Phase transitions for \(\phi_2^4\) quantum fields. (English) Zbl 0956.82501

The authors prove the existence of a phase transition for the quantum field interaction \(\lambda\phi^4+m_0{}^2\phi^2\) in two-dimensional space-time. Technically the following two results (along with a related estimate) are shown to hold for sufficiently large \(\lambda\): (1) Let \(\Delta_j\) be a unit square lattice with center at \(j\in{Z}^2\), let \(\phi(\Delta)\) denote a smeared out field \(\int_\Delta\phi(x) dx\), let \(\chi_\pm\) be the characteristic functions for the half-lines \((0,+\infty)\) and \((-\infty,0)\), respectively, and let \(\chi_\pm(\Delta_j)\) denote \(\chi_\pm(\phi(\Delta_j))\). Then \(|\langle\chi_+(\Delta_i)\chi_-(\Delta_j)\rangle-\langle\chi_+(\Delta_i)\rangle\langle\chi_-(\Delta_j)\rangle|>a>0\), which shows the existence of long-range (large \(i-j\)) correlation. (2) If a linear term \(-\mu\phi\) is added to the interaction and the limit \(\mu\rightarrow+0\) is taken, then \(\langle\chi_+(\Delta)\rangle_\mu\) has a limit greater than \({\textstyle\frac 1{2}}+\surd a\), showing asymmetry under the transformation \(\phi\rightarrow-\phi\).
Reviewer: H.Araki (Kyoto)


82B10 Quantum equilibrium statistical mechanics (general)
82B26 Phase transitions (general) in equilibrium statistical mechanics
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[1] Dobrushyn, R., Minlos, R.: Construction of a one-dimensional quantum field via a continuous Markov field. Funct. Anal. Appl.7, 324–325 (1973) · Zbl 0294.60081
[2] Fröhlich, J.: Schwinger functions and their generating functionals. II. Adv. Math. (to appear) · Zbl 0345.46057
[3] Glimm, J., Jaffe, A.: The {\(\lambda\)}({\(\phi\)}4)2 quantum field theory without cutoffs. IV. J. Math. Phys.13, 1558–1584 (1972).
[4] Glimm, J., Jaffe, A.: On the approach to the critical point. Ann. Inst. Henri Poincaré22, 13–26 (1975)
[5] Glimm, J., Jaffe, A.: {\(\phi\)} j bounds inP({\(\phi\)})2 quantum field models. Proc. of the Colloq. on Math. Methods of Quantum Field Theory, Marseille, June 1975
[6] Glimm, J., Jaffe, A.: Two and three body equations in quantum field models. Commun. math. Phys.44, 293–320 (1975)
[7] Glimm, J., Jaffe, A., Spencer, T.: A cluster expansion for the {\(\phi\)} 2 4 quantum field theory in the two phase region (in preparation)
[8] Guerra, F., Rosen, L., Simon, B.: Nelson’s symmetry and the infinite volume behavior of the vacuum inP({\(\phi\)})2. Commun. math. Phys.27, 10–22 (1972)
[9] Guerra, F., Rosen, L., Simon, B.: The vacuum energy forP({\(\phi\)})2 infinite volume limit and coupling constant dependence. Commun. math. Phys.29, 233–247 (1973)
[10] Guerra, F., Rosen, L., Simon, B.: TheP({\(\phi\)})2 Euclidean quantum field theory as classical statistical mechanics. Ann. Math.101, 111–259 (1975)
[11] Pirogov, S. A., Sinai, Ya. G.: Phase transitions of the first kind for small perturbations of the Ising model. Funct. Anal. Appl.8, 21–25 (1974) (Engl. trans.)
[12] Simon, B., Griffiths, R.: The {\(\phi\)} 2 4 field theory as a classical Ising model. Commun. math. Phys.33, 145–164 (1973)
[13] Glimm, J., Jaffe, A., Spencer, T.: Existence of phase transitions for {\(\phi\)} 2 4 quantum fields. Proc. of the Colloq. on Math. Methods of Quantum Field Theory, Marseille, June 1975 · Zbl 0956.82501
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