Where vectors in Vn describe states with n particles.
Now, here I'm going to start getting out of my depth. But I'm sure people out there can correct my errors. And I'm treading dangerously by rephrasing things a little differently from how they appear in any of the books or papers I have read.
Each of these Vns carries a representation of the Poincaré group. This means that if we apply a translation, rotation or boost to a vector in Vn we get another state Vn. So rotating, boosting or translating an n particle state just gives you another n particle state. The upshot of this is that all inertial observers can agree on how many particles a state represents.
But suppose now that we accelerate our state. We map our underlying spacetime with a function f so that if p(t) is the worldline of an inertial observer, f(p(t)) is now the worldline of an observer accelerating with constant acceleration, say a. f induces a linear mapping on V. The details of the computation are a bit messy but essentially what happens is that an n-particle state is now mapped to a state that is a linear combination of elements from all of the Vi. In particular, elements of V0 end up mapping to states with particles. Let me quote Wikipedia
the very notion of vacuum depends on the path of the observer through spacetime. From the viewpoint of the accelerating observer, the vacuum of the inertial observer will look like a state containing many particles
These particles are called Unruh radiation. But I find this notion bizarre. How can the number of particles depend on the observer? And how does this look in practice? Well suppose we're in a vacuum and someone called Fred accelerates past with a particle detector. Fred's detector will start beeping to indicate the presence of particles even though I don't get any beeps with my detector. I will in fact see Fred's detector act like it's detecting particles. In other words, although Fred and I might disagree over how many particles occupy this region of space, we can both agree on what the detector is doing. This isn't a big deal at all, we're used to the idea of non-inertial instruments acting funny, just trying using a pair of scales in an accelerating car.
So here's my conclusion from all of this: either
- It makes no sense to interpret the detection of particles by the accelerating detector as indicating that the vacuum contains particles. We can't trust the readings from a non-inertial particle detector without some kind of correction for acceleration. This is completely familiar, lots of other kinds of instruments fail when non-inertial. The actual definition of the number of particles in a region of space is chosen so as to correspond with what an inertial detector sees.
- The notion of particle, separate from that of a detector, is meaningless. We just have a Hilbert space of states and instruments that go beep under certain circumstances and we can predict when these instruments go beep by looking at properties of the Hilbert space.
Most physicsts seem to use another option:
- The number of particles depends on the frame of reference in which you are measuring
Anyway, things get even trickier. If you look at how V is split up into n-particle subspaces it turns out that the definition of this splitting depends on a choice of direction for time. In a flat spacetime we can pick any timelike direction because they're all basically related by Lorentz transforms so they all give the same results. But in a curved spacetime it's not so easy. Again we end up with an ambiguity in the number of particles, but this time (at least near a black hole) it's called Hawking radiation. I interpret this as meaning we have to take option 2. Particles simply aren't a well-defined concept, except as an approximation, in a curved spacetime. However, most physicsts still take option 3. I'm happy with option 2 because I see quantum mechanics as being primarily about vectors in a state space, not about particles. Physicists seem happy to take option 3 even though I think it's nonsensical. They're used to the idea of a quantity that is frame-dependentm eg. the x-coordinate of a vector, and feel that it's fine to extend this notion to integer valued properties such as particle number.
But what do I know? I'm not an expert in this field. I did try to study this field properly many years ago, eg. by reading Wald's book on black hole thermodynamics. I found it to be mostly clear, but I had many problems because I kept having objections to the physics, something I hadn't felt with any of the physics I had studied previously, including wacky stuff like renormalization.
Anyway, I don't have anything to contribute to this subject, I just thought I'd mention it because people might find it interesting. I discussed it a little bit with someone who knows a lot more about this subject over at Reality Conditions and I now have a bunch of papers to read on the subject. Unfortunately the setting for much of this work is C*-algebras and the like so I need to swot up on all that stuff first.