Amplitude and phase of a Feynman path

[Mike2]

Can the amplitude and phase associated with each of the paths in the Feynman path integral be connected to geometric attributes of that path? For example, is the amplitude and phase connected to how long the path is or how much it curves or how much it deviates from the geodesic?

 

[lethe]

these are paths through the space of all field configurations, which is not a Riemannian manifold, so the words "length" and "curvature" and "geodesic" are out of place here.

 

[selfAdjoint]

Originally posted by Mike2
Can the amplitude and phase associated with each of the paths in the Feynman path integral be connected to geometric attributes of that path? For example, is the amplitude and phase connected to how long the path is or how much it curves or how much it deviates from the geodesic?

The phase increases at a steady rate along each path. That is the only criterion. Then when the paths are summed up in the path integral, the phases of paths far from the stationary one cancel.

 

[Mike2]

Originally posted by selfAdjoint
The phase increases at a steady rate along each path. That is the only criterion. Then when the paths are summed up in the path integral, the phases of paths far from the stationary one cancel.

So it would seem that phase is related to the length of the path. What about amplitude?

 

[MathNerd]

1) Each separate path has an equal amplitude

2) The phase for each path is the action along the path in units of Plank’s constant

 

[lethe]

Originally posted by Mike2
So it would seem that phase is related to the length of the path.

you're not listening. you are integrating over a space that does not have a metric, so you cannot ask the length of the path, nor the curvature.

 

[selfAdjoint]

Originally posted by lethe
you're not listening. you are integrating over a space that does not have a metric, so you cannot ask the length of the path, nor the curvature.

Excuse me Lethe, but isn't there a nuance here? Each of the paths in the space-of-paths is a PATH, in euclidean space, and in that space it does have a length. And indeed it is in that space that the physics happens that is attributed to the path when you integrate over the space-of-paths. N'est-ce pas?

 

[lethe]

Originally posted by selfAdjoint
Excuse me Lethe, but isn't there a nuance here? Each of the paths in the space-of-paths is a PATH, in euclidean space, and in that space it does have a length. And indeed it is in that space that the physics happens that is attributed to the path when you integrate over the space-of-paths. N'est-ce pas?

the space of all field configurations is a Euclidean space? why do you say that? for starters, it is certainly infinite dimensional, and Euclidean space is finite dimensional.

tell me what expression you want to represent the length of a path through the space of field configurations. the action?

 

[Mike2]

Originally posted by lethe
the space of all field configurations is a Euclidean space? why do you say that? for starters, it is certainly infinite dimensional, and Euclidean space is finite dimensional.

tell me what expression you want to represent the length of a path through the space of field configurations. the action?

Isn't the phase determined from the Action Integral, and isn't the action integral proportional to the length?

 

[lethe]

Originally posted by Mike2
Isn't the phase determined from the Action Integral, and isn't the action integral proportional to the length?

the action for a relativistic point particle traveling on a Lorentzian manifold is proportional to the length of its worldline.

but the action for, say, a scalar field? as far as i can tell, this not the length.

 

[Loren Booda]

Is the amplitude of related Feynman paths normalized over actions which generally contrast between their domain magnitudes ("lengths"), but retain a constant phase difference?

 

[MathNerd]

Originally posted by Mike2
Isn't the phase determined from the Action Integral, and isn't the action integral proportional to the length?

Action

$$S = \int_{t_0}^{t_1} L \ dt$$

where S = the classical action
L = the lagrangian

Action of a scalar field

$$S = \int_{t_0}^{t_1} \int_{V} L \ d^3V \ dt$$

but in this case L = the lagrangian density.

The probability amplitude for a given path is $$A e^{i S}$$, where A is some constant that is chosen so that when you sum over all paths the total probability adds up to one and S is the action along that particular path. Remember that A is the same for ALL paths.

 

[Haelfix]

Ahh you forget the all important summation sign, in front of that expression, I insist it makes no sense physically to talk about one path, even in principle! Unless you are in a very restricted world =)

Its the coupling of all possible paths, in exactly that way, that is the fundamental quantum *thing*!
.

 

[Mike2]

Originally posted by Haelfix
Ahh you forget the all important summation sign, in front of that expression, I insist it makes no sense physically to talk about one path, even in principle! Unless you are in a very restricted world =)

Its the coupling of all possible paths, in exactly that way, that is the fundamental quantum *thing*!
.

What is the "coupling" between all possible paths?

 

[selfAdjoint]

Integration over. The paths then occupy the role that points do in elementary integration.

 

[Mike2]

Originally posted by MathNerd
Action

$$S = \int_{t_0}^{t_1} L \ dt$$

where S = the classical action
L = the lagrangian

Action of a scalar field

$$S = \int_{t_0}^{t_1} \int_{V} L \ d^3V \ dt$$

but in this case L = the lagrangian density.

The probability amplitude for a given path is $$A e^{i S}$$, where A is some constant that is chosen so that when you sum over all paths the total probability adds up to one and S is the action along that particular path. Remember that A is the same for ALL paths.

So it would seem that it is the Lagrangian that chooses the classical path out of all the possible paths by determining how the phases will add to produce the classical result. And the Lagrangian must comply with the Euler-Lagrange equation which some interpret as a vector always normal to the path. What does this all prove?