Some Exact Green Function Solutions for Non-Linear Classical Field Theories

Abstract

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function.

1. Introduction

The availability of exact solutions to field theories, both classical and quantum, is important to a general understanding of these theories. Such solutions are very rare and, in most cases, are for unphysical situations. The harnessing of the functional renormalization group can help to mitigate such a limitation significantly [1,2,3]. Still, explicit solutions are not provided in this way.

Recently, an exact solution to some quantum field theory was proposed [4,5]. This means that all the correlation functions of the theory could, in principle, be evaluated exactly. In addition, explicit solutions were provided for the one-point (1P) and two-point (2P) correlation functions by using the properties of the Jacobian elliptic functions [6]. The appearance of such transcendental functions arises by the very nature of the partial differential equations (PDEs) that characterize their equations of motion, mirrored into their quantum counterpart. Exact solutions of this kind are obtained assuming a specific property of the higher-order correlation functions that are taken to be zero when evaluated to identical arguments. This yields all of them fully given by the 1P- and 2P-correlation functions, typically in a Gaussian type.

The idea can be traced back to recently proposed exact solutions obtained for nonlinear PDEs [7]. We show that this set of solutions is very well mapped to the solutions of the corresponding classical counterparts in a functional Taylor expansion in the source. In Ref. [8], we have shown that this series can correspond to a strong coupling expansion in the inverse of the coupling.

The aim of this paper is to give a sound mathematical understanding of such solutions, providing relevant new approaches to re-derive the main results. We also yield a complete solution to a quantum SU(2) Yang–Mills theory, explicitly providing a proof of the Gaussian nature of the kind of solution we obtained. This means that, in principle, we are able to evaluate all the averages in closed form, even though there could be no need to go to very high orders. For instance, in our case, we have a Gaussian distribution and we can limit our analysis to the mean and the covariance, even if all higher order cumulants could in principle be evaluated.

The main reason to obtain a closed-form solution arises from studies on the lattice of the gluon and ghost propagators for the Yang–Mills theory, mostly in the Landau gauge [9,10,11], and the corresponding spectrum [12,13] that shows unequivocally that a mass gap appears also when the interaction with fermions is neglected. Earlier theoretical analysis supported these results [5,14,15,16,17,18] by providing closed-form formulas for the gluon propagator. Quite recently, the set of Dyson–Schwinger equations for this case was solved, for the 1P- and 2P-correlation functions, and the spectrum was computed very accurately both in three and four dimensions [5,19,20], proving also confinement as a property of the theory [19,21]. These latter results are strongly linked to the solution of the quartic scalar field theory mapped onto the Yang–Mills theory, as shown in Refs. [5,18].

Indeed, our studies rely on Dyson–Schwinger equations in a PDE shape that appears to be the most sensible approach to treat a non-perturbative quantum field theory [22,23,24]. Indeed, Bender, Milton and Savage [25] proposed to derive the Dyson–Schwinger equations and treat them in differential form. This way to manage these equations was the one used to find the exact solution [5]. The approach turns out to be quite useful when a non-trivial solution of the equation for the 1P-correlation function, or the equation of motion, is known. Therefore, a complete solution to theories that normally are considered treatable only through perturbation methods can become available.

The paper is structured as follows. In Section 2, we introduce the theories we aim to analyze. In Section 3, we solve the classical theory of a massless quartic scalar field. In Section 4, we solve the classical equations for the SU(2) Yang–Mills theory. In Section 5, we show how such classical solutions map onto the corresponding statistical field theory of the scalar field. In Section 6, we present our conclusions.

