TY - JOUR

T1 - Reflection positivity via Krein space analysis

AU - Alpay, Daniel

AU - Jorgensen, Palle

N1 - Funding Information:
Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research. The second named author (PJ) acknowledges helpful discussion with Prof Wayne Polyzou, on reflection positivity in quantum physics. It is a pleasure to thank the referee for a very careful reading of the paper and very helpful suggestions.
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/10/1

Y1 - 2022/10/1

N2 - We study reflection positivity in the context of Hilbert space, and Krein-space theory. Our context is that of triple systems (U,J,M+) assumed to satisfy the axioms for reflection positivity (also known as Osterwalder-Schrader positivity). Applications include quantum field theory and the theory of unitary representations of Lie groups. Our analysis and classification make use of Krein space theory, and of realizations of input/output systems. We present a new and explicit classification of the case when triples (U,J,M+) constitute reflection positive systems. A key point in our analysis is a specific choice of signature operator J, and Krein space, leading then to an operator theoretic and geometric classification formula for reflection positive triples (U,J,M+), with U assumed unitary and J-self-adjoint, and M+ reflecting the choices of the associated J-positive and U-invariant subspaces. We further consider the wider setting when U is only assumed J-self-adjoint, and we demonstrate how choices of U may be characterized by specific partially defined and contractive operators E. Then the corresponding admissible subspaces M+ will be realized as spaces G(X), graph of a second linear operator X, where the possibilities for such X, as operators, are decided by a non-linear equation (in X), called the Riccati equation. Our present applications include non-commutative harmonic analysis.

AB - We study reflection positivity in the context of Hilbert space, and Krein-space theory. Our context is that of triple systems (U,J,M+) assumed to satisfy the axioms for reflection positivity (also known as Osterwalder-Schrader positivity). Applications include quantum field theory and the theory of unitary representations of Lie groups. Our analysis and classification make use of Krein space theory, and of realizations of input/output systems. We present a new and explicit classification of the case when triples (U,J,M+) constitute reflection positive systems. A key point in our analysis is a specific choice of signature operator J, and Krein space, leading then to an operator theoretic and geometric classification formula for reflection positive triples (U,J,M+), with U assumed unitary and J-self-adjoint, and M+ reflecting the choices of the associated J-positive and U-invariant subspaces. We further consider the wider setting when U is only assumed J-self-adjoint, and we demonstrate how choices of U may be characterized by specific partially defined and contractive operators E. Then the corresponding admissible subspaces M+ will be realized as spaces G(X), graph of a second linear operator X, where the possibilities for such X, as operators, are decided by a non-linear equation (in X), called the Riccati equation. Our present applications include non-commutative harmonic analysis.

KW - Harmonic analysis

KW - Hilbert space

KW - J-self-adjoint

KW - Krein space

KW - Osterwalder-Schrader axioms

KW - Potapov-Ginzburg transform

KW - Realization of input/output systems

KW - Reflection positive

KW - Riccati equation

KW - Schur analysis

UR - http://www.scopus.com/inward/record.url?scp=85135691429&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2022.102411

DO - 10.1016/j.aam.2022.102411

M3 - Article

AN - SCOPUS:85135691429

VL - 141

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

M1 - 102411

ER -