×

The divergence of Van Hove’s model and its consequences. (English) Zbl 1483.81095

Summary: We study a regularized version of Van Hove’s 1952 model, in which a quantum field interacts linearly with sources of finite width lying at fixed positions. We show that the central result of Van Hove’s 1952 paper on the foundations of Quantum Field Theory, the orthogonality between the spaces of state vectors which correspond to different values of the parameters of the theory, disappears when a well-defined model is considered. We comment on the implications of our results for the contemporary relevance of Van Hove’s article.

MSC:

81T05 Axiomatic quantum field theory; operator algebras
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A40 Scattering theory of linear operators
81P05 General and philosophical questions in quantum theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Stone, MH, Linear transformations in Hilbert space. Operational methods and group theory, Proc. Nat. Acad. Sci. U.S.A., 16, 172 (1930) · JFM 56.0357.01 · doi:10.1073/pnas.16.2.172
[2] Von Neumann, J., Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann., 104, 570 (1931) · Zbl 0001.24703 · doi:10.1007/BF01457956
[3] Von Neumann, J., On infinite direct products, Compositio Math., 6, 1 (1939) · Zbl 0019.31103
[4] Van Hove, L., Les Difficultés de Divergences pour un Modelle Particulier de Champ Quantifié, Physica, 18, 145 (1952) · Zbl 0046.21307 · doi:10.1016/S0031-8914(52)80017-5
[5] Friedrichs, KO, Mathematical aspects of the quantum theory of fields part IV, Comm. Pure Appl. Math., 5, 4, 349 (1952) · Zbl 0049.27407 · doi:10.1002/cpa.3160050401
[6] Friedrichs, KO, Mathematical aspects of the quantum theory of fields (1953), New York: Interscience Publishers Inc, New York · Zbl 0052.44504
[7] Wightman, AS; Schweber, SS, Configuration space methods in relativistic quantum field theory I, Phys. Rev., 98, 812 (1955) · Zbl 0068.22602 · doi:10.1103/PhysRev.98.812
[8] Lupher, T., Who proved Haag’s theorem?, Int. J. Theor. Phys., 44, 1995 (2005) · Zbl 1095.81044 · doi:10.1007/s10773-005-8977-z
[9] Earman, J.; Fraser, D., Haag’s theorem and its implications for the foundations of Quantum Field Theory, Erkenntnis, 64, 305 (2006) · Zbl 1107.81004 · doi:10.1007/s10670-005-5814-y
[10] Haag, R., On quantum field theories, Dan. Mat. Fys. Medd., 29, 1 (1955) · Zbl 0067.21102
[11] Hall, D.; Wightman, AS, A theorem on invariant analytic functions with applications to relativistic quantum field theory, Mat. Fys. Medd. Dan. Vid. Selsk., 31, 1 (1957) · Zbl 0078.44302
[12] Sbisà, F., On Léon Van Hove’s 1952 article on the foundations of Quantum Field Theory, Rev. Bras. Ens. Fis., 42, e20200256 (2020) · doi:10.1590/1806-9126-rbef-2020-0256
[13] Streater, RF; Wightman, AS, PCT, Spin and Statistics, and All That (1964), W.A: Benjamin Inc, W.A · Zbl 0135.44305
[14] Reed, M.; Simon, B., Methods of Modern Mathematical Physics III: Scattering Theory (1979), USA: Academic Press, Inc., USA · Zbl 0405.47007
[15] Haag, R., Local Quantum Physics (1996), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg · Zbl 0857.46057 · doi:10.1007/978-3-642-61458-3
[16] Thiemann, T.; Winkler, O., Gauge field theory coherent States (GCS): IV. Infinite tensor product and thermodynamical limit, Class. Quant. Grav., 18, 4997 (2001) · Zbl 1018.83008 · doi:10.1088/0264-9381/18/23/302
[17] Loinger, A., Un semplice modello di due campi interagenti, Il Nuovo Cimento IX, 11, 1080 (1952) · Zbl 0049.27501 · doi:10.1007/BF02777590
[18] Loinger, A., Erratum to Un semplice modello di due campi interagenti, Il Nuovo Cimento X, 10, 3, 356 (1953) · Zbl 0049.27501 · doi:10.1007/BF02786209
[19] Reed, M.; Simon, B., Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (1975), USA: Academic Press Inc, USA · Zbl 0308.47002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.