Quantum field theory 1, lecture 06

4.3 Thermodynamic equilibrium

In this section we discuss the thermal equilibrium state for a single quantum harmonic oscillator. This is a first example for the approach to quantum statistical equilibrium that can later be generalised to quantum field theories with interactions. For thermodynamic equilibrium, \(Z= \text{Tr} \, e^{-\tilde{\beta }\tilde{H}}\), one identifies \(x_{N+1}\) with \(x_{-N}\) and includes no integration over \(x_{N+1}.\) The variable \(j\) is periodic, reflecting in \begin{equation*} x_{N+1} = x_{-N},\quad \quad \quad p_{N+1}=p_{-N}. \end{equation*} Formally, this can be achieved by choosing for \(\tilde{b}\) a \(\delta \) - function. For periodic boundary conditions one has \begin{equation*} \tilde{Z}= \text{Tr} \, e^{\tilde{\beta }\tilde{H}} = \int D\phi \, e^{-S}, \end{equation*} with \begin{equation*} S= -\sum _{j=N}^N \left \{ ip_j(x_{j+1}- x_j)- \delta \left [\tfrac{p_j^2}{2}+V(x_j) \right ]\right \}, \end{equation*} and \begin{equation*} \int D\phi = \left [\prod _j \int dx_j \int \frac{dp_j}{2\pi }\right ]. \end{equation*} There are a total of \(2N+1\) factors, and \(\delta \) is related to \(\tilde \beta \) by \(\delta =\frac{\tilde{\beta }}{2N+1}\).

Matsubara sum. Quantum statistics is described by the so called Matsubara formalism. We derive this formalism here for a single harmonic oscillator, with straightforward generalisations. We can diagonalize the action \(S\) by a type of Fourier transform \begin{equation*} x_j = \sum _{n=-N}^N \exp \left (\frac{2\pi inj}{2N+1}\right )\tilde{x}_n,\quad \quad \quad \tilde{x}_{-n}=\tilde{x}^{*}_{n}, \end{equation*} \begin{equation*} p_j = \sum _{n=-N}^N \exp \left (\frac{2\pi in (j+\frac{1}{2})}{2N+1}\right )\tilde{p}_n,\quad \quad \quad \tilde{p}_{-n}=\tilde{p}^*_n, \end{equation*} such that \begin{equation*} -\sum _{j=-N}^N[ip_j(x_{j+1}-x_j)]= \sum _{n=-N}^N[(2N+1) \text{sin}\left ( \frac{\pi n}{2N+1} \right )(\tilde{p}^*_n\tilde{x}_n-\tilde{p}_n \tilde{x}^*_n)]. \end{equation*} Here we use the identity (\(j=-N\) and \(j=N+1\) identified) \begin{equation*} \sum _{j=-N}^N \text{exp}\left (\frac{2\pi i (m-n)j}{2N+1}\right ) = (2N+1) \delta _{m,n}. \end{equation*} Similarly, with \(V(x_j) =x^2_j/2\), one has \begin{equation*} \frac{\delta }{2}\sum _{j=-N}^N(x^2_j + p^2_j)=\frac{(2N+1)\delta }{2}\sum _{n=-N}^N (\tilde{x}^*_n\tilde{x}_n + \tilde{p}^*_n\tilde{p}_n) = \frac{\tilde{\beta }}{2}\sum _{n=-N}^N (\tilde{x}^*_n \tilde{x}_n + \tilde{p}^*_n \tilde{p}_n). \end{equation*} The action becomes a sum over independent pieces, labelled by \(n\). The sum over \(n\) is the Matsubara sum.

Complex fields. We next introduce complex numbers \(\phi _n\) by \begin{equation*} \tilde{x}_n = \frac{1}{\sqrt{2}}(\phi _n + \phi ^*_{-n}),\quad \quad \quad \tilde{p}_n = -\frac{i}{\sqrt{2}}(\phi _n - \phi ^*_{-n}), \end{equation*} With \begin{equation*} \tilde{p}^*_n \tilde{x}_n - \tilde{x}^*_n\tilde{p}_n = i(\phi ^*_n\phi _n - \phi ^*_{-n}\phi _{-n}), \end{equation*} and \begin{equation*} \tilde{x}^*_n \tilde{x}_n + \tilde{p}^*_n\tilde{p}_n=\phi ^*_n\phi _n + \phi ^*_{-n} \phi _{-n}, \end{equation*}

we finally obtain for the action \begin{equation*} S = \sum _{n=-N}^N \left [2(2N+1)i \text{sin}\left ( \frac{\pi n}{(2N+1)} \right )+ \tilde{\beta } \right ]\phi ^*_n\phi _n. \end{equation*} The modes \(\phi _n\) are called Matsubara modes, and the sum over \(n\) is the Matsubara sum.

