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Solutions in Constructive Field Theory

Published online by Cambridge University Press:  01 January 2022

Leif Hancox-Li
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Abstract

Constructive field theory aims to rigorously construct concrete, nontrivial solutions to Lagrangians used in particle physics. I examine the relationship of solutions in constructive field theory to both axiomatic and Lagrangian quantum field theory (QFT). I argue that Lagrangian QFT provides conditions for what counts as a successful constructive solution and other information that guides constructive field theorists to solutions. Solutions matter because they describe the behavior of QFT systems and thus what QFT says the world is like. Constructive field theory clarifies existing disputes about which parts of QFT are philosophically relevant and how rigor relates to these disputes.

Type
Research Article
Information
Philosophy of Science , Volume 84 , Issue 2 , April 2017 , pp. 335 - 358
Copyright
Copyright © The Philosophy of Science Association

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Footnotes

†

I thank Bob Batterman and Michael Miller for comments and encouragement. Several anonymous referees also contributed valuable comments.

References

Aizenman, M. 1981. “Proof of the Triviality of ɸd4 Field Theory and Some Mean-Field Features of Ising Models for d > 4.” Physical Review Letters 47 (1): 1–4.CrossRefGoogle Scholar
Araki, H. 1972. “Axiomatic Field Theory.” In Proceedings of the 16th International Conference on High-Energy Physics, ed. Jackson, J. D. and Roberts, A., 1–6. Batavia, IL: National Accelerator Laboratory.Google Scholar
Baker, D. J., and Halvorson, H.. 2010. “Antimatter.” British Journal for the Philosophy of Science 61 (1): 93–121.CrossRefGoogle Scholar
Batterman, R. W. 2010. “On the Explanatory Role of Mathematics in Empirical Science.” British Journal for the Philosophy of Science 61 (1): 1–25.CrossRefGoogle Scholar
Brunetti, R., Duetsch, M., and Fredenhagen, K.. 2009. “Perturbative Algebraic Quantum Field Theory and the Renormalization Groups.” arXiv.org, Cornell University Library. https://arxiv.org/abs/0901.2038.Google Scholar
Caliceti, E., Grecchi, V., and Maioli, M.. 1986. “The Distributional Borel Summability and the Large Coupling ɸ4 Lattice Fields.” Communications in Mathematical Physics 104 (1): 163–74.CrossRefGoogle Scholar
Callaway, D. J. E. 1988. “Triviality Pursuit: Can Elementary Scalar Particles Exist?” Physics Reports 167 (5): 241–320.CrossRefGoogle Scholar
Clifton, R., and Halvorson, H.. 2001. “Are Rindler Quanta Real? Inequivalent Particle Concepts in Quantum Field Theory.” British Journal for the Philosophy of Science 52 (3): 417–70.CrossRefGoogle Scholar
Davis, P. J. 2009. When Is a Problem Solved? Washington, DC: Mathematical Association of America.Google Scholar
Detlefsen, M. 2005. “Formalism.” In The Oxford Handbook of Philosophy of Math and Logic, ed. Shapiro, S., 236–317. Oxford: Oxford University Press.CrossRefGoogle Scholar
Douglas, M. R. 2011. “Foundations of Quantum Field Theory.” Presentation slides at String-Math 2011, UPenn, Philadelphia, June 6–11.Google Scholar
Erdelyi, A. 2010. Asymptotic Expansions. New York: Dover.Google Scholar
