S -Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

Nonlocal quantum theory of a one-component scalar field in <i>D</i>-dimensional Euclidean spacetime is studied in representations of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula>-matrix theory for both polynomial and nonpolynomial interaction Lagrangians. The theory is formulated on coupling constant <i>g</i> in the form of an infrared smooth function of argument <i>x</i> for space without boundary. Nonlocality is given by the evolution of a Gaussian propagator for the local free theory with ultraviolet form factors depending on ultraviolet length parameter <i>l</i>. By representation of the <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula>-matrix in terms of abstract functional integral over a primary scalar field, the <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula> form of a grand canonical partition function is found. By expression of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula>-matrix in terms of the partition function, representation for <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula> in terms of basis functions is obtained. Derivations are given for a discrete case where basis functions are Hermite functions, and for a continuous case where basis functions are trigonometric functions. The obtained expressions for the <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula>-matrix are investigated within the framework of variational principle based on Jensen inequality. Through the latter, the majorant of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula> (more precisely, of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>-</mo> <mo form="prefix">ln</mo> <mi mathvariant="script">S</mi> </mrow> </semantics> </math> </inline-formula>) is constructed. Equations with separable kernels satisfied by variational function <i>q</i> are found and solved, yielding results for both polynomial theory <inline-formula> <math display="inline"> <semantics> <msup> <mi>&#966;</mi> <mn>4</mn> </msup> </semantics> </math> </inline-formula> (with suggestions for <inline-formula> <math display="inline"> <semantics> <msup> <mi>&#966;</mi> <mn>6</mn> </msup> </semantics> </math> </inline-formula>) and nonpolynomial sine-Gordon theory. A new definition of the <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula>-matrix is proposed to solve additional divergences which arise in application of Jensen inequality for the continuous case. Analytical results are obtained and numerically illustrated, with plots of variational functions <i>q</i> and corresponding majorants for the <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">S</mi> </semantics> </math> </inline-formula>-matrices of the theory. For simplicity of numerical calculation, the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> case is considered, and propagator for free theory <i>G</i> is in the form of Gaussian function typically in the Virton&#8315;Quark model, although the obtained analytical inferences are not, in principle, limited to these particular choices. Formulation for nonlocal QFT in momentum <i>k</i> space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process.

Abstract licence: CC BY

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