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sage

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- Thread starter sage
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sage

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- #2

mathman

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Try "QED" by Feynman. It is very readable.

- #3

jeff

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Originally posted by sage

Path-integrals are neither hypothesis nor theory, but instead powerful and indispensible tools to characterize and calculate the behaviour of quantum systems. To understand them, let's first look at a familiar classical system.

If we know where and with what velocity an ordinary baseball is thrown, it's subsequent trajectory is uniquely determined by newton's 2nd law F = ma. Putting it in a slightly strange but - as we'll see - helpful way, given it's initial position x(0) and velocity v(0) at time t = 0, the probability P{[x(0),v(0)],[x(t),v(t)]} of finding the ball at some other location x(t) with some other velocity v(t) at time t is 100% if these lie on the trajectory predicted by F = ma, and 0% if they don't.

Unlike with the baseball, according to quantum theory one cannot simultaneously know both the position and velocity - or more accurately, momentum - of an electron. In particular, we must choose whether to specify an electron's initial state in terms of either position or mometum, but not both. Suppose we choose to specify it's position x(0) at time t = 0. We then ask for the probabllity P[x(0),x(t)] of finding it at some other position x(t) at some later time t.

Now, unlike P{[x(0),v(0)],[x(t),v(t)]}, no matter what x(t) is, P[x(0),x(t)] is never 0% or 100%. Put another way, every path from x(0) to x(t), no matter how crazy, contributes to P[x(0),x(t)]. This counter-intuitive fact is a direct result of our not being able to say anything about the electron's momentum. The path-integral for this system is just the sum over the contributions to P[x(0),x(t)] from each and every path.

To explain in a worthwhile way the meaning and form of these contributions and the precise relation in this context between the classical and quantum viewpoints is difficult without a little math. Let me know if you want me to continue.

More generally, instead of particles and paths, we can study quantum fields and their evolution. This is the subject of quantum field theory. It's been applied successfully to all the known interactions (weak, strong and electromagnetic) except gravity.

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- #4

Originally posted by sage

sum over histories. Read QED.

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dumpy

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