Quantum field theory 2, lecture 21

Quantum field theory in thermal equilibrium

Quantum field theory can not only describe vacuum states and its excitations, the particles, but also states with non-zero density and temperature. We discuss now the main steps and concepts for that.

Thermal states

Thermal states are mixed states, described by density matrices. For a relativistic quantum field theory, the grand canonical ensemble is most relevant, where energt and particles number can be exchanged between subsystems. Quantum mechanically, this is described by the density matrix \[\rho = \frac{1}{Z} e^{-\frac{1}{T}(H-\mu N )} = e^{-\frac{1}{T}(H-\mu N)- \ln Z},\] where \(H\) and \(N\) are operators. The entropy is obtained by von-Neumanns formula, \[S = \text{Tr}\{ - \rho \ln \rho \} = \frac{1}{T} (\langle H \rangle - \mu \langle N \rangle) + \ln Z.\] The expectation values of energy and conserved particle number (usually proportional to a number of particles minus anti-particles) are here simply \[\begin{split} E = \langle H \rangle = \text{Tr} \{ \rho H \}, \\ N = \langle N \rangle = \text{Tr} \{ \rho N \}. \end{split}\]

Grand canonical partition function

Recall from statistical mechanis that the grand canonical partition function is given by \[Z(T, \mu, V) = \text{Tr}\left\{ e^{-\frac{1}{T}(H - \mu N)}\right\}.\] Related to the partition function is the grand canonical potential \(\Omega(T,\mu, V)\) through the relation \[Z=e^{-\Omega/T}.\] The differential of the grand potential is \[d\Omega = -SdT - N d\mu - p dV,\] and it can be expressed as \[\Omega = E-TS -\mu N = -pV,\] Thermodynamic quantities can be directly derived from \(Z\) or \(\Omega\), for example \[S=-\left. \frac{\partial \Omega}{\partial T}\right|_{\mu,V}\quad N=-\left. \frac{\partial \Omega}{\partial \mu}\right|_{T,V}.\] Other observables follow from Legendre transforms, for example \[E=\Omega+TS+\mu N=\Omega-T \frac{\partial\Omega}{\partial T}-\mu \frac{\partial\Omega}{\partial \mu}.\]

Thermodynamics for fluids

For fluids it is convenient to work with pressure \(p(T,\mu)\) as a thermodynamic potential. For constant volume \(V\) one has \[Z(T,\mu)=\exp\left(\frac{V p(T,\mu)}{T}\right)=\exp \left( \int_0^{\frac{1}{T}}d\tau \int\!d^{3}x~p(T,\mu) \right),\] and the differential Gibbs-Duhem relation \[dp=sdT+nd\mu.\] For a homogeneous state of a fluid one has constant energy density \(\epsilon=E/V\), particle density \(n=N/V\) and entropy density \(s=S/V\). With this, together with the relation \[\epsilon + p = T s + \mu n,\] one can all thermodynamic quantities of interest from the potential \(p(T,\mu)\). This also generalizes directly to situations with several conserved quantum numbers such as (net) baryon number, (net) electric charge, strangeness and so on. One can introduce a chemical potential associated to each of them.

Exercise on thermodynamics for fluids

Derive expresions for the heat capacity densities \[c_v = \frac{C_v}{V}=\frac{T}{V} \left( \frac{\partial S}{\partial T} \right)_{V,N},\quad\quad\quad c_p = \frac{C_p}{V}=\frac{T}{V} \left( \frac{\partial S}{\partial T} \right)_{p,N},\] and the thermal expansion coefficient \[\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_{p,V},\] in terms of \(p(T,\mu)\) and its derivatives.

Partition function for quantum fields

From these considerations, it becomes clear that it would be very useful to have a method to calculate the grand canonical partition function for matter described by a quantum field theory.

Note that \(e^{-\frac{1}{T}(H - \mu N)}\) resembles the kernel of transition amplitude. Transition amplitudes in Minkowski space are of the form \[\langle \phi_\text{f} | e^{-i(t_\text{f}-t_\text{in})H} | \phi_\text{in} \rangle = \int\limits_{\phi_\text{in}, \phi_\text{f}} D \phi \, e^{iS_\text{M}[\phi]}.\] The right hand side involves the Minkowski space action. For a complex scalar field \[S_\text{M}[\phi] = -\int_{t_\text{in}}^{t_\text{f}}dt\int d^3x \sqrt{g} \left\{ g^{\mu\nu}\partial_\mu\phi^*\partial_\nu\phi+m^2\phi^*\phi + \frac{\lambda}{2}(\phi^*\phi)^2 \right\}.\] The metric \(g_{\mu\nu}\) has here the signature \((-,+,+,+)\). The functional integral has the boundary conditions \[\phi(t_\text{in},\mathbf{x})=\phi_\text{in}(\mathbf{x}),\quad\quad\quad \phi(t_\text{f}, \mathbf{x})=\phi_\text{f}(\mathbf{x}),\] for the initial and final states.

