Quantum field theory 1, lecture 13

Thermal states and quantum fields in Euclidean spacetime

Thermal density operators

Now that we know how to write evolution operators as functional integrals we can do many things with them. Let us discuss the density matrix of quantum fields in a thermal state. This is not to be confused with the classical fields in a thermal state we discussed previously. Thermal states are interesting by themself and the standard vacuum state is included in the limit \(T \to 0\).

At fixed time \(t\), the thermal density matrix is formally given by \[\label{eq:thermalDensityMatrix} \begin{split} \rho = \frac{1}{Z(\beta)}e^{-\beta H},\quad\quad\quad Z(\beta)= \text{Tr}\left\{e^{-\beta H} \right\}, \end{split}\] where \(\beta=1/T\). By comparing this to the evolutionoperator \(e^{-i t H}\) we see that the operator \(e^{-\beta H}\) is in fact just an evolution operator into an imaginary time direction, for example from \(t=t_0\) to \(t=t_0 - i \beta\). Moreover, in the thermal partition function \(Z(\beta)\) one needs to take the trace which means to identify the fields at \(t=t_0\) and \(t=t_0 - i \beta\). This leads to a torus geometry with peridicity in imaginary time direction where the fields satify the condition \[\phi(t_0, \mathbf x) = \phi(t_0-i\beta, \mathbf x).\] This is called the Matsubara torus.

Analytic continuation or Wick rotation

Let us analyse what happens to the action when we evaluate it along the imaginary time contour. We introduce the imaginary time coordinate \(\tau\) through \[t=t_0 - i \tau,\] and integrate \(\tau\) from \(0\) to \(\beta = 1/T\). Note that \[\frac{\partial}{\partial t} = i \frac{\partial}{\partial\tau},\] and \(dt = -i d\tau\) The real-time action times the imaginary unit, \[i S = i \int dt \int d^3 x \left\{ \frac{1}{2} \left( \frac{\partial}{\partial t} \phi \right)^2 - \frac{1}{2} \left(\boldsymbol \nabla \phi \right)^2 - V(\phi) \right\},\] which is what enters the exponential in the transition operator \(U\), becomes then, when evaluated along the Matsubara contour, \[- S_\text{E} = - \int_0^\beta d\tau \int d^3 x \left\{ \frac{1}{2} \left(\frac{\partial}{\partial \tau} \phi\right)^2 + \frac{1}{2} \left(\boldsymbol \nabla \phi \right)^2 + V(\phi) \right\}.\] This is now an action in Euclidean space, where the metric is \[ds^2 = d\tau^2 + d\mathbf x^2,\] and the difference in sign between time and space coordinates has disapeared! The Euclidean action is of the same kind as the “actions” we have studied previously in the context of classical statistical field theories (which also explains why we called them “actions” even though formally these where parts of Hamiltonians divided by temperature). The difference is, however, that we now have one Euclidean dimension more! This additional dimension is periodic at non-zero temperature.

Zero temperature or ground state

At this point it is interesting to consider the limit \(T\to 0\) or \(\beta \to \infty\). The circumference of the Matsubara torus becomes then infinite and \(\tau\) is integrated from \(0\) to \(\infty\), or, equivalently after a change of variables, from \(-\infty\) to \(\infty\). In that limit the theory is equivalent to what we discussed previously in the context of classical fields at finite temperature, but with one dimension more. The groud state of a quantum field theory in \(d=1+3\) dimensions can be represented by a statistical field theory with \(d=4\) dimensions.

Matsubara frequencies

Taking the periodicity condition at non-zero temperature into account we can write \[\phi(\tau, \mathbf x) = T \sum_{n=-\infty}^\infty\int \frac{d^3 p}{(2\pi)^3} e^{i\omega_n \tau + i \mathbf{p} \mathbf{x}} \phi(-i\omega_n, \mathbf p),\] where \[\omega_n = 2\pi T n\] is known as the Matsubara frequeny. While \(\tau\) can be seen as an imaginary periodic time, \(\omega_n\) can be seen as an imaginary discrete frequency. In the high temperature limit only the lowest Matsubara modes with \(\omega_n=0\) contribute effectively to thermodynamic observables, all others are strongly surpressed in the correlation function \(1/[(2\pi T n)^2 + \mathbf p^2]\). Restricting to \(n=0\) leads to the classical limit of the theory we have discussed previously. In the opposite limit \(T\to 0\) we obtain an integral, \[T \sum_n \to \int \frac{d\omega}{2\pi},\] over continuous Matsubara frequencies.

Density matrix functional

The density matrix functional is given by \[\rho[\phi_+,\phi_-] = \frac{1}{Z(\beta)}\int_{\phi_+,\phi_-} D\phi~e^{-S_\text{E}[\phi]}, \label{eq:01ThermalDensityMatrix}\] where the boundary conditions for the functional integral are \[\phi(\tau=0,\mathbf x) = \phi_+(\mathbf x), \quad\quad\quad \phi(\tau=\beta, \mathbf x) = \phi_-(\mathbf x).\] One easily confirms that the density matrix is normalized correctly, \[\begin{split} \text{Tr}\{ \rho\} &= \int D\phi_+ ~\rho[\phi_+,\phi_+] = \frac{1}{Z(\beta)}\int D\phi_+\int_{\phi_+,\phi_+} D\phi~e^{-S[\phi]} = \frac{1}{Z(\beta)}\int D\phi~e^{-S[\phi]}=1, \end{split}\] where \(Z(\beta)\) is the thermodynamic partition function. The density matrix \(\eqref{eq:01ThermalDensityMatrix}\) can be combined with evolution operators as in eq. \(\eqref{eq:01EvolutionLagrangian}\) and a similar representation for \(U^\dagger\) to determine the density matrix at a later time. When the trace of such a density matrix is taken one obtains a closed time path, as a special case of the Schwinger-Keldysh double time path.