Fraser, D. 2008. “The Fate of ‘Particles’ in Quantum Field Theories with Interactions.” Studies in History and Philosophy of Science B 39 (4): 841–59.Google Scholar
Fraser, D. 2011. “How to Take Particle Physics Seriously: A Further Defence of Axiomatic Quantum Field Theory.” Studies in History and Philosophy of Science B 42 (2): 126–35.Google Scholar
Fredenhagen, K., Rehren, K.-H., and Seiler, E.. 2006. “Quantum Field Theory: Where We Are.” In Approaches to Fundamental Physics, ed. Stamatescu, I.-O. and Seiler, E., chap. 4, 61–87. Lecture Notes in Physics 721. Berlin: Springer.Google Scholar
Gallavotti, G. 1985. “Renormalization Theory and Ultraviolet Stability for Scalar Fields via Renormalization Group Methods.” Reviews of Modern Physics 57 (2): 471–562.CrossRefGoogle Scholar
Gallavotti, G., and Rivasseau, V.. 1984. “ɸ4 Field Theory in Dimension 4: A Modern Introduction to Its Unsolved Problems.” In Annales de l’IHP Physique théorique, vol. 40, 185–220. Amsterdam: Elsevier.Google Scholar
Gawedzki, K., and Kupiainen, A.. 1985. “Massless Lattice ɸ44 Theory: Rigorous Control of a Renormalizable Asymptotically Free Model.” Communications in Mathematical Physics 99 (2): 197–252.CrossRefGoogle Scholar
Glimm, J. 1969. “The Foundations of Quantum Field Theory.” Advances in Mathematics 3 (1): 101–25.CrossRefGoogle Scholar
Glimm, J., and Jaffe, A.. 1968. “A λɸ4 Quantum Field Theory without Cutoffs.” Pt. 1. Physical Review 176:1945–51.CrossRefGoogle Scholar
Glimm, J., Jaffe, A., and Spencer, T.. 1974. “The Wightman Axioms and Particle Structure in the P(ɸ)2 Quantum Field Model.” Annals of Mathematics 100 (3): 585–632.CrossRefGoogle Scholar
Halvorson, H., and Clifton, R.. 2002. “No Place for Particles in Relativistic Quantum Theories?” Philosophy of Science 69 (1): 1–28.CrossRefGoogle Scholar
Halvorson, H., and Müger, M.. 2006. “Algebraic Quantum Field Theory.” In Philosophy of Physics, ed. Butterfield, J. and Earman, J., 731–922. Handbook of the Philosophy of Science. Amsterdam: North-Holland.Google Scholar
Jaffe, A. 1969. “Whither Axiomatic Field Theory?” Reviews of Modern Physics 41 (4): 576–80.CrossRefGoogle Scholar
Jaffe, A., and Witten, E.. 2005. “Quantum Yang-Mills Theory.” http://www.claymath.org/sites/default/files/yangmills.pdf.Google Scholar
Kleinert, H., and Schulte-Frohlinde, V.. 2001. Critical Properties of ɸ4 Theories. Singapore: World Scientific.CrossRefGoogle Scholar
Kline, M. 1983. “Euler and Infinite Series.” Mathematics Magazine 56 (5): 307–14.CrossRefGoogle Scholar
Mack, G., and Pordt, A.. 1985. “Convergent Perturbation Expansions for Euclidean Quantum Field Theory.” In Quantum Field Theory, ed. Jaffe, A., Lehmann, H., and Mack, G., 267–98. Berlin: Springer.Google Scholar
Magnen, J., and Sénéor, R.. 1977. “Phase Space Cell Expansion and Borel Summability for the Euclidean ɸ34 Theory.” Communications in Mathematical Physics 56 (3): 237–76.CrossRefGoogle Scholar
Pincock, C. 2004. “A New Perspective on the Problem of Applying Mathematics.” Philosophia Mathematica 12 (2): 135–61.CrossRefGoogle Scholar
Ramond, P. 1981. Field Theory: A Modern Primer. 2nd ed. Reading, MA: Benjamin/Cummings.Google Scholar
Rivasseau, V. 1984. “Construction and Borel Summability of Planar 4-Dimensional Euclidean Field Theory.” Communications in Mathematical Physics 95 (4): 445–86.CrossRefGoogle Scholar
