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

Charles M. Newman

doi:10.1016/0022-1236(73)90029-3

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

Charles M. Newman

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

It is shown that if φ(f) ∝_R^d φ(y) f(y) dy is a Markoff random field and X_α are multiplicative functionals of φ (with E(X_α) = 1) which converge locally in L₁, then there exists a locally Markoff random field φ_* such that E(exp(iφ_*(f))) = lim_α E(X_α exp(iφ(φ))). We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take X_α proportional to exp(-λ∝_R² : P(φ(y)) : g_α(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 g_α → 1 and small λ, {X_α} does converge locally in L₁ and that the corresponding φ_* is stationary.

Original language	English (US)
Pages (from-to)	44-61
Number of pages	18
Journal	Journal of Functional Analysis
Volume	14
Issue number	1
DOIs	https://doi.org/10.1016/0022-1236(73)90029-3
State	Published - Sep 1973

ASJC Scopus subject areas

Analysis

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10.1016/0022-1236(73)90029-3

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

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