Imaginary or Euclidean time

In order to come from a transition amplitude to a thermodynamic partition function a few steps are needed. The first is to choose the time to be imaginary such that \[t_\text{f} - t_\text{in} = - i \frac{1}{T}.\] For example we may choose without loss of generality \(t_\text{in} = 0\) and \(t_\text{f} = -i \beta\) where \(\beta=1/T\). Propgation will be along imaginary time! It is convenient to introduce a variable \(\tau\) which is integrated from \(0\) to \(\beta=1/T\) with \(dt=-i d\tau\). This is actually a Euclidean time variable, as becomes clear from the invariant length element \[ds^2 = - dt^2 + d\mathbf{x}^2 = d\tau^2 + d\mathbf{x}^2.\] Accordingly, the symmetry of Minkowski space SO\((1,d-1)\) becomes simply an SO\((d)\) symmetry in Euclidean space.

Periodic boundary conditions

Taking the trace as needed for the partition function means to identify initial and final states and to sum over them. In other words, we need to set \[\phi_\text{in}(\mathbf{x})=\phi_\text{f}(\mathbf{x})=\phi(0, \mathbf{x})=\phi(t=-i\beta, \mathbf{x}),\] and include a functional integral over \(\phi(0, \mathbf{x})\). This leads to a functional integral without boundaries but with the periodic identification \[\phi(0, \mathbf{x})=\phi(t=-i/T, \mathbf{x}).\] The imaginary time dimension is periodic, the geometry is like the one of a cylinder with times \(t=0\) and \(t=-i/T\) or \(\tau=0\) and \(\tau=\beta\) identified. The geometry is thus the one of a torus, with the spatial dimensions unchanged, but time periodic. This geometry is called Matsubara torus. In the limit \(T\to 0\) the circumference of the torus becomes infinitely large, while for \(T\to \infty\) is becomes very small.

Euclidean action

With the imaginary time element \(dt = -i d\tau\) comes also a change in the action. We write \(i S_\text{M}[\phi] = - S_\text{E}[\phi]\) with Euclidean action \[S_E[\phi] = \int_{0}^{1/T}d\tau\int d^3x \left\{ \delta^{\mu\nu}\partial_\mu \phi^*\partial_\nu\phi+m^2\phi^*\phi + \frac{\lambda}{2}(\phi^*\phi)^2 \right\}\] and \[\delta^{\mu\nu}\partial_\mu \phi^*\partial_\nu\phi = \frac{\partial}{\partial \tau}\phi^* \frac{\partial}{\partial \tau}\phi + \boldsymbol{\nabla}\phi^* \boldsymbol{\nabla}\phi.\] In this form, the action is real and positive.

Chemical potential

We also need to introduce the chemical potential term. One can see this as a modification of the Hamiltonian or of the action. To introduce it properly, let us first go back to real time and let us couple the theory to an external gauge field \(A_\mu(x)\), \[S_\text{M}[\phi] = -\int_{t_\text{in}}^{t_\text{f}}dt \int d^3x \sqrt{g} \left\{ g^{\mu\nu}(\partial_\mu+iA_\mu)\phi^* (\partial_\nu-iA_\nu)\phi+m^2\phi^*\phi + \frac{\lambda}{2}(\phi^*\phi)^2 \right\}.\] The conserved current on the microscopic or classical level then follows from \[N^\mu(x) = -\frac{\delta}{\delta A_\mu(x)} S_\text{M}[\phi] = g^{\mu\nu}\left[i\phi^*(x)\partial_\nu\phi(x) - i\phi(x) \partial_\nu\phi^*(x) \right].\] The conserved particle number is \[N(t)=\int d^3x~N^0(t, \mathbf{x}).\] If we take the chemical potential to be the time component of an external gauge field, \(A_0 = \mu\), it will automatically couple to the conserved number density \(n=N^0\). One may check that signs and factors of \(i\) indeed come out correctly. After analytic continuation to Euclidean time, one obtains \[\frac{\partial}{\partial t} - iA_0 \to \frac{\partial}{\partial (-i\tau)} - iA_0 = i\left( \frac{\partial}{\partial \tau}-A_0 \right) \to i \left( \frac{\partial}{\partial \tau }-\mu \right),\] and the Euclidean action becomes \[S_\text{E}[\phi] = \int_{0}^{1/T}d\tau\int d^3x \left\{ \left(\frac{\partial}{\partial \tau}+\mu\right) \phi^* \left(\frac{\partial}{\partial \tau}-\mu \right) \phi +\boldsymbol{\nabla}\phi^* \boldsymbol{\nabla}\phi + m^2\phi^*\phi + \frac{\lambda}{2}(\phi^*\phi)^2 \right\}.\]

