LEADER 00759nam a2200241 i 4500 001 991003561619707536 005 20020509124340.0 008 000220s1965 it ||| | ita 035 $ab11180936-39ule_inst 035 $aPARLA184330$9ExL 040 $aDip.to Filosofia$bita 082 0 $a160 100 1 $aBorsari, Raffaele$063237 245 10$aLogica concreta 260 $aFirenze :$bSansoni,$c1965 300 $a243 p. ;$c23 cm 650 4$aLogica 907 $a.b11180936$b21-09-06$c28-06-02 912 $a991003561619707536 945 $aLE005IF X C 19$g1$i2005000078958$lle005$o-$pE0.00$q-$rl$s- $t0$u0$v0$w0$x0$y.i11328198$z28-06-02 996 $aLogica concreta$9274305 997 $aUNISALENTO 998 $ale005$b01-01-00$cm$da $e-$fita$git $h0$i1 LEADER 08725nam 22007093 450 001 9910915674803321 005 20231110220005.0 010 $a1-4704-7322-4 035 $a(MiAaPQ)EBC30330557 035 $a(Au-PeEL)EBL30330557 035 $a(CKB)25994207600041 035 $a(EXLCZ)9925994207600041 100 $a20230113d2023 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 14$aThe P(?) 2 Model on de Sitter Space 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2023. 210 4$d©2023. 215 $a1 online resource (282 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.281 311 08$aPrint version: C., João The Providence : American Mathematical Society,c2023 9781470455484 327 $aCover -- Title page -- List of Symbols -- Preface -- Part 1. De Sitter space -- Chapter 1. De Sitter space as a Lorentzian manifold -- 1.1. The Einstein equations -- 1.2. De Sitter space -- 1.3. The Lorentz group -- 1.4. Hyperbolicity -- 1.5. Causally complete regions -- 1.6. Complexified de Sitter space -- 1.7. The Euclidean sphere -- Chapter 2. Space-time symmetries -- 2.1. The isometry group of de Sitter space -- 2.2. Horospheres -- 2.3. The Cartan decomposition of ?(1,2) -- 2.4. An alternative decomposition of ?(1,2) -- 2.5. The Iwasawa decomposition of ?(1,2) -- 2.6. The Hannabuss decomposition of ?(1,2) -- 2.7. Homogeneous spaces, cosets and orbits -- 2.8. The complex Lorentz group -- Chapter 3. Induced representations for the Lorentz group -- 3.1. Integration on homogeneous spaces -- 3.2. Induced representations -- 3.3. Reducible representations on the light-cone -- 3.4. Unitary irreducible representations on a circle lying on the lightcone -- 3.5. Intertwiners -- 3.6. The time reflection -- 3.7. Unitary irreducible representations on two mass shells -- Chapter 4. Harmonic analysis on the hyperboloid -- 4.1. Plane waves -- 4.2. The Fourier-Helgason transformation -- 4.3. The Plancherel theorem on the hyperboloid -- 4.4. Unitary irreducible representations on de Sitter space -- 4.5. The Euclidean one-particle Hilbert space over the sphere -- 4.6. Unitary irreducible representations on the time-zero circle -- 4.7. Reflection positivity: From (3) to (1,2) -- 4.8. Time-symmetric and time-antisymmetric test-functions -- 4.9. Fock space -- Part 2. Free quantum fields -- Chapter 5. Classical field theory -- 5.1. The classical equations of motion -- 5.2. Conservation laws -- 5.3. The covariant classical dynamical system -- 5.4. The restriction of the KG equation to a (double) wedge -- 5.5. The canonical classical dynamical system. 327 $aChapter 6. Quantum one-particle structures -- 6.1. The covariant one-particle structure -- 6.2. One-particle structures with positive and negative energy -- 6.3. One-particle KMS structures -- 6.4. The canonical one-particle structure -- 6.5. Localisation -- 6.6. Standard subspaces of ? ( ¹) -- Chapter 7. Local algebras for the free field -- 7.1. The covariant net of local algebras on -- 7.2. The canonical net of local *-algebras on ¹ -- 7.3. Euclidean fields and the net of local algebras on ² -- 7.4. The reconstruction of free quantum fields on de Sitter space -- Part 3. Interacting quantum fields -- Chapter 8. The interacting vacuum -- 8.1. Short-distance properties of the covariance -- 8.2. (Non-)Commutative ^{ }-spaces -- 8.3. The Euclidean interaction -- 8.4. The interacting vacuum vector -- Chapter 9. The interacting representation of (1,2) -- 9.1. The reconstruction of the interacting boosts -- 9.2. A unitary representation of the Lorentz group -- 9.3. Perturbation formulas for the boosts -- Chapter 10. Local algebras for the interacting field -- 10.1. Finite speed of propagation for the ( )? model -- 10.2. The Haag-Kastler axioms -- Chapter 11. The equations of motion and the stress-energy tensor -- 11.1. The stress-energy tensor -- 11.2. The equations of motion -- Chapter 12. Summary -- 12.1. The conceptional structure -- 12.2. Wightman function, particle content and scattering theory -- 12.3. A detailed summary -- Appendix A. A local flat tube theorem -- Appendix B. One particle structures -- Appendix C. Sobolev spaces on the circle and on the sphere -- Appendix D. Some identities involving Legendre functions -- Bibliography -- Index -- Back Cover. 