Suppose you have some physical process you are want to describe, for example a simple situation where a few particles move around, possibly scatter from each other. Each particle has a few numbers that specify its particular state at a given time. It can have position (specified by 3 numbers), velocity (3 other numbers), electric charge, mass, energy etc. Once you specify those numbers you know precisely what particle you are discussing and what it is doing. If you have a complicated situation, with many particles moving around, you end up having to specify lots of numbers to completely describe the situation.
Let’s look at the inverse problem, suppose you are given a series of such measurements, can you reconstruct the corresponding situation? the trick is that those numbers don’t come labeled, no higher authority gives them names like charge or position, interpretation is completely up to you (as is always the case!). This is not that easy actually, and as we’ll see the results are sometimes ambiguous, so I’ll be generous and let you perform as many thought experiments as you want, even infinitely many. Why not? we are talking about theorist’s favorite experiments, the ones that cost no money.
If you are given many samples consisting of complete sets of measurements, describing different situations, you can start looking for patterns. For example, you may notice that some numbers repeat themselves. If some measurements always give the same few numbers, there must be only a few types of particles involved. The measurements then give the mass or the charge, or any other conserved numbers characterizing the type of particle in question.
On the other hand, there are other measurements which take diverse range of values. They can be small or large numbers, and can take on pretty much any value in between. Such numbers then are interpreted as either locations or velocities of the few particles we already identified. With more sophisticated analysis you can probably come up with ways of distinguishing between mass and charge, and between position and velocity, but we don’t need to get into such details. When you are done, you can map any abstract set of numbers into a mental picture. This mental picture describes the kind of simple situation we started with, a few particles moving around, possibly colliding with each other.
Once we have a mental picture, with each particle having some intrinsic properties (mass, charge, spin), and is moving around with a specific velocity in a specific direction, we can start asking basic questions about the space those particles move in. In particular, how many spatial dimensions do the particles feel? this can be defined simply as the number of different directions the particle can move, the number of components needed to specify the velocity.
Readers of my post about duality can guess what is coming next. There, I was describing how (some of the) particles in a field theory can be visualized as localized excitations of a continuous medium, and how that mental picture is ambiguous in general. Even more generally, one has to be cautious distinguishing mental pictures (or interpretations) from the physical situation, the set of measurements. Particle interpretation of a specific field configuration is one such ambiguous interpretation, one tends to use the language that is most useful, and sometimes no such useful language exists.
Similarly, even assuming particles are a meaningful concept, as I have assumed throughout, the identification of the space they explore can sometimes be ambiguous. We essentially divided all possible measurements into two kinds, some take on few discrete values, and others are unbounded continuous variables. The former are most conveniently associated with the type of particles in the system, with their intrinsic properties, and the latter with the space they move in. How can you possibly confuse the two types of quantities?
To make things appropriately confusing, we can discuss more general situations, where the two types of quantities look pretty similar, and the above division becomes more fuzzy. To start, with more and more particle species, the range of of all electric charges or masses grows, and it starts resembling an unbounded quantity. On the other hand, we can talk about momentum and position in spaces that are not just infinite flat space. If space is finite, position is not unbounded, and momentum is no longer continuous. In such situations, distinguishing spatial properties from intrinsic properties in no longer that easy.
Given a measurement that can take some moderate number of values, not too small and not too large, we now have a choice. We can say the different values correspond to many different particles of varying masses and electric charges. Alternatively we can describe the quantity measured as the momentum of a single particle in some (finite) direction (or with a position in a direction that is not continuous, but let’s not get into that). We can describe the same situation in different languages, all equally correct, but of course not always equally useful.
For example, depending on the situation, we may say that our favorite theory describes many types of particles moving in three spatial dimensions. But, depending precisely what situation we want to describe, we may find it more natural to say that we have a single type of particle moving in ten dimension (four infinite and six compact), or anything in between. It doesn’t matter, it is just two descriptions of exactly the same set of measurements.
As expected, in generic situations none of the descriptions is more useful than others, in other words there is no meaningful separation between intrinsic and spatial properties of our particles (even assuming they exist). This is perfectly OK, we don’t really need all those mental pictures to function. It just makes it difficult to give a good answer when someone asks you how many dimensions exist in string theory.
(Apologies for the long silence, I am now somewhere in the Canadian prairies. I have a feeling that one way or another, dinner will involve meat…).
Some 37 million organic and inorganic substances have CAS registry numbers. The fundamental connectivity of atoms is a Schlegel diagram – a flat graph with no crossing lines. The whole of chemistry is fundamentally flat and simply connected…
…except for some eight entries, courtesy of Kuratowski’s theorem. The simplest non-planar graph is C21H16. Submit it to a software booth way too full of itself in a sales exposition. No axiomatic system is stronger than its weakest axiom.
Chemistry or physics, excluding symmetry breakings reduces theory to heuristic. Perturbation methods are outstanding strategies for being good enough (wrong) by excluding them. Perturbational string theory is the facile approach, yes? The number of dimensions in string theory is then whatever Referees accept for publication.
Your post reminded me of a problem that I used to think about a lot back when I was working on the quantum-to-classical transition. We often talk about the “classical limit” of quantum mechanics. This has a well-defined meaning when we deal with systems that are familiar–in fact, the quantum description of many systems starts with a classical description and then “quantizes” it. But if I gave you a description of a quantum system–a state space, an initial condition, and a Hamiltonian, say, all in some arbitrary basis–could you tell me if that system has a “classical” description? That is, can you find a set of emergent, highly coarse-grained variables that have an approximately closed evolution?
It’s not obvious how you go about looking for such a thing. This makes it hard to tell if the fact that our universe has a “classical realm” is very unusual, or generic.
Hi Todd, that is indeed precisely the issue. Sometimes you have an intrinsic formulation of a quantum system (say some two dimensional CFT with no Lagrangian description), meaning a description which does not start with a classical system and includes quantum effects systematically. In those cases it is not clear what words one should be using to describe the system, because any interpretation uses a specific classical limit. That’s how you can have systems with more than one interpretation, say a four dimensional field theory, and a ten dimensional quantum gravity, which are both equally (in)accurate.
[...] just of a certain description thereof. Surprising as it sounds, this includes the question of how many dimensions spacetime has. The attempt to derive general relativity from something else, living in the same [...]