Matsubara frequencies

Bosonic fields on the Matsubara torus have periodic boundary conditions with respect to Euclidean time \(\tau\) being changed by an amout \(\beta=1/T\), \[\phi(0, \mathbf{x}) = \phi(\beta, \mathbf{x}).\] For fermionic or Grassmann fields, a careful consideration (best done with discretized time) shows that they must be anti-periodic instead, \[\psi(0, \mathbf{x}) = - \psi(\beta, \mathbf{x}).\] This has consequences for the Fourier expansion. We expand the fields as \[\chi(\tau, \mathbf{x}) = T \sum_n \int \frac{d^{d-1}p}{(2\pi)^{d-1}} e^{-i\omega_n \tau + i \mathbf{p} \mathbf{x}} \, \chi(i \omega_n, \mathbf{p}),\] where \(\chi(\tau, \mathbf{x})\) could be either bosonic or fermionic.

As a consequence of the peridicity, or anti-peridicity, the Matsubara frequencies \(\omega_n\) are discrete. They must be integer multiples of \(2\pi T\) for bosons, \[\omega_n = 2\pi T n,\] and half-integer multiples for fermions, \[\omega_n = 2\pi T (n+1/2),\] where \(n\in \mathbb{Z}\).

In the limit of \(T\to 0\), the discrete Matsubara sum becomes again an integral, \[T \sum_n = \sum_n \frac{\Delta \omega_n}{2\pi} \to \int \frac{d\omega}{2\pi}.\] In that limit we are back to the standard quantum field theory setup after Wick rotation to Euclidean space.

In the high-temperature limit \(T\to \infty\) the distance between nearby Matsubara frequncies becomes very large. We will see that a large Matsubara frequency has in the Euclidean theory a similar effect as a large mass parameter and suppresses fluctuations. Only bosonic fields have a zero mode for which the Matsubara frequency vanishes, \(\omega_0 = 0\). It is the only mode that survives in the limit \(T\to \infty\) ane leads to a theory of classical fields in termal equilibrium. Fermionic field fluctuations are not contributing in the classical limit.

Propagator on the Matsubara torus

For perturbative calculation and beyond we need the propagator \[\Delta(\tau - \tau^\prime, \mathbf{x} - \mathbf{x}^\prime)= \langle \phi(\tau, \mathbf{x}) \phi^*(\tau^\prime, \mathbf{x}^\prime) \rangle_c\] As a consequence of translational symmetry in space this connected two-point function depends only on \(\mathbf{x} - \mathbf{x}^\prime\). Similarly there is a translational invariance for Euclidean time \(\tau\) on the torus, keeping in mind that \(\tau\) and \(\tau+ n \beta\) describe the same time. This implies also that \(\Delta\tau=\tau-\tau^\prime\) can be restriced to the range \(-\beta < \Delta\tau < \beta\).

We assume now that the inverse propagator is of the form \[P(i\omega_n, \mathbf{p}) = (\omega_n - i\mu)^2 + \mathbf{p}^2 + m^2.\] This yields \[\Delta(\Delta\tau, \Delta \mathbf{x}) = T \sum_{n=-\infty}^\infty \int\frac{d^{d-1}p}{(2\pi)^{d-1}} \frac{e^{-i\omega_n \Delta\tau + i \mathbf{p} \Delta\mathbf{x}}}{(\omega_n - i \mu)^2 + \mathbf{p}^2 + m^2}.\] We now face the problem to perform the infinite sum over the Matsubara frequencies.