## An Algebraic Spin And Statistics Theorem. I (1994)

Citations: | 42 - 11 self |

### BibTeX

@MISC{Guido94analgebraic,

author = {Daniele Guido and Roberto Longo},

title = {An Algebraic Spin And Statistics Theorem. I},

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional space-time. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebras generated by Wightman fields because of the Bisognano-Wichmann theorem, is regarded as a natural primitive assumption.

### Citations

337 | Local Quantum - Haag - 1996 |

308 |
Particle creation by black holes
- Hawking
(Show Context)
Citation Context ... modular covariance property by first principles. On the physical side modular covariance manifests an interesting analogy with the Unruh effect [45] and with the Hawking black hole thermal radiation =-=[26]-=-, as first noticed by Sewell [40]. We sketch the essential ideas, see also [24]. As is known the modular group of a von Neumann algebra with respect to a given state is characterized by the Kubo-Marti... |

271 |
Local Quantum Physics
- Haag
(Show Context)
Citation Context ...covariance manifests an interesting analogy with the Unruh effect [45] and with the Hawking black hole thermal radiation [26], as first noticed by Sewell [40]. We sketch the essential ideas, see also =-=[24]-=-. As is known the modular group of a von Neumann algebra with respect to a given state is characterized by the Kubo-Martin-Schwinger condition [44] and, on the other hand, the KMS condition is peculia... |

136 |
1970] Tomita’s Theory of Modular Hilbert Algebras and its
- Takesaki
(Show Context)
Citation Context ...yclic for the algebras A(W) associated with all wedge region W in the Minkowski space, such that ∆ it W A(O)∆−it W = A(ΛW(t)O) , t ∈ R where O is any region, ∆ is the Tomita-Takesaki modular operator =-=[44,41]-=- associated with (A(W), Ω) and ΛW is the one-parameter rescaled group of pure Lorentz transformations preserving W. Reeh-Schlieder property. The vacuum vector Ω ∈ H is also cyclic for the algebras A(S... |

129 | PCT, spin and statistics, and all that - Streater, Wightman - 2000 |

122 |
An algebraic approach to quantum field theory
- Haag, Kastler
- 1964
(Show Context)
Citation Context ...e also [36,14], and a PCT theorem by Jost [29], see also [19, 39], both relying on the general holomorphic properties of the n-point functions. A spin and statistics theorem in the algebraic approach =-=[27]-=-, see also [5], was later given by Epstein [21] and has a version for (Doplicher-Haag-Roberts) DHR superselection sectors [16] and for more general topological charges [10]. All these approaches heavi... |

114 |
Index of subfactors and statistics of quantum fields
- Longo
- 1989
(Show Context)
Citation Context ...oincaré group P ↑ + (cf. also [12]). Let now ρ be a superselection sector of A in the sense of Doplicher-Haag-Roberts [15] or more generally of Buchholz-Fredenhagen [11]. An index-statistics relation =-=[34]-=- shows that Ind(ρ) = d(ρ) 2 where Ind(ρ) is the Jones index of ρ and d(ρ) is the DHR statistical dimension, namely d(ρ) = |λρ| −1 where the statistical parameter λρ ∈ R classifies the statistics in 3 ... |

111 |
Roberts Why there is a field algebra with a compact gauge group describing the superselection strucure in particle physics Commun
- Doplicher, E
- 1990
(Show Context)
Citation Context ...ose to the classical formulation. Then, in Sect. 3, we obtain our result in the context of local observables. This is done by rephrasing the statements in terms of the Doplicher-Roberts field algebra =-=[18]-=-. This last step has certain pedagogical advantages, but has to be avoided in order to extend our work to more general settings where the field algebra does not exist. In forthcoming papers [23] we sh... |

97 |
Local observables and particle statistics I
- Doplicher, Haag, et al.
- 1971
(Show Context)
Citation Context ...iance property in the net structure, providing a canonical representation of the Poincaré group P ↑ + (cf. also [12]). Let now ρ be a superselection sector of A in the sense of Doplicher-Haag-Roberts =-=[15]-=- or more generally of Buchholz-Fredenhagen [11]. An index-statistics relation [34] shows that Ind(ρ) = d(ρ) 2 where Ind(ρ) is the Jones index of ρ and d(ρ) is the DHR statistical dimension, namely d(ρ... |