One can also translate the integration measure for the variables \(x_j\) and \(p_j\) to \(\phi _n\). With \begin{equation*} \phi _n = \phi _{nR}+ i \phi _{nI}, \end{equation*} one has \begin{equation*} \int D\phi = \prod _n \left (\int _{-\infty }^{\infty } d\phi _{nR} \int _{-\infty }^{\infty } d\phi _{nI}\right ). \end{equation*} All variable transformations have been linear transformations and there is no non-trivial Jacobian. Recall that an overall constant factor of \(Z\) or additive constant in \(S\) is irrevelant.

Matsubara frequencies. At the end we take the limit \(N \to \infty .\) In this limit the neglected terms (from commutators of \(\hat x\) and \(\hat p\)) vanish. This yields the central functional integral equation for thermodynamic equilibrium, \begin{equation*} \text{Tr}\{e^{-\beta H} \} = \int D\phi \; e^{-S}. \end{equation*} For \(H= \omega (a^\dagger a + \frac{1}{2})\) one has \begin{equation*} S= \sum _{n= -\infty }^\infty (2\pi i n + \beta \omega ) \phi ^*_n\phi _n. \end{equation*} (Recall that \(\tilde{H}= a^\dagger a + \frac{1}{2}\) and \(\tilde{\beta } = \beta \omega \).) The quantities \begin{equation*} \tilde{\omega }_n = \frac{2\pi n}{\beta } = 2\pi n T \end{equation*} are called Matsubara frequencies.

Action for free quantum fields. This result extends directly to a free quantum field theory. The partition function \(Z\) factorises for the different momentum modes, \(Z=\prod _q Z_q\), and correspondingly the action for all momentum modes is simply the sum of actions for individual momentum modes, \(S= \sum _q S_q\). For a given momentum mode one has \(\tilde{\beta }= \beta \omega _q.\) Thus for \begin{equation*} H= \sum _q \omega (q) \left [a^\dagger _q a_q + \frac{1}{2} \right ], \end{equation*} one obtains \begin{equation*} \begin{split} S &= \sum _n \sum _q \left [2\pi i n + \beta \omega (q) \right ] \phi ^*_n(q) \phi _n(q)\\ &= \sum _n\sum _q \beta \left [i \tilde{\omega }_n + \omega (q) \right ]\phi ^*_n(q) \phi _n(q).\\ \end{split} \nonumber \end{equation*} One often denotes the dispersion relation by \(\omega (q)\) or by \(\varepsilon (q).\) For non-relativistic particles the Matsubara frequencies \begin{equation*} \tilde{\omega }_n = \frac{2\pi n}{\beta } = 2\pi n T \end{equation*} multiply a term quadratic in the Matsubara modes. At this point we have formulated the thermodynamics of phonons or atoms as a functional integral. It is gaussian and can easily be solved explicitely.

The solution of this functional integral is well known. It is the expression of the partition function in terms of mean occupation numbers, as derived in the course on theoretical statistical physics. It is a worthwhile exercise to reproduce this result by solving the functional integral. This involves suitable Matsubara sums. It is actually easier to compute derivatives as the mean energy.

Euclidean time. We can consider the Matsubara modes \(\phi _n\) as the modes of a discrete Fourier transformation. Indeed, making a Fourier transformations of functions on a circle yields discrete modes. Consider a function \(\phi (\tau )\), with \(\tau \) parameterizing a circle with circumference \(\beta \). Equivalently, we can take \(\tau \) to be a periodic variable with period \(\beta \) \begin{equation*} \tau + \beta \equiv \tau . \end{equation*} The Fourier expansion reads \begin{equation*} \phi (\tau ) = \sum _n \exp \left (\frac{2\pi i n \tau }{\beta }\right ) \phi _n, \end{equation*} with integer \(n\). With \begin{equation*} \begin{split} \partial _\tau \phi (\tau ) &= \sum _n\left (\frac{2\pi i n}{\beta }\right ) \exp \left (\frac{2\pi i n \tau }{\beta }\right )\phi _n \\ &= \sum _n i\tilde{\omega }_n \exp \left (\frac{2\pi i n \tau }{\beta }\right )\phi _n, \end{split} \nonumber \end{equation*} one has \begin{equation*} \int _{-\frac{\beta }{2}}^{\frac{\beta }{2}} d\tau \left \{ \phi ^*(\tau )\partial _\tau \phi (\tau ) \right \} = \sum _n i\tilde{\omega }_n \, \phi ^*_n \phi _n. \end{equation*} Here we employ the identity for discrete Fourier transforms \begin{equation*} \int _{-\frac{\beta }{2}}^{\frac{\beta }{2}} d\tau \exp \left (\frac{2\pi i (n-m)\tau }{\beta } \right ) = \beta \, \delta _{m,n}. \end{equation*} In this basis the action reads \begin{equation*} S = \int _{-\frac{\beta }{2}}^{\frac{\beta }{2}} d\tau \sum _q \left [\phi ^*(\tau ,q)\, \partial _\tau \phi (\tau ,q) + \omega (q) \, \phi ^*(\tau ,q) \phi (\tau ,q) \right ]. \end{equation*} One calls \(\tau \) the Euclidean time. The Fourier modes depend on an additional periodic variable - namely euclidean time.

