\documentstyle[aps]{revtex}
\begin{document}
\title{An \lq\lq Adiabatic" Approximation to Quantum Field Theory}
\author{J. L. Cort\'es$^{1}$\thanks{E-mail: cortes@posta.unizar.es},
J. Gamboa$^{2}$\thanks{ E-mail: jgamboa@lauca.usach.cl},
S. Lepe$^{2,3}$\thanks{E-mail: slepe@lauca.usach.cl} and
J. Lopez-Sarri\'on $^{1}$\thanks{E-mail: jlopez@dftuz.unizar.es}}
\address{$^1$Departamento de F\'{\i}sica Te\'orica,  Universidad de Zaragoza,
Zaragoza 50009, Spain.}
\address{$^2$Departamento de F\'{\i}sica, Universidad de Santiago de Chile,
Casilla 307, Santiago 2, Chile.}
\address{$^3$Instituto F\'{\i}sica, Universidad Cat\'{o}lica de
Valpara\'{\i}so, Casilla 4059, Valpara\'{\i}so, Chile.}

\maketitle
\begin{abstract}
A new approach to Quantum Field Theory (QFT), based on the extension of a 
reformulation of the adiabatic approximation to some quantum mechanical 
systems, is presented. A novel  non-analytic contribution to the efective 
fermionic action in non-Abelian gauge theories is identified. The possible 
interpretation of this contribution as a violation of the decoupling theorem 
in QFT is discussed. This approach can be an starting point for a new 
understanding of non-perturbative properties in QFT and their dependence on 
temperature and density.
\end{abstract}
\pacs{11.15.Tk, 12.38.Aw, 12.38.Lg}
\preprint{USACH-FM-02-06}
\preprint{DFTUZ-02-03}
 \section{Introduction}

 The adiabatic approximation is one of the most important methods going beyond perturbation theory in quantum mechanics. In QFT,  the necessity of non-perturbative methods is clear in many cases (low energy limit of asymptotically free theories, high energy limit of infrared safe theories). Unfortunately, the attempts to translate the adiabatic approximation to QFT have been very limited and the main results are the identification of Wess-Zumino terms and anomalies as geometric phases \cite{stone}. The complexity of QFT (a quantum mechanical system with infinite degrees of freedom) has been an obstacle to the possible use of the adiabatic approximation as the starting point to an alternative to the perturbative expansion. This paper is an attempt in this direction.
 In the next section we take as a starting point a very simple quantum mechanical system, a spin coupled to a time dependent magnetic field, and  the adiabatic approximation is reformulated in an appropriate way to be generalized to other systems. In section III, we point out the difficulties found when one tries a generalization of this reformulation in QFT. In section IV, a possible way
 to circumvent these problems is presented and our proposal is applied to an $SU(2)$ gauge invariant theory with a fermion matter field in the fundamental representation.  The leading term in the adiabatic approximation is determined and its possible relation with non-perturbative properties of the theory are discussed at a qualitative level. Finally in section V, the finite density and temperature effects are also considered.

 \section{A quantum mechanical example}

 In order to formulate the adiabatic aproximation in QFT, let us first
discuss it at the level of quantum mechanics in the most simple non-trivial
example, namely, a half-integer  spin ($j$) coupled to an external magnetic field (${\vec B}(t)$) varying periodically in time (${\vec B}(T)={\vec B}(-T)$). Let us review the well known solution of this problem \cite{berry}. One has a hamiltonian
 \begin{equation}
H \, = \, \sum_{\alpha,\beta} {\vec B}(t)\cdot {\vec J}_{\alpha\beta} a^{\dagger}_{\alpha} a_{\beta},
\label{H1}
 \end{equation}
\noindent where $\alpha,\beta = -j, -j+1,...,j-1,j$ and $a$, $a^{\dagger}$ are operators satisfying the (anti)commutation relations
\begin{equation}
\lbrace a_{\alpha}, a^{\dagger}_{\beta}\rbrace \, = \delta_{\alpha\beta}.
\label{a,adagger}
\end{equation}