95 |
The general theory of quantized fields
- Jost
- 1965
(Show Context)
Citation Context ...hall reconsider from an intrinsic point of view two well known fascinating theorems in Quantum Field Theory: the PCT theorem and the Spin and Statistics theorem. Both of them have a long history, see =-=[28,43]-=-. The spin and statistics theorem first appeared in the context of free fields in the work of Fierz [20] and Pauli [37]: one cannot second quantize particles with integer spin by anticommuting fields,... |

94 |
Notes on black hole evaporation, Phys
- Unruh
- 1976
(Show Context)
Citation Context ...opose it as a natural candidate for a derivation of the modular covariance property by first principles. On the physical side modular covariance manifests an interesting analogy with the Unruh effect =-=[45]-=- and with the Hawking black hole thermal radiation [26], as first noticed by Sewell [40]. We sketch the essential ideas, see also [24]. As is known the modular group of a von Neumann algebra with resp... |

69 |
The CPT theorem in two-dimensional theories of local observables
- Borchers
- 1992
(Show Context)
Citation Context ...ghtman fields. An algebraic version of the Bisognano-Wichmann theorem does not exist yet, except in the case of conformal theories where it holds in full generality [8]. However a theorem of Borchers =-=[6]-=- shows part of the geometric properties of the modular group for wedge regions to be always present and in particular every 1 + 1 dimensional Poincaré covariant net satisfying essential duality has th... |

69 |
On the Equilibrium States in Quantum Statistical Mechanics
- Haag, Hugenholz, et al.
- 1967
(Show Context)
Citation Context ...with respect to a given state is characterized by the Kubo-Martin-Schwinger condition [44] and, on the other hand, the KMS condition is peculiar of thermal equilibrium states in Statistical Mechanics =-=[25]-=-. By the Bisognano-Wichmann theorem the boosts satify the KMS condition with respect to the vacuum, as automorphisms of the von Neumann algebra of the corresponding wedge W and, on the other hand, the... |

57 |
Standard and split inclusions of von Neumann algebras
- Doplicher, Longo
- 1984
(Show Context)
Citation Context ...ation, while our canonical choice for the latter has the desired properties. We remark that an intrinsic way to eliminate pathological examples of the above kind comes by requiring the split property =-=[17]-=-; this indeed implies the uniqueness of the Poincaré group representation [8] and we propose it as a natural candidate for a derivation of the modular covariance property by first principles. On the p... |

56 |
Relativistic invariance and charge conjugation in quantum field theory
- Guido, Longo
- 1992
(Show Context)
Citation Context ...erefore the univalence (integer or half-integer spin alternative) is intrinsically associated with ρ. Since ρ is automatically ˜P ↑ +-covariant if d(ρ) < ∞ (assuming a regularity property for the net =-=[22]-=-), it is natural to expect a general algebraic Spin and Statistics theorem connecting these two intrinsic quantities for any sector with finite statistics. Our result in this respect will in fact show... |

54 | Modular structure and duality in conformal quantum field theory
- Brunetti, Guido, et al.
- 1993
(Show Context)
Citation Context ...al algebras are constructed from Wightman fields. An algebraic version of the Bisognano-Wichmann theorem does not exist yet, except in the case of conformal theories where it holds in full generality =-=[8]-=-. However a theorem of Borchers [6] shows part of the geometric properties of the modular group for wedge regions to be always present and in particular every 1 + 1 dimensional Poincaré covariant net ... |

54 |
Locality and the structure of particle states
- Buchholz, Fredenhagen
- 1982
(Show Context)
Citation Context ...a canonical representation of the Poincaré group P ↑ + (cf. also [12]). Let now ρ be a superselection sector of A in the sense of Doplicher-Haag-Roberts [15] or more generally of Buchholz-Fredenhagen =-=[11]-=-. An index-statistics relation [34] shows that Ind(ρ) = d(ρ) 2 where Ind(ρ) is the Jones index of ρ and d(ρ) is the DHR statistical dimension, namely d(ρ) = |λρ| −1 where the statistical parameter λρ ... |

