On quantum fields satisfying a given wave equation (Q1188466)

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 \textit{I. Segal} [J. Funct. Anal. 4, 404--456 (1969; Zbl 0187.39201); 6, 29--75 (1970; Zbl 0202.42201)]. In order to solve the Eq. (1), the study of the ground state, or vacuum of the self-adjoint operator \[ H(\lambda P,v)=H_ 0+\int_{t=0}:\lambda P(\varphi):_v\,dx \] 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.