β Scribed by Charles Burnap
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We give sufficient conditions to ensure the existence of isolated one-particle states in the joint energy-momentum spectrum of an Osterwalder-Schrader scalar boson quantum field theory. We do not require the existence of sharp-time fields or weak coupling.
We give sufficient conditions to ensure the existence of isolated one particle states in the joint energy-momentum spectrum of a scalar boson quantum field, theory. In terms of the selfadjoint mass operator M = (H2 -P2)1/2 this means the existence of a gap (0, m) in the spectrum a(M), the existence of an upper gap (m, m2) in a(M), and the existence of eigenvalues 0, m, see Theorem 2.
The proof of the existence of the lower gap relies on the exponential cluster property of Schwinger functions. The functional analysis part of this proof is straightforward.
The proof of the existence of the upper gap is more delicate. The first result relating estimates to the existence of particles was given by Glimm et al. [6]. (This appears in simplified form in [7].) As input the proof of Glimm et al. requires some refined cluster properties of the Schwinger functions which were established for weakly coupled Pi models in [6].
Here we give a somewhat different version. We relate the proof to properties of two projection operators P, and P,: The orthogonal projections onto Euclidean vectors spanned by the Euclidean vacuum Q8 and by ((1 -PO) @i(x) Q &: x E LV}, respectively. The fact that P, agrees in the mass interval (m -E, m + e) with the projection onto one particle states was established by Glimm and Jaffe [5], under the assumption that the exponential decay rate for r(x), the inverse propagator, is strictly greater than m. We define (channel) one particle irreducible (1 -PI) Schwinger functions in terms of PO and P, . We then give sufficient conditions on these 1 -PI Schwinger functions to yield the upper and lower gaps. These conditions are exponential decay properties, supplemented by a technical assumption about the distribution character of the two point Schwinger function.
We isolate the essential input as estimates which can be verified in other models, e.g., by expansion techniques or correlation inequalities. In contrast to the original
π SIMILAR VOLUMES
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The calculated elastic scattering of 1.04-GeV protons by 'OsPb, 58Ni and "'Ca is compared with experiment. The effects of the spin-orbit potential, Pauli and short-range correlations, variations in the ratio of the real to the imaginary part of the nucleon-nucleon scattering amplitude at 00 are foun