# Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories

## Abstract

**:**

## 1. Introduction

## 2. Asymptotic versus Divergent Series: Tao’S Method

#### 2.1. Asymptotic and Divergent Series

#### 2.2. Inconsistencies in the Standard Treatment of Divergent Series

#### 2.3. A New Look at Divergent Series: Terence Tao’S Method of Smoothed Sums

## 3. Application of Tao’S Method to the Casimir Effect for Perfectly Conducting Parallel Plates

**Theorem**

**1.**

**Proof.**

## 4. General Aspects of Nonperturbative Quantum Fields: Wightman Axioms for Interacting Quantum Fields, Dressed Particles in a Charged Sector, and Unstable Particles

- $(a.)$
- the relativistic transformation law;
- $(b.)$
- the spectral condition;
- $(c.)$
- hermiticity;
- $(d.)$
- local commutativity;
- $(e.)$
- positive-definiteness,

- $(f.)$
- interacting fields are assumed to satisfy the singularity hypothesis (the forthcoming Definition).

#### 4.1. The Källén-Lehmann Representation

#### 4.2. The Singularity Hypothesis

#### 4.2.1. Steinmann Scaling Degree and A Theorem

**Definition**

**1.**

**Theorem**

**2.**

**Corollary**

**1.**

#### 4.2.2. The ETCR Hypothesis and Its Consequences for the Singularity Hypothesis

## 5. A Proposal for the Meaning of the Condition $\mathbf{Z}=\mathbf{0}$: The Presence of Massless and Unstable Particles

**Corollary**

**3.**

## 6. Models for Atomic Resonances, Unstable and “Dressed” Particles: What Distinguishes Quantum Field Theory from Many-Body Systems?

