No attempt will be made here to explain what mathematical physics is about. There is no general agreement even among the experts. Moreover, this field of research is regarded as somewhat dubious by many physicists. However, the following words of Maxwell are right on target:
J.C. MAXWELL
On Faraday's lines of force
Nobody has explained why we should worry about the foundations of quantum mechanics better than John Bell:
J.S. BELL
Speakable and Unspeakable in Quantum Mechanics
The first charge against "measurement", in the fundamental axioms of quantum mechanics, is that it anchors the shifty split of the world into "system" and "apparatus". A second charge is that the word comes loaded with meaning from everyday life, meaning which is entirely inappropriate in the quantum context. When it is said that something is "measured" it is difficult not to think of the result as referring to some preexisting property of the object in question. This is to disregard Bohr's insistence that in quantum phenomena the apparatus as well as the system is essentially involved. If it were not so, how could we understand, for example, that "measurement" of a component of "angular momentum" ... in an arbitrarily chosen direction ... yields one of a discrete set of values? When one forgets the role of the apparatus, as the word "measurement" makes all too likely, one despairs of ordinary logic ... hence "quantum logic". When one remembers the role of the apparatus, ordinary logic is just fine.
In other contexts, physicists have been able to take words from ordinary language and use them as technical terms with no great harm done. Take for example the "strangeness", "charm", and "beauty" of elementary particle physics. No one is taken in by this "baby talk". ... Would that it were so with "measurement". But in fact the word has had such a damaging effect on the discussion, that I think it should now be banned altogether in quantum mechanics.
J.S. BELL
Against "Measurement"
Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger's equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. However, this deterministic theory of particles in motion completely accounts for all the phenomena of nonrelativistic quantum mechanics, from spectral lines to interference effects, and it does so in a completely ordinary manner. It was first presented in 1927 by Louis de Broglie, just after the inception of quantum mechanics itself. It was soon abandoned and utterly ignored until rediscovered a quarter century later by David Bohm, who showed how it resolved the measurement problem and accounted for the reduction of the wave packet. Its principal advocate for the past three decades was John Bell:
J.S. BELL
Speakable and Unspeakable in Quantum Mechanics
J.S. BELL
Speakable and Unspeakable in Quantum Mechanics
For a little more on Bohmian mechanics click here; for more detail, click here.
*The picture of the two-slit interference patterm is courtesy of Martin Daumer*