2. Preliminaries

We consider a scalar field $\varphi$ with a single component and $Z_2$ symmetry with the action

$$S_{\varphi }=\int d^{4}x\left[{\displaystyle \frac{1}{2}}{(\partial \varphi )}^2-{\displaystyle \frac{\lambda }{4}}{\varphi }^4+j\varphi \right],$$

(1)

where j is an arbitrary source. Our aim is to solve the equation of motion

$${\partial }^2\varphi +\lambda {\varphi }^3=j,$$

(2)

where the constant $\lambda$ is kept to obtain an understanding of the solutions on the strength of the given coupling. Working in $d=4$ dimensions has the advantage of keeping such a constant dimensionless. We will show that this model can be completely solved using elliptic functions. Similarly, we evaluate the corresponding partition function

$$Z\left[j\right]=\int \left[d\varphi \right]e^{-S_{\varphi }},$$

(3)

showing how the classical solution extends to the quantum field theory through the evaluation of the correlation functions. The structure is of Gaussian type.

For the Yang–Mills theory, we limit our analysis to the SU(2) group. We will have the action

$$S_A=\int d^{4}x\left(-{\displaystyle \frac{1}{4}}F·F+j·A\right)$$

(4)

where $j_{\mu }$ is a generic source and an element of the su(2) algebra. The equation of motion is given by (Latin letters $a,b,c,\dots$ represent group indices taking the values $1,2,3$)

$$F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{ϵ}^{abc}{A}_{\mu }^b{A}_{\nu }^c$$

(5)

where g plays the same role as $\lambda$ before, leading to the equations of motion

$${\partial }^{\mu }{F}_{\mu \nu }^a+g{ϵ}^{abc}{A}^{b\mu }{F}_{\mu \nu }^c=j_{\nu }^a,$$

(6)

where ${ϵ}^{abc}=1$ for even permutations of the indexes $abc$, ${ϵ}^{abc}=-1$ for odd permutations, and ${ϵ}^{abc}=0$ if two or more indexes are equal. In the following, we will provide a solution to Equation (6). We will show how the solution can be obtained through the equations of the scalar field given a mapping theorem between these two theories [17,18].

3. Solution for the Scalar Field

3.1. Homogeneous Equation

Our ability to solve the full problem relies heavily on the solution of the homogeneous equation

$${\partial }^2{\varphi }_0+\lambda {\varphi }_0^3=0.$$

(7)

To solve this equation, we introduce the wave-like coordinate $\xi =p·x$, where p is the momentum four-vector. In terms of this coordinate, we have

$$p^2{\displaystyle \frac{d^2{\varphi }_0\left(\xi \right)}{d{\xi }^2}}+\lambda {\varphi }_0^3\left(\xi \right)=0.$$

(8)

The solution of this equation can be easily written down as

$${\varphi }_0\left(\xi \right)=asn(\xi +\theta ,i)$$

(9)

with $a^2\lambda /p^2=2$, where a and $\theta$ are integration constants and $sn(z,i)$ is Jacobi’s elliptic sine of elliptic modulus i. Due to its extensive use in literature, we fix $a^2=\sqrt{2/\lambda }{\mu }^2$ and are left with the dispersion relation

$$p^2={\mu }^2\sqrt{{\displaystyle \frac{\lambda }{2}}}.$$

(10)

Therefore, our solution of the homogeneous equation takes the form

$${\varphi }_0\left(x\right)=\mu {\left({\displaystyle \frac{2}{\lambda }}\right)}^{1/4}sn(p·x+\theta ,i).$$

(11)

From a physical point of view, the solution represents a non-linear wave moving at a speed $v=\left|p\right|/p_0<1$ with dispersion relation provided by Equation (10).

We note that the technique we used to obtain our solution is similar to that applied to obtain the Volkov solution of the Dirac equation [26]. The kind of solution we have obtained is indeed a Fubini–Lipatov instanton [27,28], as pointed out in our preceding works (see Ref. [29] and references therein) and firstly noted in Ref. [30] for the ground state of the Yang–Mills theory, quite different in its physical interpretation to the Volkov solution. Anyway, the choice of this solution is arbitrary. In our case, it grants an exact analytical solution for the given equation and for the Green function, which is all we need for our aims.