\noindent We want to determine the probability amplitude that the system remains in the ground state ($|0>$) during time evolution, {\it i.e.}, the matrix element $<0(T)|0(-T)>$. In order to do this calculation it is convenient to use at each time,  the direction of the magnetic field as the  spin quantization axis to rewrite the hamiltonian as
\begin{equation}
H(t) \, = \, \sum_{m=-j}^{j} m B(t) a^{\dagger}_m(t) a_m(t).
\label{H2}
\end{equation}
The ground state of the system, $|0(t)>$ is the state satisfying the conditions
\begin{equation}
a_m(t) |0(t)> = 0 \:\:\: \mbox{for} \:\:\: m>0
{\hskip 2cm}
a^{\dagger}_m(t) |0(t)> = 0 \:\:\: \mbox{for} \:\:\: m<0.
\label{0(t)}
\end{equation}
\noindent In the adiabatic approximation one has two contributions to the matrix element $<0(T)|0(-T)>$, one due to the energy of the ground state
\begin{equation}
E_0 (t) \, = \, \sum_{m<0} m B(t) \, = \, - \frac{(j+1/2)^2}{2} B(t),
\label{E0}
\end{equation}
\noindent and a contribution due to the time evolution of the phase of the ground state $|0(t)>$
\begin{equation}
\int_{-T}^{T} dt <0(t)|i \partial_t|0(t)> \, = \, - {\cal I}m \int d{\vec S} \sum_{|k(t)>\neq|0(t)>} \frac{<0(t)|\nabla_{{\vec B}} H |k(t)> \wedge
<k(t)|\nabla_{{\vec B}} H |0(t)>}{(E_k - E_0)^2}.
\end{equation}
\noindent Stoke's theorem has been applied to rewrite the time integral as a surface integral in magnetic field space and the matrix elements of the gradient of the hamiltonian can be directly read from the matrix representation of the angular momentum operator.

After some straightforward algebra one finds
\begin{equation}
\int_{-T}^{T} dt <0(t)|i \partial_t|0(t)> \, = \, \frac{(j+1/2)^2}{2} \Omega[{\hat B}],
\label{Berry}
\end{equation}
\noindent where
\begin{equation}
\Omega [{\hat B}] \, = \, \int d{\vec S}\cdot \frac{{\vec B}}{B^3},
\end{equation}
\noindent is the solid angle that ${\vec B}$ subtends. This is the standard derivation of the adiabatic approximation including
Berry's phase (\ref{Berry}) \cite{berry}.

\noindent Let us now see how these results can be rederived in an alternative formulation based on a Grassmann path integral representation of the evolution,  which will be useful in the discussion of QFT in section IV. One can represent the operators $a$, $a^\dagger$ by Grassmann variables $\psi$, ${\bar \psi}$ and the probability amplitude that the system remains in the ground state by a Euclidean Grassmann path integral,
\begin{equation}
\lim_{T\to \infty} <0(T)|0(-T)> \, = \, \frac{\int d\psi d{\bar \psi} e^{-S}}{\int d\psi d{\bar \psi} e^{-S}|_{{\vec B}=0}} \, = \, e^{-\Gamma[{\vec B}]},
\label{GammaB}
\end{equation}
\noindent where the Euclidean action for the fermionic system is
\begin{equation}
S \, = \, \int d\tau {\bar \psi} (\partial_{\tau} + {\vec J}.{\vec B}) \psi.
\label{Se}
\end{equation}
In order to simplify the action, one introduces new Grassmann variables $c_m$, ${\bar c}_m$ through the expansion
\begin{equation}
\psi = \sum_{m=-j}^{j} c_m f_m
{\hskip 2cm}
{\bar \psi}= \sum_{m=-j}^{j} f_m^{\dagger} {\bar c_m},
\end{equation}
\noindent where $f_m(t)$ are the eigenspinors defined by,
\begin{equation}
{\vec J}\cdot {\vec B} f_m \, = \, m B f_m.
\label{fmB}
\end{equation}
\noindent The action as a function of the new variables is given by,
\begin{equation}
S \, = \, \int d\tau \lbrace\sum _m \left[{\bar c}_m (\partial_{\tau} + m B) c_m \right] + \sum_{m,m^{'}}  {\bar c}_m i {\cal A}_{m,m^{'}}  c_{m^{'}} \rbrace,
\end{equation}
\noindent where $i {\cal A}_{m,m^{'}} =  f_m^{\dagger} \partial_{\tau} f_{m^{'}}$. In the adiabatic approximation, one neglects off-diagonal terms
($m\neq m^{'}$) and the path integral becomes a product of independent integrals
\begin{equation}
e^{- \Gamma^{(ad)}[{\vec B}]}\,  = \, \prod_{m=-j}^{j} \int dc_m d{\bar c}_m e^{- \int_{-\beta/2}^{\beta/2} d\tau
{\bar c}_m (\partial_{\tau} + i {\cal A}_m + m B) c_m},
\label{GammaadB}
\end{equation}
\noindent with ${\cal A}_m = {\cal A}_{m,m}$ and the limits on the Euclidean time ($\tau =  i t$) incorporate finite temperature ($1/\beta$) effects in the imaginary time formalism (\cite{kapusta}).

