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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)

MSC:
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
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[1] Baez, J, Wick products of the free Bose field, J. funct. anal., 86, 211-225, (1989) · Zbl 0694.46052
[2] Baez, J; Seoal, I; Zhou, Z, An introduction to algebraic and constructive quantum field theory, (1992), Princeton Univ. Press New Jersey
[3] {\scJ. Baez and Z. Zhou}, Renormalized oscillator Hamiltonians, Adv. Math., to appear.
[4] Friedman, C, Renormalized oscillator equations, J. math. phys., 14, 1378-1380, (1973)
[5] Jaffe, A; Lesniewski, A; Wieczerkowski, C, A priori quantum field equations, Ann. physics, 192, 2-20, (1989) · Zbl 0676.35078
[6] Kato, T, Perturbation theory for linear operators, (1980), Springer-Verlag Berlin
[7] Segal, I; Segal, I, Nonlinear functions of weak processes, II, J. funct. anal., J. funct. anal., 6, 29-75, (1970) · Zbl 0202.42201
[8] Segal, I; Segal, I, Construction of non-linear local quantum process, II, Ann. math., Invent. math., 14, 211-241, (1971) · Zbl 0221.47023
[9] Simon, B; Hoegh-Krohn, R, Hypercontractive semigroups and two dimensional self-coupled Bose fields, J. funct. anal., 9, 121-180, (1972) · Zbl 0241.47029
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