quantum interpretations - Why do people still talk about bohmian mechanics/hidden variables

I was reading the Feynman lectures in physics and after thinking about it for a while it seems particularly unreasonable to talk about hidden variables. Let us say that the electron has some internal variables as yet unknown which determine its trajectory given a set of initial conditions just like in classical mechanics. But since these hidden variables are unobserved, coupling it with a classical system should make their effect unchanged. This is what Feynman says, I think, in the last paragraph of Ch1 Vol 3, that if in the double slit experiment, if these inner variables dictate that the electron goes through the upper slit and land at a particular place on the opposite screen, and some other place for the lower screen, then the probability must neccesarily be the sum of two Gaussian like peaks, which does not agree with experiment.

So if I concluded that inner workings of an electron had some additional hidden variables, then it should yield, as they should be independent of the classical apparatus, mutually exclusive probabilities that do not quiet add up the way as observed. But then I do a hidden variables search on the archive and a lot of smart guys still write about it, as late as Feb 2011.

So the argument I have used might be somehow incomplete, can anyone explain how?

EDIT: Sorry for editing this question almost three years later. I tried to locate the exact reference from the Feynman lectures I was referring to and this is the updated source, Sec 7 Ch 1 Vol 3

We make now a few remarks on a suggestion that has sometimes been made to try to avoid the description we have given: “Perhaps the electron has some kind of internal works—some inner variables—that we do not yet know about. Perhaps that is why we cannot predict what will happen. If we could look more closely at the electron, we could be able to tell where it would end up.” So far as we know, that is impossible. We would still be in difficulty. Suppose we were to assume that inside the electron there is some kind of machinery that determines where it is going to end up. That machine must also determine which hole it is going to go through on its way. But we must not forget that what is inside the electron should not be dependent on what we do, and in particular upon whether we open or close one of the holes. So if an electron, before it starts, has already made up its mind (a) which hole it is going to use, and (b) where it is going to land, we should find P1 for those electrons that have chosen hole 1, P2 for those that have chosen hole 2, and necessarily the sum P1+P2 for those that arrive through the two holes. There seems to be no way around this. But we have verified experimentally that that is not the case. And no one has figured a way out of this puzzle. So at the present time we must limit ourselves to computing probabilities. We say “at the present time,” but we suspect very strongly that it is something that will be with us forever—that it is impossible to beat that puzzle—that this is the way nature really is.

particle physics - Energy of the electron-muon reaction

Lets see the reaction:

$e^- \mu^- \to e^- \pi^- \nu_\mu \;\;\;\;\;\;\;\;\;\;\; {(1)}$

I suppose, that this reaction occurs as follows

$e^- \mu^- \to e^- \mu^- \pi^+ \pi^- \to e^- \pi^- \nu_\mu$

Is it possible at energy less than 2*140 MeV?

The same is for the analogous proton-muon reaction

$p^+ \mu^+ \to p^+ \pi^+ \bar{\nu}_\mu \;\;\;\;\;\;\;\;\;\;\;{(2)}$

Once more reaction:

$p^+ p^+ \to p^+ p^+ \pi^- \pi^+ \to p^+ n \pi^+ \;\;\;\;\;\;\;\;\;\;\;{(3)}$

What are the experimental data?

---

P.S. This question is important enough. The main solar reaction is

$p^+ p^+ \to d^+ e^+ \nu_e $

If this reaction occurs as follows

$p^+ p^+ \to p^+ p^+ e^- e^+ \to p^+ n \nu_e e^+ \to d^+ e^+ \nu_e$

then it would require the energy of more than 2*0.511 MeV to take place. Cross section of this reaction will be much less, so the main solar reaction should be

$p^+ p^+ e^- \to d^+ \nu_e $

---

---

Update 20.02.11

Why I think so? I suppose that new particles are created in pairs particle-antiparticle. So the reaction

$e^- e^- \to e^- e^- \pi^- \pi^+ $

requires the energy of more than 2*140 MeV to take place

As well as the reaction

$e^- e^- \to e^- e^- \pi^- e^+ \nu_e $

from symmetry considerations, since there is a decay

$\pi^+ \to e^+ \nu_e$

And the result is

$e^- e^- \to e^- e^- \pi^- e^+ \nu_e \to e^- \pi^- \nu_e $

contrary to

$e^- e^- \to e^- \nu_e W^- \to e^- \pi^- \nu_e$

with the minimum reaction energy of 140 MeV

The same is valid for the reactions (1) (2) (3)

So what is the experimental data on minimum energy of these reactions?

homework and exercises - Solving the Liouville equation for classic harmonic oscillator via method of characteristics

I'm interested in solving Liouville's equation

$$\frac{\partial W}{\partial t} + \{ H,W\}=0$$

with $$H=\frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$

using the method of characteristics. However I can't seem to find any resources on how to do it. Would appreciate any pointers.