\noindent The integral on each pair of variables ($c_m$, ${\bar c}_m$),  is a standard quantum mechanical determinant (\cite{Dunne})
\begin{equation}
\det [\partial_{\tau} + i {\cal A}_m(\tau) + m B(\tau)] \, = \, {\cal N} \, \cosh \left[\frac{\beta}{2} (m {\tilde B} + i {\tilde{\cal A}}_m\right],
\label{detqm}
\end{equation}
\noindent where ${\cal N}$ is an infinite constant that will cancel in the ratio of Grassmann integrals in (\ref{GammaB}) and  we have introduced the notation
\begin{equation}
{\tilde f} \, = \, \frac{1}{\beta} \int _{-\beta/2}^{\beta/2} d\tau \, f(\tau).
\label{tilde}
\end{equation}
\noindent In the zero temperature limit the quantum mechanical determinants (\ref{detqm}) become exponentials and the effective fermionic action in the adiabatic approximation takes the simple form
\begin{equation}
- \Gamma^{(ad)}[{\vec B}] \, = \, - \frac{i}{2} \sum_m \left( |m| \int dt B(t) + \frac{m}{|m|} \int dt {\cal A}_m(t) \right),
\label{GammaadB2}
\end{equation}
\noindent where the first term can be recognized as the dynamical phase $-i\int dt E_0(t)$ with $E_0$ the energy of the ground state of the quantum mechanical system (\ref{E0}). The second term in (\ref{GammaadB2}) reproduces Berry's phase as one can show by using once more Stokes theorem and the definition of $f_m$ in (\ref{fmB}), {\it i.e.}
\begin{equation}
\int dt i {\cal A}_m (t) \, = \, \int dt f_m^{\dagger} \partial_t f_m \, = \, - i m \Omega[{\hat B}],
\label{Am}
\end{equation}
\noindent with $\Omega[{\hat B}]$ the solid angle that the magnetic field subtends in its evolution.

The adiabatic approximation to this simple quantum mechanical system in the Grassmann path integral representation,  will reappear as an ingredient in some approximation to a QFT with fermionic fields as we will see later. Also this reformulation of the adiabatic approximation  is interesting because it allows to go beyond the zero temperature limit by using the quantum mechanical determinants in (\ref{detqm}).

 \section{Formal direct approach and its difficulties}

 The purpose of this section is to introduce a direct extension to QFT of the reformulation of the previous section. Before doing that we will show the problems of a direct implementation of the adiabatic approximation.

 The most natural way to formulate the adiabatic approximation and the related Born-Oppenheimer approach in QFT is based on the use of the Schr\"odinger representation,  where the wave function of quantum mechanics is replaced by a functional in the space of field configurations. After this replacement,  all the standard results of the adiabatic quantum mechanical expansion can be applied directly in QFT \cite{Ralston}. This formulation gives a new perspective of the anomaly in chiral gauge theories which appears as a geometric phase in the space of gauge field configurations related to the gauge non-invariance of the phase of the fermionic Fock states \cite{Niemi}. Unfortunately,  a quantum mechanical system with
infinite degrees of freedom is too complicated to go beyond the study of a few topological properties and the adiabatic approximation remains as a reformulation of the theory at a formal level.

An alternative way to implement the adiabatic formulation is based on a direct use of the reformulation of the quantum mechanical example of the previous section in a QFT system. For definiteness,  let us consider the action in Euclidean space
for a fermionic field $\Psi$,

\begin{equation}
{\cal S} = \int d^D x \, \left\lbrace {\bar \Psi} \gamma_\mu
\left(\partial_\mu + i e A_\mu\right) \Psi \right\rbrace,
\label{S1}
\end{equation}
\noindent where $A_\mu$ is a vector field which can be dynamical (gauge theory) or an auxiliary field (introduced to linearize a four-fermion interaction).

