An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 9.2

Functional Methods — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · functional-methods-quantum-statistical-mechanics

Problem

Quantum statistical mechanics. (a) Evaluate the quantum statistical partition function Z=tr[e−βH]Z = \text{tr}[e^{-\beta H}] (where β=1/kT\beta = 1/kT) using the strategy of Section 9.1 for evaluating the matrix elements of e−iHte^{-iHt} in terms of functional integrals. Show that one finds a functional integral, over functions defined on a domain that is of length β\beta and periodically connected in the time direction. Note that the Euclidean form of the Lagrangian appears in the weight. (b) Evaluate this integral for a simple harmonic oscillator, LE=12x˙2+12ω2x2,L_E = \frac{1}{2}\dot{x}^2 + \frac{1}{2}\omega^2 x^2, by introducing a Fourier decomposition of x(t)x(t): x(t)=∑nxn⋅1βe2πint/β.x(t) = \sum_n x_n \cdot \frac{1}{\sqrt{\beta}} e^{2\pi i n t/\beta}. The dependence of the result on β\beta is a bit subtle to obtain explicitly, since the measure for the integral over x(t)x(t) depends on β\beta in any discretization. However, the dependence on ω\omega should be unambiguous. Show that, up to a (possibly divergent and β\beta-dependent) constant, the integral reproduces exactly the familiar expression for the quantum partition function of an oscillator. [You may find the identity sinh⁡z=z⋅∏n=1∞(1+z2(nπ)2)\sinh z = z \cdot \prod_{n=1}^{\infty}\left(1 + \frac{z^2}{(n\pi)^2}\right) useful.] (c) Generalize this construction to field theory. Show that the quantum statistical partition function for a free scalar field can be written in terms of a functional integral. The value of this integral is given formally by [det⁡(−∂2+m2)]−1/2,\left[\det(-\partial^2 + m^2)\right]^{-1/2}, where the operator acts on functions on Euclidean space that are periodic in the time direction with periodicity β\beta. As before, the β\beta dependence of this expression is difficult to compute directly. However, the dependence on m2m^2 is unambiguous. (More generally, one can usually evaluate the variation of a functional determinant with respect to any explicit parameter in the Lagrangian.) Show that the determinant indeed reproduces the partition function for relativistic scalar particles. (d) Now let ψ(t)\psi(t), ψˉ(t)\bar{\psi}(t) be two Grassmann-valued coordinates, and define a fermionic oscillator by writing the Lagrangian LE=ψˉψ˙+ωψˉψ.L_E = \bar{\psi}\dot{\psi} + \omega\bar{\psi}\psi. This Lagrangian corresponds to the Hamiltonian H=ωψˉψ,with {ψˉ,ψ}=1;H = \omega\bar{\psi}\psi, \quad \text{with } \{\bar{\psi}, \psi\} = 1; that is, to a simple two-level system. Evaluate the functional integral, assuming that the fermions obey antiperiodic boundary conditions: ψ(t+β)=−ψ(t)\psi(t + \beta) = -\psi(t). (Why is this reasonable?) Show that the result reproduces the partition function of a quantum-mechanical two-level system, that is, of a quantum state with Fermi statistics. (e) Define the partition function for the photon field as the gauge-invariant functional integral Z=∫DA  exp⁡ ⁣(−∫[14(Fμν)2])Z = \int \mathcal{D}A \; \exp\!\left(-\int [\frac{1}{4}(F_{\mu\nu})^2]\right) over vector fields AμA_\mu that are periodic in the time direction with period β\beta. Apply the gauge-fixing procedure discussed in Section 9.4 (working, for example, in Feynman gauge). Evaluate the functional determinants using the result of part (c) and show that the functional integral does give the correct quantum statistical result (including the correct counting of polarization states).

Related theory

Browse the problem library