The construction of stationary two-dimensional Markoff fields with an application to quantum field theory

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Abstract

It is shown that if φ(f) ∝Rd φ(y) f(y) dy is a Markoff random field and Xα are multiplicative functionals of φ (with E(Xα) = 1) which converge locally in L1, then there exists a locally Markoff random field φ* such that E(exp(iφ*(f))) = limα E(Xα exp(iφ(φ))). We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take Xα proportional to exp(-λ∝R2 : P(φ(y)) : gα(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of gα → 1 and small λ, {Xα} does converge locally in L1 and that the corresponding φ* is stationary.

Original languageEnglish (US)
Pages (from-to)44-61
Number of pages18
JournalJournal of Functional Analysis
Volume14
Issue number1
DOIs
StatePublished - 1973

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Quantum Field Theory
Random Field
Converge
Multiplicative
Choose
Directly proportional
Polynomial
Generalization

ASJC Scopus subject areas

  • Analysis

Cite this

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title = "The construction of stationary two-dimensional Markoff fields with an application to quantum field theory",
abstract = "It is shown that if φ(f) ∝Rd φ(y) f(y) dy is a Markoff random field and Xα are multiplicative functionals of φ (with E(Xα) = 1) which converge locally in L1, then there exists a locally Markoff random field φ* such that E(exp(iφ*(f))) = limα E(Xα exp(iφ(φ))). We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take Xα proportional to exp(-λ∝R2 : P(φ(y)) : gα(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of gα → 1 and small λ, {Xα} does converge locally in L1 and that the corresponding φ* is stationary.",
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T1 - The construction of stationary two-dimensional Markoff fields with an application to quantum field theory

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N2 - It is shown that if φ(f) ∝Rd φ(y) f(y) dy is a Markoff random field and Xα are multiplicative functionals of φ (with E(Xα) = 1) which converge locally in L1, then there exists a locally Markoff random field φ* such that E(exp(iφ*(f))) = limα E(Xα exp(iφ(φ))). We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take Xα proportional to exp(-λ∝R2 : P(φ(y)) : gα(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of gα → 1 and small λ, {Xα} does converge locally in L1 and that the corresponding φ* is stationary.

AB - It is shown that if φ(f) ∝Rd φ(y) f(y) dy is a Markoff random field and Xα are multiplicative functionals of φ (with E(Xα) = 1) which converge locally in L1, then there exists a locally Markoff random field φ* such that E(exp(iφ*(f))) = limα E(Xα exp(iφ(φ))). We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take Xα proportional to exp(-λ∝R2 : P(φ(y)) : gα(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of gα → 1 and small λ, {Xα} does converge locally in L1 and that the corresponding φ* is stationary.

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