3.2. Functional Taylor Expansion

In order to find a general solution, we assume that the solution is a functional of the source j, $\varphi =\varphi \left[j\right]$. This functional is expanded in a functional Taylor series

$$\varphi \left[j\right]={\varphi }_0\left(x\right)+\sum _{k=1}^{\infty }\int C_k(x,x_1,\dots ,x_k)\prod _{l=1}^{k}j\left(x_l\right)d^4{x}_l.$$

(12)

Our aim will be to obtain the Green functions $C_k(x_1,\dots ,x_k)$, the analogous of the correlation functions in statistical field theory, in closed form. It is not difficult to see that

$$C_k(x,x_1,\cdots ,x_k)={\displaystyle \frac{{\delta }^k\varphi \left[j\right]}{\delta j\left(x_1\right)\dots \delta j\left(x_k\right)}}.$$

(13)

The general solution $\varphi \left[j\right]$ satisfies the equation of motion (2). Accordingly, for the Green functions $C_k$ at vanishing source $j=0$, we obtain the set of equations

$$\begin{array}{c}\hfill {\displaystyle {\partial }^2{\varphi }_0\left(x\right)+\lambda {\varphi }_0^3\left(x\right)=0,}\end{array}$$

(14)

$$\begin{array}{c}\hfill {\displaystyle {\partial }^2{C}_1(x,x_1)+3\lambda {\varphi }_0^2\left(x\right)C_1(x,x_1)={\delta }^4(x-x_1),}\end{array}$$

(15)

$$\begin{array}{l}{\displaystyle {\partial }^2{C}_2(x,x_1,x_2)+3\lambda {\varphi }_0^2\left(x\right)C_2(x,x_1,x_2)+6\lambda {\varphi }_0\left(x\right)C_1(x,x_1)C_1(x,x_2)=0,}\end{array}$$

(16)

$$\begin{array}{l}{\displaystyle {\partial }^2{C}_3(x,x_1,x_2,x_3)+3\lambda {\varphi }_0^2\left(x\right)C_3(x,x_1,x_2,x_3)}\\ \hspace{1em}\hspace{1em}+6\lambda {\varphi }_0\left(x\right)C_1(x,x_3)C_2(x,x_1,x_2)+6\lambda C_1(x,x_1)C_1(x,x_2)C_1(x,x_3)\hfill \\ \hspace{1em}\hspace{1em}+6\lambda {\varphi }_0\left(x\right)C_2(x,x_1,x_3)C_1(x,x_2)+6\lambda {\varphi }_0\left(x\right)C_1(x,x_1)C_2(x,x_2,x_3)=0,\dots \end{array}$$

(17)

From this set of equations, we easily recognize that $C_1(x,x_1)$ is the Green function of this non-linear problem. The corresponding PDE is linear and, therefore, in principle amenable to an analytic solution. It is also clear that the functions $\left\{C_k\right|k>1\}$ are all given by combinations of ${\varphi }_0$ and $C_1$. Therefore, the problem is completely solved if we are able to get a closed-form solution for these two functions. We already know ${\varphi }_0$. Thus, our aim is to determine $C_1$.

It is interesting to note the relevant difference to the linear case. In such a case, all the functions $C_n$ with $n>1$ are 0. Thus, it is easy to conclude that nonlinear wave propagation brings into existence all these higher order Green functions. This is very similar to the application of the Fourier series to a nonlinear wave equation that gives rise to higher order harmonics from a single mode plane wave. Higher order correlation functions acquire a well-defined meaning in a statistical sense, as we will see below.

3.3. Green Function

We have to solve the equation

$${\partial }^2{C}_1(x,x_1)+3\lambda {\varphi }_0^2\left(x\right)C_1(x,x_1)={\delta }^4(x-x_1),$$

(18)

where

$${\varphi }_0\left(x\right)=\mu {\left({\displaystyle \frac{2}{\lambda }}\right)}^{1/4}sn(p·x+\theta ,i).$$

(19)

