Quantum field theory 2, lecture 23

Re-deriving theromdynamics

Thermodynamics is usually developed in the context of classical or quantum mechanics. It is interesting to re-derive it here directly using relativistic and field theoretic concepts. This will help to clarfy what are the detailed conditions for global thermal equilibrium states and how they can be described in general coordinates.

The starting point for the derivation of thermodynamics are conservation laws, which for a quantum field theory are best stated in local form. The conservation of energy and momentum is a consequence of diffeomorphism symmetry if the theory is formulated in general coordinates with Riemannian metric \(g_{\mu \nu}(x)\), \[\nabla_\mu T^{\mu \nu}(x) = 0. %\label{eq:EnergyMomentumConservation}\] In addition the theory may exhibit a U\((1)\) symmetry leading to a covariantly conserved particle number current, \[\nabla_\mu N^{\mu}(x) = 0. %\label{eq:ParticleNumberConservation}\] One should read here \(T^{\mu\nu}(x)\) and \(N^\mu(x)\) as beging the expectation values with respect to some quantum state.

Furthermore, one also introduces an entropy current \(s^{\mu}(x)\). In a phenomenological approach it is postulated to be governed by a local form of the second law \[\nabla_\mu s^{\mu}(x) \ge 0, %\label{eq:LocalSecondLaw}\] where equality is reached in global thermal equilibrium. Unlike the two former equations the local second law does not follow from symmetry considerations and needs a more careful justification. Moreover, it is not clear whether a local entropy current is well-defined in out-of-equilibrium situations.

One should note that the above equations could be supplemented by additional conservation laws or equations for additional order parameters \(\Phi(t)\), which we assume here to vanish for simplicity.

Vanishing entropy production

With the conservation relations as well as the local form of the second law one can discuss relativistic thermodynamics. Thermal equilibrium states are supposed to be fully specified by the energy-momentum tensor and conserved particle number expectation values. Accordingly, one can assume the entropy current to be a function of the conserved energy-momentum tensor and particle current \(s^\mu(T^{\lambda\nu}, N^\sigma)\), and write the second law \[\nabla_\mu s^{\mu} = \frac{\partial s^{\mu}}{\partial T^{\lambda \nu}} \nabla_\mu T^{\lambda \nu} + \frac{\partial s^{\mu}}{\partial N^{\sigma}} \nabla_\mu N^{\sigma} \geq 0.\] Because this should reduce to an equality in thermal equilibrium as a consequence of the two covariant conservation laws for energy-momentum and particle number, one should have \[\frac{\partial s^{\mu}}{\partial T^{\lambda \nu}} = - \beta_\nu \, \delta^{\mu}_{\lambda}, \hspace{1cm} \frac{\partial s^{\mu}}{\partial N^{\sigma}}= - \alpha \, \delta^{\mu}_{\sigma},\] or, in other words, the differential shoud be \[ds^\mu = - \beta_\nu d T^{\mu\nu} - \alpha dN^\mu.\] Here \[\beta^\nu(x) = \frac{u^\nu(x)}{T(x)}\] is a vector field corresponding to the ratio of fluid velocity and temperature. The fluid velocity is a time-like vector field normalized with the metric to \(g_{\mu\nu}(x)u^\mu(x) u^\nu(x)=-1\), and points in time-direction, in the fluid rest frame.

Similarly, \[\alpha(x)=\frac{\mu(x)}{T(x)}\] is a scalar field corresponding to the ratio of chemical potential and temperature. One should understand these relations as definitions, very similar as the standard definitions of temperature and chemical potential from the differential of entropy \(S(E,N)\) in the microcanonical ensemble, \[dS = \frac{1}{T} dE - \frac{\mu}{T} dN.\]

Stationary entropy production

The divergence of the entropy current \(\nabla_\mu s^\mu\) is non-negative. Accordingly, it must not only vanish in thermal equilibrium, but also be stationary. One finds for its differential \[d(\nabla_\mu s^\mu) = \nabla_\mu d s^\mu = - (\nabla_\mu \beta_\nu) d T^{\mu\nu} - (\partial_\mu \alpha) \, dN^\mu = 0,\] which leads to the condition that \(\beta^\nu(x)\) must be a Killing vector field and \(\alpha\) a constant, \[\nabla_\mu \beta _\nu(x) + \nabla_\nu \beta _\mu(x) = 0, \quad\quad\quad \partial_\mu \alpha(x) = 0. %\label{eq:KillingEquation}\] These are two conditions for global thermal equilibrium states, and they hold in any coordinate system. In Minkoski space, a simple possibility is for example that the fluid velocity \(u^\mu\), temperature \(T\) and chemical potential \(\mu\) are all constant.

