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AIP conference proceedings

Quantum Many-Body Virial Theorem And Matsubara Green’s Function

大輔 安間

Published Jan 1, 2004• 0 citations
Publisher

Abstract

We discuss the quantum field theoretical formulation of the virial theorem on the basis of the canonical field theory of the generalized coordinate transformation and show the equation of motion of a charged Fermion system coupled to an electromagnetic field. Possible application to Fermion‐Boson mixtures is also discussed.
DOI
10.1063/1.1764295

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The virial theorem, introduced by Clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics, is studied by making use of symplectic formalism as an approach, in the case of both the Hamiltonian and Lagrangian systems. The possibility of establishing virial-like theorems from one-parameter groups of non-strictly canonical transformations is analysed; the case of systems with a position-dependent mass is also discussed. Using the modern symplectic approach to quantum mechanics, we arrive at the quantum virial theorem in full analogy with the classical case.
The theory of many-Boson systems was considered. A canonical transformation of the creation and annihilation operators was introduced in order to eliminate the singleparticle zero-momentam state and allow the application of field-theoretic penturbation techniques. These techniques were used to give a simpler derivation of a result due to Belyaev and to prove a theorem about the excitation spectrum of the system. General techniques for calculating the Green's functions are discussed. The excitation spectrum was calculated to second order for small wave number for the case of a repulsive square-well potential.
We formulate equations of motion and conservation laws for a quantum many-body system in a co-moving Lagrangian reference frame. It is shown that generalized inertia forces in the co-moving frame are described by Green's deformation tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$ and a skew-symmetric vorticity tensor ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$, where $\mathbf{\ensuremath{\xi}}$ in the Lagrangian coordinate. Equations of motion are equivalent to those for a quantum many-body system in a space with time-dependent metric ${g}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$ in the presence of an effective magnetic field ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$. To illustrate the general formalism we apply it to the proof of the harmonic potential theorem. As another example of application we consider a fast long wavelength dynamics of a Fermi system in the dynamic Hartree approximation. In this case the kinetic equation in the Lagrangian frame can be solved explicitly. This allows us to formulate the description of a Fermi gas in terms of an effective nonlinear elasticity theory. We also discuss a relation of our results to time-dependent density functional theory.