Feyman's sum over path hypothesis

[sage]

can you give me a primer on feyman's sum over path hypothesis.no maths please, just what it tells and its current standing with respect to other theories on this matter.

 

[mathman]

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

 

[jeff]

Originally posted by sage
can you give me a primer on feyman's sum over path hypothesis.no maths please, just what it tells and its current standing with respect to other theories on this matter.

Path-integrals are neither hypothesis nor theory, but instead powerful and indispensable 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.

 

[theEVIL1]

Originally posted by sage
can you give me a primer on feyman's sum over path hypothesis.no maths please, just what it tells and its current standing with respect to other theories on this matter.

sum over histories. Read QED.

 

[dumpy]

Feynman expanded the wave functions that you get from Schrödinger's equation in terms of particle paths that the particle might follow. With each path will be associated a phase, and so the are able to interfere just like waves. Unfortunately there are huge numbers of trajectories to sum over, and most are not those determined by classical physics. However, when you approach a semi-classical limit you discover that only the classical paths contribute significantly to the expansion. All other paths interfere destructively. Therefore you can begin to see just how classical physics results from quantum physics.