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• 1
Electronic Resource
Springer
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract Various aspects of Markov field theory are treated. We give a Fock space description of the scalar free field with Nelson's Markov property formulated in terms of projections. We consider conditions imposed on analytically continued Wightman distributions at Euclidean points so that a Euclidean Markov field theory will result. Euclidean theories in higher dimensional imaginary times are considered. We show how the generalized free field theory can be interpreted as a Markov Euclidean field theory. The spatially cutoff linear perturbation model is solved in arbitrary space-time dimensions and the Wightman distributions are obtained explicitly in the limit as the cutoff is removed. The appendices contain a discussion and derivation of the Segal isomorphism and we give some generalizations of Feynman-Kac formulas inR n and in the Fock space of Euclidean field theory.
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
Communications in mathematical physics 122 (1989), S. 233-247
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract Block renormalization group transformations (RGT) for lattice and continuum Euclidean Fermions in d dimensions are developed using Fermionic integrals with exponential and “δ-function” weight functions. For the free field the sequence of actionsD k generated by the RGT from D, the Dirac operator, are shown to have exponential decay; uniform ink, after rescaling to the unit lattice. It is shown that the two-point functionD −1 admits a simple telescopic sum decomposition into fluctuation two-point functions which for the exponential weight RGT have exponential decay. Contrary to RG intuition the sequence of rescaled actions corresponding to the “δ-function” RGT do not have uniform exponential decay and we give examples of initial actions in one dimension where this phenomena occurs for the exponenential weight RGT also.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
Communications in mathematical physics 138 (1991), S. 487-505
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We obtain low temperature properties of the classical vector model in a hierarchical formulation in three or more dimensions. We consider the lattice model in a zero or non-zero magnetic field, where the single site spin variable ϕ∈R v has a density proportional to $$e^{ - \lambda (\phi ^2 - 1)^2 }$$ for large λ≦∞. Using renormalization group methods we obtain a convergent expansion for the free energy with zero magnetic field. For non-zero fields a shift formula is used to obtain the effective action generated by the renormalization group transformation (RGT). To obtain the pure state zero field free energy and spontaneous magnetization we take the thermodynamic limit together with the zero field limit at a specified rate. The spontaneous magnetization,m, is calculated, is non-zero and the pure state free energy coincides, as expected, with the zero field free energy. Also the sequence of zero field actions does not have a limit but we show that the sequence of actions generated from the original action shifted bym does; the limiting action corresponds to a non-canonical Gaussian fixed point of the RGT.
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• 4
Electronic Resource
Springer
Communications in mathematical physics 199 (1999), S. 493-520
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract: We obtain convergent multi-scale expansions for the one-and two-point correlation functions of the low temperature lattice classical N– vector spin model in d≥ 3 dimensions, N≥ 2. The Gibbs factor is taken as where , , , are large and 0 〈 v≤ 1. In the thermodynamic and limits, with h=e 1, and Δ≡∂★∂, the expansion gives (spontaneous magnetization), , (Goldstone Bosons), , and , where , for some ρ 〉 0, and c 0 is aprecisely determined constant.
Type of Medium: Electronic Resource
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• 5
Electronic Resource
Springer
Communications in mathematical physics 84 (1982), S. 153-170
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract From a Feynman-Kac formula in a Fermion Fock space for the Schwinger functions of the infinite lattice periodic two-dimensional Ising model, scaled and scaling limit Schwinger functions are defined and shown to admit an absolutely convergent series representation. As the critical temperature is attained, it is shown that the scaled Schwinger functions converge and that the resulting scaling limit Schwinger functions obey the Osterwalder-Schrader axioms.
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• 6
Electronic Resource
Springer
Communications in mathematical physics 97 (1985), S. 429-442
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract For a 2+1 strongly coupled (β=2/g 2 small) Wilson action lattice gauge theory with complex character we analyze the mass spectrum of the associated quantum field theory restricted to the subspace generated by the plaquette function and its complex conjugate. It is shown that there is at least one but not more than two isolated masses and each mass admits a representation of the formm(β)=−4lnβ+r(β), wherer(β) is a gauge group representation dependent function analytic inβ 1/2 orβ atβ=0. For the gauge group SU(3) there is mass splitting and the two massesm ± are given by $$m_ \pm (\beta ) = - 41n\beta + 16r^4 + \tfrac{1}{2}(2 \pm 1)\beta + \left( {d_ \pm (\beta )\sum\limits_{n = 2}^\infty {c_n^ \pm } \beta ^n } \right)$$ wherer=3 is the dimension of the representation andd ±(β) is analytic atβ=0.c n ± can be determined from a finite number of theβ=0 Taylor series coefficients of finite lattice truncated plaquette-plaquette correlation function at a finite number of points.
