Science > Physics > Entanglement, foundations of QM, and the sum-over-paths picture
| Topic: |
Science > Physics |
| User: |
"Giorgio Torrieri" |
| Date: |
04 Dec 2003 06:44:12 PM |
| Object: |
Entanglement, foundations of QM, and the sum-over-paths picture |
I have been thinking for a while about the different formulations of
QM (canonical and sum-over-paths), and there are several issues which
I do not see covered in the standard textbooks.
Is there anyone (textbook, paper ecc.) which discussed the topics
below at
length in a rigorous and quantitative way?
Thanks for your help!
i) An exposition of an EPR/Entangled system in terms of the path
integral approach.
Qualitatively, I think this would be a very useful exercise
because the
"paradox" disappears when the problem is tackled this way:
There is no wave function which instantly collapses when the
"observation"
is made.
Instead there are paths which interfere with neighbouring paths
in a local
way, and cancel out unless a symmetry of the Lagrangian is
preserved.
More complicated entanglement (eg cryptography, Bell inequality)
could be
handled in a much more intuitive way using some over-paths than
canonical quantum mechanics.
But this is just a naive first impression. I am looking for a
textbook
which actually does this. Or does it get horrendously
complicated?
ii) More generally, I am looking for a discussion on how the
postulates of
quantum mechanics relate to the sum-over-paths picture.
Usually, textbooks cover how to derive Shrodinger's equation from
the sum
over paths with the phase given by the classical action (more
generally,
how to derive the field equations from the Lagrangian), and then
just
do a couple of examples (two slit & the Aharonov Bohm effect).
I am looking for a textbook that discusses the mathematical
foundations a bit deeper: In canonical quantum mechanics , you
can
always transform the wave function into a
function of your chosen observable (eg FTs to get a wavefunction
in
momentum space from position space),
and the probability of observation can be easily related, AS A
POSTULATE,
to the form of the hamiltonian and the Eigenfunction.
Is there a book/paper which discusses how these assumptions
translate into
the sum-over-paths formalism?
(Some of the problems in Shankar's textbook discuss solving the
SHO in the
sum-over-paths approach and getting energy levels and PDFs from
the
propagator. I guess I am looking for more of that)
Finally, most textbooks are extrmely un-rigorous about the
treatment of
fermions in the sum-over-paths approach:
Fermions (+the exclusion principle), together with entanglement,
are THE
defining features of QM which are completely absent in the
classical world
(if only because geometrically they are very weird objects)
In the canonical QM formulation, spin 1/2 states arise naturally
as
mathematical objects in the solution of spherically symmetric
hamiltonians, and Fermions arise naturally
when considering how many-body wavefunctions can behave under
particle
interchange.
Yet I have never seen a treatment of fermions in the
sum-over-paths
picture which approaches this "naturalness", or shows how a
quantum
Fermionic system smoothly goes
into a classical one as the planck constant goes to 0 (supposedly
the best
feature of the sum-over-paths approach is the way the
QM-classical
transition follows). Instead, most QFT textbooks just show
that Grassman coefficients capture the anti-commutation rules and
arm-wave
the path integral over grassman variables as a matter of
definition.
Once again, thanks for any feedback/suggestions
GT
.
|
|
|
Related Articles |
|
|