It is easy to write down a solution for the associate homogeneous equation given by

$${\varphi }_h\left(x\right)=acn(p·x+\theta ,i)dn(p·x+\theta ,i),$$

(20)

where cn and dn are Jacobian elliptic functions of elliptic modulus i. Observing that

$${\displaystyle \frac{d}{dz}}sn(z,i)=cn(z,i)dn(z,i)$$

(21)

and

$$sn(z,i)={\displaystyle \frac{2\pi }{K\left(i\right)}}\sum _{n=0}^{\infty }{(-1)}^n{\displaystyle \frac{e^{-\left(n+\frac{1}{2}\right)\pi }}{1+e^{-(2n+1)\pi }}}sin\left((2n+1){\displaystyle \frac{\pi z}{2K\left(i\right)}}\right),$$

(22)

the solution of the homogeneous solution can be written as a Fourier series,

$${\varphi }_h\left(x\right)={\displaystyle \frac{{\pi }^2}{K^2\left(i\right)}}\sum _{n=0}^{\infty }{(-1)}^n(2n+1){\displaystyle \frac{e^{-\left(n+\frac{1}{2}\right)\pi }}{1+e^{-(2n+1)\pi }}}cos\left((2n+1){\displaystyle \frac{\pi }{2K\left(i\right)}}(p·x+\theta )\right).$$

(23)

The choice $\theta =(4m+1)K\left(i\right)$ with $m\in \mathbb{Z}$ for the phase grants that the function ${\varphi }_0\left(x\right)$ is zero on the light cone. In quantum field theory, this solution would immediately give the Feynman propagator. In classical field theory, we have to use the Laplace–Fourier transform to get the initial conditions right. We want to preserve the causal behavior of the Feynman propagator in quantum field theory by fixing boundary conditions such that positive energy solutions propagate forward in time and negative energy solutions propagate backward in time, as our aim is to put this Green function to work in a fully quantum formulation of the theory. In order to work out this aim, we calculate

$$C_1(\omega ,x)={\int }_0^{\infty }dt{e}^{-i(\omega +i\epsilon )t}{\varphi }_h\left(x\right),$$

(24)

where $\epsilon >0$ is a small quantity, which is sent to zero in the end in order to obtain the correct (Feynman) integration path. The integration gives

$$C_1(\omega ,x)={\displaystyle \frac{1}{2}}\sum _{n=0}^{\infty }{A}_n\left[{\displaystyle \frac{e^{i(2n+1)\frac{\pi }{2K\left(i\right)}p·x}}{(\omega -(2n+1)\frac{\pi }{2K\left(i\right)}{p}_0)+i\epsilon }}-{\displaystyle \frac{e^{-i(2n+1)\frac{\pi }{2K\left(i\right)}p·x}}{(\omega +(2n+1)\frac{\pi }{2K\left(i\right)}{p}_0)+i\epsilon }}\right],$$

(25)

where

$$A_n={\displaystyle \frac{{\pi }^2}{K^2\left(i\right)}}(2n+1){\displaystyle \frac{e^{-\left(n+\frac{1}{2}\right)\pi }}{1+e^{-(2n+1)\pi }}}.$$

(26)

We can perform a Fourier transform in space and use the condition $p^2=m^2$ to obtain

$$\begin{array}{ccc}\hfill C_1\left(k\right)& =& {\displaystyle \frac{1}{2}}\sum _{n=0}^{\infty }{A}_n[{\displaystyle \frac{{\delta }^3(k+(2n+1)\frac{\pi }{2K\left(i\right)}p)}{(\omega -(2n+1)\frac{\pi }{2K\left(i\right)}\sqrt{p^2+m^2})+i\epsilon }}\hfill \\ & & \phantom{(}-{\displaystyle \frac{{\delta }^3(k-(2n+1)\frac{\pi }{2K\left(i\right)}p)}{(\omega +(2n+1)\frac{\pi }{2K\left(i\right)}\sqrt{p^2+m^2}))+i\epsilon }}].\hfill \end{array}$$

(27)

