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31 Aug 2007

What are dimensions?

Filed under: Physics — paulcook @ 6:57 pm

Since I’ve hardly been filling this blog with posts recently, I thought I might post an email I wrote recently, in reply to a question I received about the nature of dimensions. Mine is by no means a complete answer, but maybe it’s interesting. Follow-up questions welcome!

The question:

I have been reading a lot of books and web sites on string theory. It all seems very interesting, all these extra dimension and so fourth. I was just curious, it is supposed that these extra dimensions could be real, I have yet to read how the first three dimensions that we take for granted in this universe are real physical things. I know that the term dimension is used in plotting locations and trajectories of objects in space on paper, but are they real physical things that exist in the real universe?
I would really appreciate your help in this.

That’s a tricky question. To start with the three dimensions of space (and one of time) that we’re well aware of: they’re real in that they are what makes space, well, space. The idea of “space” is that it provides somewhere that things can be — without dimensions, there’d be no way to talk about where something is, how far apart things are, and so forth. Motion is merely the movement of things within the space.

Now that’s all there was to space, before relativity. Einstein’s General Relativity shows that space itself is a “dynamical” object, which means essentially that it is something that can change. Basically, matter and space interact — space is how you define where something is, but the presence of something (matter and/or energy) in turns affects the lengths of nearby pieces of space. So light travelling near a star is “bent” by the gravity of the star, as a result of the mass of the star affecting the definition of coordinates and motion nearby. So in this respect dimensions and space become a physical entity on which matter has an effect.

The additional dimensions predicted by string theory are no different to the three (plus time) that we’re used to, at least conceptually. When specifying the position of something, we just need to specify locations in each of nine directions, as well as a time. However, the fact that in our daily lives we only experience three of those spacial dimensions means that the other six are somehow irrelevant on large length scales. This might because they’re “rolled up” really small, by which I mean that the possible range of positions in that dimension is very small, and so everything is so close to everything else in that direction that we can’t even tell that there is another dimension. Alternatively, the particles from which we’re made might be in some sense “trapped” on the surface of a three-dimension object in the nine spacial dimensions.

However, in a very real sense we still don’t actually know what spacetime really is. Quantum gravity considerations strongly suggest that space is not continuous on the smallest scales — there should exist a smallest possible length, the Planck length. Any length smaller than this makes no sense. Furthermore, gravity and the effect of mass on spacetime arise in the string theory in an exactly analogous way to that in which particles arise — as specific vibrational modes of strings. So this seems to complete our growing re-interpretation of spacetime from being merely a fixed measurement apparatus on which physics happens, to being itself a part of physics. This makes physics very difficult, however, as most of our current techniques rely on the existence of concepts like detectors at spacial infinity, or being able to define a universal starting time for an interaction. So in my opinion certainly one of the interesting areas that string theory will be exploring in coming years is “emergent geometry”, where concepts that look at large scales like spacetime will turn out to arise from quite different interactions in some theory that resembles string theory in certain regimes.

23 Jun 2006

Strings 2006

Filed under: Personal,Physics — paulcook @ 7:06 am

One more day of the Strings 2006 conference to go! It’s been a very intense conference, with many ideas and talks. I’ve gained some ideas which are at least somewhat related to things I’ve been thinking about, but I’m not sure how they’ll turn out.

I have been a little disappointed that there have been no “big picture” talks — string theory has had a slow year, and I would have been interested to hear what some of the big names were thinking about for the coming year. Though I suppose if anyone did have a big idea, they would have already published.

I’ve been contributing a little to Jonathan Shock’s very ambitious efforts to summarise proceedings at the conference, over at his blog. It’s turning into a useful reference (judging by his page hits, if nothing else), and it’s certainly be useful for me to review the talks. Though it is hard trying to write down succinct summaries of talks in other areas of string theory, especially when there are 5.5 hours of talks a day!

26 May 2005

Asymmetries in scientific fields

Filed under: Physics,Studies — paulcook @ 8:40 pm

I’ve just finished reading Michelle’s candidacy report, on some of the research she’s been doing into catalysts for certain reactions in organic chemistry. It’s highlighted for me some of the differences between different fields of research — and in particular, between theoretical physics, and the more experimental work that most other people at Caltech do.

Michelle’s report was very easy to read. That’s partially because it’s well written, but I’d say it’s also because the difficulty is in a different place to that of papers that I’m used to reading. “Reading” a longish string theory paper can take literally months — and even then, I can’t claim to understand everything that is being said. Reading Michelle’s report took a few hours, and while I didn’t understand all the jargon, I think I got the gist of the issues and approach that her research took.

On the other hand, having read Michelle’s report has brought me no nearer to actually being able to synthesise anything more interesting than a bowl of pasta and sauce. “Reading” the string theory paper, however, has involved working through the mathematics behind each step, at great length — sometimes hours for a single line. So once I’m finished, I’ve done a substantial portion of the work that was required to write the paper in the first place. The analogy would be me reading Michelle’s paper, and then making some of the reagents too — and that would take months too.

15 Apr 2005

Lines, intersections and dimensions

Filed under: Physics — paulcook @ 11:00 pm

Building on the interest in my post on knots and different dimensions, I thought I’d say a few words on some interesting issues on lines in different dimensions, as raised by a recent seminar. It’s an interesting mind game!

We explicitly consider only generic situations from here on, which means that we ignore special cases as physically unlikely. So parallel lines need not be considered — all lines will have some random angle between them.

1 Apr 2005

The problem with point particles

Filed under: Physics — paulcook @ 1:08 am

Quantum mechanics is often viewed as a “weird” theory, with all sorts of non-intuitive predictions. However, there are more serious conceptual problems with classical mechanics, at least in its simpler formulations. One of these is what happens near point particles — in fact, point particles in classical mechanics lead to all sorts of infinities. I’ll say a little about this below, as well as talk about what quantum mechanics and string theory have to say about these short-distance infinities.

10 Feb 2005

Knots and different dimensions

Filed under: Physics,Studies — paulcook @ 11:53 pm

It seems that most of the time these days, I being asked to think about things in a different number of spacial dimensions than 3 (up-down, left-right, forwards-backwards). Now, some properties follow through to additional dimensions quite easily, but I was thinking recently about one that doesn’t: knots — it’s a great example of the tricks that dimensions can play on one.

What’s a knot? Take a piece of string (an object extended in one dimension). Form a loop and feed it through the loop. The knot so formed cannot, by continuous deformation of the string, be “undone” without moving an end of the string back through the loop.

But knots only exist in three spacial dimensions.

8 Feb 2005

Blog: Luboš Motl’s reference frame

Filed under: Blogosphere gems,Physics — paulcook @ 11:45 am

One of my favourite physics-related blogs: Luboš Motl’s reference frame.

The content is about evenly split between very insightful discussions on the content and broader relevance of recent papers in string theory, and posts on politics and current affairs, from the perspective of a Harvard-based string theorist.

Be warned, though, that the technical discussions are fairly technical. For instance, here’s a great post on the recent Lin, Lunin and Maldacena paper, on a general solution for supergravity solutions, with a nice duality map to conformal field theory fermions.

Now if only he can get rid of his resident crackpot, Quantoken, who seems obliged to prove his ignorance with every one of his (many) comments…