49 |
On the duality condition for quantum fields
- Bisognano, Wichmann
- 1976
(Show Context)
Citation Context ...d Uρ(2π) denotes the corresponding rotation by 2π. Modular covariance also implies that the anti-unitary involution Θ, definable by the modular theory according to the Bisognano-Wichmann prescription =-=[3]-=-, implements a complete space-time reflection. As shown in [22], this entails that Θ intertwines a sector with its conjugate. We therefore obtain a PCT symmetry. We come now back to the origin of the ... |

48 |
Lectures on von Neumann algebras
- Strătilă, Zsidó
- 1979
(Show Context)
Citation Context ...yclic for the algebras A(W) associated with all wedge region W in the Minkowski space, such that ∆ it W A(O)∆−it W = A(ΛW(t)O) , t ∈ R where O is any region, ∆ is the Tomita-Takesaki modular operator =-=[44,41]-=- associated with (A(W), Ω) and ΛW is the one-parameter rescaled group of pure Lorentz transformations preserving W. Reeh-Schlieder property. The vacuum vector Ω ∈ H is also cyclic for the algebras A(S... |

40 |
On the duality condition for a Hermitian scalar field
- Bisognano, Wichmann
- 1975
(Show Context)
Citation Context ...r with its conjugate. We therefore obtain a PCT symmetry. We come now back to the origin of the modular covariance property. Its main justification certainly comes from the Bisognano-Wichmann theorem =-=[2,3]-=-, to the effect that this property holds if the local algebras are constructed from Wightman fields. An algebraic version of the Bisognano-Wichmann theorem does not exist yet, except in the case of co... |

39 |
The theory of quantïzed fields
- Schwinger
- 1953
(Show Context)
Citation Context ...an axiomatic approach [46] to establish model independent results: the connection between spin and statistics was obtained by Burgoyne [13], see also [36,14], and a PCT theorem by Jost [29], see also =-=[19, 39]-=-, both relying on the general holomorphic properties of the n-point functions. A spin and statistics theorem in the algebraic approach [27], see also [5], was later given by Epstein [21] and has a ver... |

33 |
Quantum Field Theory in Terms of Vacuum Expectation Values, Phys. Rev
- Wightman
- 1956
(Show Context)
Citation Context ... product of the charge and the time-inversion symmetry CT. Pauli proved in [38] that PCT is always a symmetry of Lorentz invariant field equations. It was a success of the Wightman axiomatic approach =-=[46]-=- to establish model independent results: the connection between spin and statistics was obtained by Burgoyne [13], see also [36,14], and a PCT theorem by Jost [29], see also [19, 39], both relying on ... |

29 | Group cohomology, modular theory and spacetime symmetries
- Brunetti, Guido, et al.
- 1995
(Show Context)
Citation Context ... indication that the spin-statistics relation might be different in contexts like Quantum Gravity where a sharp causality principle does not occur. 2of the Poincaré group P ↑ +, with positive energy =-=[9]-=-. Indeed we shall prove here that it is actually covariant with respect to P ↑ + as a special case of our general Spin and Statistics theorem and taking to completion our previous work. Therefore modu... |

27 |
The connection between spin and statistics
- Pauli
- 1940
(Show Context)
Citation Context ...em and the Spin and Statistics theorem. Both of them have a long history, see [28,43]. The spin and statistics theorem first appeared in the context of free fields in the work of Fierz [20] and Pauli =-=[37]-=-: one cannot second quantize particles with integer spin by anticommuting fields, i.e. fields obeying Fermi statistics, nor particles with half-integer spin by local fields, i.e. fields obeying Bose s... |

26 | An algebraic characterization of vacuum states in Minkowski space
- Buchholz, Summers
- 1993
(Show Context)
Citation Context ...s work. Therefore modular covariance is a way to intrinsically encode the Poincaré covariance property in the net structure, providing a canonical representation of the Poincaré group P ↑ + (cf. also =-=[12]-=-). Let now ρ be a superselection sector of A in the sense of Doplicher-Haag-Roberts [15] or more generally of Buchholz-Fredenhagen [11]. An index-statistics relation [34] shows that Ind(ρ) = d(ρ) 2 wh... |

19 | A Note on Essential Duality
- Yngvason
- 1994
(Show Context)
Citation Context ...nt net satisfying essential duality has the modular covariance property. At the present time no counter-example to modular covariance is known to exist within Poincaré covariant theories (see however =-=[33,48]-=-). There are nevertheless counter-examples to the spin-statistics theorem [42]: these are constructed by infinite multiplicity fields where the Poincaré group representation is not unique. It turns ou... |

