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

Obtendo uma representação fechada para a função de Green térmica dual bosônica usando métodos de integração sobre o plano complexo

ARTIGOS GERAIS

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

Obtendo uma representação fechada para a função de Green térmica dual bosônica usando métodos de integração sobre o plano complexo

Leonardo Mondaini¹

Grupo de Física Teórica e Experimental, Departamento de Ciências Naturais, Universidade Federal do Estado do Rio de Janeiro, Rio de Janeiro, RJ, Brasil

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.

RESUMO

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 Cauchy-Riemann 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.

1. Introduction

As remarked in Ref. [1], despite the fact that quantum field theories are usually formulated in coordinate space, calculations, in both T = 0 and T ≠ 0 cases, are almost always performed in momentum space. However, when we are faced with the exact calculation of correlation functions we are naturally led to the problem of finding closed-form expressions for Green functions in coordinate space [2,3].

A closed-form representation for the thermal Green function of the free massless scalar field² in 2D was firstly presented in Ref. [2], where attention has been focused on its relation with the bosonization of the massive Thirring model (MTM)³ using the imaginary-time formalism for finite temperature quantum field theory [4]. Unfortunately, the authors omit valuable details of the calculations, which would be useful for graduate students and researchers working on this subject.

In the present work we present a simple and yet appealing step-by-step derivation of an exact closed-form representation for the thermal Green function of 2D free massless scalar theory in the coordinate space, at a level accessible to usual graduate students in physics. This has been obtained by using the imaginary-time formalism along with methods of integration on the complex plane and the software Mathematica. The peculiar form of this, allows us to easily recognize it as the real part of an analytic function, a fact that leads us to determine the corresponding dual thermal Green function as the imaginary part of that function, according to the Cauchy-Riemann conditions. This dual thermal Green function turns out to be a key ingredient for the obtainment of fermion correlators of the MTM at finite temperature, as shown in Ref. [5].

2. Two-dimensional thermal Green function

The Euclidean thermal Green function of the 2D free massless scalar theory can be written in the coordinate space (r ≡ (x,τ)) as [4]

where ω_n = 2πn/β, β = 1/k_BT. At T = 0, G_T(r) reduces to the 2D Coulomb potential.

The sum appearing in Eq. (1) may be evaluated by considering the following integral on the complex plane [6]

where

and the integration contour C is defined in Fig. 1-(a).

The function f(z) has poles at z = ±k, being therefore outside the contour C. Those of δ_BE(βz), are situated at z = i2πn/β = iω_n, (n = 0, ±1, ±2,...) hence inside the contour C.

It is easy to see, using the residue theorem [7] that the sum coincides with the integral I_C and, therefore, we have

Deforming the contour C into C' shown in Fig. 1-(b) we have, using the residue theorem again:

Hence, since I_C = I_C', we have

which allows us to rewrite the expression (1) for G_T(r) as

Observe that only the real part of the integral is non-vanishing. Making the change of variable k → (2π/β)y, defining a ≡ (2π/β)x, θ ≡ [(2π/β)τ – π], and introducing the regulator b we may rewrite Eq. (7) as

This integral, which can be found in Ref. [8], gives

The sum above may be evaluated in terms of hypergeometric functions ₂F₁ with the help of Mathematica [9]. The result is

Taking the b → 0 limit in the expression above, we obtain (since ₂F₁(1, 1, 2; -z) = )

By inserting the expressions for a and θ and defining the regulator mass µ₀≡ (2π/β)e^-1/2b, we may write the scalar thermal Green function, after some algebra, as

which coincides with the result presented for the first time in Ref. [2].

Finally, notice that the Eq. (12) may be also written as

where (z = x + iτ)

3. The dual thermal Green function

From Eq. (13) we can also see that the thermal Green function may be written as the real part of an analytic function of a complex variable ζ, namely

where F(ζ) ≡ –(1/2π)ln[µ₀ζ(r)].

The imaginary part of F(ζ) may be written as

Now, from the analyticity of F(ζ), then, it follows that its imaginary and real parts must satisfy the Cauchy-Riemann conditions, which are given by

This property characterizes as the dual thermal Green function.

