# Measure on Path Integral not defined

#### friend

Where can I find an On-line exposition of the undefined nature of the measure in the Feynman path integral? Thanks.

#### AEM

Just curious, what do you mean by the phrase "the undefined nature of the measure" ?

#### friend

Just curious, what do you mean by the phrase "the undefined nature of the measure" ?
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 guessing here, but maybe it has something to do with the inabilitiy to form larger measures from the measure of disjoint unions of smaller subsets of paths. Or maybe the infinite dimensions violates the requirement to have only a finite intersection of subsets. Or maybe it has something to do with the product of distributions not generally being well defined. I really don't know.

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#### Fra

I think that maybe "ambigous" would be an different word.

Without furhter qualifiers the usual notion
$$\int Df e^{iS[f]}$$
where $$f: X \mapsto f(X)$$, and a measure is defined on X.

is not well defined, because it's only a symbolic notation. It doesn't mean it can't be defined, but it's at minimum ambigous. There is a CHOICE involved to make sense of this.

Then the question is what measure to assign/construct to the "function space" f belongs to so that this intergral makes sense?

Just like before you introduced the riemmann integral in calculus, transition from discrete sum to an infinite sum is ambigous.

It is very clear what "summing over a finite set of discrete path" means, but what does it actually mean when the paths form an infinite uncountable set? You need to define a MEASURE on the now continous space of paths.

This is actually also at the core of my personal objection for the continuum probability. It's simply because I do not see the physical basis for DISTINGUISHING and continuum of real numbers. This objection is very close at least conceptually to the problem of "defining a measure of physically distinguishable paths" in the path integral, when you are using a language that permitts an uncountable infinite set of paths.

If you see this as a mathematical problem, then it's a matter of constructing these measures for certain classes of functions.

From the physics point of view, it's also a matter of knowing what we are trying to do. What is the action? What is a transition amplitude?

/Fredrik

#### strangerep

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.
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.

(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.)
(The last bit is important in the case of the path integral over a nonseparable space of paths.)

#### Fra

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.
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.

As I see it, the ambigousness, is that to make sense out of this, one constrains the space of paths, to something in order to be able to define the integral by means of a countable sequences. But the CHOICE of countable sequence, implies as far as I understand that we impose and arbitrary constraint on the "space of paths".

Instead of such what I persoanlly think is physically ambigous way, I think we should define that the space of physically distinguishable paths is. And I am sure if will end up managable, since I can't imagine the physical meaning in a physical observer beeing able to distinguish an uncountable set. Even the intinite countable, is suspect, but it could be reached in the large complexity limit, but when it's uncountable things get suspect.

Another way of phrasing the ambigous choice of constraining the spce, is that it amounts to a non-unique choice of ergodic hypothesis with a equiprobable set of microstates over the space of paths. I think this way of solving this problem is strange. I think there may be another way, which implies reconstructing the entire "path integral" instead of trying to invent a way for it to make sense. Even if we invent a mathematical sense, it may be physically ambigous, since we need to not only be able to calculate it, the calculations should also match nature.

I think the set of "physically distinguishable paths" IS the "natural" constraint we need. The question then seems reduce to figuring out what physically distinguishing a path, from the point of view of an observer, really means? What does it mean for an observer to "distinguish" different possibilities?

I think somehow this "distinguishability" i the observer-dependent MEASURE on the space of paths we are looking for.

/Fredrik

#### strangerep

I'm not see the nonseparable nature of the space of paths. [...]
A space of well-behaved continuous paths is separable.
See, e.g.

http://en.wikipedia.org/wiki/Classical_Wiener_space

Separability follows from the Stone-Weierstrass theorem.
(Any continuous function can be approximated arbitrarily
closely by a polynomial.)

By my previous remark about nonseparability I only meant that even
if one considers a space of more pathological paths, one still has
trouble defining a suitable measure.

#### friend

So what about the dx1dx2...dxn. Is this a well defined measure, even if n is infinite? Are we talking about an ill-defined measure only when we interpret this as a space of paths?

#### strangerep

So what about the dx1dx2...dxn. Is this a well defined measure, even if n is infinite?
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.

Are we talking about an ill-defined measure only when we interpret this as a space of paths?
No. A path is just a particular kind of function. If we have a space of such functions which
is also a vector space (ie closed under addition and scalar multiplication) on which a norm is
defined, then we have (almost) a Banach space. (A true Banach space is what you get if the
space contains all its limit points.)

(We work with vectors in a finite dimensional space in the form of ordered tuples.
Similarly, we work with vectors in an infinite dimensional space in the form of functions.)

#### Fra

When gravity is addes, this very issue gets even worse, for reference one could note how Loll writes in

"The Emergence of Spacetime or Quantum Gravity on Your Desktop"
http://arxiv.org/abs/0711.0273

which reasons likes this:

"At the heart of the approach lies an explicit realization of the infamous “Sum over Histories”, also known as the gravitational path integral,
...
Before one has specified the integration space, the integration measure and the conditions under which the integration leads to a meaningful (i.e. non-infinite) result, it should be regarded as a statement of intent rather than a well-defined mathematical quantity."

Their CHOICE is to make a specific construction

"The integration space G is a space of causal, Lorentzian geometries, obtained from a certain limiting process, which will be described below."

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

#### friend

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
I see an inconsistency here. They are assuming before hand what is physical and not letting the math guide them to what is physical.

#### AEM

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 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."

The history of physics is full of examples where physical intuition guided the choice of mathematical concepts and how they were to be interpreted.

When it comes to deciphering nature methinks one must keep a balance between one's intuition about the physical and rigor of the mathematicians. A noted mathematician, Rene Thom once quipped "The word rigor often preceeds the word mortis".

#### jostpuur

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.
This 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 $[0,1]\times [0,1]\times\cdots$ and $\mathbb{R}\times\mathbb{R}\times\cdots$. Often product topologies are used, and measures are Borel with respect to them.

#### Fra

I see an inconsistency here. They are assuming before hand what is physical and not letting the math guide them to what is physical.

1) I now know that you are looking to derive physics from logic or math. On that point I think we can agree to disagree, although I partially appreciate your quest :) I've already presented some of my arguments why physics from pure logic is still ambigous. It lies in the choice of the axioms that defines your logic. If you consider the "set of all possible logical systems" then I have a hard time to see how that can be constructive and lead to physical predictions.

2) I do not like CDT in itself, but I just injected that paper as an example of reasoning. But the general idea to lead the physical insight choose guide the choice of mathematical abstractions, instead of the other way around is more natural to me at least.

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

#### AEM

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
I want to voice my agreement with this statement.

#### Fra

I see an inconsistency here. They are assuming before hand what is physical and not letting the math guide them to what is physical.
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.

So I really don't defend the CDT paper. But it's because I think they are not radical enough.

/Fredrik

#### strangerep

strangerep said:
[...] it's ill-defined if n is 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?
It can make a big difference depending on whether N is "huge but finite" compared to "infinite".
Sometimes, properties which hold for all finite N no longer hold in the limit (at least, not the
way you'd like them to). Such is the case here.

#### strangerep

This 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 $[0,1]\times [0,1]\times\cdots$ and $\mathbb{R}\times\mathbb{R}\times\cdots$. Often product topologies are used, and measures are Borel with respect to them.
Are you talking about Gaussian/Wiener measures? If so, yes they're used in path
integrals of course, but entail some extra assumptions about analytic continuation, etc.

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