We should take the Feynman Postulate of quantum mechanics (sum on all ways) seriously. We should also take the Action Principle seriously, because the action is richer than the equations it generates, since we can vary it with respect to various parameters and find new equations (eg. the action for the field yields the equations for the sources). (for example, see Kaluza where I talk about the way action-reaction is implicit in the action principle)
It now seems we can derive the Dirac equation (in 2d, anyway) from the amplitude postulate.
From the Feynman Postulate : don't worry about how to compactify string theory - just sum on all ways. One can also find the vacuum of the yang-mills theory quite easily in this way : just sum up all possible vacua (ie. all possible instanton populations). As is well known, we are free to include a phase in the Feynman sum which must be determined in some other way.
More from the Feynman postulate : "many worlds" and measurement collapse follow directly from the Feynman amplitudes. Measurement is simply a Feynman transition amplitude in which the initial state goes into one of several final states which are superselection sectors. That is, those final states never recombine. This is "decoherence" (and appears to be collapse to someone "inside" one of those states). The way these final states never recombine is because they have many degrees of freedom, each of which has some small probability to switch to another sector, but taken together this probability goes to zero.
In article <352C9FDB.DF885FFB@erols.com>, firstname.lastname@example.org says... >On page 1-10 of "The Feynman lectures on Physics", Feynman says: > >"If an experiment is performed which is capable of determining >whether one or another alternative is actually taken, the >probability of the event is the sum of the probabilities for each >alternative. The interference is lost: P = P_1 + P_2." > >Does this part of Feynman's _presentation_ of quantum theory need >to be changed? When _do_ you get approximately correct answers >when you add probabilities? If a change in the prose is needed, >what might Feynman have written (especially considering his >strong desire to be always precisely correct)? I personally think Feynman would change this a lot in the light of modern considerations. In particular, the whole idea of measurement collapse can be understood in terms of the decoherence idea. In fact, adding amplitudes and adding probabilities are both approximations to the general case which is some mix thereof. Amplitudes are exact when the system is isolated (the quantum limit), while probabilites are exact when the system is in heavy contact with the outside world (the classical limit). In this sense we now realize that the classical Bohr limit hbar -> 0 is *wrong* : you do not get classical mechanics that way, because you still have superpositions, EPR pairs and whatnot floating around. You must also take the limit (decoherence -> infinity). Let me talk about these thing semi-mathematically. The exact rule for quantum is to start with some initial state, compute the amplitude to get to a final state, A(i,f) ; amplitudes add, and we can get A from path integrals. Now divide the initial and final states into "system" and "environment" (S and E). Let's consider two limits : a = a(S) * x(E) b = b(S) * x(E) in a process like this the environment has not interacted with the system at all. When we integrate out the environment, A(i,f) will be a simple amplitude sum in the system, and we get ordinary quantum, in ket notation, we see the factorization a+b = ( |a(S)> + |b(S)> ) * |E> Now consider the opposite severe case: a = a(S) * a(E) b = b(S) * b(E) implied here is that the environment is very large, so that the amplitude to go from a(E) to b(E) is almost zero, that is, they are "superselection" sectors (perhaps the dial on an apparatus). So, lets compute an obersvable in these cases. We want
= <(a+b)|x|(a+b)> In the "quantum" case : = = (a) + (b) + 2 (ab) In the "classical" case : = + same for b cross terms in are zero. = (a) + (b) This is addition of probabilities = classical statistics. Feynman's prescription of "if it is possible to detect what has happened" is an approximation to this, because if any sort of interaction has happened with something which is in the environment, then we will get something like this superselection. There are several fine points in this which it is useful to explore : 1. is these environmental sectors are so hard to get between, then how did we get them in the first place? A measurement is something like (a(S)+b(S))*null(E) -> a(S)*a(E) + b(S)*b(E) so what we require is a special hamiltonian that can induce this transition and then turn off, with very little probability of fluctuations in the environment. 2. can we really not see what's happened? You might try to put a spectator spin (or something) in the middle of your electron diffraction experiment. Obviously, if the spin goes into one of two orthogonal states, the a(E) and b(E) environment states will kill the interference effect. You might try to sneak by by letting the spin go into non-orthogonal states so you get some information about which path the particles took. It's useful to convince yourself that whenever you get information, that's exactly when there's no interference effect. It should be obvious from this construction that it is necessarily impossible to "see" the inner workings of the interference.
Charles Bloom / cb at my domain Send Me Email
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