The DOI matches its Crossref record; 18 of the 18 references it cites were found in a registry or an index.
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.
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“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]
Citations
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
Flagged first, then by citations
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
Cited by
Sorted by citations
Items 301–400 of 1,000
A priori probability and localized observers
Donald, M. J. (1992) · Foundations of Physics · doi:10.1007/bf00732696 · Cited by 26
On a new definition of quantal states
Misra, B. (1974) · Springer eBooks · doi:10.1007/978-94-010-2274-3_27 · Cited by 26
Was Von Neumann Right After All?
(2016) · BENTHAM SCIENCE PUBLISHERS eBooks · doi:10.2174/9781681081793116010013 · Cited by 25
Phenomenon of mobility in non-linear theories
Mielnik, B. (1985) · Communications in Mathematical Physics · doi:10.1007/bf01216093 · Cited by 25
Emch, G. G. (2007) · Elsevier eBooks · doi:10.1016/b978-044451560-5/50013-0 · Cited by 25
Indefinite causal structures for continuous-variable systems
Giacomini, F., Castro-Ruiz, E., & Brukner, Č. (2016) · New Journal of Physics · doi:10.1088/1367-2630/18/11/113026 · Cited by 29
Operads for algebraic quantum field theory
Benini, M., Schenkel, A., & Woike, L. (2020) · Communications in Contemporary Mathematics · doi:10.1142/s0219199720500078 · Cited by 25
The consistent histories interpretation of quantum fields in curved spacetime
Blencowe, M. P. (1991) · Annals of Physics · doi:10.1016/0003-4916(91)90193-c · Cited by 23
Braid Statistics in Three-Dimensional Local Quantum Theory
Fröhlich, J. M., Gabbiani, F., & Marchetti, P. A. (1990) · NATO ASI series. Series B : Physics · doi:10.1007/978-1-4615-3802-8_2 · Cited by 25
A theory of causality: Causality=interaction (as defined by a suitable quantum field theory)
Heathcote, A. (1989) · Erkenntnis · doi:10.1007/bf01239130 · Cited by 25
No-Go Theorem for ‘Free’ Relativistic Anyons in d=2+1
Mund, J. (1998) · Letters in Mathematical Physics · doi:10.1023/a:1007429606788 · Cited by 24
The charged sectors of Quantum Electrodynamics in a framework of local observables
Fröhlich, J. M. (1979) · Communications in Mathematical Physics · doi:10.1007/bf01197187 · Cited by 25
C*-algebras and Mackey's axioms
Plymen, R. (1968) · Communications in Mathematical Physics · doi:10.1007/bf01645801 · Cited by 25
Eliminating the ‘Impossible’: Recent Progress on Local Measurement Theory for Quantum Field Theory
Papageorgiou, M., & Fraser, D. (2024) · Foundations of Physics · doi:10.1007/s10701-024-00756-8 · Cited by 24
Models of Free Quantum Field Theories on Curved Backgrounds
Benini, M., & Dappiaggi, C. (2015) · Mathematical physics studies · doi:10.1007/978-3-319-21353-8_3 · Cited by 24
Bagarello, F., & Sewell, G. L. (1998) · Journal of Mathematical Physics · doi:10.1063/1.532417 · Cited by 24
Fermion current algebras and Schwinger terms in (3+1)-dimensions
Edwin Langmann, A. (1994) · Communications in Mathematical Physics · doi:10.1007/bf02105184 · Cited by 24
Intrinsic algebraic characterization of space-time structure
Bannier, U. (1994) · International Journal of Theoretical Physics · doi:10.1007/bf00671024 · Cited by 24
On the implementability of automorphism groups
Borchers, H. J. (1969) · Communications in Mathematical Physics · doi:10.1007/bf01645386 · Cited by 24
Some remarks on effects, operations, and unsharp measurements
Busch, P., Grabowski, M., & LAHTI, P. J. (1989) · Foundations of Physics Letters · doi:10.1007/bf00690299 · Cited by 23
A new approach to spin & statistics
Kuckert, B. (1995) · Letters in Mathematical Physics · doi:10.1007/bf00750839 · Cited by 24
Axiomatic Foundations of Quantum Theories
Guenin, M. (1966) · Journal of Mathematical Physics · doi:10.1063/1.1704929 · Cited by 24
Quantum-Classical Hybrid Systems and their Quasifree Transformations
Dammeier, L., & Werner, R. F. (2023) · Quantum · doi:10.22331/q-2023-07-26-1068 · Cited by 23
C *-algebraic quantization and the origin of topological quantum effects
Landsman, N. P. (1990) · Letters in Mathematical Physics · doi:10.1007/bf00417225 · Cited by 23
