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An Algebraic Approach to Quantum Field Theory
Abstract
It is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors ℌ1 and ℌ2 are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equivalent or not. This allows a purely algebraic formulation of the theory. The framework of an algebraic version of quantum field theory is discussed and compared to the customary operator approach. It is pointed out that one reason (and possibly the only one) for the existence of unitarily inequivalent faithful, irreducible representations in quantum field theory is the (physically irrelevant) behavior of the states with respect to observations made infinitely far away. The separation between such ``global'' features and the local ones is studied. An application of this point of view to superselection rules shows that, for example, in electrodynamics the Hilbert space of states with charge zero carries already all the relevant physical information.
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Cites
Mathematical Foundations of Quantum Mechanics
von Neumann, J. (2018) · Princeton University Press eBooks · doi:10.1515/9781400889921 · Cited by 5,072
General Theory of Banach Algebras.
Batho, E. H., & Rickart, C. E. (1961) · American Mathematical Monthly · doi:10.2307/2312028 · Cited by 1,178
Quantum Field Theories with Composite Particles and Asymptotic Conditions
Haag, R. (1958) · Physical Review · doi:10.1103/physrev.112.669 · Cited by 525
Self-citationQuantum Field Theory in Terms of Vacuum Expectation Values
Wightman, A. S. (1956) · Physical Review · doi:10.1103/physrev.101.860 · Cited by 473
Postulates for General Quantum Mechanics
Segal, I. E. (1947) · Annals of Mathematics · doi:10.2307/1969387 · Cited by 426
Irreducible representations of operator algebras
Segal, I. E. (1947) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1947-08742-5 · Cited by 381
The dual spaces of 𝐶*-algebras
Fell, J. M. G. (1960) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1960-0146681-0 · Cited by 233
Distributions in Hilbert Space and Canonical Systems of Operators
Segal, I. E. (1958) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1958-0102759-x · Cited by 213
Postulates of Quantum Field Theory
Haag, R., & Schroer, B. (1962) · Journal of Mathematical Physics · doi:10.1063/1.1703797 · Cited by 199
Self-citationLes difficultés de divergences pour un modèle particulier de champ quantifié
Hove, L. V. (1952) · Physica · doi:10.1016/s0031-8914(52)80017-5 · Cited by 198
Research works printed but not linked: 9
- 1962, Helv. Phys. Acta, 35, 147
- 1938, Comp. Math., 6, 1
- 1955, Kgl. Danske Videnskab. Selskab Mat.-Fiz. Medd., 29
- 1962, Illinois J. Math., 6, 500
- 1960, Trans. Am. Math. Soc., 94, 365
- 1963, Ann. Physik, 11, 29
- 1943, Mat. Sb., 12, 197
- 1948, Isvertija Ser. Mat., 12, 445
- 1954, Tôhoku Mat. J., 6, 212
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Self-citation
Metadata
- Authors
- Haag, R., & Kastler, D.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 1964-07-01
- Volume
- 5
- Issue
- 7
- Pages
- 848-861
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W1994275197
- Cited by
- 1,250
- References
- 18
- Citations by year
- 2026: 41 · 2025: 26 · 2024: 31 · 2023: 28 · 2022: 26 · 2021: 33 · 2020: 28 · 2019: 63 · 2018: 21 · 2017: 28 · 2016: 28 · 2015: 28 · 2014: 22 · 2013: 10 · 2012: 42
- FWCI
- 16.95
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Mathematical Physics (publishedVersion)
- Funders
- National Science Foundation
- Argonne National Laboratory
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Keywords
- algebraic quantum field theory, faithful representation, unitarily inequivalent representations, irreducible representations, global and local features, superselection rules, electrodynamics, operator approach
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