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Author Simon, Barry
Source CiteSeerX
Content type Text
File Format PDF
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract It is both a pleasure and an honor to write the introduction of this issue in honor of the (recent) sixtieth birthdays of Jürg Fröhlich and Tom Spencer and to be able to place their joint work in some perspective. Tom and Jürg have about twenty-five joint papers, several with additional authors including two with me. I want to focus here on two sets of methods: infrared bounds and multiscale analysis, which are surely among the most significant developments in rigorous statistical physics in the last quarter of the last century. Infrared bounds ([29, 30]), discovered in 1975 and proven using reflection positivity, provide upper bounds on the Fourier transform of the spin-spin correlation at nonzero momentum and force a macroscopic occupation of zero momentum at low temperature (aka Bose–Einstein condensation of spin waves). This implies long-range order, and so, a phase transition. The method was used for quantum spin antiferromagnets by Dyson– Lieb–Simon [20, 21]. Remarkably, after more than thirty years, it remains the only method known to rigorously prove breaking of nonabelian symmetry—even for the abelian case, there is only one other approach to the short-range case using multiscale analysis (see below). For slow decay two-dimensional plane rotors, there are also results of Kunz–Pfister [49]. Among later applications of infrared bounds are Sokal’s specific heat bounds [59], Aizenman’s [1] and Fröhlich’s [24] proofs of the triviality of φ 4 theories in five or more dimensions, the Aizenman–Fernández analysis of long-range models [4], Helffer’s estimates of eigenvalue splitting for certain Schrödinger operators in the thermodynamic limit [43, 44], and the work of Biskup–Chayes on mean-field driven phase transitions [7, 8]. Reflection positivity was introduced in Euclidean field theory by Osterwalder–Schrader [54] and was a key element, albeit implicitly,
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Publisher Date 2008-01-01