4 ms·
From the paper, at the top of page 10 of the PDF, at the end of section A: "Some authors conclude, incorrectly, that the countability of quanta implies a parti
by gpsx 10y ago
From the paper, at the top of page 10 of the PDF, at the end of section A:
"Some authors conclude, incorrectly, that the countability of quanta implies a
particle interpretation of the quantized system. Discreteness is a necessary but
not sufficient condition for particles. Quanta are countable, but they are spatially
extended and certainly not particles. Eq. (3) implies that a single mode's spatial
dependence is sinusoidal and fills all space, so that adding a monochromatic
quantum to a field uniformly increases the entire field's energy (uniformly
distributed throughout all space!) by hf. This is nothing like adding a particle.
Quanta that are superpositions of different frequencies can be more spatially
bunched and in this sense more localized, but they are always of infinite extent. So
it's hard to see how photons could be particles."
As mentioned above, you can take linear combinations of these different single particle states at different energies and come up with various energy/location spreads. Doesn't one such combination have a spatial spread of zero? This would correspond to a single quanta at a single location in space.
My physics may be a bit rusty since I have been out for a while. Combining the different frequency components from the different field configurations is not _exactly_ the same as simple Fourier analysis, on the face of it. However, the individual contribution from a given field configuration (meaning a single frequency) is very small since there are so many different field configurations contributing (an infinite number). I believe the Fourier result does apply to the expectation value of the particles location here.
If I am thinking correctly this seems to be a very critical error in the paper. Someone correct me if I am wrong.
* * *
EDIT: I believe I said something incorrectly. Where I said "I believe the Fourier result does apply to the expectation value of the particle's location here." I meant to make a stronger statement, "I believe the Fourier result does apply to the effective value of the particles wave function in this location in this case." (The expectation value being zero would not bean the field does not extend to that location.)
- sova 10y agoI suppose the whole crux and beauty of the article is that we can talk about physics without numbers. As soon as numbers enter the picture, we have discretized something that is beyond discretization.
- rrmm 10y ago(Not a physics person, but) As I read it, he is just arguing against the implication that Countability implies Particles. So he is saying that adding a quantum that is monochromatic has an effect across all space (which is clearly a non-particle like effect). Then he admits that you can get localization through superpositions of quanta. Since he is arguing against a logical implication he only needs to show a counterexample exists. The problematic part in choosing a single mode like this is that it can lead to a non-normalizable integral at some point (ie, it won't be a member of the Hilbert space). QFT always made more since to me. Everything flows more naturally than in QM, where out of the blue, they introduce wave packets. The paper does feel a bit shaky though especially as it is essentially taking a pedagogical point (start teaching from a QFT viewpoint) and making it into an argument about the foundations of physics. The non-relativistic QM view point is taught because it represents a smooth departure point from classical thinking without requiring the full mathematical complexity of field theory.
- gpsx 10y agoI think the general assumption is that QFT is true, it is just whether you can call the physics it creates as being particles. At least that is how I interpret it. QFT is QM, but it just has a lot more degrees of freedom (infinite) than what you are talking here about when you say QM. You can write a Schroedinger's equation for a field. Rather than the wave function being a function of X, the wave function is a function of F(X) where F(x) is a field configuration over space. I would say some of his argument comes down to semantics because by his definition in order for something to be a particle it can't have spatial extent. I think a lot of people don't include this in the definition of a particle. They do associate the number of "Quanta" with the number of particles, at least in a field theory. What I was saying above is that I believe you can create an object by the "Quanta" definition in QFT that has no spatial extent. His argument is based on the fact that this is not possible. Adding these Quanta together to form a zero extent object is similar to saying in Fourier analysis you can take a number of different frequencies (like the quanta at different wave lengths) and create a delta function (which has no spatial extent). The thing is, even though we are dealing with adding a bunch of frequencies together, it is not exactly the same as a Fourier series because we are adding them together in a different way. You make a good point about people first teaching the simple Schroedinger's equation of particles because it is a lot easier. I think what is so hard about quantum mechanics is relating it to the real world, and not the axioms themselves. The simple form is much easier to relate to the real world.