Summary
Starting from a given factorizing S-matrix S in two space-time dimensions, we review a novel strategy to rigorously construct quantum field theories describing particles whose interaction is governed by S. The construction procedure is divided into two main steps: Firstly certain semi-local Wightman fields are introduced by means of Zamolodchikov’s algebra. The second step consists in proving the existence of local observables in these models. As a new result, an intermediate step in the existence problem is taken by proving the modular compactness condition for wedge algebras.
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References
H. Araki: Mathematical Theory of Quantum Fields. Oxford University Press, New York, 1999.
H. Babujian and M. Karowski: The “Bootstrap Program” for Integrable Quantum Field Theories in 1 + 1 Dim. Preprint (2001). [arXiv: hep-th/0110261].
B. Berg, M. Karowski and P. Weisz: Construction of Green’s functions from an exact S matrix. Phys. Rev. D 19:2477 (1979).
J.J. Bisognano and E.H. Wichmann: On the duality condition for a hermitian scalar field. J. Math. Phys. 16:985 (1975).
H.-J. Borchers: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143:315 (1992).
H.-J. Borchers, D. Buchholz and B. Schroer: Polarization-Free Generators and the S-Matrix. Commun. Math. Phys. 219:125 (2001). [arXiv: hep-th/0003243].
J. Bros: A Proof of Haag-Swieca’s Compactness Property for Elastic Scattering States. Commun. Math. Phys. 237:289 (2003).
R. Brunetti, D. Guido and R. Longo: Modular localization and Wigner particles. Rev. Math. Phys. 14:759 (2002). [arXiv: math-ph/0203021].
D. Buchholz: Product States for Local Algebras. Commun. Math. Phys. 36:287 (1974).
D. Buchholz, C. D’Antoni and R. Longo: Nuclear Maps and Modular Structures I: General Properties. J. Funct. Anal. 88:233 (1990).
D. Buchholz, C. D’Antoni and R. Longo: Nuclear Maps and Modular Structures II: Applications to Quantum Field Theory. Commun. Math. Phys. 129:115 (1990).
D. Buchholz and P. Jacobi: On the nuclearity condition for massless fields. Lett. Math. Phys. 13:313 (1987).
D. Buchholz and P. Junglas: On The Existence of Equilibrium States in Local Quantum Field Theory. Commun. Math. Phys. 121:255 (1989).
D. Buchholz and G. Lechner: Modular Nuclearity and Localization. Ann. H. Poincaré 5:1065 (2004). [arXiv: math-ph/0402072].
D. Buchholz and E. H. Wichmann: Causal Independence and the Energy-Level Density of States in Local Quantum Field Theory. Comm. Math. Phys. 106:321 (1986).
O.A. Castro-Alvaredo: Bootstrap Methods in 1+1-Dimensional Quantum Field Theories: The Homogeneous Sine-Gordon Models. PhD thesis, 2001. [arXiv: hep-th/0109212].
S. Doplicher and R. Longo: Standard and split inclusions of von Neumann algebras. Commun. Math. Phys 75:493 (1984).
H. Epstein: Some analytic properties of scattering amplitudes in quantum field theory. In: Particle Symmetries and Axiomatic Field Theory, Brandeis Summer School 195, Gordon and Breach, New York, 1966.
R. Haag: Local Quantum Physics. Springer Verlag, Berlin, 2nd ed., 1996.
R. Haag and J.A. Swieca: When does a quantum field theory describe particles?. Commun. Math. Phys. 1:308 (1965).
R.V. Kadison and J.R. Ringrose: Fundamentals of the Theory of Operator Algebras, Vol. II. Academic Press, Orlando, 1986.
G. Lechner: Polarization-Free Quantum Fields and Interaction. Lett. Math. Phys. 64:137 (2003). [arXiv: hep-th/0303062].
G. Lechner: On the existence of local observables in theories with a factorizing S-matrix. To appear in J. of Phys. A (2005). [arXiv: math-ph/0405062].
A. Liguori and M. Mintchev: Fock spaces with generalized statistics. Commun. Math. Phys. 169:635 (1995). [arXiv: hep-th/9403039].
R. Longo: Notes on algebraic invariants for noncommutative dynamical systems. Commun. Math. Phys. 69:195 (1979).
B.M. McCoy, C.A. Tracy and T.T. Wu: Two Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas for n-Point Functions. Phys. Rev. Lett. 38:793 (1977).
P. Mitra: Elasticity, Factorization and S-Matrices in (1 + 1)-Dimensions. Phys. Lett. B 72:62 (1977).
M. Müger: Superselection structure of massive quantum field theories in (1+1)-dimensions. Rev. Math. Phys. 10:1147 (1998). [arXiv: hep-th/9705019].
J. Mund: The Bisognano-Wichmann theorem for massive theories. Annales Henri Poincaré 2:907 (2001). [arXiv: hep-th/0101227].
A. Pietsch: Nuclear locally convex spaces. Springer Verlag, Berlin, Heidelberg, New York, 1972.
W. Rudin: Real and complex analysis. McGraw-Hill Book Company, 1987.
S. Sakai: C*-Algebras and W*-Algebras. Springer Verlag, 1971.
B. Schroer: Modular localization and the bootstrap form-factor program. Nucl. Phys. B 499:547 (1997). [arXiv: hep-th/9702145].
B. Schroer and H.W. Wiesbrock: Modular constructions of quantum field theories with interactions. Rev. Math. Phys 12:301 (2000). [arXiv: hep-th/9812251].
F.A. Smirnov: Formfactors in completely integrable models in quantum field theory. Advanced Series in Mathematical Physics 14, World Scientific, 1992.
M. Reed and B. Simon: Methods of modern mathematical physics I: Functional Analysis. Revised and enlarged edition, Academic Press, 1980.
M. Reed and B. Simon: Methods of modern mathematical physics II: Fourier Analysis, Self-Adjointness. Academic Press, 1975.
M. Reed and B. Simon: Methods of modern mathematical physics III: Scattering Theory. Academic Press, 1979.
S.J. Summers: On the Independence of Local Algebras in Quantum Field Theory. Rev. Math. Phys. 2:201 (1990).
A. Zamolodchikov: Factorized S-matrices as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. 120:253 (1979).
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Lechner, G. (2007). Towards the Construction of Quantum Field Theories from a Factorizing S-Matrix. In: de Monvel, A.B., Buchholz, D., Iagolnitzer, D., Moschella, U. (eds) Rigorous Quantum Field Theory. Progress in Mathematics, vol 251. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7434-1_13
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