Finally, we integrate out the arbitrary three-momentum p in order to get to the Feynman propagator (which is explicitly given only in full momentum space), using the covariant integration $\int d^{3}p/2E_p$ with $E_p=\sqrt{p^2+m^2}$, to obtain

$$\begin{array}{ccc}\hfill C_1\left(k\right)& =& {\displaystyle \frac{1}{4\sqrt{k^2/{\left((2n+1)\frac{\pi }{2K\left(i\right)}\right)}^2+m^2}}}\sum _{n=0}^{\infty }{A}_n\times \phantom{(}\hfill \\ & & \phantom{(}[{\displaystyle \frac{1}{(\omega -(2n+1)\frac{\pi }{2K\left(i\right)}\sqrt{k^2/{\left((2n+1)\frac{\pi }{2K\left(i\right)}\right)}^2+m^2})+i\epsilon }}\hfill \\ & & \phantom{(}-{\displaystyle \frac{1}{(\omega +(2n+1)\frac{\pi }{2K\left(i\right)}\sqrt{k^2/{\left((2n+1)\frac{\pi }{2K\left(i\right)}\right)}^2+m^2})+i\epsilon }}].\hfill \end{array}$$

(28)

For the Feynman propagator, we have

$$G(x-y)={\displaystyle \frac{1}{4\pi }}\delta \left({\tau }_{xy}^2\right)+{\varphi }_h(x-y),$$

(29)

where ${\tau }_{xy}^2={(x_0-y_0)}^2-{(x_1-y_1)}^2-{(x_2-y_2)}^2-{(x_3-y_3)}^2$. A simple verification of this result, normally not reported in textbooks, can be found at, e.g., https://physics.stackexchange.com/q/615102 (accessed on 8 November 2024). Using $\square \delta \left({\tau }_{xy}^2\right)=4\pi {\delta }^4(x-y)$ and $\theta$ such that ${sn}^2(p·(x-y)+\theta ,i)\delta \left({\tau }_{xy}^2\right)=\delta \left({\tau }_{xy}^2\right)$, that is zero outside of the light cone, we can indeed show that this propagator solves the PDE for the Green function. As both $G(x-y)$ and $C_1(x,y)$ are solving the same PDE, we can conclude that $C_1(x,y)=G(x-y)$.

3.4. Higher Order Green Functions

Having calculated the Green function $C_1$ makes it easy to get all the higher order Green functions. For completeness, we can write a couple of them as

$$\begin{array}{l}{\displaystyle C_2(x,y,z)=-6\lambda \int d^{4}w{C}_1(x,w){\varphi }_0\left(w\right)C_1(w,y)C_1(w,z),}\end{array}$$

(30)

$$\begin{array}{l}{\displaystyle C_3(x,y,z,w)=-6\lambda \int d^{4}v{C}_1(x,v){\varphi }_0\left(v\right)C_1(v,y)C_2(v,z,w)}\\ \hspace{1em}\hspace{1em}-6\lambda \int d^{4}v{C}_1(x,v)C_1(v,y)C_1(v,z)C_1(v,w)-6\lambda \int d^{4}v{C}_1(x,v){\varphi }_0\left(v\right)C_2(v,y,z)C_1(v,w)\\ \hspace{1em}\hspace{1em}-6\lambda \int d^{4}v{C}_1(x,v){\varphi }_0\left(v\right)C_1(v,y)C_2(v,z,w),\dots \hfill \end{array}$$

(31)

As previously stated, these higher order Green functions are obtained by the combination of ${\varphi }_0$ and $C_1$ and nothing else. Therefore, we conclude that the problem is completely solved by knowing these latter functions.

4. Solution for the Yang–Mills Field

If we consider the current expansion

$$A_{\mu }^a\left[j\right]=A_{\mu }^{a\left(0\right)}\left(x\right)+\sum _{k=1}^{\infty }\int C_{{\mu }_1\dots {\mu }_k}^{a_1\dots a_k\left(k\right)}(x,x_1,\dots ,x_k)\prod _{l=1}^k{j}^{a_l{\mu }_l}\left(x_l\right)d^4{x}_l,$$