Rivasseau, V. 1991. From Perturbative to Constructive Renormalization. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Rivasseau, V. 1999. “Constructive Renormalization Theory.” arXiv.org, Cornell University Library. http://arxiv.org/abs/math-ph/9902023.Google Scholar
Rosen, L. 1973. “The P(ɸ)2 Model.” In Fundamental Interactions in Physics and Astrophysics, ed. Iverson, G., Perlmutter, A., and Mintz, S., 86–107. Studies in the Natural Sciences 3. New York: Plenum.CrossRefGoogle Scholar
Ruetsche, L. 2011. Interpreting Quantum Theories. Oxford: Oxford University Press.CrossRefGoogle Scholar
Salmhofer, M. 2007. Renormalization: An Introduction. Berlin: Springer.Google Scholar
Sénéor, R. 1988. “Some Remarks for the Construction of Yang-Mills Field Theories.” In Renormalization of Quantum Field Theories with Non-linear Field Transformations, ed. Breitenlohner, P., Maison, D., and Sibold, K., 22–31. Lecture Notes in Physics 303. Berlin: Springer.CrossRefGoogle Scholar
Simon, B. 1982. “Large Orders and Summability of Eigenvalue Perturbation Theory: A Mathematical Overview.” International Journal of Quantum Chemistry 21 (1): 3–25.CrossRefGoogle Scholar
Steinmann, O. 2000. Perturbative Quantum Electrodynamics and Axiomatic Field Theory. Berlin: Springer.CrossRefGoogle Scholar
Strocchi, F. 2013. An Introduction to the Non-perturbative Foundations of Quantum Field Theory. Oxford: Oxford University Press.CrossRefGoogle Scholar
Summers, S. J. 2012. “A Perspective on Constructive Quantum Field Theory.” arXiv.org, Cornell University Library. https://arxiv.org/abs/1203.3991.Google Scholar
Tao, T. 2016. “Generalized Solutions.” UCLA. http://www.math.ucla.edu/~tao/preprints/generalized_solutions.pdf.Google Scholar
Teller, P. 1997. An Interpretive Introduction to Quantum Field Theory. Princeton, NJ: Princeton University Press.Google Scholar
Wallace, D. 2006. “In Defence of Naiveté: The Conceptual Status of Lagrangian Quantum Field Theory.” Synthese 151 (1): 33–80.CrossRefGoogle Scholar
Wallace, D. 2011. “Taking Particle Physics Seriously: A Critique of the Algebraic Approach to Quantum Field Theory.” Studies in History and Philosophy of Science B 42 (2): 116–25.Google Scholar
Wightman, A. S. 1969. “What Is the Point of So-Called ‘Axiomatic Field Theory’?” Physics Today 22 (9): 53–58.CrossRefGoogle Scholar
Wightman, A. S. 1973. “Constructive Field Theory: Introduction to the Problems.” In Fundamental Interactions in Physics and Astrophysics, ed. Iverson, G., Perlmutter, A., and Mintz, S., 1–85. Studies in the Natural Sciences 3. New York: Plenum.Google Scholar
Iverson, G., Perlmutter, A., and Mintz, S. 1976. “Hilbert’s Sixth Problem: Mathematical Treatment of the Axioms of Physics.” In Mathematical Developments Arising from Hilbert Problems, ed. Browder, F. A., 147–240. Providence, RI: American Mathematical Society.Google Scholar
Browder, F. A. 1986. “Some Lessons of Renormalization Theory.” In The Lesson of Quantum Theory, ed. de Boer, J., Dal, E., and Ulfbeck, O., 201–26. Amsterdam: North-Holland.Google Scholar
Wilson, M. 2008. Wandering Significance: An Essay on Conceptual Behaviour. Oxford: Oxford University Press.Google Scholar
Wüthrich, A. 2012. “Interpreting Feynman Diagrams as Visual Models.” Spontaneous Generations: A Journal for the History and Philosophy of Science 6 (1): 172–81.Google Scholar