sci.physics.research

Could you please elaborate. Lets consider an electron that is emitted
from an electron gun towards a series of three screens. each screen
has three slits( try imagining the double/triple slit experiments from
feynman vol 3). Now we want an amplitude, f(b)= sum over i,j( F_ai
H_ij G_jb) to represent the amplitude for the electron to start from
the electron gun placed at "a" and pass over all i=1,2,3 slits in the
first screen, moving on to the second screen which has j=1,2,3 marked
slits and finally hitting the screen at b. assume that we have
detectors at only 3 finite places on the screen for simplicity.


electron i j detector screen
gun
1 1 1
a 2 2 2
3 3 3

-KM

this is not my real name wrote:
> This post is a little dreamy and may not make sense, so please bear with
> it. On the other hand, there might be something to it.
>
> The other day I noticed that the determinant of a matrix looks like a
> sum-over-histories. You can think of it as the sum of all the paths from
> the top to the bottom, and if you use complex numbers then you even have
> interference, but with the problem that you have the alternating {+1 and -1}
> in the products. There is also a thing called the Permanent [see wikipedia]
> that doesn't have alternating signs, it just uses +1 the entire time. You
> can also define all sorts of things with other numbers on the unit circle
> like {1,i,-1,-i}, etc.
>
> At first I thought it was interesting but not-so-great, until i remembered
> that the determinant is a group homomorphism. That means the kernel is
> mapped to the identity element of the image group. Now I often call the
> identity element the "do nothing" element, because it does just that:
> nothing. For instance, 2*1 = 2, or 2+0=2, etc. Now the shortest "distance"
> between 2 points is a geodesic, and if its light, then its a "null
> geodesic": it does nothing, and everything can be defined in reference to
> it. So fermats principle of least time, the concept of the geodesic, and of
> the fundamental frequency, and of quickest path to a win in a stochastic
> game, and least action all seem to be describable by the identity element.
> For instance, you can think of the fourier transform of the number 1: You
> have all negative and positive frequencies and they cancel out to a dirac
> delta function. Its the identity element of an additive group. You wouldnt
> get much energy coming over the radio at that frequency.
>
> But the path integral cancels out to those same things. Perhaps the
> determinant is some kind of path integral that is related to fermions, the
> permanent is one for bosons, and the other similar structures are for
> anyons. I found some references on the internet, but they seemed sort of
> lofty and mysterious. Here are 3 questions that are possible a little more
> practical: if you have a system with a continuous spectrum, then
>
> 1. The matrix has a dimension that is
> continuous, so can the determinant be defined?
> If it converged, it would be an
> integral of infinite products.
>
> 2. If it could be defined, would the
> resulting continuous-dimensional structure still be a group? [set of
> matrices
> with certain types of determinant
> = group.]
>
> 3. Do you need lebesgue theory or just
> the more familiar riemann integrals?
>
> If you are an expert on this stuff, please explain.
>
> JHS