(32)

the zeroth order term solves exactly the homogeneous equation

$$\begin{array}{l}{\displaystyle {\partial }^{\mu }({\partial }_{\mu }{A}_{\nu }^{a\left(0\right)}-{\partial }_{\nu }{A}_{\mu }^{a\left(0\right)}+g{ϵ}^{abc}{A}_{\mu }^{b\left(0\right)}{A}_{\nu }^{c\left(0\right)})}\\ \hspace{1em}\hspace{1em}+g{ϵ}^{abc}{A}^{b\left(0\right)\mu }({\partial }_{\mu }{A}_{\nu }^{c\left(0\right)}-{\partial }_{\nu }{A}_{\mu }^{c\left(0\right)}+g{ϵ}^{cde}{A}_{\mu }^{d\left(0\right)}{A}_{\nu }^{e\left(0\right)})=0,\hfill \end{array}$$

(33)

once we select the Lorenz gauge [5]. The connection to the specific set of solutions can be accomplished by introducing the mixed symbols for SU(2) ($a=1,2,3$ is the group index)

$${\eta }_{\mu }^1=(0,1,0,0),{\eta }_{\mu }^2=(0,0,1,0),{\eta }_{\mu }^3=(0,0,0,1),$$

(34)

and using a mapping between a scalar field and the Yang–Mills theory [18]

$$A_{\mu }^{a\left(0\right)}\left(x\right)={\eta }_{\mu }^a{\varphi }_0\left(x\right),$$

(35)

where ${\varphi }_0\left(x\right)$ is the solution provided for the scalar field. By calculating the variation with respect to $j_{\rho }^f$, we derive the equations of motion (6), in Lorenz gauge yielding

$$\begin{array}{l}{\displaystyle {\partial }^2{C}_{\nu \rho }^{af\left(1\right)}(x,y)+g{ϵ}^{abc}{C}_{\mu \rho }^{bf\left(1\right)}(x,y){\partial }^{\mu }{A}_{\nu }^{c\left(0\right)}\left(x\right)+g{ϵ}^{abc}{A}_{\mu }^{b\left(0\right)}\left(x\right){\partial }^{\mu }{C}_{\nu \rho }^{cf\left(1\right)}(x,y)}\\ \hspace{1em}\hspace{1em}+g{ϵ}^{abc}{C}_{\rho }^{bf\left(1\right)\mu }(x,y){\partial }_{\mu }{A}_{\nu }^{c\left(0\right)}\left(x\right)-g{ϵ}^{abc}{A}^{b\left(0\right)\mu }\left(x\right){\partial }_{\nu }{C}_{\mu \rho }^{cf\left(1\right)}(x,y)\hfill \\ \hspace{1em}\hspace{1em}+g^2{ϵ}^{abc}{ϵ}^{cde}{C}_{\rho }^{bf\left(1\right)\mu }(x,y)A_{\mu }^{d\left(0\right)}\left(x\right)A_{\nu }^{e\left(0\right)}\left(x\right)+g^2{ϵ}^{abc}{ϵ}^{cde}{A}^{b\left(0\right)\mu }\left(x\right)C_{\mu \rho }^{df\left(1\right)}(x,y)A_{\nu }^{e\left(0\right)}\left(x\right)\hfill \\ \hspace{1em}\hspace{1em}+g^2{ϵ}^{abc}{ϵ}^{cde}{A}^{b\left(0\right)\mu }\left(x\right)A_{\mu }^{d\left(0\right)}\left(x\right)C_{\nu \rho }^{ef\left(1\right)}(x,y)={\eta }_{\nu \rho }{\delta }_{af}{\delta }^4(x-y).\hfill \end{array}$$

(36)

Using the mapping theorem for SU(2), this equation collapses to

$${\partial }^2{C}_1(x,y)+6g^2{\varphi }_0^2\left(x\right)C_1(x,y)={\delta }^4(x-y),$$

(37)

and we are back to the scalar field case, provided we choose $\lambda =2g^2$, the factor 2 being a Casimir operator of SU(2). Therefore, from this point on, everything is obtained in an identical way. Just in order to complete our derivation, we write down the final formula for the classical SU(2) Yang–Mills theory in Lorenz gauge,

$$C_{\mu \nu }^{ab}\left(k\right)={\delta }_{ab}\left({\eta }_{\mu \nu }-{\displaystyle \frac{k_{\mu }{k}_{\nu }}{k^2}}\right)C_1\left(k\right),$$

(38)

where $C_1\left(k\right)$ is the Green function we computed for the scalar field theory. We emphasize that sets of mixed symbols like those given in Equation (34) can be provided in principle for any gauge group, making this kind of classical solution easily generalized.