 Using the decomposition
 \begin{equation}
 \Psi = \left(\matrix{\Psi_R\cr \Psi_L}\right)
  {\hskip 2cm}
 {\bar \Psi} = \left( \Psi^{\dagger}_L \;\; \Psi^{\dagger}_R \right),
 \label{psiLR}
 \end{equation}
 \noindent for the fermionic field in a representation where all gamma matrices are off-diagonal, one has
 \begin{equation}
 {\cal S} = \int d^4 x \, \left\lbrace \Psi^{\dagger}_L \left(\partial_4 +
 ieA_4\right) \Psi_L
 + \Psi^{\dagger}_R \left(\partial_4 + ieA_4\right) \Psi_R
 + \Psi^{\dagger}_L {\vec \sigma}.\left(-i{\vec \nabla} + e{\vec A}\right)
 \Psi_L - \Psi^{\dagger}_R {\vec \sigma}.\left(-i{\vec \nabla} +
 e{\vec A}\right) \Psi_R \right\rbrace.
 \label{S2}
 \end{equation}

 Following the steps of the previous section,  we introduce the eigenfunctions $\Phi_n ({\vec x})$
\begin{equation}
\left[{\vec \sigma}.\left(-i{\vec \nabla} + e{\vec A}\right)\right]
\;  \Phi_n ({\vec x}) = \epsilon_n \Phi_n ({\vec x}).
\label{phi_n}
\end{equation}
\noindent These eigenfunctions and the eigenvalues $\epsilon_n$ are,
in fact,  functionals of the gauge field at a given time and a more
precise notation for them is $\Phi_n[{\vec A}(\tau)]({\vec x})$ and
$\epsilon_n[{\vec A}(\tau)]$.

Next step is to use the decomposition of the fermionic fields in terms
of the eigenfunctions in (\ref{phi_n}),
\begin{equation}
\Psi_L({\vec x},\tau) =  \sum_n c_{L_n} (\tau) \;
 \Phi_n[{\vec A}(\tau)]({\vec x}) {\hskip 2cm}
\Psi_R({\vec x},\tau) =  \sum_n  c_{R_n} (\tau) \;
 \Phi_n[{\vec A}(\tau)]({\vec x})
\label{c}
\end{equation}
\begin{equation}
\Psi_L^{\dagger}({\vec x},\tau) = \sum_n c_{L_n}^{\dagger} (\tau)   \;
 \Phi_n^{\dagger}[{\vec A}(\tau)]({\vec x}) {\hskip 2cm}
\Psi_R^{\dagger}({\vec x},\tau) =  \sum_n  c_{R_n}^{\dagger} (\tau) \;
 \Phi_n^{\dagger}[{\vec A}(\tau)]({\vec x}).
\label{cdagger}
\end{equation}

Using the orthogonality of the eigenfunctions $\Phi_n$, the action
takes a compact form in terms of the (Grassman) coeficients $c_L$, $c_R$
\begin{eqnarray}
 {\cal S} \, = \int d\tau &\lbrace & \sum_n \left[c_{L_n}^{\dagger} (\tau)
\partial_{\tau} c_{L_n} (\tau) +
 c_{L_n}^{\dagger} (\tau)\epsilon_n c_{L_n} (\tau) +
i  c_{L_n}^{\dagger} (\tau) {\cal A}_n c_{L_n} (\tau)\right]
\nonumber \\
&+& \sum_n  \left[c_{R_n}^{\dagger} (\tau)
\partial_{\tau} c_{R_n} (\tau) -
 c_{R_n}^{\dagger} (\tau)\epsilon_n c_{R_n} (\tau) +
i  c_{R_n}^{\dagger} (\tau) {\cal A}_n c_{R_n} (\tau)\right]
\nonumber \\
&+& \sum_{n\neq m} i \left[c_{L_n}^{\dagger} (\tau) {\cal A}_{nm} c_{L_m} (\tau) +
c_{R_n}^{\dagger} (\tau) {\cal A}_{nm} c_{R_m} (\tau)\right] \rbrace,
\label{S3}
\end{eqnarray}
\noindent where we have introduced the connection ${\cal A}_n$,
\begin{equation}
{\cal A}_n [A(\tau)] = \int d{\vec x} \;
\Phi_n^{\dagger}[{\vec A}(\tau)]({\vec x}) \left(-i \partial_{\tau} + eA_4\right)
\Phi_n[{\vec A}(\tau)]({\vec x})
\label{calA_n}
\end{equation}
\noindent and ${\cal A}_{nm}$ for $n\neq m$,
\begin{equation}
{\cal A}_{nm} [A(\tau)] = \int d{\vec x} \;
\Phi_n^{\dagger}[{\vec A}(\tau)]({\vec x}) \left(-i \partial_{\tau} + eA_4\right)
\Phi_m[{\vec A}(\tau)]({\vec x})
\label{calA_nm}
\end{equation}

In the adiabatic approximation one neglects the off-diagonal terms ($n\neq m$) and then one has infinite copies of quantum mechanical systems each of them similar to the one discussed in section II. However for a general vector field configuration the spectrum ($\epsilon_n$) will be continous and the difference of energy levels  can be arbitrarily small rendering the adiabatic expansion  out of control. Besides that, the eigenvalues $\epsilon_n$ and eigenfunctionals $\Phi_n$ are not known and the formulation remains once more at a formal level.

