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arXiv:1404.6336 · Physica Scripta

Observables and density matrices embedded in dual Hilbert spaces

Tomaž Prosen, Laura F. Martignon, T. H. Seligman

Published Jun 1, 2015• 1 citation

Abstract

The introduction of operator states and of observables in various fields of quantum physics has raised questions about the mathematical structures of the corresponding spaces. In the framework of third quantization it had been conjectured that we deal with Hilbert spaces although the mathematical background was not entirely clear, particularly, when dealing with bosonic operators. This in turn caused some doubts about the correct way to combine bosonic and fermionic operators or, in other words, regular and Grassmann variables. In this paper we present a formal answer to the problems on a simple and very general basis. We illustrate the resulting construction by revisiting the Bargmann transform and finding the known connection between L 2 ( R ) and the Bargmann–Hilbert space. We pursue this line of thinking one step further and discuss the representations of complex extensions of linear canonical transformations as isometries between dual Hilbert spaces. We then use the formalism to give an explicit formulation for Fock spaces involving both fermions and bosons thus solving the problem at the origin of our considerations.

DOI: 10.1088/0031-8949/90/7/074036

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The introduction of operator states and of observables in various fields of quantum physics has raised questions about the mathematical structures of the corresponding spaces. In the framework of third quantization it had been conjectured that we deal with Hilbert spaces although the mathematical background was not entirely clear, particularly, when dealing with bosonic operators. This in turn caused some doubts about the correct way to combine bosonic and fermionic operators or, in other words, regular and Grassmann variables. In this paper we present a formal answer to the problems on a simple and very general basis. We illustrate the resulting construction by revisiting the Bargmann transform and finding the known connection between L 2 ( R ) and the Bargmann–Hilbert space. We pursue this line of thinking one step further and discuss the representations of complex extensions of linear canonical transformations as isometries between dual Hilbert spaces. We then use the formalism to give an explicit formulation for Fock spaces involving both fermions and bosons thus solving the problem at the origin of our considerations.

The basic ideas of second quantization and Fock space are extended to density operator states, used in treatments of open many‐body systems. This can be done for fermions and bosons. While the former only requires the use of a non‐orthogonal basis, the latter requires the introduction of a dual set of spaces. In both cases an operator algebra closely resembling the canonical one is developed and used to define the dual sets of bases. We here concentrated on the bosonic case where the unboundedness of the operators requires the definitions of dual spaces to support the pair of bases. Some applications, mainly to non‐equilibrium steady states, will be mentioned.

We discuss the numerical implementation of two related representations of fermionic density matrices which have been introduced in Dalton et al. (2016). In both of them, the density matrix is expanded in a basis of Bargmann coherent states with weights given by the two phase space distributions. We derive the equations of motion for the distributions when imaginary time evolution is generated by the Hubbard Hamiltonian. One of them is a Grassmann Fokker–Planck equation that can be re-cast into a remarkably simple Itô form involving solely complex variables. In spite of this simple form, we demonstrate that complications arise in numerically computing the expectation value of any observable. These are due to exponential growth in the matrix elements of the stochastic propagator, delicate numerical sensitivity in performing primitive linear algebra operations, and the re-appearance of a sign problem.

Recent work by Moshinsky et al. on the role and applications of canonical transformations in quantum mechanics has focused attention on some complex extensions of linear transformations mapping the position and momentum operators x̂ and p̂ to a pair η̂ and ζ̂ of canonically conjugate, but not necessarily Hermitian, operators. In this paper we show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η̂, ζ̂ can be realized in the Schrödinger representation η and −id/dη. We provide a unitary mapping onto the ordinary Hilbert space of square-integrable functions over the real line through an integral transform. The transform kernels provide a representation of a subsemigroup of SL(2,C). The well-known Bargmann transform is the special case when η̂ and iζ̂ are the harmonic oscillator raising and lowering operators. The Moshinsky-Quesne transform is regained in the limit when the canonical transformation becomes real, a case which contains the ordinary Fourier transform. We present a realization of these transforms through hyperdifferential operators.

We discuss the numerical implementation of two related representations of fermionic density matrices which have been introduced in Annals of Physics 370, 12 (2016). In both of them, the density matrix is expanded in a basis of Bargmann coherent states with weights given by the two phase space distributions. We derive the equations of motion for the distributions when imaginary time evolution is generated by the Hubbard Hamiltonian. One of them is a Grassmann Fokker-Planck equation that can be re-cast into a remarkably simple Itô form involving solely complex variables. In spite of this simple form, we demonstrate that complications arise in numerically computing the expectation value of any observable. These are due to exponential growth in the matrix elements of the stochastic propagator, delicate numerical sensitivity in performing primitive linear algebra operations, and the re-appearance of a sign problem.

The reduced

k

-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of

k

-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of

k

-body operators for all

k

.