5. Partition Function of the Scalar Field

We have seen that we are able to solve exactly some non-linear classical theory, provided we are able to compute the solution of the homogeneous equation and the Green function of the first order. A functional series expansion gives the solution one is looking for at any desired order. This way to find exact solutions is very similar to what is achieved in statistical or quantum field theory. If the whole set of correlation functions is computable in closed form at any order one could claim in principle to have solved a given theory exactly. In most common applications, generally provided by realistic models that are mostly given by non-linear PDEs, this is not possible except for the free case. Therefore, one could ask how to extend the above exact solution for the scalar field to statistical field theory. In order to understand this, let us consider a generic scalar field theory having a partition function

$$Z\left[j\right]=\int \left[d\varphi \right]e^{-S\left[\varphi \right]-\int j\varphi d^{4}x}.$$

(39)

We will have a set of Schwinger functions

$$S_n(x_1,\dots ,x_n)={\displaystyle \frac{1}{Z}}{\displaystyle \frac{{\delta }^{n}Z}{\delta j\left(x_1\right)\dots \delta j\left(x_n\right)}}{|}_{j=0}=Z^{-1}\int \left[d\varphi \right]\varphi \left(x_1\right)\dots \varphi \left(x_n\right)e^{-S\left[\varphi \right]-\int j\varphi d^{4}x}{|}_{j=0}.$$

(40)

These functions yield the connected correlation functions

$$G_n(x_1,\dots ,x_n)={\displaystyle \frac{{\delta }^{n}lnZ}{\delta j\left(x_1\right)\dots \delta j\left(x_n\right)}}{|}_{j=0}.$$

(41)

The relations

$$\begin{array}{ccc}\hfill S_1\left(x_1\right)& =& G_1\left(x_1\right),\hfill \\ \hfill S_2(x_1,x_2)& =& G_2(x_1,x_2)+G_1\left(x_1\right)G_1\left(x_2\right),\hfill \\ \hfill S_3(x_1,x_2,x_3)& =& G_3(x_1,x_2,x_3)+G_2(x_1,x_2)G_1\left(x_3\right)+G_2(x_1,x_3)G_1\left(x_2\right)\hfill \\ & & \phantom{(}+G_2(x_2,x_3)G_1\left(x_1\right)+G_1\left(x_1\right)G_1\left(x_2\right)G_1\left(x_3\right),\dots \hfill \end{array}$$

(42)

hold between Schwinger and connected functions. We say that a theory will admit a Gaussian solution if all the connected functions (cumulants) of order greater than 2 can be computed through the cumulants of order 1 and 2. This is very well known from statistics. These Gaussian solutions completely map onto those we have found for the classical theories. A trivial example of a Gaussian solution is found in a free massive scalar theory. In our case, we aim to provide some non-trivial examples.

If we do not care too much about translation invariance that might or might not be broken by nature, a straightforward way to obtain such Gaussian solutions for a given statistical field theory is by solving the Dyson–Schwinger set of equations, assuming that higher-order connected correlation functions ($n>2$) are zero if two or more arguments coincide. In such a case, we are in principle able to completely solve the set of equations, as shown in Ref. [5], because each equation becomes independent from the higher order ones in the set. This class of Gaussian solutions of the Dyson–Schwinger set of equations for a statistical field theory could prove to be relevant in physics in the case where agreement with lattice and/or experimental data is achieved. In all other cases, it is just a technique to obtain some closed-form analytical solutions, which also have significant pedagogical meaning for realistic theories where such solutions are currently missing. The solutions of the Dyson–Schwinger equations completely map the classical solutions.

