zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Ghost busting: making sense of non-Hermitian Hamiltonians. (English) Zbl 1143.81312
Aoki, Takashi (ed.) et al., Algebraic analysis of differential equations. From microlocal analysis to exponential asymptotics. Festschrift in honor of Takahiro Kawai. Containing papers presented at the conference on algebraic analysis of differential equations--from microlocal analysis to exponential asymptotics, Kyoto, Japan, July 7--14, 2005. Tokyo: Springer (ISBN 978-4-431-73239-6/hbk). 55-66 (2007).
Summary: The Lee model is an elementary quantum field theory in which mass, wave-function, and charge renormalization can be performed exactly. Early studies of this model in the 1950’s found that there is a critical value of $g^2$, the square of the renormalized coupling constant, above which $g^2_0$ the square of the unrenormalized coupling constant, is negative. For $g^2$ larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. In this non-Hermitian regime a new state appears whose norm is negative. This state is called a ghost. It has always been thought that in this ghost regime the Lee model is an unacceptable quantum theory because unitarity appears to be violated. However, in this regime while the Hamiltonian is not Hermitian, it does possess $\cal{PT}$ symmetry. It has recently been discovered that a non-Hermitian Hamiltonian having $\cal{PT}$ symmetry may define a quantum theory that is unitary. The proof of unitarity requires the construction of a time-independent operator called $\cal C$. In terms of $\cal C$ one can define a new inner product with respect to which the norms of the states in the Hilbert space are positive. Furthermore, it has been shown thalt time evolution in such a theory is unitary. In this talk the $\cal C$ operator for the Lee model in the ghost regime is constructed in the $V/N\theta$ sector. It is then shown that the ghost state has a positive norm and that the Lee model is an acceptable unitary quantum field theory for all values of $g^2$. For the entire collection see [Zbl 1126.34002].

81T10Model quantum field theories
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis
81T05Axiomatic quantum field theory; operator algebras