects interwoven, which to me would suit the SpaceTime I see, very ‘plastic’. And then perhaps the real object wouldn’t be to define further ‘forces’ and ‘smaller constituents’ as much as to explain the ‘symmetries’ coming into existence.

And I can’t see how it would invalidate what we already are doing and ‘knowing’ either. It’s not as much an trial to exchange parameters as ‘reinvent them’ if you see what I mean? As everything would be as an ‘mirror explanation’ it seems to me, not so much invalidating what we already know as to try to look at it from a new aspect.

Don’t know if this made any sense, one can try though

By: yoron

yoron — Mon, 08 Feb 2010 01:33:08 +0000

Just curious here. Is there any experimental proofs for existing one and two dimensional ‘systems’ in our 3D+times arrow?

Naively imagining a two dimensional object as observed in SpaceTime I come to the conclusion that from some angles it will exist, from others it won’t. as it will have f.ex a lengt and a width but no height.

And if it doesn’t exist as proof-able experiments would it be wrong looking at it as ‘emergences’ instead? Which then won’t exclude different ‘dimensionalities’ as there easily could be ‘bubbles’ consisting of more and possibly even fewer dimensions too, but not of the ‘copy & paste’ variety. They would all be ‘self consistent’ systems if so as I see it, and ‘whole objects’, like our SpaceTime seems to be to me?

Any views of that?

By: skullsinthestars

skullsinthestars — Tue, 01 Sep 2009 19:44:02 +0000

Wade: Your story reminds me of a number of times I’ve “rediscovered” important results. I tried to console myself with the idea that, although I hadn’t found anything new, I was at least as clever as the person who first made that result!

By: skullsinthestars

skullsinthestars — Tue, 01 Sep 2009 19:42:32 +0000

Mike: Very good point! As I’m sure you’re aware, such definitions of ‘small’ and ‘large’ are significant in using perturbation theory and asymptotic analysis.

By: skullsinthestars

skullsinthestars — Tue, 01 Sep 2009 19:41:48 +0000

Mary: What’s with the anger at high-energy these days? I actually jump between MKS and CGS from time to time, depending on what I’m trying to calculate. If I’m interested only in electromagnetic wave propagation, then CGS is most elegant; if I’m trying to actually calculate physical forces/momenta with my results, CGS works.

Incidentally, I’ll have more to say on units in the future; the subject is more fascinating than one might think. Unfortunately, I first have about 500 pages of background reading to do before I feel comfortable writing on the subject…

By: skullsinthestars

skullsinthestars — Tue, 01 Sep 2009 19:38:16 +0000

IM: Yep; as Mike notes below, dimensionless parameters can be used to not only make a model more general but also give a unitless measure of what constitutes ‘large’ and ‘small’ in a problem. This can be very important in making simplifying approximations…

By: skullsinthestars

skullsinthestars — Tue, 01 Sep 2009 19:25:45 +0000

Grad: Dimensional analysis can often be the theorist’s best friend, as these examples hopefully illustrate. The problems described here can’t readily be solved by any rigorous mathematical formalism, but nice results can be found by the dimensional case.

I’m not really against any particular system of units — each of them has evolved for the convenience of a specific collection of researchers. In high-energy physics, where much of the time the end result of the calculation is a dimensionless branching fraction, dimensionless units save a lot of headaches. In theoretical optics, I find cgs units to be sanity-saving for long calculations. As long as a set is well-defined, it isn’t that much of a headache.

By: Wade Walker

Wade Walker — Mon, 31 Aug 2009 21:33:17 +0000

This technique recently helped me find an expression for the “speed of sound” in a system of nonlinear conservation equations. Since I knew the primitive variables’ dimensionalities, I could show that the speed of sound was probably proportional to sqrt(pressure / density).

Of course, right after deriving this, I realized I could have looked up the exact same expression in any fluid dynamics textbook, since my primitive variables (density, velocity, and pressure) were identical

By: Mike Fairchild

Mike Fairchild — Sun, 30 Aug 2009 16:04:12 +0000

I’ll also point out that making things unitless gives you an unambiguous definition of “small.” Once everything is unitless, “small” means much much less than one, and that’s that. This can be useful in figuring out what effects might be important or not, and this information might not be immediately clear from an equation with units.

By: Mary

Mary — Fri, 28 Aug 2009 14:16:39 +0000

You’re an *optical* theorist and not a high energy theorist, otherwise you’d only work in units where c=hbar=1 and time has dimensions of length. Or something. I got so confused by the different “simplifying” unit conventions used in my different classes that I gave up on figuring out which simplifications were consistent with one another and decided to stick to MKS units. Sure, it makes for some uglier numerical calculations, but that’s why God (oops, I mean Wolfram) gave us Mathematica. Plus, MKS are my initials.

Incidentally, I can remember being a senior in high school and suddenly deciding to do a unit analysis on E=mc^2. When I realized it worked, I wondered why no one came up with that equation *before* Einstein. It was so obvious! Later it dawned on me that they were perfectly happy with E=.5mv^2, which does, of after all, have exactly the same units…