330 $a"In 1975 Figari, Høegh-Krohn and Nappi constructed the P(Phi)2 model on the de Sitter space. Here we complement their work with new results, which connect this model to various areas of mathematics. In particular, i.) we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs of SO0(1, 2) for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface. We describe the free classical dynamical system in both its covariant and canonical form, and present the associated quantum one-particle KMS structures in the sense of Kay (1985). Furthermore, we discuss the localisation properties of one-particle wave functions and how these properties are inherited by the algebras of local observables. ii.) we describe the relations between the modular objects (in the sense of Tomita-Takesaki theory) associated to wedge algebras and the representations of the Lorentz group. We connect the representations of SO(1,2) to unitary representations of SO(3) on the Euclidean sphere, and discuss how the P(Phi)2 interaction can be represented by a rotation invariant vector in the Euclidean Fock space. We present a novel Osterwalder-Schrader reconstruction theorem, which shows that physical infrared problems are absent on de Sitter space. As shown in Figari, Hoegh-Krohn, and Nappi (1975), the ultraviolet problems are resolved just like on flat Minkowski space. We state the Haag-Kastler axioms for the P(Phi)2 model and we explain how the generators of the boosts and the rotations for the interacting quantum field theory arise from the stress-energy tensor. Finally, we show that the interacting quantum fields satisfy the equations of motion in their covariant form. In summary, we argue that the de Sitter P(Phi)2 model is the simplest and most explicit relativistic quantum field theory, which satisfies basic expectations, like covariance, particle creation, stability and finite speed of propagation"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aOperator algebras 606 $aGeneralized spaces 606 $aLorentz groups 606 $aManifolds (Mathematics) 606 $aQuantum field theory 606 $aQuantum theory -- Quantum field theory; related classical field theories -- Axiomatic quantum field theory; operator algebras$2msc 606 $aQuantum theory -- Quantum field theory; related classical field theories -- Constructive quantum field theory$2msc 606 $aQuantum theory -- Quantum field theory; related classical field theories -- Quantum field theory on curved space backgrounds$2msc 606 $aTopological groups, Lie groups -- Lie groups -- Structure and representation of the Lorentz group$2msc 606 $aOperator theory -- Linear spaces and algebras of operators -- Applications of operator algebras to physics$2msc 615 0$aOperator algebras. 615 0$aGeneralized spaces. 615 0$aLorentz groups. 615 0$aManifolds (Mathematics) 615 0$aQuantum field theory. 615 7$aQuantum theory -- Quantum field theory; related classical field theories -- Axiomatic quantum field theory; operator algebras. 615 7$aQuantum theory -- Quantum field theory; related classical field theories -- Constructive quantum field theory. 615 7$aQuantum theory -- Quantum field theory; related classical field theories -- Quantum field theory on curved space backgrounds. 615 7$aTopological groups, Lie groups -- Lie groups -- Structure and representation of the Lorentz group. 615 7$aOperator theory -- Linear spaces and algebras of operators -- Applications of operator algebras to physics. 676 $a530.152/556 676 $a530.152556 686 $a81T05$a81T08$a81T20$a22E43$a47L90$2msc 700 $aC$b João$01778419 701 $aJäkel$b Christian D$01778420 701 $aMund$b Jens$01778421 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910915674803321 996 $aThe P(?) 2 Model on de Sitter Space$94301272 997 $aUNINA