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Publications of the Research Institute for Mathematical Sciences

The General Form of non-Fock Coherent Boson States

Reinhard Honegger, Alfred Rieckers

Published Apr 30, 1990• 34 citations
Publisher

Abstract

Specifying their (normally ordered) characteristic functions we determine all states of the boson C *-Weyl algebra which satisfy Glauber's coherence condition and are not realizable as density operators in Fock space. The pure ones are shown to be just the eigenstates of the annihilation operators in their GNS-representations (in contrast to the Fock case) and are characterized in many equivalent manners. The central decomposition of an arbitrary coherent state has the macroscopic phase variable as parameter and is supported by the pure coherent states, which is in fact the only way for a maximal decomposition. The set of all coherent states with the same absolute factorizing function is proven to be a Bauer simplex. The appearence of a classical coherent field part is studied in detail in the GNS-representations and shown to correspond to an enlargement of the set of one boson states by just one additional mode.

DOI
10.2977/prims/1195171085

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In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particles with integer spin and have symmetric wave functions; Fermions are quantum particles with half-integer spin and are represented by anti-symmetric wave functions. The functional setting is given by a symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references [23, 42]. Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions. In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as \(\hbar \searrow 0\) of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity. As we have done for finite systems, we use here Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers [114–116] Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations. Finally, following the paper [225] one can show that, in the limit \(\hbar \rightarrow 0\), the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical fields equation (Hartree equation) with a uniform remainder estimates in time.

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Specifying their (normally ordered) characteristic functions we determine all states of the boson C *-Weyl algebra which satisfy Glauber's coherence condition and are not realizable as density operators in Fock space. The pure ones are shown to be just the eigenstates of the annihilation operators in their GNS-representations (in contrast to the Fock case) and are characterized in many equivalent manners. The central decomposition of an arbitrary coherent state has the macroscopic phase variable as parameter and is supported by the pure coherent states, which is in fact the only way for a maximal decomposition. The set of all coherent states with the same absolute factorizing function is proven to be a Bauer simplex. The appearence of a classical coherent field part is studied in detail in the GNS-representations and shown to correspond to an enlargement of the set of one boson states by just one additional mode.
We analyze the first order coherent, states (and various sub-classes) over an arbitrary C*-Weyl algebra. Their normally ordered characteristic functions are expressed in terms of an infinite, positive-definitite matrix, which exhibits an additional positive-definiteness condition if the states are classical. In the latter case the matrix elements are identified as moments of probability measures on C. By going over to measures on bC, the Bohr compactification of C, there arises a Bauer simplex of generalized classical coherent states
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.
Abstract This chapter defines and determines the properties of the important (though non-physical) boson and fermion coherent states, which are specified in terms of c-number (bosons) or Grassmann (fermions) variables and their complex conjugates, and defined via unitary displacement operators acting on the vacuum state. Important non-orthogonality, overcompleteness and coherence properties of coherent states are obtained. Coherent states are shown to be eigenstates of mode annihilation operators, with eigenvalues given by the c-number or Grassmann variable specifying the state. Related (unnormalised) Bargmann states are defined, depending only on the variable and not its complex conjugate, and are used to obtain useful representations of physical states and canonical forms for operators in terms of Bargmann state projectors, as well as obtaining important results such as for traces of operator products in terms of phase space integrals. Mode creation operators acting on Bargmann states are shown as equivalent to differentiation.
Abstract A decomposition theory for positive sesquilinear forms densely defined in Hilbert spaces is developed. On decomposing such a form into its closable and singular part and using Bochner's theorem it is possible to derive the central decomposition of the associated gauge-invariant quasifree state on the boson C*-Weyl algebra. The appearance of a classical field part of the boson system is studied in detail in the GNS-representation and shown to correspond to the so-called singular subspace of a natural enlargement of the one-boson testfunction space. In the example of Bose-Einstein condensation a non-trivial central decomposition (or equivalently a non-trivial classical field part) is directly related to the occurrence of the condensation phenomenon.
In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particles with integer spin and have symmetric wave functions; Fermions are quantum particles with half-integer spin and are represented by anti-symmetric wave functions. The functional setting is given by a symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references [23, 42]. Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions. In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as ℏ↘0\hbar \searrow 0ℏ↘0 of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity. As we have done for finite systems, we use here Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers [114–116] Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations. Finally, following the paper [225] one can show that, in the limit ℏ→0\hbar \rightarrow 0ℏ→0, the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical fields equation (Hartree equation) with a uniform remainder estimates in time.