**Theorem**

**3.**

**Proof.**

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zyla, P.A.; Barnett, R.M.; Beringer, J.; Dahl, O.; Dwyer, D.A.; Groom, D.E.; Lin, C.-J.; Lugovsky, K.S.; Pianori, E.; Robinson, D.J.; et al. Review of Particle Physics. Prog. Theor. Exp. Phys.
**2020**, 8, 083C01. [Google Scholar] - Weinberg, S. The Quantum Theory of Fields Volume I—Foundations; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Thirring, W. On the divergence of perturbation theory for quantized fields. Helv. Phys. Acta
**1953**, 26, 33. [Google Scholar] - Hooft, G. Reflections on the renormalization procedure for gauge theories. Nucl. Phys. B
**2016**, 912, 4. [Google Scholar] [CrossRef] [Green Version] - Wightman, A.S. Should we believe in quantum field theory? In The Whys of Subnuclear Physics; Zichichi, A., Ed.; Plenum Press: New York, NY, USA, 1979. [Google Scholar]
- Wreszinski, W.F. Unstable states in a model of nonrelativistic quantum electrodynamics. J. Stat. Phys.
**2021**, 182, 26. [Google Scholar] [CrossRef] - Gavrilov, S.P.; Gitman, D.M.; Dmitriev, V.D.; Panferov, A.D.; Smolyansky, S.A. Radiation problems accompanying carrier production by an electric field in graphene. Universe
**2020**, 6, 205. [Google Scholar] [CrossRef] - Thirring, W.E. A soluble relativistic field theory. Ann. Phys.
**1958**, 3, 91. [Google Scholar] [CrossRef] - Mattis, D.C.; Lieb, E.H. Exact solution of a many-Fermion system and its associate Bose field. J. Math. Phys.
**1965**, 6, 304–312. [Google Scholar] [CrossRef] [Green Version] - Verbeure, A.; Zagrebnov, V.A. About the Luttinger model. J. Math. Phys.
**1993**, 34, 785. [Google Scholar] [CrossRef] - Mastropietro, V.; Mattis, D.C. Luttinger Model—The First 50 Years and Some New Directions; World Scientific Publ. Co.: Singapore, 2013. [Google Scholar]
- Jaekel, C.D.; Wreszinski, W.F. A criterion to characterize interacting theories in the Wightman framework. Quantum Stud. Math. Found.
**2021**, 8, 51. [Google Scholar] [CrossRef] [Green Version] - Buchholz, D. Gauss’ law and the infraparticle problem. Phys. Lett. B
**1986**, 174, 331. [Google Scholar] [CrossRef] - Streater, R.F.; Wightman, A.S. PCT, Spin and Statistics and All That; W. A. Benjamin, Inc.: Princeton, NJ, USA, 1964. [Google Scholar]
- Sakurai, J.J. Advanced Quantum Mechanics; Addison Wesley: Boston, MA, USA, 1967. [Google Scholar]
- Hardy, G.H. Divergent Series; Oxford at the Clarendon Press: Oxford, UK, 1949. [Google Scholar]
- Tao, T. The Euler-Maclaurin Formula, Bernoulli Numbers, the Zeta Function and Real Variable Analytic Continuation. Available online: https://terrytao.wordpress.com/2010/04/10/ (accessed on 4 July 2021).
- Casimir, H.B.G. On the Attraction Between Two Perfectly Conducting Plates. Indag. Math.
**1948**, 10, 261. [Google Scholar] - Buck, R.C. Advanced Calculus; McGraw Hill: New York, NY, USA, 1965. [Google Scholar]
- Abramowitz, M.; Stegun, I. Handbook of Mathematical Functions; Dover, 1965; Available online: https://www.math.hkbu.edu.hk/support/aands/intro.htm (accessed on 4 July 2021).
- Wightman, A.S. Progress in the foundations of quantum field theory. In Proceedings of the 1967 International Conference on Particles and Fields; Hagen, C.R., Guralnik, G., Mathur, V.A., Eds.; Wiley-Interscience: Hoboken, NJ, USA, 1967. [Google Scholar]
- Candelpergher, B. Ramanujan summation of divergent Series. In Lecture Notes in Mathematics 2185; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Hochstadt, H. The Functions of Mathematical Physics; John Wiley: Hoboken, NJ, USA, 1971. [Google Scholar]
- Itzykson, C.; Zuber, J.B. Quantum Field Theory; McGraw-Hill Book Co.: New York, NY, USA, 1980. [Google Scholar]
- Milonni, P.W. The Quantum Vacuum: An Introduction to Quantum Electrodynamics; Academic Press: Cambridge, MA, USA, 1994. [Google Scholar]
- Milton, K.A. The Casimir Effect: Physical Manifestations of Zero Point Energy; World Scientific: Singapore, 2001. [Google Scholar]
- Plunien, G.; Muller, B.; Greiner, W. The Casimir effect. Phys. Rep.
**1986**, 134, 87. [Google Scholar] [CrossRef] - Bordag, M.; Klimchitskaya, G.L.; Mohideen, U.; Mostepanenko, V.M. Advances in the Casimir Effect; Oxford Science Publications: Oxford, UK, 2009. [Google Scholar]
- Farina, C. The Casimir effect: Some aspects. Braz. J. Phys.
**2006**, 36, 1137. [Google Scholar] [CrossRef] [Green Version] - Martin, P.A.; Buenzli, P.R. The Casimir effect. Acta Phys. Pol.
**2006**, 37, 2503. [Google Scholar] - Spohn, H. Dynamics of Charged Particles and Their Radiation Field; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Jaffe, A.M. Nine Lessons of My Teacher Arthur Strong Wightman. A Talk at Princeton University 10-3-2013. IAMP News Bulletin. April 2013. Available online: https://www.arthurjaffe.com/Assets/pdf/Arthur%20Wightman%2010%20March%202013.pdf (accessed on 4 July 2021).
- Lieb, E.H. Some problems in statistical mechanics that I would like to see solved. Physica A