 \section{ A new approach and its application to an $SU(2)$ gauge theory}

 The only way we have found to use the reformulation of the adiabatic approximation to get a useful expansion in QFT,  is based on the introduction of variables independently at each point. In order to do that,  one has to select an operator at each point and use its eigenfunctions in the  expansion of some of the fields at this point. We can then identify two ingredients in the formulation of the new approach. The first one is a separation of the fields in two sets, one of them corresponding to the spin degrees of freedom of the quantum mechanical example. The second one  is the choice of the operator at each point whose eigenfunctions are used to introduce new variables for the spin-like fields.

 Several requirements constraint the ambiguities in this two ingredients. The action should be quadratic in the spin-like fields either directly or after the introduction of appropriate auxiliary fields. The expression for some of the terms in the action as a function of the new variables should be as simple as possible. The contribution from the remaining terms in the action (including the space derivatives) as well as the corrections to the \lq\lq adiabatic" approximation (off-diagonal contributions in the new variables) should be small. The search of a good set of fields and local operators defining the new variables has to be done, however case by case. The usefulness of the approach will be established if one finds examples where all these requirements are satisfied yielding to a dominant contribution with interesting results.

In order to illustrate our approach let us consider the Lagrangean of a fermionic system coupled to a vector field
 \begin{equation}
 {\cal L} = {\bar \Psi} i \gamma^{\mu} \partial_{\mu} \Psi -
 g {\bar \Psi} \gamma^{\mu} A_{\mu}^aT^{a} \Psi - m {\bar \Psi} \Psi,
 \label{Ls1}
\end{equation}
 \noindent where $T^a$ are the generators of $SU(2)$ acting on the fermionic fields in the fundamental representation. It is convenient to use the Dirac representation for the $\gamma$ matrices with
 \begin{equation}
 \gamma^0 = \left(\matrix{ I & 0\cr 0 & - I} \right)
 \label{gamma0}
\end{equation}
 \noindent and the decomposition in bispinors of the Dirac field
 \begin{equation}
 \Psi = \left(\matrix{\varphi \cr \chi}\right).
 \label{Psi}
\end{equation}
 We neglect for a moment, the terms proportional to the space components of the vector field (${\vec A}^a$) and to space derivatives of the fermionic field. The remaining terms take a simple form if we use the eigenvectors $f_{\pm}$ of the operator ${\hat A}_0^a T^a$
  \begin{equation}
 \left({\hat A}_0^a T^a\right) \, f_{\pm} \, = \pm \frac{1}{2} \, f_{\pm},
 \label{f}
\end{equation}
\noindent where we have used the parametrization $A_0^a = A_0 {\hat A}_0^a$ with $\sum_a {\hat A}_0^a {\hat A}_0^a =1$.
With these eigenvectors,  one can introduce the new fermionic variables
$\varphi_{n,i}$, $\chi_{n,i}$
\begin{equation}
\varphi \, = \sum_{n=\pm} \sum_{i=1,2} \varphi_{n,i} f_{n,i},
\label{varphi}
\end{equation}
\begin{equation}
\chi \, = \sum_n \sum_{i=1,2} \chi_{n,i} f_{n,i},
\label{chi}
\end{equation}
where the bispinors $f_{n,i}$ are given by
\begin{equation}
f_{n,1} \, = \, \left(\matrix{ f_n \cr 0}\right),
{\hskip 2cm}
f_{n,2} \, = \, \left(\matrix{ 0 \cr f_n }\right).
\label{fni}
\end{equation}
\noindent Note that the new fermionic variables have been introduced independently at each point in space. We then have Grassmann variables $\varphi_{n,i}$, $\chi_{n,i}$ at each space-time point.

In the new representation for the fermionic variables,  one has
\begin{equation}
\Psi^{\dagger} \partial_{\tau} \Psi \, =
\sum_{n,i} \left[\varphi_{n,i}^{\dagger} \partial_{\tau} \varphi_{n,i} \, + \,
\chi_{n,i}^{\dagger} \partial_{\tau}\chi_{n,i} \right] \, - \,
\sum_{n,n^{'},i} \left[\varphi_{n,i}^{\dagger} \varphi_{n^{'},i} \, + \,
\chi_{n,i}^{\dagger} \chi_{n^{'},i} \right] \,  i {\cal A}_{n,n^{'}}
\label{partialtau}
\end{equation}
\noindent with $i {\cal A}_{n,n^{'}} = f_n^{\dagger} \partial_{\tau} f_{n^{'}}$ and $\tau= i t$ is the Euclidean time.

