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Quantum Mechanics and Quantum Field Theory can both be formulated in terms of the so-called functional integrals.

The point is that intuitively it is an "integral over all possible paths" or rather "integral over all possible field configurations", traditionaly denoted as

$$\int A[\gamma(t)]\mathcal{D}\gamma(t)$$

$$\int A[\phi(x)]\mathcal{D}\phi(x)$$

for respectively paths and fields. It seems however that this is not well defined. I really don't understand how can one manipulate something that isn't defined, so I'm searching for the right way to understand these things.

Is there some way to make sense of these objects? I heard that as traditional measures it is not possible, but is there any other alternative way to make this be defined? If there is no way, how can someone work with one object that has no meaning associated with it and compute things with it?

Andrews
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Gold
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2 Answers2

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There are two versions of such path integrals: the Minkowskian and the Euclidean. The first one is of the form $$ \int\ F(\phi)\ e^{iS(\phi)}\ D\phi $$ and the second one is $$ \int F(\phi)\ e^{-S(\phi)}\ D\phi\ . $$ Here $S$ is the action functional and $F$ is another functional corresponding to observables. The Minkowskian case lies outside ordinary measure theory, even in the Gaussian case (a Theorem by Cameron https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1960391126 correcting an earlier mistake by Gelfand and Yaglom who thought that $\sigma$-additive complex measures would work). One instead has to use a limit of a time slicing procedure to make sense of it.

For the Euclidean case, ordinary measure theory is perfectly adequate. The "I heard that as traditional measures it is not possible" is a common misconception that unfortunately gets repeated ad infinitum. The part $e^{-S(\phi)}\ D\phi$ should be a Borel probability measure on a space of distributions like $\mathscr{S}'(\mathbb{R}^d)$ seen as an ordinary topological space. The most canonical topology to use is the strong topology. When dealing with concrete models, it is a highly nontrivial task to construct this probability measure which a priori should be the weak limit of a sequence of well defined probability measures obtained by introducing ultraviolet and infrared cut-offs.

Edit: For more details on how to construct the measure as weak limits of (finite) lattice measures in the free field case see:

https://mathoverflow.net/questions/362040/reformulation-construction-of-thermodynamic-limit-for-gff

https://mathoverflow.net/questions/364470/a-set-of-questions-on-continuous-gaussian-free-fields-gff?noredirect=1&lq=1

https://mathoverflow.net/questions/384124/bochner-minlos-for-moment-generating-functions?noredirect=1&lq=1

Abdelmalek Abdesselam
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  • Thanks @AbdelmalekAbdesselam. Could you point out some reference showing how to rigorously define the measure (for instance in free theory) following this approach you mention of taking a weak limit of a sequence of measures obtained introducing UV and IR cut-offs? I believe that in free theory at least this is well understood, isn't it? – Gold Dec 29 '18 at 18:21
  • @user1620696: In the free case you don't need to take weak limits of removing cutoffs. You can define the probability measure directly using the Bochner-Minlos Theorem. The most accessible reference I know is https://arxiv.org/abs/1706.09326 Note that pedagogically it also makes sense to define the free measure by taking a weak limit starting from a lattice with finite volume. The proof is easy using the Levy continuity theorem also proved in the reference I mentioned. – Abdelmalek Abdesselam Jan 02 '19 at 11:32
  • Thanks for the reference. I think that will help. I'm also trying to connect with the usual (unrigorous) approach presented in most QFT books (like Peskin). I think the connection lies in this approach you mention of defining the free measure by taking a weak limit starting from a lattice with finite volume. At least this looks like what is done in these books, right? – Gold Jan 02 '19 at 11:45
  • I can't say since I don't have the S&P QFT book in front of me, but regardless of the choice of pedagogical introduction the Euclidean free measure obtained in the end is the one I hinted at via weak limits. – Abdelmalek Abdesselam Jan 02 '19 at 11:59
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Usually these are defined by a limiting process, splitting the QM path into lots of infinitesimal segments (or in Fourier space, by summing over a finite number of modes) and examining the continuous limit (if it exists). See for example the Feynman-Hibbs book. For field theory the path integral over field configurations is even more abstract and difficult to define.

If you don't like that, one may simply define the Gaussian functional integral in analogy to the discrete case (real variables and matrix in the exponent) and manipulate the path integral into moments of the Gaussian (essentially perturbation theory).

Incidentally, the worldline formalism to QFT replaces the functional field integral into a less abstract QM path integral.

lux
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