Abstract
In this chapter, we authorize ourselves to use the physicist’ notations and some of their problematic computations, describing where are the mathematical problems, so that one has a good idea of what needs to be done to solve them. We start by describing the basic computations of finite dimensional Gaussian integrals with source, the defining equation of the Fourier-Laplace transform, and perturbative Gaussians because they are the main source of inspiration for the definition of the (infinite dimensional) functional integral. We derive the finite dimensional Feynman rules for some simple models, and explain Feynman’s interpretation of the functional integral. We proceed with the study of functional derivatives, and with a perturbative definition of functional integrals, followed by the description of Schwinger’s quantum variational principle. We finish by a discussion of the problems that occur with quantum field theories with gauge freedom.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Cartier, P., DeWitt-Morette, C.: Functional Integration: Action and Symmetries. Cambridge Monographs on Mathematical Physics, p. 456. Cambridge University Press, Cambridge (2006). ISBN 978-0-521-86696-5; 0-521-86696-0. Appendix D contributed by Alexander Wurm
Costello, K., Gwilliam, O.: Factorization algebras in perturbative quantum field theory (2010, preprint)
Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society Colloquium Publications, vol. 55, p. 785. Am. Math. Soc., Providence (2008). ISBN 978-0-8218-4210-2
Folland, G.B.: Quantum Field Theory: A Tourist Guide for Mathematicians. Mathematical Surveys and Monographs, vol. 149, p. 325. Am. Math. Soc., Providence (2008). ISBN 978-0-8218-4705-3
Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems, p. 520. Princeton University Press, Princeton (1992). ISBN 0-691-08775-X; 0-691-03769-8
LaChapelle, J.: Functional integration for quantum field theory. arXiv Mathematical Physics e-prints (2006). arXiv:math-ph/0610035
Phillips, T.: Finite dimensional Feynman diagrams. AMS (2001). http://www.ams.org/featurecolumn/archive/feynman1.html, 1–7
Rivers, R.J.: Path Integral Methods in Quantum Field Theory, 2nd edn. Cambridge Monographs on Mathematical Physics, p. 339. Cambridge University Press, Cambridge (1990). ISBN 0-521-25979-7; 0-521-36870-7
Zeidler, E.: Quantum Field Theory. I. Basics in Mathematics and Physics, p. 1020. Springer, Berlin (2006). doi:10.1007/978-3-540-34764-4. ISBN 978-3-540-34762-0; 3-540-34762-3. A bridge between mathematicians and physicists
Zeidler, E.: Quantum Field Theory. II. Quantum Electrodynamics, p. 1101. Springer, Berlin (2009). ISBN 978-3-540-85376-3. A bridge between mathematicians and physicists
Zeidler, E.: Quantum Field Theory. III. Gauge Theory, p. 1126. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22421-8. ISBN 978-3-642-22420-1. A bridge between mathematicians and physicists
Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena, 2nd edn. International Series of Monographs on Physics, vol. 85, p. 996. Clarendon/Oxford University Press, London/New York (1993). Oxford Science Publications. ISBN 0-19-852053-0
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Paugam, F. (2014). Mathematical Difficulties of Perturbative Functional Integrals. In: Towards the Mathematics of Quantum Field Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-319-04564-1_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-04564-1_18
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04563-4
Online ISBN: 978-3-319-04564-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)