nLab constructive quantum field theory

Idea

In the broad sense of the word constructive quantum field theory refers to the mathematically rigorous construction of full (i.e. non-perturbative) quantum field theories. More specifically the term has come to be used mainly for attempts to rigorously construct path integral measures for Wick rotated Euclidean field theories on Minkowski spacetime (Jaffe). This approach has led to a construction of scalar field theory in spacetime dimension 3, and Yang-Mills theory on a four-dimensional torus.

While a mathematically rigorous construction of perturbative quantum field theory is given by causal perturbation theory/perturbative AQFT, construction of non-perturbative quantum field theories has remained by and large elusive, except for toy example of free field theories or low spacetime dimension (e.g. 2d CFTs or scalar field theory in 3d) or topological quantum field theories. In fact the non-perturbative quantization of Yang-Mills theory(QCD) in 4d is listed as one of the open “Millennium Problems” by the Clay Mathematics Institute (see here).

It might be noteworthy that for the established rigorous construction of perturbative QFT via causal perturbation theory/perturbative AQFT a) the path integral or any measures that could go with it plays no role at all (instead the causal additivity of the S-matrix is axiomatized directly) and b) the construction is a formal deformation quantization (Collini 16). This might suggest that rigorous construction of non-perturbative quantum field theory ought to analogously proceed via strict deformation quantization.

Related concepts

• AQFT, FQFT

References

Review:

• Arthur Jaffe, Constructive quantum field theory (pdf)

• Stephen J. Summers, A Perspective on Constructive Quantum Field Theory (arXiv:1203.3991)

See also:

• Garth Warner: Bosonic Quantum Field Theory, EPrint Collection, University of Washington (2008) [hdl:1773/3710, pdf, pdf]

Discussion specifically of constructive path integrals:

• James Glimm, Arthur Jaffe, Quantum physics – A functional integral point of view, 535 pages, Springer

• Simon, Functional Integration in Quantum Physics (AMS, 2005)

• Sergio Albeverio, Raphael Høegh-Krohn, Sonia Mazzucchi. Mathematical theory of Feynman path integrals - An Introduction, 2 nd corrected and enlarged edition,

  Lecture Notes in Mathematics, Vol. 523. Springer, Berlin, 2008 (ZMATH)

• Sonia Mazzucchi, Mathematical Feynman Path Integrals and Their Applications, World Scientific, Singapore, 2009.

• Magnen, Rivasseau, Seneor, Construction of YM4 with an infrared cutoff (CMP)

Discussion of the problem of quantization of Yang-Mills theory from the point of view of constructive field theory is in

• Sourav Chatterjee, Yang-Mills for probabilists (arXiv:1803.01950)