Inequivalent coherent state representations in Group Field Theory
Kegeles, A., Oriti, D., & Tomlin, C. (2018) · MPG.PuRe (Max Planck Society) · Cited by 23
Physical uniformities on the state space of nonrelativisitic quantum mechanics
Werner, R. F. (1983) · Foundations of Physics · doi:10.1007/bf01906273 · Cited by 23
On the question of temperature transformations under Lorentz and Galilei boosts
Sewell, G. L. (2008) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/41/38/382003 · Cited by 23
Global Observables in Local Quantum Physics
Fredenhagen, K. (1993) · Springer eBooks · doi:10.1007/978-94-017-2823-2_4 · Cited by 24
Quantum stochastic processes I
Lewis, J. T. (1981) · Physics Reports · doi:10.1016/0370-1573(81)90081-8 · Cited by 25
Edwards, C. M. (1972) · Communications in Mathematical Physics · doi:10.1007/bf01878476 · Cited by 23
Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories
Müger, M. (1998) · Communications in Mathematical Physics · doi:10.1007/s002200050264 · Cited by 27
Ergodic states in a non commutative ergodic theory
Doplicher, S., & Kastler, D. (1968) · Communications in Mathematical Physics · doi:10.1007/bf01651215 · Cited by 23
Self-citationInfinity Operads and Monoidal Categories with Group Equivariance
Yau, D. (2021) · WORLD SCIENTIFIC eBooks · doi:10.1142/12687 · Cited by 22
The locality axiom in quantum field theory and tensor products of C*-algebras
Brunetti, R., Fredenhagen, K., Imani, P., & Rejzner, K. (2014) · Reviews in Mathematical Physics · doi:10.1142/s0129055x1450010x · Cited by 22
A New Light on Nets of C*-Algebras and Their Representations
Ruzzi, G., & Vasselli, E. (2012) · Communications in Mathematical Physics · doi:10.1007/s00220-012-1490-3 · Cited by 22
Approximate ternary quadratic derivations on ternary Banach algebras and C*-ternary rings
Bodaghi, A., & Alias, I. A. (2012) · Advances in Difference Equations · doi:10.1186/1687-1847-2012-11 · Cited by 22
Differential cohomology and locally covariant quantum field theory
Becker, C., Schenkel, A., & Szabo, R. J. (2016) · Reviews in Mathematical Physics · doi:10.1142/s0129055x17500039 · Cited by 22
State-dressed local operators in the AdS / CFT correspondence
Bahiru, E., Belin, A., Papadodimas, K., Sárosi, G., & Vardian, N. (2023) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.108.086035 · Cited by 27
The Hilbert Space Axiom in Quantum Mechanics
Gudder, S. (1983) · Springer eBooks · doi:10.1007/978-1-4684-8830-2_9 · Cited by 22
Semibounded representations and invariant cones in infinite dimensional lie algebras
Neeb, K. (2010) · Confluentes Mathematici · doi:10.1142/s1793744210000132 · Cited by 22
Remarks on the GNS Representation and the Geometry of Quantum States
Chruściński, D., & Marmo, G. (2009) · Open Systems & Information Dynamics · doi:10.1142/s1230161209000128 · Cited by 22
W *-dynamics of infinite quantum systems
Sewell, G. L. (1982) · Letters in Mathematical Physics · doi:10.1007/bf00403273 · Cited by 22
Observables in W*‐Algebraic Quantum Mechanics
Amann, A. (1986) · Fortschritte der Physik · doi:10.1002/prop.19860340402 · Cited by 24
Higher Structures in Algebraic Quantum Field Theory
Benini, M., & Schenkel, A. (2019) · Fortschritte der Physik · doi:10.1002/prop.201910015 · Cited by 31
Quantum generalization of Kolmogorov entropy
Srinivas, M. D. (1978) · Journal of Mathematical Physics · doi:10.1063/1.523916 · Cited by 19
Transformations of a wave packet in a Penning trap
C., D. J. F. (1992) · Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B · doi:10.1007/bf02899290 · Cited by 21
Equilibrium states of a class of quantum mean-field theories
Bóna, P. (1989) · Journal of Mathematical Physics · doi:10.1063/1.528489 · Cited by 21
Construction of Field Algebras with Quantum Symmetry from Local Observables
Schomerus, V. (1994) · Cited by 21
Zum kausalen Verhalten eines relativistischen quantenmechanischen Systems
Schlieder, S. (1971) · Vieweg+Teubner eBooks · doi:10.1007/978-3-322-83700-4_11 · Cited by 22
Characterizing invariants for local extensions of current algebras
