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On quantum fields satisfying a given wave equation. (English) Zbl 0777.35064
The authors treat here the quantum fields on \(\mathbb{R}\times S^ 1\) satisfying the nonlinear Klein-Gordon equation \[ (\square+m^ 2)\varphi+\lambda:P'(\varphi):_ v=0. \tag{1} \] Here \(P\) is a given real polynomial, bounded below, and \(:P'(\varphi):_ v\) means the one by the renormalized powers developed by I. Segal [J. Funct. Anal. 4, 404- 456 (1969; Zbl 0187.392); 6, 29-75 (1970; Zbl 0202.422)]. In order to solve the Eq. (1), the study of the ground state, or vacuum of the self- adjoint operator \[ H(\lambda P,v)=H_ 0+\int_{t=0}:\lambda P(\varphi):_ vdx \] is required. If the vacuum is equal to v itself, up to a scalar multiple, \[ \varphi_{int}(t,x)=e^{itH(\lambda P,v)}\varphi(0,x)e^{-itH(\lambda P,v)} \] is a solution of the Eq. (1). The state \(u(\lambda)\), equal to the vacuum of \(H(\lambda P,u(\lambda))\) for \(\lambda\in[0,\lambda_ 0]\), is uniquely defined under the two conditions “norm-continuous” and “boundedness”. When \(P(x)=x^ 4\), the authors give two different states \(u_ i(\lambda)\), \(i=1,2\), equal to the vacua of \(H(\lambda P,u_ i(\lambda))\) for \(\lambda\in(0,\lambda_ 0]\), respectively. Here \(H(\lambda P,u_ i(\lambda))\), \(i=1,2\), are not unitary equivalent. Finally they show the similar known results for the one dimensional anharmonic oscillator.
Reviewer: H.Yamagata (Osaka)

35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35L75 Higher-order nonlinear hyperbolic equations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI
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