In-out formalism

For many problems in quantum field theory one does not need the Schwinger-Keldysh or in-in formalism. Instead one can work in a situation where the ingoing as well as the out-going state are actually vacuum or ground states. This describes in particular situations with just a few particles in the initial and final state for which one can specify convenient creation and annihilation operators acting on the vaccum state. Much of the scattering physics needed to describe collider experiments can be described this way.

Vacuum-to-vacuum transition amplitude

A contour where the incoming state and the outgoing state are the vacuum, but that nevertheless goes along real times can be achieved by rotating the real time slightly into the complex plane. By integratig fom \(t_0 \to -\infty (1-i \epsilon)\) to the final time \(t_\text{f} \to \infty (1- i \epsilon)\) we have at both ends of the integration contour terms \(e^{-i \infty (1-i\epsilon) H} \sim e^{-\epsilon \infty H}\), which effectively project to the ground state. For states above the minimal energy, the exponential supression is so strong that only the ground state remains.

The integration contour will play a role in deciding which Greens functions of a differential operator to take. Recall that Greens functions are non unique but depend on the boundary conditions. A simple prescription, equivalent to the above rotation in time integration contour, is to multipy the Hamiltonian with \((1-i\epsilon)\), or, even simpler and equivalently in practice, to replace \(m^2\) with \(m^2-i\epsilon\). We will take this \(i\epsilon\) prescription into account later on when calculating Greens functions.

Feynman propagator

Let us consider now a two-point correlation function of the type \[\frac{1}{i}G(x-y) = \frac{1}{Z} \int D\phi \, \phi(x) \phi(y) e^{iS_2[\phi]},\] where time \(t\) is integrated along the vaccuum-to-vacuum or in-out contour and we work with the quadratic action \[S_2[\phi] = \int d^4 x \left\{ - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi(x) \partial_\nu \phi(x) - \frac{1}{2}m^2 \phi(x)^2 \right\} = -\frac{1}{2}\int \frac{d^4 p}{(2\pi)^4} \left\{ \phi^*(p) \left[ p^2 + m^2 \right] \phi(p) \right\}.\] We are using now a four-dimensional notation with \(x=(t,\mathbf x)\), \(p=(\omega, \mathbf p)\) and \(p^2=-\omega^2 + \mathbf p^2\). In the second equation we have introduced the fields in Fourier space through \[\phi(x) = \int \frac{d^4 p}{(2\pi)^4} e^{ipx} \phi(p),\] where \(px=-\omega t+\mathbf p \mathbf x\). For real fields one has \(\phi^*(p) = \phi(-p)\). The two-point function follows now from the standard receipe of Gaussian integration and we obtain \[\begin{split} G(x-y) = & \int \frac{d^4 p}{(2\pi)^4} e^{ip(x-y)} \frac{1}{p^2 + m^2 - i \epsilon} \\ = & \int \frac{d^4p}{(2\pi)^4} e^{-ip^0 (x^0-y^0) + i \mathbf{p}(\mathbf x - \mathbf y)} \frac{-1}{(p^0-\sqrt{\mathbf p^2 + m^2}+i\epsilon)(p^0+\sqrt{\mathbf p^2 + m^2}-i\epsilon)}. \end{split}\label{eq:01FeynmanPropagator}\] Here we have inserted the \(i\epsilon\) which will help us to pick the right integration contour. Note that \(G(x-y)\) is a Greens function to the inverse propagator in the sense \[\left[-g^{\mu\nu} \partial_\mu \partial_\nu + m^2 \right] G(x-y) = \delta^{(4)}(x-y).\] One may perform the integration over the frequency \(p^0\) in eq. \(\eqref{eq:01FeynmanPropagator}\). Note first that there are poles at \[p^0 = \sqrt{\mathbf p^2 + m^2} - i \epsilon, \quad\quad\quad p^0 = -\sqrt{\mathbf p^2 + m^2} + i \epsilon.\] For \(x^0-y^0 >0\) we can close the \(p^0\) integration contour in the lower half of the complex plane and get a contribution from the residue at \(p^0=\sqrt{\mathbf p^2 + m^2} = E_\mathbf{p}\). In contrast, for \(x^0-y^0 < 0\) the contour must be closed in the upper half of the complex plane, and we pick up a contribbution from the residue at \(p^0 = - \sqrt{\mathbf p^2 + m^2} = - E_\mathbf{p}\). Taken together this yields \[\begin{split} G(x-y) = & \theta(x^0 - y^0) \int \frac{d^3 p}{(2\pi)^3} \frac{i}{2 E_\mathbf{p}}e^{-i E_\mathbf{p} (x^0 - y^0) + i\mathbf{p} (\mathbf x - \mathbf y)} \\ & + \theta(y^0 - x^0) \int \frac{d^3 p}{(2\pi)^3} \frac{i}{2 E_\mathbf{p}}e^{+i E_\mathbf{p} (x^0 - y^0) + i\mathbf{p} (\mathbf x - \mathbf y)} \\ \end{split}\] Depending on the time ordering we find either a term with positive frequency or one with negative frequency. The Feynman propagator \(G(x-y)\) is also called time-ordered propagator.

Exercise

By choosing different contours of the frequency integration, derive expressions for a retared propagtor that vanishes when \(x^0-y^0<0\) and an advanced propagator that vanishes when \(x^0-y^0>0\).