Local action and transfer matrix. This action is a local action in the sense of lectures 2 and 3. Discretizing \(\tau \) on a lattice with distance \(\varepsilon \), and with \(\tau = j \varepsilon , \; j= -N\cdots N\; \text{periodic}, \; \epsilon = \tfrac{\beta }{2N+1},\) the partial derivative is replaced by a lattice derivative \begin{equation*} \partial _\tau \phi (\tau ) = \frac{1}{\varepsilon } \left [\phi (\tau +\varepsilon ) - \phi (\tau ) \right ], \end{equation*} One can write (with \(\sum _\tau \equiv \sum _j\)) \begin{equation*} S= \sum _\tau \mathscr{L}(\tau ), \end{equation*} with \begin{equation*} \mathscr{L}(\tau )= \frac{1}{2} \sum _q \left \{ \phi (\tau +\varepsilon )\phi ^*(\tau )- \phi ^*(\tau +\varepsilon )\phi (\tau )+\varepsilon \omega (q) \left [ \phi ^*(\tau +\varepsilon )\phi (\tau ) + \phi (\tau +\varepsilon )\phi ^*(\tau ) \right ] \right \}. \end{equation*} Here we omit the label \(q\) for the momentum modes. Note that \(\mathscr{L}(\tau )\) is a complex function of complex variables \(\phi (\tau )\) and \(\phi (\tau +\varepsilon ).\) With respect to \(\tau \) the action involves next neighbour interactions, similar to the Ising model. We could go the inverse way and compute the transfer matrix. We know already the answer in the bosonic occupation number basis \begin{equation*} \hat{T} = \exp \left [-\frac{\beta }{2N+1}\sum _q \omega (q) \left (a^\dagger _q a_q + \frac{1}{2}\right )\right ], \end{equation*} with \(2N+1\) the number of time points. This is compatible with \begin{equation*} Z= \text{Tr} \left \{\hat{T}^{2N+1}\right \}. \end{equation*}

This closes the circle to our first approach. We could start with the functional integral, derive the transfer matrix, and define the partition function as a product of transfer matrices.

Quantum gas of bosonic atoms. For free bosonic atoms (without internal degrees of freedom) the dispersion relation is \begin{equation*} \epsilon (q) = \frac{\vec{q}^2}{2M}-\mu , \end{equation*} with \(\mu \) the chemical potential. We can make a Fourier-transform to three-dimensional position space, \begin{equation*} S = \int _{-\frac{\beta }{2}}^{\frac{\beta }{2}} d\tau \int d^3 x \{\phi ^*(\tau ,\vec{x})\partial _\tau \phi (\tau ,x) + \frac{1}{2M} \vec \nabla \phi ^*(\tau ,\vec{x})\vec{\nabla }\phi (\tau ,\vec{x}) - \mu \phi ^*(x)\phi (x)\} \end{equation*} This is the action of a field theory in Euclidean time.

For a quantum field theory the action defines the weight factor in a functional integral. The extremum of the action yields the ”classical field equation”. This classical field equation is, however, a ”microscopic” object. The field equations that are valid for a quantum field theory have to include the effects of fluctuations!

Interactions. So far we have discussed models that represent quantum fields without interactions. This is a very good approximation for phonons if the energy is not too high. Free quantum field theories can be represented in momentum space as uncoupled harmonic oscillators. For them the description is simple both in the functional integral formalism (gaussian integration) and in the operator formalism. The situation changes in the presence of interactions.

Consider a pointlike interaction between bosonic atoms. \begin{equation*} H = H_0 + H_{int} \end{equation*} \begin{equation*} H_0 = \sum _q \omega (q) \left (a^\dagger _q a_q + \frac{1}{2}\right ) \end{equation*} \begin{equation*} H_{int} = \frac{\lambda }{2}\sum _{q_1,q_2,q_3,q_4} a^\dagger _{q_4}a^\dagger _{q_3}a_{q_2}a_{q_1} \delta (q_1+q_2-q_3-q_4). \end{equation*} Two atoms with momentum \(q_1\) and \(q_2\) are annihilated, two atoms with momenta \(q_3\) and \(q_4\) are created. Momentum conservation is guaranteed by the \(\delta \)-function.

