V=V_{0}⊕V_{1}⊕V_{2}…

Where vectors in V

_{n}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 V_{n}s carries a representation of the Poincaré group. This means that if we apply a translation, rotation or boost to a vector in V_{n} we get another state V_{n}. 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 V_{i}. In particular, elements of V_{0} 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.

## 6 comments:

I tend to agree with the operationalist viewpoint, but I am not sure why you say the "particles are relative" approach is nonsense. For each timelike direction which follows a Killing vector of the spacetime you can define a notion of particles. Then for a given state of the quantum field you can calculate how many of each of these particles are present. The numbers will be in general different except in special cases such as Minkowski space when both directions are related by a Lorentz transformation. Why is it "nonsense" to say that "having x particles of kind X" and "having y particles of kind Y" are two properties of the same quantum state? (For example, X = Minkowski, Y = Rindler => x = 0, y = thermal spectrum, if the state is the standard Minkowski vacuum)

If I read you correctly you're making particles a function of a timelike killing vector. So you're saying that the answer "how many particles associated to this vector are there" is well defined even though the question "how many particles are there?" isn't. That makes sense in its own right. But don't you find that to be a bizarre notion of "particle"? If this is what you mean then the word seems a bit too dissimilar to the usual notion of the word "particle" for me to be comfortable with calling it a "particle".

Note: I'm not in any way criticising the physics itself. The theory looks like it might be good (though in the absence of a solid quantum theory of gravity it's a bit uncertain). I'm only talking about semantics here. In fact, I had another look at Wald yesterday and I notice that at one point he stops doing the mathematics for a bit so he can step back to talk "in English" about the theory he's been developing. At this point he no longer writes "particle" but ""particle"", ie. he puts scare quotes around the word. In other words, he's happy to use the word particle as a technical term to refer to a property of elements of a certain type of vector space, but he's uncomfortable with identifying this notion with the ordinary use of the word particle.

Oh yes, it is very dissimlar to the common notion of "particle". But then the ordinary quantum-mechanical (non QFT) notion of particle is not very close to it either. It's just that we've had more time to get used to it, and that we see it apply in experiment every day. If we could easily do experiments on the Unruh and Hawking effects, then perhaps the observer-relative definition of particle would not sound strange but natural.

But yes, at the end it's just a semantical problem.

So now I'm confused by this different issue:

A particle at rest in a gravitational field (eg. on the surface of the Earth) is non-inertial. So surely it radiates EM energy. But obviously it can't, as that wouldn't conserve energy. I'm confused!

After puzzling over your question for some hours, I resorted to Google and found (with some relief) that the issue is indeed a highly nontrivial one, so my failure to find an answer imediately was not evidence of poor understanding of GR and EM on my part. :-) Here are a couple of links:

http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000074000002000154000001&idtype=cvips&gifs=yes

(AJP article; appearently it shows that the EM radiation by a

uniformlyaccelerated charge transforms to a static field for comoving observers; this implies via the equivalence principle that a charge at rest on a static gravitational field does not radiate from the point of view of other static observers)http://www.mathpages.com/home/kmath528/kmath528.htm

(An informal discussion fo the question; according to it the question is really unresolved and connected to the perennial "self-force" problem in classical electrodynamics. But I'm not sure of the reliability of this source. And the vagueness/lack of rigour of the discussion is likely to make a mathematician shudder!

Alas, I can't access the ajp article :-(

Also, does it have anything to say on why a particle 'at rest' in a gravitational well should appear not to radiate EM to a comoving observer? The difference between these two sitautions seem analogous to the difference between Unruh and Hawking radiation.

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