A time-like Killing vector does not exist for all spacetimes, the condition is equivalent to the spacetime being stationary. Evolving spacetimes as they are needed in cosmology do not have any time-like Killing vectors, for example, and accordingly no equilibrium states.

In terms of the fluid velocity \(u^\mu\), and the projector orthogonal to the fluid velocity \(\Delta^{\mu\nu}=u^\mu u^\nu + g^{\mu\nu}\), one can decompose the Killing vector equilibrium condition as a set of identities, \[\begin{split} u^\mu \partial_\mu T = & 0,\\ T \Delta^\mu_{~\rho} u^\nu \nabla_\nu u^\rho + \Delta^{\mu\rho}\partial_\rho T = & 0, \\ \sigma^{\mu\nu} = \frac{1}{2} \left[ \Delta^{\mu\rho} \Delta^{\nu\sigma} + \Delta^{\mu\sigma} - \frac{1}{d-1} \Delta^{\mu\nu} \Delta^{\rho\sigma} \right] \nabla_\rho u_\sigma = & 0, \\ \nabla_\mu u^\mu = & 0. \end{split}\] We introduced here \(\sigma^{\mu\nu}\) as a combination of derivatives of the fluid velocity that is symmetric, trace-less, and orthogonal to the fluid velocity. In deriving these equations we used the identity \(u_\rho \nabla_\mu u^\rho = 0\), which follows from the normalization condition \(u_\rho u^\rho = -1\).

Global equilibrium states

We can now formulate what are global thermal equilibrium states in a relativistic quantum field theory. Global states can be defined as density matrices or density matrix functionals \(\rho[\phi_1, \phi_2]\) on Cauchy hypersurfaces \(\Sigma\) of spacetime. In the grand canonical ensemble they are given by \[\begin{split} \rho = & \frac{1}{Z} \exp\left( \int d\Sigma_\mu \left\{ - \beta_\nu(x) T^{\mu\nu}(x) - \alpha(x) N^\mu(x) \right\} \right) \\ = & \exp\left( \int d\Sigma_\mu \left\{ - \beta_\nu(x) T^{\mu\nu}(x) - \alpha(x) N^\mu(x) + \beta^\mu(x) p(x)\right\} \right) \end{split}\] We use here the surface element \(d\Sigma_\mu = \sqrt{g} n_{\mu}(x) d^{d-1}x = \sqrt{g}\epsilon_{\mu\alpha\beta\gamma} dx^\alpha dx^\beta dx^\gamma/3!\), with the last equation holding for \(d-1=3\). The time-like normal vector \(n^\mu(x)\) is assumed to be oriented towards the future direction, which together with our choice of metric signature \((-,+,+,+)\) explains the minus signs.

In the second equation we wrote the partition function for the thermal state as \[Z = \exp\left( - \int d\Sigma_\mu \beta^\mu(x) p(x) \right),\] which generalizes our previous expression \(Z=\exp\left( p V / T\right)\) on a constant time hypersurface of Minkowski space.

Similar to what we have discussed before, one can understand the density matrix as an evolution operatore from points \(x^\mu\) on the hypersurface \(\Sigma\) to \(x^\mu-i\beta^\mu(x)\). The chemical potential term can be conveniently rewritten in terms of an external gauge field \(A_\rho(x) = \mu(x) u_\rho(x)\). The density matrix functional becomes \[\rho[\phi_1, \phi_2] = \frac{1}{Z} \int\limits_{\phi_1, \phi_2} D\phi \exp\left( -S[\phi] \right),\] where \(\phi(x)=\phi_1(x)\) and \(\phi(x-i\beta) = \phi_2(x)\) is kept fixed at the boundary, and \(S[\phi]\) is the action with Euclidean conventions in a space with Euclidean metric \(g_{\mu\nu}(x)+2u_\mu(x) u_\nu(x)\).