Rehren, K.-H., Stanev, Y. S., & Todorov, I. T. (1996) · Communications in Mathematical Physics · doi:10.1007/bf02101529 · Cited by 21
The energy-momentum spectrum in local field theories with broken Lorentz-symmetry
Borchers, H. J., & Detlev Buchholz, A. (1985) · Communications in Mathematical Physics · doi:10.1007/bf01206185 · Cited by 21
Classical Representations of Quantum Mechanics Related to Statistically Complete ObservablesPreprint
Stulpe, W. (2006) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0610122 · Cited by 23
An Analysis of Completely-Positive Trace-Preserving Maps on 2x2 MatricesPreprint
Ruskai, M. B., Szarek, S. J., & Werner, E. M. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0101003 · Cited by 21
Compactly supported linearised observables in single-field inflation
Fröob, M. B., Hack, T.-P., & Higuchi, A. (2017) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2017/07/043 · Cited by 20
Inequivalent coherent state representations in group field theory
Kegeles, A., Oriti, D., & Tomlin, C. (2018) · Classical and Quantum Gravity · doi:10.1088/1361-6382/aac39f · Cited by 29
Locally convex quasi ⁎-algebras with sufficiently many ⁎-representations
Fragoulopoulou, M., Trapanı, C., & Triolo, S. (2011) · Journal of Mathematical Analysis and Applications · doi:10.1016/j.jmaa.2011.11.013 · Cited by 20
Can locally Hadamard quantum states have non-local singularities?
Gonnella, G., & Kay, B. S. (1989) · Classical and Quantum Gravity · doi:10.1088/0264-9381/6/10/013 · Cited by 20
Some Aspects of Quantum Entanglement for CAR Systems
Moriya, H. (2002) · Letters in Mathematical Physics · doi:10.1023/a:1016158125660 · Cited by 20
An operational analysis of the space-time structure
Toller, M. (1977) · Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B · doi:10.1007/bf02739179 · Cited by 20
Level 1 WZW superselection sectors
Fuchs, J., Ganchev, A., & Vecsernyés, P. (1992) · Communications in Mathematical Physics · doi:10.1007/bf02097016 · Cited by 20
Dynamical semigroups in the algebraic formulation of statistical mechanics
Majewski, W. A. (1984) · Fortschritte der Physik · doi:10.1002/prop.19840320302 · Cited by 20
Logically independent von Neumann lattices
R�dei, M. (1995) · International Journal of Theoretical Physics · doi:10.1007/bf00676284 · Cited by 20
?Almost? mean-field ising model: An algebraic approach
Bagarello, F., & Trapanı, C. (1991) · Journal of Statistical Physics · doi:10.1007/bf01053740 · Cited by 20
Particle localization in field theory
Steinmann, O. (1968) · Communications in Mathematical Physics · doi:10.1007/bf01648330 · Cited by 20
Integral and differential structures for quantum field theoryPreprint
Labuschagne, L. E., & Majewski, W. A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1702.00665 · Cited by 20
Wavefront sets in algebraic quantum field theoryPreprint
Verch, R. (1999) · Cited by 20
The star product in interacting quantum field theory
Hawkins, E., & Rejzner, K. (2020) · Letters in Mathematical Physics · doi:10.1007/s11005-020-01262-4 · Cited by 19
Some people and some problems met in half a century of commitment to mathematical physics
Haag, R. (2010) · The European Physical Journal H · doi:10.1140/epjh/e2010-10032-4 · Cited by 21
Self-citationA Philosophical Approach to Quantum Field Theory
Öttinger, H. C. (2017) · Cambridge University Press eBooks · doi:10.1017/9781108227667 · Cited by 26
Quantum field theory in curved space-times a general mathematical framework
Isham, C. J. (1978) · Lecture notes in mathematics · doi:10.1007/bfb0063686 · Cited by 20
The quantum theory of free automorphic fields
Banach, R. (1980) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/13/6/039 · Cited by 19
Exponentiating derivations of quasi ∗‐algebras: possible approaches and applications
Bagarello, F., Inoue, A., & Trapanı, C. (2005) · International Journal of Mathematics and Mathematical Sciences · doi:10.1155/ijmms.2005.2805 · Cited by 19
Vector states on algebras of observables and superselection rules
Sushko, V. N., & Khoruzhii, S. S. (1970) · Theoretical and Mathematical Physics · doi:10.1007/bf01038302 · Cited by 19
The Schwinger model: A view from the temporal gauge