For the interaction term one has
\begin{equation}
- g \Psi^{\dagger} A_0^{a} T^{a} \Psi \, = \, - \sum_n g_n A_0 \sum_i
\left[\varphi_{n,i}^{\dagger} \varphi_{n,i} \, + \,
\chi_{n,i}^{\dagger} \chi_{n,i} \right],
\label{int0}
\end{equation}
\noindent with $g_{\pm} = \pm \frac{g}{2}$.

The Lagrangian density before including space derivatives and the space components of the vector field as a function of the new variables is
\begin{eqnarray}
{\cal L} (A_0^a) \,& = & \, \sum_{n,i} \left[\varphi_{n,i}^{\dagger}
\left(\partial_{\tau}- g_n A_0 - m + i {\cal A}_n\right) \varphi_{n,i}
\, + \, \chi_{n,i}^{\dagger}
\left(\partial_{\tau}- g_n A_0 + m + i {\cal A}_n\right) \chi_{n,i} \right]
\nonumber \\ \,& + & \,
\sum_{n\neq n^{'},i} \left[\varphi_{n,i}^{\dagger} \varphi_{n^{'},i} \, + \,
\chi_{n,i}^{\dagger} \chi_{n^{'},i} \right] \, i {\cal A}_{n,n^{'}}.
\label{L0}
\end{eqnarray}

\noindent We then have at each point,  a generalization of the quantum mechanical example in section II with $A_0^a$ playing the role of the magnetic field ${\vec B}$ (the direction in the internal $SU(2)$ space is the analog to the orientation of the magnetic field) and four instead of one spin variable
(corresponding to the four components of the Dirac spinor). Then,  the energies associated to the new variables are
\begin{equation}
E(\varphi_{n,i}) = g_n A_0 + m,
{\hskip 3cm}
E(\chi_{n,i}) = g_n A_0 - m.
\label{E}
\end{equation}

The corrections to the adiabatic approximation due to the off-diagonal terms ($n\neq n^{'}$) in (\ref{L0}) involve levels separated by a gap $g A_0$. Then, the adiabatic approximation will be justified when the product of the coupling and  the time component of the gauge field is large compared to the time derivative of the orientation of the gauge field (${\dot {\hat A}}_0^a$).

With respect to the consistency of treating the space derivatives and space components of the gauge field as a correction, we have a sum of two contributions,
\begin{eqnarray}
\Psi^{\dagger} {\vec \alpha}.(-i\vec{\nabla}+g\vec{A}^aT^a) \Psi \, & = & \,
\sum_{n,i,i^{'}} \left[\varphi_{n,i}^{\dagger} (-i\vec{\nabla}) \chi_{n,i^{'}}
\, + \, \chi_{n,i}^{\dagger} (-i\vec{\nabla}) \varphi_{n,i^{'}}\right] \,
\left(f_{n,i}^{\dagger} {\vec \sigma} f_{n,i^{'}}\right) \nonumber \\ &+& \,
\sum_{n,n^{'},i,i^{'}} \left[\varphi_{n,i}^{\dagger} \chi_{n^{'},i^{'}} \, +
\chi_{n,i}^{\dagger} \varphi_{n^{'},i^{'}}\right] \,
\left(f_{n,i}^{\dagger} {\vec \sigma}.(-i {\vec \nabla}+g\vec{A}^aT^a) f_{n^{'},i^{'}}
\right).
\label{nabla}
\end{eqnarray}
\noindent The first term is a non-diagonal contribution between energy levels separated by $2m$. Since the eigenvectors $f_n$ depend only on the direction in the internal space of $A_0$, then  the corrections due to these terms will be proportional to $\vec{\nabla}{\hat A}_0^a/2m$. The second term has non-diagonal contributions between energy levels separated by $2m$, $2m+gA_0$ or $|2m-gA_0|$ and there are two types of terms, ones proportional to $\vec{\nabla}{\hat A}_0^a$ and the others proportional to $g\vec{A}^a$. From these simple arguments one can see what are the conditions on the vector field and the fermion mass  in order to treat (\ref{nabla}) as a small correction to (\ref{L0}). It should be 
noted that considering spacial derivatives as corrections does not mean that
we are making use of the usual derivative expansion method~\cite{WZ}.