14 |
Exclusion principle, Lorentz group, and reflection of space-time and charge
- Pauli
- 1955
(Show Context)
Citation Context ...PCT theorem originated in [35] as a relation between the existence of the space-inversion symmetry P and the existence of the product of the charge and the time-inversion symmetry CT. Pauli proved in =-=[38]-=- that PCT is always a symmetry of Lorentz invariant field equations. It was a success of the Wightman axiomatic approach [46] to establish model independent results: the connection between spin and st... |

13 | Local observables and particle statistics II - Doplicher, Roberts - 1974 |

13 |
Takesaki M., Extensions of KMS states and chemical potential
- Araki, Haag, et al.
- 1977
(Show Context)
Citation Context ...heorem 3.1(b). The translations along W commute with cW(t), since they commute both with U(ΛW(t))and ∆it W , and act ergodically on F(W) by the irreducibility of the net. By the Galois correspondence =-=[1]-=-, it follows that cW(·) is a one-parameter subgroup of V (G). We show now that cW(t) = I. Indeed, on the one hand, by twisted duality (Proposition 2.5) cW ′(t) = cW(−t), W ∈ W, t ∈ R. On the other han... |

12 |
A comment on a recent work of
- Wiesbrock
- 1992
(Show Context)
Citation Context ...The role of the modular covariance assumption may be also understood by its consequences. Among other things, it implies the positivity of the energy for the constructed Poincaré group representation =-=[47,9]-=-. As is known the positivity of the energy is lost on a curved space-time, and the modular covariance seems to be the appropriate substitute in this case. Moreover, as already mentioned, it gives rise... |

11 |
On the converse of the Reeh-Schlieder theorem
- Borchers
- 1968
(Show Context)
Citation Context ...eeh Schlieder property is known to hold for the vacuum vector in a Poincaré covariant theory as a consequence of the positivity of the energy and the weak additivity assumption for the local algebras =-=[7]-=-. Modular covariance needs however further comments. Postponing for a while the justification for such an assumption, we recall that this entails the net to be covariant with respect to the universal ... |

9 |
Local fields with the wrong connection between spin and statistics
- Streater
- 1967
(Show Context)
Citation Context ...resent time no counter-example to modular covariance is known to exist within Poincaré covariant theories (see however [33,48]). There are nevertheless counter-examples to the spin-statistics theorem =-=[42]-=-: these are constructed by infinite multiplicity fields where the Poincaré group representation is not unique. It turns out that the wrong connection between spin and statistics depends on the wrong c... |

6 |
Spin and statistics of quantum topological charges”, Fizika 17
- Buchholz, Epstein
- 1985
(Show Context)
Citation Context ...m in the algebraic approach [27], see also [5], was later given by Epstein [21] and has a version for (Doplicher-Haag-Roberts) DHR superselection sectors [16] and for more general topological charges =-=[10]-=-. All these approaches heavily rely on arguments of analytic continuation, whose nature give some mysterious effectiveness to the results. Moreover they make use of certain detailed structures, either... |

6 | Relativity of temperature and the Hawking effect - Sewell - 1980 |

6 |
Relativity of Temperature and the Hawking Effect”, Phys
- Sewell
- 1980
(Show Context)
Citation Context ...irst principles. On the physical side modular covariance manifests an interesting analogy with the Unruh effect [45] and with the Hawking black hole thermal radiation [26], as first noticed by Sewell =-=[40]-=-. We sketch the essential ideas, see also [24]. As is known the modular group of a von Neumann algebra with respect to a given state is characterized by the Kubo-Martin-Schwinger condition [44] and, o... |

5 |
On local functions of fields
- Landau
- 1974
(Show Context)
Citation Context ...nt net satisfying essential duality has the modular covariance property. At the present time no counter-example to modular covariance is known to exist within Poincaré covariant theories (see however =-=[33,48]-=-). There are nevertheless counter-examples to the spin-statistics theorem [42]: these are constructed by infinite multiplicity fields where the Poincaré group representation is not unique. It turns ou... |