Becher, P. (1983) · Annals of Physics · doi:10.1016/0003-4916(83)90033-7 · Cited by 18
The dynamical entropy of the quantum Arnold cat map
Andries, J., Fannes, M., Tuyls, P., & Alicki, R. (1995) · Letters in Mathematical Physics · doi:10.1007/bf00750844 · Cited by 19
C*-Algebra Approach in Quantum Field Theory
Araki, H. (1981) · Physica Scripta · doi:10.1088/0031-8949/24/5/013 · Cited by 19
Primas, H., & Müller‐Herold, U. (1978) · Advances in chemical physics · doi:10.1002/9780470142578.ch1 · Cited by 19
Wightman, A. S. (1976) · Springer eBooks · doi:10.1007/978-94-010-1490-8_1 · Cited by 18
Kudler-Flam, J., Leutheusser, S., Rahman, A. A., Satishchandran, G., & Speranza, A. J. (2025) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.111.105001 · Cited by 38
Semantics for a Quantum Programming Language by Operator Algebras
Cho, K. (2014) · Electronic Proceedings in Theoretical Computer Science · doi:10.4204/eptcs.172.12 · Cited by 18
Foundations for Quantum Mechanics
Giles, R. (1979) · Springer eBooks · doi:10.1007/978-94-009-9351-8_13 · Cited by 18
Linear Yang–Mills Theory as a Homotopy AQFT
Benini, M., Bruinsma, S., & Schenkel, A. (2019) · Communications in Mathematical Physics · doi:10.1007/s00220-019-03640-z · Cited by 18
A Universe of Processes and Some of Its Guises
Coecke, B. (2011) · Cambridge University Press eBooks · doi:10.1017/cbo9780511976971.004 · Cited by 25
Algebraic classical and quantum field theory on causal sets
Dable-Heath, E., Fewster, C. J., Rejzner, K., & Woods, N. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.101.065013 · Cited by 18
Infrared Problem, Higgs Phenomenon and Long Range Interactions
Morchio, G., & Strocchi, F. (1986) · NATO ASI series. Series B : Physics · doi:10.1007/978-1-4757-0363-4_7 · Cited by 18
Canonical quantization of gauge invariant field theories
Kundt, W. (1966) · Springer tracts in modern physics · doi:10.1007/bfb0045895 · Cited by 19
Maxwell meets Reeh–Schlieder: The quantum mechanics of neutral bosons
Hawton, M., & Debierre, V. (2017) · Physics Letters A · doi:10.1016/j.physleta.2017.04.004 · Cited by 17
Time and the Foundations of Quantum Mechanics
Pashby, T. (2014) · D-Scholarship@Pitt (University of Pittsburgh) · Cited by 19
Measurements and mathematical formalism of quantum mechanics
Slavnov, D. A. (2007) · Physics of Particles and Nuclei · doi:10.1134/s1063779607020013 · Cited by 17
The Quantum Sine-Gordon model in perturbative AQFT
Bahns, D., & Rejzner, K. A. (2018) · White Rose Research Online (University of Leeds, The University of Sheffield, University of York) · Cited by 17
Soliton mass and surface tension in the (λ|Ø|4)2 quantum field model
Bellissard, J., Fröhlich, J. M., & Gidas, B. (1978) · Communications in Mathematical Physics · doi:10.1007/bf01609473 · Cited by 17
The superposition of the states and the logic approach to quantum mechanics
Zecca, A. (1981) · International Journal of Theoretical Physics · doi:10.1007/bf00669795 · Cited by 17
Quantum soliton operators for vortices and superselection rules
Ezawa, Z. F. (1978) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.18.2091 · Cited by 17
Schrodinger revisited: an algebraic approachPreprint
Brown, M. R., & Hiley, B. J. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0005026 · Cited by 17
A Class of Almost Equilibrium States in Robertson-Walker SpacetimesPreprint
Küskü, M. (2008) · arXiv (Cornell University) · doi:10.3204/desy-thesis-2008-020 · Cited by 17
Gill, T. L., & Zachary, W. W. (2011) · Afrika Matematika · doi:10.1007/s13370-011-0003-4 · Cited by 16
Dynamical Locality of the Free Maxwell Field
Fewster, C. J., & Lang, B. (2015) · Annales Henri Poincaré · doi:10.1007/s00023-015-0398-9 · Cited by 18
On the quantum logic approach to quantum mechanics
Gudder, S. (1969) · Communications in Mathematical Physics · doi:10.1007/bf01646431 · Cited by 16
The foundations of Quantum Field Theory
Glimm, J. G. (1969) · Advances in Mathematics · doi:10.1016/0001-8708(69)90004-8 · Cited by 16
Köster, S. (2004) · Reviews in Mathematical Physics · doi:10.1142/s0129055x0400200x · Cited by 16
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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
