I hear that the measure of the path integral, which seems to be a measure on the space of paths is not well defined. Or that maybe the infinite dimensional measure, dx1*dx2*dx3*...*dxn is not well defined. But that's all the explanation I seem to be able to get. Most of the time "undefined" means it is infinite or has more than one value depending on how you evalute it. I'd like clarification on the meaning of "undefined" as well as you do. For it seems impossible for the path integral formulation to be fundamental if it is based on undefined measures in mathematical analysis.Just curious, what do you mean by the phrase "the undefined nature of the measure" ?
Wiki has a bit more:I hear that the measure of the path integral, which seems to be a measure on the space of paths is not well defined. Or that maybe the infinite dimensional measure, dx1*dx2*dx3*...*dxn is not well defined. But that's all the explanation I seem to be able to get.
(The last bit is important in the case of the path integral over a nonseparable space of paths.)Let (X, || ||) be an infinite-dimensional, separable Banach space. Then the only locally finite and translation-invariant Borel measure μ on X is the trivial measure, with μ(A) = 0 for every measurable set A. Equivalently, every translation-invariant measure that is not identically zero assigns infinite measure to all open subsets of X.
(Many authors assume that X is separable. This assumption simplifies the proof considerably, since it provides a countable basis for X, and if X is a Hilbert space then the basis can even be chosen to be orthonormal. However, if X is not separable, one is still left with the undesirable property that some open sets have zero measure, so μ is not strictly positive even if it is not the trivial measure.)
I'm not see the nonseparable nature of the space of paths. It sounds like you may have more information than you are telling us here. This may be a nice start. But at least for me it needs to be fleshed out a bit more. Thank you.Wiki has a bit more:
http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure
(The last bit is important in the case of the path integral over a nonseparable space of paths.)
I'm not sure if this helps, but I tend to see the physical perspective and from that perspective, failue or non-separability, means there is no sensible way to find the countable set to "approximate" the uncountable set of paths, and thus no way to perform the "sum over path" as and ordinary sum over natural numbers.I'm not see the nonseparable nature of the space of paths. It sounds like you may have more information than you are telling us here. This may be a nice start. But at least for me it needs to be fleshed out a bit more. Thank you.
A space of well-behaved continuous paths is separable.I'm not see the nonseparable nature of the space of paths. [...]
No, it's ill-defined if n is infinite. That's the main point of the extract fromSo what about the dx1dx2...dxn. Is this a well defined measure, even if n is infinite?
No. A path is just a particular kind of function. If we have a space of such functions whichAre we talking about an ill-defined measure only when we interpret this as a space of paths?
I see an inconsistency here. They are assuming before hand what is physical and not letting the math guide them to what is physical.What they attempt is to constrain the integration space to what they think are the set of PHYSICAL possibilities, so that all so called non-physical paths have a zero measure, so these possibilities need not even be included in the integral. This as a general idea is IMHO PHYSICALLY sound, but I think they fail to produce are argument to show the uniqueness and observer independence of this choice.
/Fredrik
Yes, but we are only talking about processes that allow the dimensionality to approach infinity; the dimensionality is never allowed to actually be infinity. Does that make a difference?No, it's ill-defined if n is infinite. That's the main point of the extract from
http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure
which I quoted earlier.
I have to interject here that this is where physicists and mathematicians diverge. Remember it was Hilbert who said "Every school boy in Gottingen knows more about four dimensional geometry than Einstein, yet it was he who created General Relativity not the mathematicians."I see an inconsistency here. They are assuming before hand what is physical and not letting the math guide them to what is physical.
This is not relevant for the topic because path integrals do not rely on translation-invariant measures.Wiki has a bit more:
http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure
Let (X, || ||) be an infinite-dimensional, separable Banach space. Then the only locally finite and translation-invariant Borel measure μ on X is the trivial measure, with μ(A) = 0 for every measurable set A. Equivalently, every translation-invariant measure that is not identically zero assigns infinite measure to all open subsets of X.
Some more comments.I see an inconsistency here. They are assuming before hand what is physical and not letting the math guide them to what is physical.
I want to voice my agreement with this statement.Some more comments.
Part of the measure issue IMO, is exactly that all "mathematically possible paths" aren't necessarily "measureable" from a physical point of view. the question is of course, out of the set of all mathematics, which possibilites are the physical ones (ie the ones we should COUNT)?
/Fredrik
If you refer to them assuming too much structure in their sample space, like lorentian causality, then I fully agree. But that's I think a different point since I think the causal structure of spacetime is emergent with spacetime is. So I would like to instead see a different sampling space even without spacetime topology and instead somehow just sample over physical complexions, and hopefully see that spacetime structure including the lorenzian structure is a result of an underlying selforganisation. I also reject the manual choice of the EH-action.I see an inconsistency here. They are assuming before hand what is physical and not letting the math guide them to what is physical.
It can make a big difference depending on whether N is "huge but finite" compared to "infinite".Yes, but we are only talking about processes that allow the dimensionality to approach infinity; the dimensionality is never allowed to actually be infinity. Does that make a difference?strangerep said:[...] it's ill-defined if n is infinite.
Are you talking about Gaussian/Wiener measures? If so, yes they're used in pathThis is not relevant for the topic because path integrals do not rely on translation-invariant measures.
It is possible to define well defined measures on sets such as [itex][0,1]\times [0,1]\times\cdots[/itex] and [itex]\mathbb{R}\times\mathbb{R}\times\cdots[/itex]. Often product topologies are used, and measures are Borel with respect to them.