For the functional integral this adds to the action a piece \begin{equation*} S_{int} =\tfrac{\lambda }{2} \int d\tau \int d^3 x [(\phi ^*(\tau ,\vec{x}) \phi (\tau ,\vec{x}))^2 -2\delta \mu \phi ^*(\tau ,\vec{x}) \phi (\tau ,\vec{x})] \end{equation*} with \(\delta \mu \sim \lambda \) a counterterm that corrects \(\mu \). The additional interaction term is is the only modification needed for the functional integral! Euclidean time remains periodic with period \(\beta \), and this is the only point where the value of the temperature enters. We will not perform here a derivation of the Matsubara formalism in the presence of interactions. Starting from the operator formalism one can divide \(\beta \) into small pieces and work with a basis of ”coherent states”. This cuts short the various transformations that we have performed for the free theory. We will simply take the functional integral in euclidean time as a starting point.

For an interacting gas of bosonic atoms the functional integral permits us to investigate phenomena as the Bose-Einstein condensation and the associated superfluidity in dependence on temperature and particle number density or chemical potential. For atoms at ultracold temperature this is a very interesting topic both for experiment and theory.

A systematic treatment of interactions beyond a perturbative expansion in small \(\lambda \) is rather hard in the operator formalism. For the functional integral formulation powerful methods are available. This is one of the main reasons why we concentrate on the functional integral.

Zero temperature limit. For \(T\longrightarrow 0\) one has \(\beta \longrightarrow \infty \). The circumference of the circle goes to infinity. Instead of discrete Matsubara modes one has continuous modes with frequency \(\tilde{\omega } = q_0\) and therefore a continuous four-dimensional momentum integral. The momenta \(q_0\) and \(\vec{q}\) appear, however differently in the action. The same holds for the dependence of \(S\) on \(\tau \) and \(\vec{x}\). There is a first derivative with respect to \(\tau \), but a squared first derivative or second derivative with respect to \(\vec{x}\). This difference will go away for relativistic particles. For bosonic atoms with a pointlike interaction one finds for the action in Fourier space for the \(T\longrightarrow 0\) limit of the thermal equilibrium state \begin{equation*} \begin{split} S &= \int _q \phi ^*(q)\left (i \tilde{\omega } + \frac{\vec{q}^2}{2M}-\mu +\lambda \delta \mu \right )\phi (q) \\ &\qquad + \frac{\lambda }{2}\int _{q_1}\int _{q_2}\int _{q_3}\int _{q_4} \phi ^*(q_4)\phi ^*(q_3)\phi _(q_2)\phi (q_1)\delta (q_4 + q_3-q_2-q_1), \end{split} \end{equation*} where we have chosen an appropriate continuum normalization of \(\phi (q)\), with \begin{equation*} \phi (q) \equiv \phi (\tilde{\omega },\vec{q}) \end{equation*} \begin{equation*} \int _q = \frac{1}{(2\pi )^4} \int d\tilde{\omega } d^3 \vec{q} \end{equation*} \begin{equation*} \delta (q) = (2\pi )^4 \delta (\omega )\delta (q_1)\delta (q_2)\delta (q_3). \end{equation*} The \(\delta \) function expresses conservation of the euclidean four momentum \(q\). It reflects translation symmetry in space and euclidean time \(\tau \). The limit \(T\longrightarrow 0\) can be associated in some sense with the vacuum, if one chooses \(\mu \) such that the mean particle number vanishes.

Summary. At this stage we have established an important starting point for our lecture based on the functional integral. The functional integral can describe both classical statistical thermodynamic equilibrium and quantum statistical thermodynamic equilibrium. Different models or different microphysical laws are encoded in the particular form of the action. This form is often largely dictated by symmetry. The ”fundamental laws” are formulated in terms of the action. It is often not necessary to know the precise form of the Hamiltonian in the operator formalism for quantum systems, or the precise form of the transfer matrix for classical probabilistic systems. This is an important advantage, since the operator formalism can become quite complicated for interacting many body systems.

The lecture is called ”quantum field theory”, but you may realise that the quantum aspects are actually not crucial. What counts are the presence of fluctuations. The origin of the fluctuations, be it quantum fluctuations or thermal fluctuations or both, is not important. A more adapted name for our lecture could be ”probabilistic field theory”. We stick to the traditional name of quantum field theory for historical reasons. It should also be clear that our treatment applies to arbitrary settings with fluctuations. Fluctuations may be market fluctuations in economy or fluctuations in the reproduction of species in biology. Whenever a system is described by a probability distribution there exists an associated action.