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 -