You could glue them together. In which case you wouldn’t get any net magnetic field anywhere (in the ideal case, at least). In the real-world case, the magnetization probably wouldn’t be quite uniform, so you’d have small leaks, but it would fall off very quickly with distance.
Yes and no… All of the magnetic monopoles in our Universe[sup]*[/sup] must have a charge of over a hundred times greater than the charge of the electron. If we had a bunch of those, but no electrons or the equivalent, then the world would be very different, because all of the interactions would be much stronger, and we couldn’t study them with perturbative field theories. If, however, you had a different world where magnetic monopoles were small and common, and electric monopoles were large and scarce, it would behave the same as our own.
As my advisor explains it, we know that magnetic monopoles exist; they’s just a very small number of them, which may be zero. That is to say, it is known that a magnetic monopole could be formed under the right conditions, but those conditions are very difficult to achieve, and may have never occurred in the observable Universe.
No. The photon field is actually describes the potential of the electromagnetic field, not the field itself. Remember that we write the electric field E as the (spatial) gradient of some scalar function ϕ – the “scalar potential” – and the magnetic field B as the (spatial) curl of some vector field A – the “vector potential”.
We collect these into a 4-dimensional vector field by using ϕ as the time component of the 4-vector and A as the spatial components. We call this the “4-potential”, and its four components (suitably quantized) form the photon field.
Be very careful here. I think your advisor was making a joke that casual readers on the board wouldn’t get. We “know” that they exist because the existence of one of them would topologically quantize electric charge. Since electric charge is observed to be quantized, and we have no other good explanation for that fact, physicists are willing to say “we know that magnetic monopoles exist”.
I thought that a magnetic monopole was observed once, albeit rather fleetingly, around 1979. I read about it 4th grade and more recently I read a reference to this same observation. However, I gather that it was a case of, “Did we just see what I think we saw?”
Shrodinger originally said every material object has a wave function, regardless of size or nature, which gave the probability amplitudes for finding the object at location x,y,z. But now I hear that the photon has no wave function. Why not?
I said “suitably quantized”. I never said that the photon has no wave function. I just didn’t go into the horrific mess that is quantum field theory for massless particle fields.
At first there was a debate between De broglie, Schrodinger et al about the physical significance of the wave function. The electrons wave function was not as “real” as the electron; it was only a map of the electrons liklihood to be found at various places in space. It was not a wave in 3D physical space, but in a conceptual space of any number of dimensions. for example, a system of two electrons had a wave function in a six-dimensional space. Born said the wave function only gave liklihoods of finding the object at x,y,z.
Then along came second quantization saying an electrons orbital was a real three dimensional wave undergoing harmonic oscillations. Like all oscillators its energy levels ware quantized. It could only contain discrete units of energy. For electrons, only two states were possible, but for bosons any number were possible.
Why was the Born interpretation abandoned? What justifies the reinterpretation of the wave function into a real field in 3-d space?
It’s pretty solidly settled among physicists, though it’s on theoretically shaky ground. Still, as part of quantum electrodynamics it makes fantastically accurate predictions, so that’s good enough for them.
Basically you start by treating each of the four component functions separately. Then you use Fourier analysis to break them down (“analyze”) into basic pieces, sort of like sine functions. Each basic piece is in a sense independent of each of the others, so you can treat them as separate, non-interacting degrees of freedom. And there’s four sets, one for each component.
Next you have to quantize this system, turning variables measuring the size of the basic pieces into operators on some appropriate Hilbert space. This is pretty straightforward, except that we’ve got an infinite number of variables to pay attention to here.
And then things get really messed up when you realize that the particle was supposed to be massless. That is, the four directions aren’t really independent – there’s a relation between them. This happens for massive particles too, but for massless ones the relation is a real bitch to deal with, and took a whole new technique to find a way of handling it. Basically, there’s a quantity like a determinant that you divide by for the massive particle case, but it’s zero in the massless case so you can’t do the same thing.
So the people working on QED figured out a way to fake it and give predictions, and the predictions are good, but to explain it I’d need to basically repeat four or five chapters from the middle of Weinberg.
I don’t know where you’re reading this, but you’re really mangling it. Or the author you’re cribbing from is. The photon field I’m talking about is a classical field, which must be quantized to make physical predictions. The wavefunction is an inherently quantum phenomenon, which we can reconstruct from the quantized photon field.
I know QED starts from classical electromagnetic waves. I know that the normal modes are described by the same Lagrangian as material harmonic oscillators and therefore can be quantized just like oscillators can. But QFT doesn’t stop there. It takes all first-quantized wave functions, like electronic orbitals in atoms, and treats them as classical waves to be quantized. Thats called second quantization. My question is, what is the justification for this? Why was the Born interpretation abandoned?
I did answer the question. The quantum field is not the wavefunction, though you seem to think it is. Given the field, you can construct the S-matrix, and from that you can construct the wavefunction. I already said this.
OK so walk me through this. You have a hydrogen atom. Its electron has a wave function, obtained by solving the Schrodinger equation. Undergraduates are taught that this wave function is a complete description of the electrons state, not the electron itself. It is a “map” of the probability of finding the electron at point x,y,z if you attempt to locate it there. Later, QFT interpreted this orbital as an actual field in space. Heisenberg even called it a “classical field.” Since it has a time dependence of exp-iwt it can be described by the same type of Lagrangian used to describe classical waves. It can be considered as a normal mode of a vibration of some kind, the generalized position coordinate of which can be interpreted as an operator. The rest follows just as a classical light wave is quantized. All I want to know is, why can we regard this orbital as a classical wave field that can be quantized just as a light wave can?
Because in both situations the Fourier modes of the field behave like an infinite collection of harmonic oscillators, and we know we can quantize harmonic oscillators.
Incidentally, the Fourier modes are the orbitals you were talking about. Part of the upshot is that the second-quantized system allows us to talk about one and several field excitations (electrons) on the same footing.
