Probability paradox
Thursday 2007 December 27
A result that amazed me when I first heard of it is that Gaussian measure on infinite dimensional Hilbert spaces does not exist.
At first sight it seems obvious that it exists: on any finite dimensional quotient one can define a Gaussian probability measure, and these are all “compatible”. So their “inverse limit” should give a Gaussian probability measure on the original infinite dimensional Hilbert space.
In fact this cannot work: the ball of radius R in 1 dimensional space has Gaussian volume V<1, so the ball of radius R in n dimensions has Gaussian measure at most 
In fact it is possible to construct Gaussian measure on a Hilbert space H, but it has support larger than H. More precisely, if S is a Hilbert-Schmidt operator from H to K, then although Gaussian measure is not well defined, its image under S is a well defined probability measure on K (Sazonov’s theorem).
The non-existence of Gaussian measure on infinite dimensional Hilbert spaces is one of the things that makes quantum (or rather Euclidean) field theory hard: roughly speaking the functions one wants to integrate are only defined on H and not on the larger space K.
Friday 2007 December 28 at 7:06 pm
Dear Richard,
It seems to me that the root problem is that a sequence of iid Gaussian random variables is almost surely going to not be square-summable. If tries to one normalises it to be so by dividing out by the l^2 norm, then every coefficient is almost surely zero, which is presumably not what one wants. The problem seems analogous to why one cannot extract a meaningful probability measure on the integers by taking inverse limits of the uniform probability measure on, say, {-N,…,N}. (Amusingly, ultrafilters partially solve this problem also, but that is another story.)
Perhaps what this means is that one has to work with some sort of finite-dimensional approximation to field theories, and keep track of all the dependencies on dimension that subsequently arise. After all, the theory of Gaussian random matrices has progressed quite happily without any need for an infinite dimensional limit (though one could argue that the theory of free probability serves to some extent as a partial proxy for such a limit).
Saturday 2007 December 29 at 2:25 am
Working with some sort of finite dimensional approximation to field theories and taking a limit is in fact more or less what Glimm and Jaffe actually do to construct field theories: they approximate space by a finite lattice, so the space of states is a finite dimensional Hilbert space, then show that there is a limit as you add more and more points to the lattice.
Tuesday 2008 January 8 at 6:11 pm
Here’s another way of looking at more or less the same comment Terry made. From the point of view of measure theory (as opposed to say, functional analysis), the most straightforward way to construct an “infinite dimensional Gaussian measure” is to take the countable product of copies of R with 1-dimensional Gaussian measure. One then gets a probability measure on the space of sequences of real numbers. But then l^2 (and indeed l^\infty) is a set of measure 0.
Of course there are people who work with “Gaussian measures” on infinite dimensional Hilbert (or Banach) spaces, but I suppose the point is that for field theory you wish you had a notion of “standard Gaussian measure” which would be unitarily invariant?
Also, do you find the nonexistence of such a measure more surprising than the nonexistence of “Lebesgue measure” on an infinite Banach space?
Thursday 2008 January 10 at 5:32 pm
If one replaces space by a finite lattice to define a measure there is difficulty with fermions. The continuum limit introduces extra modes which are unphysical. This is called fermion doubling. Elaborate methods are introduced in lattice field theories to address this.
Sunday 2008 February 3 at 12:50 am
Do you know of anything that ‘looks like’ a Gaussian measure on Banach spaces in general (or especially seperable Banach spaces)? I ask because it is useful to have the following quantity defined on seperable Hilbert spaces: Let x_1, x_2, … be a complete orthnormal sequence, and then let, for a measurable function f and vector y:
\[\lim_{n \arrow \infty} \int_{R^n} f(\) e^{-\pi(a_1 ^2 + \cdots a_n ^2)} da_1 da_2 \cdots da_n\].
The resulting quantity won’t depend on the orthonormalization chosen, and can be used to prove all sorts of interesting results (uniform boundedness in Hilbert spaces, for instance). The question (maybe too vague of one) is whether you can replicate this idea on Banach spaces.
I have to apologize in advance if there is a standard construction yielding a ‘pseudo-measure’ like this; I know almost nothing about the subject.
Sunday 2008 February 3 at 12:53 am
hrm; I don’t know how to include TeX. The quantity I wanted to type is
\lim_{n \arrow \infty} \int_{R^n} f(<y,a_1 x_1 + a_2 x_2 + … a_n x_n) e^{-\pi(a_1 ^2 + … + a_n ^2)} da_1 da_2 … da_n
Thursday 2008 June 19 at 9:08 am
Somehow i missed the point. Probably lost in translation
Anyway … nice blog to visit.
cheers, Porridge.
Friday 2008 August 15 at 2:42 pm
This is somthing I have been looking for a long time. Thanks!!!
