nLab constructive quantum field theory

Idea

In the broad sense of the term, 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 in 3 dimensions.

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

• nonperturbative quantum field theory

• lattice gauge field theory

• AQFT, FQFT

References

General

Introduction and review:

• Arthur S. Wightman: Constructive Field Theory Introduction to the Problems, in: Fundamental Interactions in Physics and Astrophysics, Studies in the Natural Sciences 3, Springer (1973) [doi:10.1007/978-1-4613-4586-2_1]

• Stephen J. Summers: A Perspective on Constructive Quantum Field Theory [arXiv:1203.3991, pdf]

• Arthur Jaffe, Constructive quantum field theory [pdf, pdf]

• Wikipedia: Constructive quantum field theory

On constructive gauge field theory:

• David Brydges, Jürg Fröhlich, Erhard Seiler: On the construction of quantized gauge fields. I. General results, Annals of Physics 121 1–2 (1979) 227-284 [doi:10.1016/0003-4916(79)90098-8]

• Erhard Seiler: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Physics 159, Springer (1982) [doi:10.1007/3-540-11559-5]

See also:

• John Baez, Irving Segal, Zhengfang Zhou: Introduction to Algebraic and Constructive Quantum Field Theory, Princeton University Press (1992) [ISBN:9780691605128, doi:10.1515/9781400862504, pdf]

• 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, Springer (1981, 1987) [doi:10.1007/978-1-4612-4728-9]

• Barry Simon: Functional integration and quantum physics, AMS Chelsea Publishing 351, AMS (2005) [ISBN:978-0-8218-3582-1, pdf]

• Sergio Albeverio, Raphael Høegh-Krohn, Sonia Mazzucchi. Mathematical theory of Feynman path integrals - An Introduction, Lecture Notes in Mathematics 523, Springer (2008) [ZMATH]

• Sonia Mazzucchi, Mathematical Feynman Path Integrals and Their Applications, World Scientific (2009)

• Magnen, Rivasseau, Seneor: Construction of $YM_4$ with an infrared cutoff [euclid:cmp/1104253284]

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

• Sourav Chatterjee, Yang-Mills for probabilists [arXiv:1803.01950]

See also:

• Jiasheng Lin: Constructive Quantum Field Theory on Curved Surfaces and Related Topics [arXiv:2507.21655]

Formalization in Lean of the construction of the free scalar quantum field:

• Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim: Formalization of QFT [arXiv:2603.15770]

For 2D YM theory

For D=2 Yang-Mills theory:

• Leonard Gross, Christopher King, Ambar N. Sengupta: Two dimensional Yang-Mills theory via stochastic differential equations, Annals of Physics 194 1 (1989) 65–112 [doi:10.1016/0003-4916(89)90032-8]

• Ambar Sengupta, The Yang-Mills measure for $S^2$, Journal of Functional Analysis 108 2 (1992) 231–273 [doi;10.1016/0022-1236(92)90025-E]

• Ambar Sengupta: Quantum Gauge Theory on Compact Surfaces, Annals of Physics 221 1 (1993) 17–52 [doi:10.1006/aphy.1993.1002]

• Ambar Sengupta: Quantum Gauge Theory on Compact Surfaces, Memoirs of the AMS 126 (1997) [ISBN:978-1-4704-0185-6, doi:10.1006/aphy.1993.1002]

On rigorous construction of path integral measures for 2D Yang-Mills theory (at least for 2D Maxwell theory) invariant under area-preserving diffeomorphisms:

• Doug Pickrell: On $YM_2$ measures and area-preserving diffeomorphisms, Journal of Geometry and Physics 19 4 (1996) 315-367 [doi:10.1016/0393-0440(95)00034-8]

For 3D MCS theory

A construction of what looks like Maxwell-Chern-Simons theory:

• Doug Pickrell: On the Action of the Group of Diffeomorphisms of a Surface on Sections of the Determinant Line Bundle, Pacific Journal of Mathematics 193 1 (2000) 177-199 [doi:10.2140/pjm.2000.193.177, pdf]

For 3D YMCS theory

No rigorous construction of non-abelian 3D Yang-Mills-Chern-Simon theory is currently known.

Comments on what is known and what is not, in comparison to the 2D case:

• Doug Pickrell: Notes on invariant measures for loop groups [arXiv:2207.09913]