Obtaining a closed-form representation for the dual bosonic thermal Green function by using methods of integration on the complex plane

Abstract

We derive an exact closed-form representation for the Euclidean thermal Green function of the two-dimensional (2D) free massless scalar field in coordinate space. This can be interpreted as the real part of a complex analytic function of a variable that conformally maps the infinite strip −∞ < x < ∞ (0 < τ < β) of the z = x + iτ (τ : imaginary time) plane into the upper-half-plane. Use of the Cauchy-Riemann conditions, then allows us to identify the dual thermal Green function as the imaginary part of that function. Keywords: thermal Green function, massless scalar field, residue theorem.Nós deduzimos uma representação fechada exata para a função de Green térmica euclidiana do campo escalar livre sem massa bidimensional no espaço das coordenadas. Esta pode ser interpretada como a parte real de uma função complexa analítica de uma variável que realiza um mapeamento conforme da faixa infinita −∞ < x < ∞ (0 < τ < β) do plano z = x + iτ (τ : tempo imaginário) no semiplano superior. O uso das condições de CauchyRiemann nos permite então identificar a função de Green térmica dual como a parte imaginária daquela função. Palavras-chave: função de Green térmica, campo escalar sem massa, teorema dos resíduos.

The Rise of Solitons in Sine-Gordon Field Theory: From Jacobi Amplitude to Gudermannian Function

“…Notice that since Equation ( 3) is invariant under Lorentz transformations (x µ → x µ = Λ µ ν x ν ) [9], its solutions may be obtained through the solutions of the corresponding equation for the static case (φ ≡ φ(x)) by a simple Lorentz boost, namely [10,11]. Thus, in what follows, we will focus on the solutions of the equation…”

Thermal Soliton Correlation Functions in Theories with a &lt;i&gt;Z&lt;/i&gt;(N) Symmetry

“…This has also been obtained by using methods of integration on the complex plane [10]. At T = 0, G T (r) reduces to the 2D Coulomb potential and we retrieve the usual mapping onto the Coulomb gas [2].…”

“…Then using the connection between the SG theory and the 2D neutral CG along with the representation for the relevant soliton creation operators introduced in [7][8][9], and both the Euclidean thermal Green function of the 2D free massless scalar field in coordinate space and its dual [5,10], we obtain our expression for the corresponding solitonic correlation function at finite temperature.…”