Then, in the approximation where we keep only the terms diagonal in (\ref{L0}) and using the result (\ref{GammaadB2}-\ref{Am}) we have a new approximation to the effective fermionic action,

\begin{eqnarray}
\Gamma_{ad} \,& = & \, \int d^3{\vec x} \, \sum_n \,
sgn (g_n A_0 + m)\left[\int d\tau ( g_n A_0 + m ) +
\frac{g_n}{g} i \Omega[{\hat A}_0^a]\right] \nonumber \\ \, & + & \,
\int d^3{\vec x} \, \sum_n \,
sgn (g_n A_0 - m)\left[\int d\tau ( g_n A_0 - m ) +
 \frac{g_n}{g} i \Omega[{\hat A}_0^a]\right],
\label{Gamma_ad}
\end{eqnarray}
\noindent where $\Omega[{\hat A}_0^a]$ is the solid angle that ${\hat A}_0^a$ subtends in internal space in its time evolution.

We can distinguish two different regions. In the weak coupling region ($g A_0 \ll 2 m$) there is a cancellation of contributions from the two sets of new variables $\varphi_{n,i}$ and $\chi_{n,i}$ and $\Gamma_{ad}$ reduces to a constant. On the other hand, in the strong coupling region ( $g A_0 \gg 2 m$)  the effective fermionic action in the adiabatic approximation is
\begin{equation}
\Gamma_{ad}^{(s)} \, = \, 2 g  \int d\tau d^3 {\vec x} \, A_0  \, + 2 i \Omega[{\hat A}_0^a].
\label{Gammas}
\end{equation}
\noindent The presence of a contribution in the effective fermionic action which dissapears in the weak coupling region,  suggest a possible relation between the presence of such contribution and the non-pertubative properties of the theory.

One should note that the limit $m\to \infty$ --for fixed $g A_0$-- corresponds
to the weak coupling region,  where the new non-perturbative contribution 
dissapears as expected from the decoupling theorem \cite{appel}. On the other 
hand for an arbitrarily large mass,  one can be in the strong coupling limit  
if one has a sufficiently large coupling $g$ and/or gauge field $A_0$ and 
then the non-perturbative trace of the fermionic system remains.

At this point, it is worth noting that the final result (\ref{Gamma_ad}) 
is clearly not gauge invariant. It is so because the adiabatic approximation, 
even at the quantum mechanics level, does not preserve all symmetries of the 
theory~\cite{Jac}. 
Indeed, at the beginning of our calculations we made use of a particular 
gauge choice where the temporal gauge field components behave in a very 
different way to the spacial ones. If one is able to find a field and a gauge 
choice that fulfill that requirements, one can figure out some particular 
non-perturbative features of this background and, if one were able to get 
every order in this approximation, one would recover explicit gauge and Lorentz
invariance.
 
\section{Finite density and temperature effects}

The generalization of the adiabatic approximation to the $SU(2)$ gauge theory in the case of finite density is trivial. All one has to do is to include the chemical potencial ($\mu$) through a term $\mu {\bar \Psi} \gamma^0 \Psi$ in the Lagrangian which modifies the energies associated to the new variables
\begin{equation}
E(\varphi_{n,i}) = g_n A_0 + m + \mu,
{\hskip 3cm}
E(\chi_{n,i}) = g_n A_0 - m + \mu.
\label{Emu}
\end{equation}
\noindent The energy differences are not modified and  all the estimates of the corrections due to space derivatives and the space components of the vector field are not changed.

The effective fermionic action in the adiabatic approximation is now,

\begin{eqnarray}
\Gamma_{ad} (\mu)\,& = & \, \int d{\vec x} \, \sum_n \,
sgn (g_n A_0 + m + \mu)\left[\int dt ( g_n A_0 + m + \mu) +
\frac{g_n}{g}  i \Omega[{\hat A}_0^a]\right] \nonumber \\ \, & + & \,
\int d{\vec x} \, \sum_n \,
sgn (g_n A_0 - m + \mu)\left[\int dt ( g_n A_0 - m + \mu) +
 \frac{g_n}{g}  i \Omega[{\hat A}_0^a]\right].
\label{Gamma_admu}
\end{eqnarray}

We consider $\mu >0$ for definiteness. In this case,  we can consider three different regions.  The first one corresponds to $gA_0 < 2|m-\mu|$ where once more there is a cancellation of contributions and one has a trivial adiabatic effective action as in the weak coupling region of the case without chemical potencial. There is also an analog of the strong coupling region where $gA_0 > 2|m+\mu|$ and the adiabatic effective action is
(\ref{Gammas}). Finally,  there is an intermediate region, $2|m-\mu| < gA_0 < 2|m+\mu|$ where there is a cancellation of the contributions of half of the new fermionic variables and the result for the effective action is
\begin{equation}
\Gamma_{ad}^{(i)} \, = \,  g  \int d\tau d^3 {\vec x} \, A_0  \, +  i \Omega[{\hat A}_0^a] + \mbox{constant}.
\label{Gammai}
\end{equation}

