<p>The (elliptic) stochastic quantization equation for the (massive) <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40072_2025_374_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$\cosh (\beta \varphi )_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>cosh</mo> <msub> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> model, for the charged parameter in the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40072_2025_374_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> regime (i.e. <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40072_2025_374_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta ^2 < 4 \pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>β</mi> <mn>2</mn> </msup> <mo><</mo> <mn>4</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>), is studied. We prove the existence, uniqueness and the properties of the invariant measure of the solution to this equation. The proof is obtained through a priori estimates and a lattice approximation of the equation. For implementing this strategy we generalize some properties of Besov spaces in the continuum to analogous results for Besov spaces on the lattice. As a final result we show how to use the stochastic quantization equation to verify the Osterwalder-Schrader axioms for the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40072_2025_374_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$\cosh (\beta \varphi )_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>cosh</mo> <msub> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> quantum field theory, including the exponential decay of correlation functions.</p>

Elliptic stochastic quantization of Sinh-Gordon QFT

期刊论文
发表日期：

2025年7月9日

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Nikolay Barashkov,
Francesco Carlo De Vecchi

摘要

The (elliptic) stochastic quantization equation for the (massive) $\cosh (\beta \varphi )_2$ $cosh {(β φ)}_{2}$ model, for the charged parameter in the $L^2$ $L^{2}$ regime (i.e. $\beta ^2 < 4 \pi $ $β^{2} < 4 π$ ), is studied. We prove the existence, uniqueness and the properties of the invariant measure of the solution to this equation. The proof is obtained through a priori estimates and a lattice approximation of the equation. For implementing this strategy we generalize some properties of Besov spaces in the continuum to analogous results for Besov spaces on the lattice. As a final result we show how to use the stochastic quantization equation to verify the Osterwalder-Schrader axioms for the $\cosh (\beta \varphi )_2$ $cosh {(β φ)}_{2}$ quantum field theory, including the exponential decay of correlation functions.