Theory of Phase Transitions. Rigorous Results by Y. G. Sinai and D. Ter Haar (Auth.)

By Y. G. Sinai and D. Ter Haar (Auth.)

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10) \fth)-2U Π,(φ)ΗΨ,)\ s [(25+i)''-i]||^ll where Α Π , and n^'y^ (2· 12) π lin, 77, (), GROUND STATES OF PERTURBED HAMILTONIAN 39 and Η{φ)=Ηο(φ) + ίί(φ) imply ^ ' ί ο ( < ^ ) + / 7 ο ( φ ) [ ο - [ ( 2 5 + ι / - ΐ ] | | ^ Ι Ι ] + 2 ; = i Π,{ψ)ΗΦ,) = = h{Φ)-2¡=oΠ,(ψ)HΦH2UΠ,(ψ)^(Φ<)+ + /7o(^)[^-[(25+0'-i]ll^ll]. 1 implies the statement. d. of the basic Now we are in a position to specify the perturbation Jif=J^o+^ Hamiltonian J^q we shall consider later.

I n t r o d u c t i o n The most fundamental problem in the theory of phase transitions (equilibrium statistical mechanics) is certainly that of giving a description of limit Gibbs distribu­ tions. As discussed in Chapter I, limit Gibbs distributions for a Hamiltonian ^ are constructed as infinite volume limits of conditional Gibbs distributions in finite volumes with different boundary conditions. 3). Therefore the characteriza­ tion problem of ^ { ^ ) reduces t o that of its extremal points. Extremal points of the convex set ^ ( ^ ) are called indecomposable limit Gibbs distributions; statistical properties of pure thermodynamic phases are described just by indecomposable limit Gibbs distributions.

Then there appears one of the most important examples of limit Gibbs distributions arising in two-dimen- 26 LIMIT GIBBS DISTRIBUTIONS sional quantum field theory. >, νφ) + ml φ^] dx^ dx^ = dxi dx2. This random field can only be determined as a generalized stationary Gaussian field whose correlation operator is given by the Fourier transform of the function {Xli^Xl+ml)-\ Consider, for an arbitrary domain V, the r a n d o m variable fv2ti^k'y(XuX2)'^dx,dx, = JÉV<-^>(^). This random variable can be determined either by taking the limit of lattice models of q u a n t u m field theory, described above, or by the aid of multiple stochastic integrals of the Hermite-Ito type (cf.

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