Action for equilibrated fluid in general coordinates

It is usful to express the action of a fluid in thermal equilibrium in terms of general coordinates. For vanishing field expectation values we have an action that still depends on the matric \(g_{\mu\nu}(x)\) and the external gauge field \(A_\mu(x)\), \[\Gamma[g, A] = - \int d^d x \sqrt{g} \left\{ p(T,\mu) + \text{const} \right\}.\] This follows by simple analytic continuation of the action in Matsubara space to real times. Thermal equilibrium states are fixed by the periodicity condition \[\chi(x^\mu) = \pm \chi(x^\mu-i\beta^\mu(x)),\] The temperature can thus be written as \[T(x) = \frac{1}{\sqrt{-g_{\mu\nu}(x)\beta^\mu(x)\beta^\nu(x)}},\] and the chemical potential in terms of \(A_\rho(x) =\mu(x) u_\rho(x)\) as \[\mu(x) = \frac{-A_\rho(x)\beta^\rho(x)}{\sqrt{-g_{\mu\nu}(x)\beta^\mu(x)\beta^\nu(x)}}.\] We have intentionally expressed everything in terms of the metric \(g_{\mu\nu}(x)\), the external gauge field \(A_\rho(x)\) and the vector field \(\beta^\mu(x)\), because we can then easily vary the action with respect to the metric and the gauge field to obtain the energy-momentum tensor and conserved number current. The inverse temperature vector \(\beta^\mu(x)\), that defines the periodicity condition of fields, is kept fixed in this variantions.

Energy-momentum tensor and current from action

From the quantum effective one can get the energy-momentum tensor and conserved number current through the variations \[\delta \Gamma = \int d^d x \sqrt{g} \left\{ -\frac{1}{2} T^{\mu\nu}(x)\delta g_{\mu\nu}(x) + N^\mu(x) \delta A_\mu(x) \right\}.\] where the action is here defined such that the effective potential apears with positive sign, and the pressure accordingly with negative sign. To do the variations, recall \(g=-\det(g_{\mu\nu})\) and \[\delta \sqrt{g} = \frac{1}{2} \sqrt{g} g^{\mu\nu} \delta g_{\mu\nu},\] such that \[\begin{split} T^{\rho\sigma} = & p \, g^{\rho\sigma} + 2 \frac{\partial p}{\partial T} \frac{\partial T}{\partial g_{\rho\sigma}} + 2 \frac{\partial p}{\partial \mu} \frac{\partial \mu}{\partial g_{\rho\sigma}} \\ = & p \, g^{\rho\sigma} + s T^3 \beta^\rho \beta^\sigma + n \mu T^2 \beta^\rho \beta^\sigma \\ = & p \, g^{\rho\sigma} + (\epsilon+p) u^\rho u^\sigma. \end{split}\] We have used here thermodynamic identities such as \(dp=sdT + n d\mu\) and \(\epsilon + p = sT + \mu n\). The resulting energy-momentum tensor is indeed of the expected form for a thermal equilibrium state.

In a similar way we find the conserved current \[N^\rho = - \frac{\partial p}{\partial \mu} \frac{\partial \mu}{\partial A_\rho} = n T \beta^\rho = n u^\rho.\] This is also of the form expected.

Ideal fluid dynamics

In thermal equilibrium, the energy-momentum tensor is fixed by the fluid velocity, or the frame where the fluid is at rest, and two thermodynamic variables which can be \(T\) and \(\mu\) or \(\epsilon\) and \(n\) for example. The idea behind ideal fluid dynamics is to postulate that thermal equilibrium states can be made local, such that the fluid velocity \(u^\mu(x)\), the energy density \(\epsilon(x)\) and particle number density \(n(x)\) can become general functions of space and time but that the energy-momentum tensor and number current still have locally the same form as in global equilibrium. It is not guarantied that this works, because terms proportional to gradients of the fluid fields emerge in general out-of-global equilibrium. In this sense, ideal fluid dynamics can only be the leading order of a derivative expansion.

Assuming now that the global equilibrium expressions also hold at local thermal equilibrium, we find for the conservation of energy and momentum \[\nabla_\mu T^{\mu\nu} = \nabla_\mu \left( (\epsilon+p) u^\mu u^\nu + p g^{\mu\nu} \right) = 0.\] This equation can be contracted with \(u_\nu\), which yields \[u^\mu\partial_\mu \epsilon + (\epsilon + p) \nabla_\mu u^\mu =0,\] and it can be contracted with the projector orthogonal to the fluid velocity \(\Delta^\rho_{~\sigma} = u^\rho u_\sigma + \delta^\rho_{~\sigma}\), which yields \[(\epsilon+p) u^\mu \nabla_\mu u^\rho + \Delta^{\rho\sigma}\partial_\sigma p = 0.\] These two equations can be seen as spefiying the time evolution of energy density and fluid velocity, respectively. They get supplemented by the conservation law for particle number, \[u^\mu \partial_\mu n + n \nabla_\mu u^\mu = 0,\] which can be seen as a time evolution equation for the density \(n\).

Let us emphasize again that ideal fluid dynamics in only an approximation and higher order corrections in a derivative expansion are needed when one goes further away from global equilibrium.