Once more we can discuss the decoupling of the fermion degrees of freedom. There are once more cases with arbitrarily large mass and density where for sufficiently large $gA_0$ a fermionic signal remains. On the other hand for fixed mass and $gA_0$ the fermion decouples in the limit $\mu \to \infty$. This shows that the non-perturbative properties of the non-abelian gauge theory related to the presence of a non-trivial adiabatic effective action  dissapear in the infinite density limit.

It is also very easy to generalize the adiabatic approximation to the case of finite temperature. As we have seen in section II,  the modification of the fermionic integral for each quantum mechanical system is very simple and the effective fermionic action in the adiabatic approximation at finite temperature (and also including a chemical potencial) is given by
\begin{eqnarray}
- \Gamma_{ad} (\mu,\beta)\,& = & \, \int d{\vec x} \, \sum_n \,
\ln \cosh \left[\int_{-\beta/2}^{\beta/2} d\tau ( g_n A_0 + m + \mu) +
i \Omega[{\hat A}_0^a]\right] \nonumber \\ \, & + & \,
\int d{\vec x} \, \sum_n \,
 \ln \cosh \left[\int_{-\beta/2}^{\beta/2} d\tau ( g_n A_0 - m + \mu) +
i \Omega[{\hat A}_0^a]\right].
\label{Gamma_admuT}
\end{eqnarray}
\noindent where $\beta$ is the inverse of the temperature \cite{coment}. If
one takes the high temperature limit, $\beta \to 0$ for fixed ($\mu$, $m$, $gA_0$), then the adiabatic effective action is proportional to $\beta^2$ and then the new non-perturbative  contribution dissapears.

In all cases where the effective action in the adiabatic approximation is non trivial one has a term proportional to $\int d\tau d^3 {\vec x} A_0$ and then, in order to have a finite action, the gauge field $A_0$ should be concentrated in a finite region in space-time. This fact, together with the dissapereance of the new contribution in the high temperature or high density limits suggests a relation of the presence of a non-trivial adiabatic approximation and the confinement in the non-abelian gauge theory.  Note that if this relation holds,  then the confinement at low energies would be due to the presence of heavy quarks which do not decouple as one would naively expect due to non-perturbative effects.
\acknowledgments This work has been partially supported by the
 7010516  Fondecyt- Chile, by AECI (Programa de
Cooperaci\'on con Iberoam\'erica) and by MCYT (Spain), grant FPA2000-1252. JLS
thanks the Spanish Ministerio de Educaci\'on y Cultura for support.


\begin{references}
\bibitem{stone} The literature is very extense and some importants references are;  P. Nelson and L. Alvarez-Gaum\'e, {\it Comm. Math. Phys.} {\bf 99}, 103 (1985); E. Witten, {\it Nucl. Phys.} {\bf B223}, 422 (1983); M. Stone, {\it Phys. Rev.} {\bf D33}, 1191 (1986). Others importants references are reprinted in A. Shapere and F. Wilczek in {\it Geometric Phases in Physics}, World Scientific (1989).
\bibitem{berry} M.V. Berry, {\it Proc. R. Lond} {\bf A392}, 45 (1984).
\bibitem{kapusta} See {\it e.g.}, J. Kapusta, {\it Finite-temperature field theory}, Cambridge University Press (1989).
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\bibitem{Ralston} For a first attempt to formulate gauge theory along these lines see J. P. Ralston, {\it Phys. Rev.} {\bf D51} 2018 (1995).
\bibitem{Niemi} A. Niemi and G. Semenoff, {\it Phys. Rev. Lett.} {\bf 55}, 927 (1985); {\bf 56}, 1019 (1986); H. Sonoda, {\it Phys. Lett.} {\bf B156}, 220
(1985); {\it Nucl. Phys.} {\bf B266}, 410 (1986).
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\bibitem{appel} T. Appelquist and J. Carazzone, {\it Phys. Rev.}{\bf D11}, 2856 (1975).
\bibitem{Jac} R. Jackiw, {\it Int. J. Mod. Phys.} {\bf A3}, 285 (1988).
\bibitem{coment} Note that $A_0$ can be considered as a dynamical field
instead of a time dependent external field.
\end{references}

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