4. Concluding remarks

We would like to make a few comments about Eq. (13). Firstly, we note that in the zero temperature limit (T → 0, β → ∞), we have ζ(z) → z and ζ^*(z) → z^* and, therefore, we recover the well-known Green function at zero temperature, namely

Comparing Eq. (13) with Eq. (18) we can also see that the only effect of introducing a finite temperature is to exchange the complex variable z for ζ(z). Since ζ(z) is analytic, we conclude that the thermal Green function is obtained from the one at zero temperature by the following conformal mapping [7]: the infinite strip 0 < τ < β and –∞ < x < ∞ is mapped into the region within the upper-half-ζ-plane. Notice that only the values [0,β] of τ are relevant because this variable is periodic in β, as it should at finite T.

Acknowledgments

The author would like to thank E.C. Marino for his valuable comments during the calculations.

This work has been supported in part by Fundação CECIERJ.

References

[1] H.A. Weldon, Phys. Rev. D 62, 056003 (2000); H.A. Weldon, Phys. Rev. D 62, 056010 (2000).

[2] D. Delépine, R. González Felipe, and J. Weyers, Phys. Lett. B 419, 296 (1998).

[3] L. Mondaini, E.C. Marino and A.A. Schmidt, J. Phys. A: Math. Theor. 42, 055401 (2009).

[4] A. Das, Finite Temperature Field Theory (World Scientific, Singapore, 1997).

[5] L. Mondaini and E.C. Marino, Mod. Phys. Lett. A 23, 761 (2008).

[6] A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems (Dover Publications, Inc., New York, 2003).

[7] H.J. Weber and G.B. Arfken, Essential Mathematical Methods for Physicists (Elsevier Academic Press, San Diego, 2004).

[8] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, San Diego, 2000).

[9] Wolfram Research, Inc., Mathematica Edition: Version 5.2 (Wolfram Research, Inc., Champaign, 2005).

Recebido em 19/8/2011; Aceito em 6/8/2012; Publicado em 22/10/2012

2

A field ϕ(

x) is called a

scalar field, in contrast to a

vector or

tensor field, when under a Lorentz transformation

x

^µ →

x'

^µ = Λ

^µ

_ν

x

^ν, it transforms, trivially, according to ϕ(

x) → ϕ'(

x) = ϕ(Λ

^-1

x). The quantization of such fields gives rise to spin-0 particles (

scalar bosons), like the famous Higgs boson (a proposed elementary particle in the Standard Model of particle physics). However, it is a well-known fact that most particles in the universe have a nonzero intrinsic spin. These particles arise in field theory when we consider fields which transform non-trivially under Lorentz transformations. Indeed, fields with spin have more complicated transformation laws, since the various components of the fields rotate into one another under Lorentz transformations. A good example of such a field is the vector field A

^µ (

x) of electromagnetism, which transforms as A

^µ (

x) → Λ

^µ

_ν

A

^ν(Λ

^-1

x). In field theory, spin-1 particles are described as the quanta of vector fields. Such

vector bosons play a central role as the mediators of interactions (

force carriers) in particle physics. Another good example comes from the Dirac equation, whose quantization gives rise to spin-1/2 particles (

fermions).

Last but not least, we must stress that, for each kind of field, we may conceive field theories describing massive and massless particles. For instance, in the case of vector (spin-1) fields, we may cite as important examples the gauge fields of the electromagnetic, the weak, and the strong interactions, whose corresponding vector bosons are, respectively, massless photons, massive

W

^± and

Z

⁰ bosons, and massless gluons.

3

The MTM is described by the Lagrangian density

L =

(

iγ

_µ∂

^µ –

m)ψ –

(

γ

_µψ)(

γ

^µ ψ), where ψ is a two-component Dirac fermion field in (1+1)-dimensions, and γ

^µ are Dirac gamma matrices. Notice that the interaction is the only local interaction possible since the model involves only four fermion variables. It is well known that it can be mapped, by a method called bosonization, into the sine-Gordon model of a scalar field, whose dynamics is determined by

L =

∂

_µϕ ∂

^µ ϕ + 2αcosηϕ, where the couplings in the two models are related as

g = π(4π/η

² – 1);

m

ψ = - 2αcosηϕ. Both models have been extensively studied.

Publication Dates

• Publication in this collection
  04 Dec 2012
• Date of issue
  Sept 2012

History

• Received
  19 Aug 2011
• Accepted
  06 Aug 2012