**1999**, 263, 491. [Google Scholar] [CrossRef] - Lieb, E.H.; Loss, M. Stability of a model of relativistic quantum electrodynamics. Comm. Math. Phys.
**2002**, 228, 561. [Google Scholar] [CrossRef] [Green Version] - Jaekel, C.D.; Wreszinski, W.F. Stability of relativistic quantum electrodynamics in the Coulomb gauge. J. Math. Phys.
**2018**, 59, 032303. [Google Scholar] [CrossRef] [Green Version] - Bohr, N.; Rosenfeld, L. Zur Frage der Messbarkeit der elektromagnetischen Feldgroessern. Kgl. Danske Vidensk. Selsk. Mat. Fys. Med.
**1933**, 12, 8. [Google Scholar] - Wightman, A.S. Quantum field theory in terms of vacuum expectation values. Phys. Rev.
**1956**, 101, 860. [Google Scholar] [CrossRef] - Scharf, G.; Wreszinski, W.F. On the Casimir effect without cutoff. Found. Phys. Lett.
**1992**, 5, 479. [Google Scholar] [CrossRef] - Brown, L.S.; Maclay, G.J. Vacuum stress between conducting plates: An image solution. Phys. Rev.
**1969**, 184, 1272. [Google Scholar] [CrossRef] - Niekerken, O. Quantentheoretische und Klassische Vakuum—Kraefte bei Temperatur Null und bei endlicher Temperatur; Diplomarbeit Hamburg Februar. 2009. Available online: https://www.osti.gov/etdeweb/biblio/21196771 (accessed on 4 July 2021).
- Schwartz, L. Mathematics for the Physical Sciences; Dover Publ. Inc.: Mineola, NY, USA, 2008. [Google Scholar]
- Blanchard, P.; Bruening, E. Mathematical Methods in Physics; Birkhaeuser: Boston, MA, USA, 2003. [Google Scholar]
- Barton, G. An Introduction to Advanced Field Theory; Interscience: Hoboken, NJ, USA, 1963. [Google Scholar]
- Weinberg, S. The Quantum Theory of Fields Volume 2; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Steinmann, O. Perturbation Expansions in Axiomatic Field Theory. Lect. Notes. Phys.
**1971**, 11. [Google Scholar] [CrossRef] - Wightman, A.S. Introduction to some aspects of quantized fields. In High Energy Electromagnetic Interactions and Field Theory; Lévy, M., Ed.; Gordon and Breach: Philadelphia, PA, USA, 1967. [Google Scholar]
- Haag, R. Local Quantum Physics, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- Wreszinski, W.F. Equal time limits of anticommutators in relativistic theories. Nuovo. Cim. A
**1971**, 1, 691–705. [Google Scholar] [CrossRef] - Jaekel, C.D.; Mund, J. Canonical interacting quantum fields in two-dimensional de Sitter space. Phys. Lett. B
**2017**, 772, 786–790. [Google Scholar] [CrossRef] - Strocchi, F.; Wightman, A.S. Proof of the charge superselection rule in local relativistic quantum field theory. J. Math. Phys.
**1974**, 15, 2198. [Google Scholar] [CrossRef] - Lehmann, H. Über Eigenschaften von Ausbreitungsfunktionen und Renormierunskonstanten quantisierter Felder. Nuovo Cimento
**1954**, 11, 342. [Google Scholar] [CrossRef] - Lowenstein, J.H.; Swieca, J.A. Quantum electrodynamics in two dimensions. Ann. Phys.
**1971**, 68, 172–195. [Google Scholar] [CrossRef] - Steinmann, O. Perturbative QED in terms of gauge invariant fields. Ann. Phys.
**1984**, 157, 232–254. [Google Scholar] [CrossRef] - Houard, J.; Jouvet, B. Étude d’un modèle de champs quantifiés à constante de renormalisation nulle. Nuovo Cim.
**1960**, 18, 466. [Google Scholar] [CrossRef] - Araki, H.; Munakata, Y.; Kawaguchi, M.; Goto, T. Quantum field theory of unstable particles. Progr. Theor. Phys.
**1957**, 17, 419. [Google Scholar] [CrossRef] [Green Version] - Landsman, N.P. Non-shell unstable particles in thermal field theory. Ann. Phys.
**1988**, 186, 141–205. [Google Scholar] [CrossRef] - Martin, P.; Rothen, F. Many Body Problems and Quantum Field Theory, 2nd ed.; An Introduction; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Ynduráin, F.J. Definition of the Hamiltonian in a simple field-theoretical model. J. Math. Phys.
**1966**, 7, 1133. [Google Scholar] [CrossRef] - Sewell, G.L. Quantum Theory of Collective Phenomena; Oxford University Press: Oxford, UK, 1986. [Google Scholar]
- Casher, A.; Kogut, J.; Susskind, L. Vacuum polarization and the quark-parton puzzle. Phys. Rev. Lett.
**1973**, 31, 792. [Google Scholar] [CrossRef] - Swieca, J.A. Solitons and confinement. Fortschr. Phys.
**1977**, 25, 303. [Google Scholar] [CrossRef] - Buchholz, D.; Fredenhagen, K. From path integrals to dynamical algebras: A macroscopic view of quantum physics. Found. Phys.
**2020**, 50, 727. [Google Scholar] [CrossRef] - Alicki, R.; Fannes, M.; Verbeure, A. Unstable particles and the Poincaré semigroup in quantum field theory. J. Phys. A Math. Gen.
**1986**, 19, 919. [Google Scholar] [CrossRef] - Sewell, G.L. Quantum Mechanics and Its Emergent Macrophysics; Princeton University Press: Princeton, NJ, USA, 2002. [Google Scholar]
- Mund, J.; Rehren, K.H.; Schroer, B. Gauss’ law and string-localised quantum field theory. J. High Energy Phys.
**2020**, 1, 1. [Google Scholar] [CrossRef] [Green Version] - Narnhofer, H.; Peter, I.; Hirring, W.T. How hot is de Sitter space? Int. J. Mod. Phys. B
**1996**, 10, 1507. [Google Scholar] [CrossRef] [Green Version] - Bahns, D.; Fredenhagen, K.; Rejzner, K. Local nets of von Neumann algebras in the Sine-gordon model. Commun. Math. Phys.
**2021**, 383, 1975. [Google Scholar] [CrossRef] - Alazzawi, S.; Lechner, G. Inverse scattering and locality in integrable field theories. arXiv
**2016**, arXiv:1608.02359. [Google Scholar] - Fröhlich, J.; Seiler, E. The massive Thirring-Schwinger model: Convergence of perturbation theory and particle structure. Helv. Phys. Acta
**1976**, 49, 889. [Google Scholar] - Wightman, A.S.; Schweber, S. Configuration-space methods in relativistic quantum field theory. Phys. Rev.
**1955**, 98, 812. [Google Scholar] [CrossRef] - Fröhlich, H. Interactions of electrons with lattice vibrations. Proc. R. Soc. A
**1952**, 215, 291. [Google Scholar]

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Wreszinski, W.F.
Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories. *Universe* **2021**, *7*, 229.
https://doi.org/10.3390/universe7070229

**AMA Style**

Wreszinski WF.
Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories. *Universe*. 2021; 7(7):229.
https://doi.org/10.3390/universe7070229

**Chicago/Turabian Style**

Wreszinski, Walter Felipe.
2021. "Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories" *Universe* 7, no. 7: 229.
https://doi.org/10.3390/universe7070229