For the sake of completeness, we list here the first few Dyson–Schwinger equations for a massless quartic scalar theory [5], showing how these equations match the classical equations we presented before.

$$\begin{array}{c}{\displaystyle {\partial }^2{G}_1\left(x\right)+\lambda \left[G_1^3\left(x\right)+3G_2(x,x)G_1\left(x\right)+G_3(x,x,x)\right]=0,}\end{array}$$

(43)

$$\begin{array}{l}{\displaystyle {\partial }^2{G}_2(x,y)+\lambda \left[3\right(G_1^2\left(x\right)G_2(x,y)+3G_2(x,x)G_2(x,y)}\\ \hspace{1em}\hspace{1em}+3G_3(x,x,y)G_1\left(x\right)+G_4(x,x,x,y)]={\delta }^4(x-y),\end{array}$$

(44)

$$\begin{array}{l}{\displaystyle {\partial }^2{G}_3(x,y,z)+\lambda [6G_1\left(x\right)G_2(x,y)G_2(x,z)+3G_1^2\left(x\right)G_3(x,y,z)}\\ \hspace{1em}\hspace{1em}+3G_2(x,z)G_3(x,x,y)+3G_2(x,y)G_3(x,x,z)+3G_2(x,x)G_3(x,y,z)\hfill \\ \hspace{1em}\hspace{1em}+3G_1\left(x\right)G_4(x,x,y,z)+G_5(x,x,x,y,z)]=0,\hfill \end{array}$$

(45)

$$\begin{array}{l}{\displaystyle {\partial }^2{G}_4(x,y,z,w)+\lambda [6G_2(x,y)G_2(x,z)G_2(x,w)+6G_1\left(x\right)G_2(x,y)G_3(x,z,w)}\\ \hspace{1em}\hspace{1em}+6G_1\left(x\right)G_2(x,z)G_3(x,y,w)+6G_1\left(x\right)G_2(x,w)G_3(x,y,z)+3G_1^2\left(x\right)G_4(x,y,z,w)\hfill \\ \hspace{1em}\hspace{1em}+3G_2(x,y)G_4(x,x,z,w)+3G_2(x,z)G_4(x,x,y,w)+3G_2(x,w)G_4(x,x,y,z)\hfill \\ \hspace{1em}\hspace{1em}+3G_2(x,x)G_4(x,y,z,w)+3G_1\left(x\right)G_5(x,x,y,z,w)+G_6(x,x,x,y,z,w)]=0,\dots \hfill \end{array}$$

(46)

It is easy to see that except for the quantum correction arising from $G_2(x,x)$, the choice $\{G_k(x,x,\dots )=0|k>2\}$ completely maps the above set to the classical solution. In quantum field theory, the quantum correction entails a renormalization technique to be evaluated. The final result is consistent and yields a Gaussian solution for the theory.

It should be pointed out that non-Gaussian solutions are also possible, even if they are not exact like this one. This crucially depends on the choice of the ground state solution of $G_1$, which is the choice of the solutions of the PDE for the 1P-correlation function. A well-known example is the perturbative solution that chooses $G_1$ as a constant. Our exact solution can also be taken as an unperturbed starting solution to evaluate further non-Gaussian corrections as for quantum chromodynamics.

6. Conclusions

We have provided exact Green function solutions to classical nonlinear field theories and have shown that such solutions map very well onto the corresponding analysis through Dyson–Schwinger equations in statistical field theory. These special solutions represent non-trivial Gaussian solutions for the latter. It is enough to know the first two correlation functions (the exact solution of the homogeneous equation and the Green function for the classical theory) to perform in principle whatever computation in the given theory. This also provides a generic way to perform computations in classical field theory in the case where nonlinear PDEs are involved and some solutions of the corresponding homogeneous equations are known.

An interesting field of applications for this technique would be black hole physics [31,32,33]. Indeed, something in this direction has been carried out by one of us (M.F.) [34,35]. We will exploit this application further in our future works.