Wednesday 2008 September 10 at 6:32 pm
Quantum field theory is certainly hard, but I sincerely doubt that the considerations noted in the blog entry and in Prof. Tao’s reply are to the point. Consider the one-dimensional harmonic oscillator, perhaps the simplest quantum mechanical system there is. Despite it’s simplicity, its Hilbert space is infinite-dimensional, so all the considerations in the blog entry attach. Whatever the root of the problem with quantum field theory, it cannot be the non-existence of a Gaussian measure, for the very good reason that the measure does not exist for much simpler systems, such as the one-dimensional harmonic oscillator.
Moreover, the result, sketched in the blog entry, that the Gaussian measure of the whole infinite-dimensional Hilbert space vanishes, has a natural analog for the harmonic oscillator. Paths with finite action have zero measure. Sydney Coleman gives a nice argument for this result in an appendix to his lecture “The Uses of Instantons” (reprinted in Aspects of Symmetry by Cambridge University Press).
For what its worth, my own perspective on what makes quantum field theory hard is motivated by the work of Oded Schramm and collaborators that describe and define the self-similiar curves arising in quantum field theories of two-dimensional percolation and other critical phenomena (Please see Prof. Tao’s blog entry for Sept 3 for a description of this work and a note regarding the recent tragedy concerning Oded). There is good numerical evidence to suggest that self-similarity arises in higher-dimensions as well. Thus, making sense of quantum field theory in general probably entails describing and defining random, self-similar surfaces in two and higher dimensions, a topic of independent interest, and by no means trivial.
Friday 2009 May 1 at 7:08 am
To answer the question about real separable Banach spaces: the answer is yes there is notion of a Gaussian measure. The idea is to 1) exploit the fact that a real sep Banach space can be shown to be an “abstract Wiener space” i.e. contains a Hilbert space H dense in it and “measurable” wrrt to the Banach norm. (Notion of measurable norm needs to be made precuse) and 2) to then use a suitable generalization of the construction of the Wiener measure over C[0,1]: i.e. defining a Gaussian measure as a measure that is Gaussian in the normal sense when looking at cylinder sense over finite dimesional orthogonal projections of H … one then shows this has a unique sigma-additive extension..
A good reference on all this is Kuo’s Gaussian Measures in Banach Spaces (!) in the Lecture Notes in mathematics series by Springer
Saturday 2009 May 16 at 8:45 pm
I want give a little comment on Mark Meckes’s
question:
“Also, do you find the nonexistence of such a measure more surprising than the nonexistence of “Lebesgue measure” on an infinite Banach space?”
In this context, I want inform you that there exist some translation-invariant Borel measures (which are not $\sigma$-finite) on the separable Banach space U with basis such that they get finite non-zero values on some compact subsets
and infinite values on every non-degenarate balls.
If $\mu$ is such measure then $\mu(X)=0$ implies that $X$ is “shy”. They are called as “generators of shy sets” on $U$.
It interesting also that in some non-separable Banach spaces(for example, on l^{\infty}-the vector space of all bounded real-valued sequences) there exists a translation-invariant (non-$\sigma$-finite) measures which are defined on the minimal $\sigma$-algebra generated by all balls and which get a numerical value 1 on the unite closed ball.
In Solovay model(ZF+DC+”every subset of $R$ is lebesgue measurable”) we have more strict result:
There exists a Borel measure on l^{\infty} with above mentioned properties.
Sunday 2009 May 17 at 5:24 am
Here is a certain construction of a Gaussian probability measure $\gamma$ on infinite-dimensional Hilbert space $l^2$:
Let $\mu$ be a standard Gaussian measure of $R^{\infty}$.
Let $(A_k)_{k \in N}$ be such a sequence of positive numbers that
$0<\mu(\prod_{k \in N}[-A_k,+A_k])<1$.
We define a mapping $S:R^{\infty}\to R^{\infty}$ as follows
$S((x_k)_{k \in N})=(\frac{x_k}}{2^kA_k})_{k \in N}$.
Then $S(\prod_{k \in N}[-A_k,+A_k])\subset l_2$.
We set $\gamma(X)=\mu(S^{-1}(X))$ for every $X \in B(l_2)$, where $B(l_2)$ denotes a Borel $\sigma$-algebra of subsets of $l_2$.
Remark. Note that in your “Probability paradox” you try to construct a Gaussian measure $\gamma$ on $l_2$ by the identity mapping $f:R^{\infty} \to R^{\infty}$ (i.e., $f((x_k)_{k \in N}= (x_k)_{k \in N}$ for $(x_k)_{k \in N} \in R^{\infty}$) as follows:
$\gamma(X)=\mu(f^{-1}(X))$ for every $X \in B(l_2)$.
But your measure is identically zero on $l_2$ because
$\gamma(l_2)=\mu(f^{-1}(l_2))=\mu(l_2)=0$.
Friday 2009 October 30 at 4:22 am
My comment is well behind the making of this post, but all I have to say is that this is an excellent discussion.