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May I ask is it true that all the interacting 4 dimension qft couldn't be constructed and defined consistently and rigorously? If we are able to rigorously constructed lower dimension qft, what are the usage of those algebraic qft since real qft is 4 dimension and have interaction. Any experts can clarify?

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We are not able, at least for the moment, to define in a rigorous mathematical fashion a meaningful interacting QFT in $3+1$ dimensions that is coherent with the perturbative theory utilized by physicists (in more precise words, that satisfies the Wightman axioms).

On the contrary, some rigorous interacting QFTs can be defined in lower (spatial) dimensions. Take for example the $\phi^4$ model: it is not well-defined in $3+1$ dimensions, but it is in $1+1$ and $2+1$ (the latter for sure on finite volume, I am not sure about the infinite volume limit) as proved by Glimm and Jaffe in the sixties. This simplified models, even if they may not be physically interesting, has been analyzed in the hope that the same tools may be utilized in the meaningful theories. Unluckily, this has not been the case so far (anyways, there is still ongoing work on the subject, especially concerning the Yang-Mills theory).

However there is no rigorous "no-go theorem" that says that the interacting quantum field theories in $3+1$ dimensions cannot be constructed, but it may be possible that for some model the limitations are more fundamental and not only related to the lack of mathematical tools (see the comment by A.A. below).

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    $\begingroup$ $\phi^4$ in 2+1 has been done in infinite volume by Feldman and Osterwalder as well as Magnen and Seneor. Also, there is a non-rigorous "no-go theorem" called the Coleman-Gross Theorem in the physics literature which excludes for instance scalar models in 3+1 (as well as pretty much anything except YM plus eventually not too many Fermions). In 4+1 there is a true theorem excluding scalar theories by Aizenman and Froehlich. $\endgroup$ – Abdelmalek Abdesselam May 7 '15 at 14:39
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    $\begingroup$ @AbdelmalekAbdesselam: Doesn't the Aizenman-Froehlich result only exclude a certain typpe of constructions for a $\phi^4$ theory in $>4$ dimensions? It does not prove that a scalar Wightman QFT satisfying a renormalized quartic field equations does not exist. $\endgroup$ – Arnold Neumaier May 13 '15 at 15:26
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    $\begingroup$ @Arnold: yes your are right. I guess a reasonable conjecture is that a Wightman scalar field in four or more dimensions must be in the Borchers class of a generalized free field. The AF result does not quite prove that. I somewhat disagree with "a certain type". While it is correct, it might suggest that AF only excluded some kind of exotic attempt at constructing a scalar QFT in >4 dimensions. The approach which they rule out via continuum limits of Euclidean lattice theories, I think, is the most standard one. $\endgroup$ – Abdelmalek Abdesselam May 19 '15 at 14:07
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    $\begingroup$ @AbdelmalekAbdesselam: What do you think of Klauder's construction attempts, e.g., in arxiv.org/abs/1405.0332 ? Has anyone checked whether this would satisfy the Wightman axioms? Or proved that it does not? $\endgroup$ – Arnold Neumaier May 20 '15 at 15:52
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    $\begingroup$ @Arnold: I know of the existence of the article but I did not read it, so I cannot say. From a quick glance, it seems to be written in the style of physics papers instead of math ones, and it looks more like a brief research announcement. Unless JK followed it up with a paper containing complete mathematical proofs, I am afraid your question might be undecidable... $\endgroup$ – Abdelmalek Abdesselam May 20 '15 at 16:02
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The tools and techniques for constructing QFTs are the same whatever the dimension. However you cannot prove conjectures which are false, no matter how powerful your tools are. The issue with 3+1 is that the class of models for which the conjecture "yeah it can be constructed" is in all likelihood true is much more narrow than in 1+1 and 2+1. Basically the only candidate is Yang-Mills theory which is a rather difficult beast. Yet there has been work on the finite volume construction by Balaban, Federbush as well as Magnen-Rivasseau-Seneor.


Edit: Just to clarify, Wightman axioms are just a definition of what a QFT is. Namely, it is a precise formulation of what the end goal of "constructing a QFT" is, but it does not tell you how to get there. Besides, these axioms are not quite appropriate for $YM_4$, not because it is in 4 dimensions but because it is a gauge theory. Nevertheless, there is a precise formulation of the $YM_4$ end goal in the $1M problem description by Jaffe and Witten (see also the follow up by Douglas, all available here).

As for methods being the same, essentially you need to construct a probability measure on a space of distributions as a weak limit of well defined measures as you remove the UV and IR cut-offs, then you get the QFT in Minkowski space using the Osterwalder-Schrader Theorem. The difficult part is the construction of the limit probability measure. The methods for doing that are the rigorous renormalization group theory or its variant called the multiscale cluster expansion method. These methods work for scalar models in $1+1$, $2+1$ as well as $YM_4$ as treated by Balaban and the other authors I mentioned above.

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  • $\begingroup$ Do you mean that the only 4 dimensional "constructable" and which " satisfy wightman axioms" is only yang mills? It seems that there are no satisfying 4 dimension interacting qft at all . Is it guaranteed that wightman axioms (especially the positivity condition) must be correct in 4 dimension ? Can anyone elaborate on "the tools and techniques for constructing qft in whataever dimension are the same? Thanks $\endgroup$ – user41508 May 8 '15 at 8:35
  • $\begingroup$ @user41508: hope my edit answers your questions. $\endgroup$ – Abdelmalek Abdesselam May 8 '15 at 15:16
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    $\begingroup$ We can't be sure. These axioms were abstracted from what was known at the time about QFT, primarily on the basis of perturbation theory and basic physical principles like relativistic invariance. Then Nelson, Glimm and Jaffe constructed nontrivial examples in $1+1$ and $2+1$ which showed the axioms are not vacuous. What else do you want? or rather what does "correct" mean for you? $\endgroup$ – Abdelmalek Abdesselam May 8 '15 at 17:12
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    $\begingroup$ @user41508: All conditions in the Wightman axioms with exception of the uniqueness of the vacuum are essential for any meaningful Hilbert space interpretation of a relativistic QFT. They express very basic properties of unitarity, causality and covariance of observable local fields. They are known to be inadequate for unobservable fields such as gauge fields - but even in the presence of gauge fields, the observable local fields such as currents and squared curvature must satisfy the axioms; see my review at physicsoverflow.org/21846 $\endgroup$ – Arnold Neumaier May 13 '15 at 16:04
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    $\begingroup$ Note: observable local fields = gauge invariant local fields. These are sums of products of local fields at the same point with all gauge indices and spinor indices contracted, and products regularized by subtracting the singular part of the operator product expansion. $\endgroup$ – Arnold Neumaier May 13 '15 at 16:46
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QFTs in spacetime of dimension $<4$ have their use in real applications - 3D theories to quantum surfaces, and 2D theories to quantum wires. There much of the exceptional behavior of lower-dimensional QFTs can be observed.

A famous example is the fractional Hall effect with anyonic (rather than Bose or Fermi) statistics.

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