On P(φ)_2 interactions at positive temperature

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Robl, Florian;
  • Subject: QA

The Schwinger functions of the thermal P (φ)2 model on the real line and the vacuum P (φ)2\ud model on the circle are equal up to interpretation of their time and space coordinates.\ud This is called Nelson symmetry. In the present work this correspondence is exploited\... View more
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    13 references, page 1 of 2

    1 Preliminaries 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Three Approaches to Quantum Field Theory . . . . . . . . . . . . . . . . . 6 1.2.1 Some Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Hamiltonian and Algebraic Approach . . . . . . . . . . . . . . . . . 9 1.2.3 Euclidean Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Free Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.1 The Free Euclidean Field . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.2 The Free, Scalar Vacuum Field on the Circle Sβ . . . . . . . . . . . 25 1.3.3 The Free, Scalar Thermal Field on R . . . . . . . . . . . . . . . . . 28

    2 The H¨older Inequality for KMS States 31 2.1 The Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Non-commutative Lp-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.1 Relative Modular Operators . . . . . . . . . . . . . . . . . . . . . . 34 2.2.2 Positive Cones and Lp-Spaces for von Neumann Algebras . . . . . . 35 2.3 Proof of the Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    3 Construction of the Thermal P(φ)2 Model 40 3.1 Euclidean Fields on the Cylinder . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.1 The Interacting Measure on the Cylinder . . . . . . . . . . . . . . . 40 3.1.2 Sharp-time Fields, Existence of the Euclidean Measure in the Thermodynamic Limit and Nelson Symmetry . . . . . . . . . . . . . . . 42 3.2 The Osterwalder-Schrader Reconstruction . . . . . . . . . . . . . . . . . . 45 3.3 The Wightman Functions on the Circle . . . . . . . . . . . . . . . . . . . . 50 3.4 The Thermal Wightman Functions . . . . . . . . . . . . . . . . . . . . . . 58 3.4.1 An Application of the Strong Disk Theorem . . . . . . . . . . . . . 59 3.4.2 Products of Sharp-time Fields and their Domains . . . . . . . . . . 61 3.4.3 Temperedness of the Thermal Wightman Distributions . . . . . . . 67

    4 Properties of the Thermal P(φ)2 Model 73 4.1 Verification of the Wightman Axioms . . . . . . . . . . . . . . . . . . . . . 73 4.2 Exponential Decay of Correlation Functions . . . . . . . . . . . . . . . . . 77 4.3 On the Ka¨ll´en-Lehmann Representation . . . . . . . . . . . . . . . . . . . 80 4.3.1 The Ka¨ll´en-Lehmann Representation for Vacuum Models on a Circle 81 4.3.2 Properties of Weight Functions for Vacuum Models on the Circle . 85 4.3.3 Ramifications for the Thermal P(φ)2 Model . . . . . . . . . . . . . 86 Figure 4.2: The regions excluded by Equation (4.51) for mass M and |n| = 1, 2, 3 (|n| = 1 darkest, positive n-values on the right, negative n-values on the left, the maxima of the respective curves are at n).

    [13] J. Bros. Les probl`emes de construction d'envelppoes d'holomorphie en th´eorie quantique de champs. S´eminaire Lelong Analyse, tome 4(exp. n◦ 8):1-23, 1962.

    [14] J. Bros and D. Buchholz. Towards a relativistic KMS condition. Nucl Phys. B, 429:291-318, 1994.

    [28] J. Glimm and A. Jaffe. The Wightman axioms and particle structure in the P (φ)2 quantum field model. Ann. Math., 100(3):585-632, 1974.

    [60] M. Reed and B. Simon. Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-adjointness. Academic Press, 1975.

    [61] L. Rosen. The (φ2n)2 quantum field theory: Higher order estimates. Communications in Pure and Applied Mathematics, 24:417-457, 1971.

    [63] I. Segal. A non-commutative extension of abstract integration. Annals of Mathematics, 57:401-457, 1953. Correction to the Paper “A non-commutative extension of abstract integration”, Ann. of Math. 58 (1953) 595-596.

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