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• 7
Electronic Resource
College Park, Md. : American Institute of Physics (AIP)
Journal of Mathematical Physics 26 (1985), S. 2342-2345
ISSN: 1089-7658
Source: AIP Digital Archive
Topics: Mathematics , Physics
Notes: It is known that the mass spectrum of a strongly coupled (β=2/g2 small) 2+1 Wilson action lattice gauge theory contains a mass m0∼−4 ln β and two excited masses m1, m2∼−6 ln β and that m0+4 ln β has a convergent expansion in powers of β. We show that m1, m2 admit convergent expansions of the form −6 ln β+r(β), where r(β) is analytic at β=0. Furthermore, a finite lattice algorithm is given for determining cn, the nth β=0 Taylor coefficient of r(β). Here, cn only depends on a finite number of β=0 Taylor series coefficients of the plaquette–plaquette, plaquette–double plaquette, and double plaquette–double plaquette truncated correlation functions at a finite number of points. For the gauge group Z2, by duality, m1, m2 map to bound states of the low-temperature Ising model; a possible relation between an increasing number of bound states and roughening is discussed.
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• 8
Electronic Resource
Springer
Communications in mathematical physics 27 (1972), S. 137-145
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We show that the non-relativistic quantum mechanicaln-body HamiltoniansT(k)=T+kV andT, the free particle Hamiltonian, are unitarily equivalent in the center of mass system, i.e.,T(k)=W ± (k)TW ± (k) −1 fork sufficiently small and real. $$V = \sum\limits_i {V_i }$$ , a sum ofn(n−1)/2 real pair potentials,V i, depending on the relative coordinatex i ∈R 3 of the pairi, whereV i is required to behave like |xi|− 2 −ε as |x i |→∞ and like |xi|− 2 +ε as |x i |→0.T(k) is the self-adjoint operator associated with the form sumT+kV. There are no smoothness requirements imposed on theV i . Furthermore $$W_ \pm (k) = \mathop {s - \lim }\limits_{t \to \pm \infty } e^{iT(k)t} e^{ - iTt}$$ are the wave operators of time dependent scattering theory and are unitary. This result gives a quantitative form of the intuitive argument based on the Heisenberg uncertainty principle that a certain minimum potential well depth and range is needed before a bound state can be formed. This is the best possible long range behavior in the sense that ifkV i ≦C i |x i |−b , 0〈b≦2 for |x i |〉R i (0〈R i 〈∞) and allC i are negative thenT(k) has discrete eigenvalues andW ±(k) are not unitary.
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• 9
Electronic Resource
Springer
Communications in mathematical physics 103 (1986), S. 569-597
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We investigate the mass spectrum of a 2+1 lattice gauge-Higgs quantum field theory with Wilson action βA P+λA H, whereA P(A H) is the gauge (gauge-Higgs) interaction. We determine the complete spectrum exactly for all β, λ〉0 by an explicit diagonalization of the gauge invariant “transfer matrix” in the approximation that the interaction terms in the spatial directions are omitted; all gauge invariant eigenfunctions are generated directly. For fixed momentum the energy spectrum is pure point and disjoint simple planar loops and strings are energy eigenfunctions. However, depending on the gauge group and Higgs representations, there are bound state energy eigenfunctions not of this form. The approximate model has a rich particle spectrum with level crossings and we expect that it provides an intuitive picture of the number and location of bound states and resonances in the full model for small β, λ〉0. We determine the mass spectrum, obtaining convergent expansions for the first two groups of masses above the vacuum, for small β, λ and confirm our expectations.
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• 10
Electronic Resource
Springer
Letters in mathematical physics 25 (1992), S. 29-37
ISSN: 1573-0530
Keywords: 81E15 ; 82A25
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We consider a class of scalar field lattice models with action 1/2(∂ϕ)+V(ϕ), V small. After n block renormalization group transformations, new formulas are obtained for the finite lattice generating and correlation functions. For some infrared asymptotic-free models in the thermodynamic and n → ∞ limits, the formulas for correlation functions are especially simple, isolate the correct dominant long-distance behavior, and can be used to control the subdominant contributions.
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