5 |
Connection between spin and statistics
- Lüders, Zumino
- 1958
(Show Context)
Citation Context ...nt field equations. It was a success of the Wightman axiomatic approach [46] to establish model independent results: the connection between spin and statistics was obtained by Burgoyne [13], see also =-=[36,14]-=-, and a PCT theorem by Jost [29], see also [19, 39], both relying on the general holomorphic properties of the n-point functions. A spin and statistics theorem in the algebraic approach [27], see also... |

5 |
On the connection of spin with statistics”, Nuovo Cim
- Burgoyne
- 1958
(Show Context)
Citation Context ...Lorentz invariant field equations. It was a success of the Wightman axiomatic approach [46] to establish model independent results: the connection between spin and statistics was obtained by Burgoyne =-=[13]-=-, see also [36,14], and a PCT theorem by Jost [29], see also [19, 39], both relying on the general holomorphic properties of the n-point functions. A spin and statistics theorem in the algebraic appro... |

4 |
A uniqueness result for quasi-free KMS
- Kay
- 1985
(Show Context)
Citation Context ...lation operators in any space-time direction. By the one-particle version of the Bisognano Wichmann theorem (which follows of course from the Bisognano-Wichmann theorem in the free field setting, see =-=[30]-=- for a short direct verification of this special case) we may identify the rescaled boost transformations with the modular group of the real Hilbert subspace of the one-particle Hilbert space associat... |

3 | Group representations - Lipsman - 1974 |

3 | A comment on a recent work of Borchers - Wiesbrock - 1992 |

3 |
S.D.: Relativistic Quantum Fields
- Bjorken, Drell
- 1965
(Show Context)
Citation Context ...nd quantization of Bose particles by anti-commuting fields, microscopic causality is still asymptotically present and its violation is sizeable only at distances comparable to the Compton wave length =-=[4]-=-. This is perhaps an indication that the spin-statistics relation might be different in contexts like Quantum Gravity where a sharp causality principle does not occur. 2of the Poincaré group P ↑ +, w... |

2 |
Uber die relativische Theorie kräftfreir Teilchen mit beliebigem spin
- Fierz
- 1939
(Show Context)
Citation Context ...: the PCT theorem and the Spin and Statistics theorem. Both of them have a long history, see [28,43]. The spin and statistics theorem first appeared in the context of free fields in the work of Fierz =-=[20]-=- and Pauli [37]: one cannot second quantize particles with integer spin by anticommuting fields, i.e. fields obeying Fermi statistics, nor particles with half-integer spin by local fields, i.e. fields... |

2 |
CTP invariance in a theory of local observables
- Epstein
- 1967
(Show Context)
Citation Context ... see also [19, 39], both relying on the general holomorphic properties of the n-point functions. A spin and statistics theorem in the algebraic approach [27], see also [5], was later given by Epstein =-=[21]-=- and has a version for (Doplicher-Haag-Roberts) DHR superselection sectors [16] and for more general topological charges [10]. All these approaches heavily rely on arguments of analytic continuation, ... |

2 |
R.: “The Conformal Spin and Statistics
- Guido, Longo
(Show Context)
Citation Context ...gebra [18]. This last step has certain pedagogical advantages, but has to be avoided in order to extend our work to more general settings where the field algebra does not exist. In forthcoming papers =-=[23]-=- we shall indeed provide a more intrinsic approach in terms of local observables only, that will cover low dimensional and conformal theories in particular. The general picture will be clarified by ex... |

1 | Takesaki M., "Extensions of KMS states and chemical potential - Araki, Haag, et al. - 1977 |

1 | Drell S.D., "Relativistic Quantum Fields - Bjorken - 1965 |

1 |
Local rings and the connection between spin and statistics
- Borchers
- 1965
(Show Context)
Citation Context ... and a PCT theorem by Jost [29], see also [19, 39], both relying on the general holomorphic properties of the n-point functions. A spin and statistics theorem in the algebraic approach [27], see also =-=[5]-=-, was later given by Epstein [21] and has a version for (Doplicher-Haag-Roberts) DHR superselection sectors [16] and for more general topological charges [10]. All these approaches heavily rely on arg... |

1 | Fredenhagen K., "Locality and structure of particle states - Buchholz - 1982 |

1 | On the connection of spin with statistics", Nuovo Cimento 8 - Burgoyne - 1958 |