Answer

You can start by reading the wikipedia article on the method of characteristics. You will see that in our case the tangent of the characteristic is $(1,-\partial H/\partial q,\partial H/ \partial p)$ where the components are in order $t,p,q$. When you formulate the equation of the characteristic, you will actually find out you get equations of motion of a single particle with this Hamiltonian.

Once you solve these equations for $t(s),p(s),q(s)$ with a parameter $s$, you will know that a solution will satisfy $W(t(s),p(s),q(s))=const.$. This is actually very intuitive - the probability distribution $W$ flows with the particles evolving according to their equations of motion.

The solutions will be determined by two constants corresponding to e.g. phase $\phi$ and amplitude $A$ of the oscillation, you will have $t^{\phi, A}(s),p^{\phi,A}(s),q^{\phi,A}$. Depending on the kind of initial data or problem you are solving, you should be able to get the solution by integrating over the family of characteristics parametrized by $A,\phi$ with a certain weight function $f(A,\phi)$ at a certain $s$ and then get the solution on the rest of the $t,p,q$ space by going through all values of $s$.

Say you are given an initial $W_{0}(p,q)$ and want to evolve it through time. Then you have to find the map $p,q \to \phi,A$ substitute it into $W_0$ and this will be your weight function over the $\phi, A$.

quantum field theory - scattering singularity

In QFT when one works out the cross section between two colliding electrons one gets a formula which is proportional to $\theta^{-4}$ where $\theta$ is the scattering angle which is due to a nearly on-shell photon with momentum $\approx 0$. See for example Peskin and Schroeder pg 155. This is also seen in ordinary Rutherford scattering and apparently is not a problem experimentally as one can't detect particles along the collision beam. Also, in Rutherford scattering the singularity gets removed if one takes into account the screening of the nucleus by the electron cloud (see QM by Messiah (1961), pg 422) which modifies the Coulomb nature of the potential.

I was wondering if this singularity is indicating some problem with the calculation? Is there some effect in the calculation which has been neglected which when included would remove the singularity? Could renormalization effects maybe have a similar effect to the electron cloud in the Rutherford case?

electromagnetism - Maxwell equations invariant under Lorentz transformation but not Galilean transformations

Why Maxwell equations are not invariant under Galilean transformations, but invariant under Lorentz transformations? What is the deep physical meaning behind it?

units - To what extent are quantities fundamental?

Arguably the most well-known and used system of units is the SI-system. It assigns seven units to seven ‘fundamental’ quantities (or dimensions). However, there are other possible options, such as Gaussian units or Planck units. Until recently, I thought that these different systems differed only in scale, e.g. inches and metres are different units, but they both measure length. Recently though, I discovered that it is not simply a matter of scale. In the Gaussian system for example, charge has dimensions of $[mass]^{1/2} [length]^{3/2} [time]^{−1}$, whereas in the SI-system it has dimensions of $[current] [time]$. Also, I have always found it a bit strange that mass and energy have different units even though they are equivalent, but I find it hard to grasp that a quantity can be ‘fundamental’ in one system, and not in an other system.

Does this mean that all ‘fundamental’ quantities are in fact arbitrary? Would it be possible to declare a derived SI-unit fundamental, and build a consistent system with more base units? What is the physical meaning of this?

quantum interpretations - If the only observer dies, could observer 2 ever have see a different result?

My layman's understanding of QFT is that once a system is observed, any future observation will always yield the same result. Would this hold true if the original-and-only observer, without any physical recording other than human memory, and not communicated to any other person, died before any subsequent observation?

If the plot of the film was, "the spy sent the vital, world-saving data back to headquarters, encoded with a quantum key, only to be intercepted and viewed by the antagonist of the story; but the dashing hero catches and kills him before he could reveal what he learned, and the data arrives still encrypted back at HQ", what would the physicists be muttering to each other as they left the theater?

Or put another way, If I flip a coin into a box, observe it, then bury the box until after my death, does the coin exist in a "superimposed" state of heads/tails/on the edge until observed again, or will any future observation always be what I saw; or would it be that "superimposed" state but always resolve to the same answer? (apologies if I'm using terms incorrectly).

My intuition says information is information, so even if lost to our access within decaying brain cells, HQ would know the data had been viewed, no matter the state of decomposition of the "person who did the measuring" but I've run into a road block in phrasing the questions to dive deeper down the rabbit hole.

What resources would one recommend --to a well-read layperson who has little trouble following most Royal Institute lectures, to give you an idea of my reading level --to further pursue this odd question?