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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Holevo introduced such channels and showed that they suffice to demonstrate the quantum advantage, although both minimal entropy and capacity are achieved with orthogonal inputs.
“Holevo [7,8] introduced such channels and showed that they suffice to demonstrate the”
From An analysis of completely-positive trace-preserving maps on M2 (2002) · cites this paper as [7]
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- 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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