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For a class of Bardeen-Cooper-Schrieffer (BCS)-models, with complex, weakly momentum dependent interaction coefficients, the representation dependent effective Hamiltonians and their spectra are reconsidered in order to obtain a consistent physical picture by means of operator algebraic methods. The starting point is the limiting dynamics, the existence of which had been proved in a previous work, in terms of a C*-dynamical system acting in a classically extended, electronic Canonical Anticommutation Relations (CAR)-algebra. The C*-algebraic KMS-theory, including the low temperature limit, specifies the order parameters. These appear as classical observables, which commute with all other observables, constituting elements of the center of the algebra. The algebraic spectral theory, in the sense of Arveson, is first applied to the dynamics in general pure energy state representations. The spectra of the finite temperature representations are analyzed, identifying the gap as the lowest of those energy values, which are stable under local perturbations. Further insights are obtained by decomposing the thermal dynamical systems into the pure energy state Heisenberg dynamics, after having first extended them to more comprehensive W*-dynamical systems. The decomposing orthogonal measure is transferred to the infinite product space of quasi-particle occupation numbers and its support is characterized in terms of 0-1-laws leading to an asymptotic ratio of quasi-particles and holes, which depends on the temperature. This ratio is connected with an algebraic invariant of the representation dependent observable algebra. Energy renormalization aspects and pair occupation probabilities are discussed. The latter reveal, beside other things, the difference between macroscopic term occupation and coherent macroscopic term occupation for a condensate.

Starting from a (possibly infinite dimensional) pre-symplectic space (E, ), we study a class of modified Weyl quantizations. For each value of the real Planck parameter ~ we have a C*-Weyl algebra W(E,~ ), which altogether constitute a con- tinuous field of C*-algebras, as discussed in previous works. For ~ = 0 we construct a Frechet-Poisson algebra, densely contained in W(E,0), as the classical observables to be quantized. The quantized Weyl elements are decorated by so-called quantiza- tion factors, indicating the kind of normal ordering in specific cases. Under some assumptions on the quantization factors, the quantization map may be extended to the Frechet-Poisson algebra. It is demonstrated to constitute a strict and continu- ous deformation quantization, equivalent to the Weyl quantization, in the sense of Rieel and Landsman. Realizing the C*-algebraic quantization maps in regular and faithful Hilbert space representations leads to quantizations of the unbounded field expressions.

After having set up a concise frame for pure and mixed all-order coherent states, we let them undergo a rather general squeezing transformation. These squeezed states are always non-classical, but some—apparently experimentally preparable—of them are still second-order coherent. They combine peculiar properties of their quantum fluctuations with the ordering features of coherence, and deserve special attention. Quite generally, several criteria for non-classicality are discussed and applied to the considered class of squeezed states. The connection between non-classicality and anti-bunching resp. sub-Poissonian photon statistics is shown to be more subtle than quantum optical usage suggests.

For an arbitrary (possibly infinite-dimensional) pre-symplectic test function space \( (E, \sigma) \) the family of Weyl algebras \( \{\mathcal{W}(E, \hbar\sigma)\}_{\hbar\in\mathbb{R}} \) , introduced in a previous work [1], is shown to constitute a continuous field of C*-algebras in the sense of Dixmier. Various Poisson algebras, given as abstract (Frechet-) *-algebras which are C*-norm-dense in \( \mathcal{W}(E, 0) \) , are constructed as domains for a Weyl quantization, which maps the classical onto the quantum mechanical Weyl elements. This kind of a quantization map is demonstrated to realize a continuous strict deformation quantization in the sense of Rieffel and Landsman. The quantization is proved to be equivariant under the automorphic actions of the full affine symplectic group. The relationship to formal field quantization in theoretical physics is discussed by suggesting a representation dependent direct field quantization in mathematically concise terms.

A closed, entirely quantum mechanical Josephson oscillator model is considered, consisting of two superconductors in tunneling contact, which weakly interact with the photon field. For each superconductor we use, for notational simplicity, the BCS model in the strong coupling approximation and restrict ourselves to Anderson’s quasi-spin formulation. The thermodynamic limit of the global (non-equilibrium) dynamics is formulated for a large variety of states. It arises a generalization of previously developed cocycle equations, connecting the collective behaviour of the two superconductors with the photon field dynamics. In the physically most interesting situations, where the two superconductors are in thermal equilibrium (below the critical temperature) at voltage difference V, we show, that for arbitrary initial states the outgoing multi-photon states are quantum optically all-order coherent and constitute an almost monochromatic radiation of frequency $ 2eV/\hbar $. Furthermore, we deduce in detail, how the collective behaviour of the superconductors influences the collective (that are the optical) features of the emitted microwave photons.

Employing positive-definiteness arguments we analyse Boson field states, which combine classical and quantum mechanical features (signal and noise), in a constructive manner. Mathematically, they constitute Bauer simplexes within the convex, weak-*-compact state space of the C*-Weyl algebra, defined by a presymplectic test function space (smooth one-Boson wave functions) and are affinely homeomorphic to a state space of a classical field. The regular elements are expressed in terms of weak distributions (probability premeasures) on the dual test function space. The Bauer simplex arising from the bare vacuum is shown to generalize the quantum optical photon field states with positive P-functions.

We establish the limiting dynamics of a class of inhomogeneous bipolaronic models for superconductivity which incorporate deviations from the homogeneous models strong enough to require disjoint representations. The models are of the Hubbard type and the thermodynamics of their homogeneous part has been already elaborated by the authors. Now the dynamics of the systems is evaluated in terms of a generalized perturbation theory and leads to a C*-dynamical system over a classically extended algebra of observables. The classical part of the dynamical system, expressed by a set of 15 nonlinear differential equations, is observed to be independent from the perturbations. The KMS states of the C*-dynamical system are determined on the state space of the extended algebra of observables. The subsimplices of KMS states with unbroken symmetries are investigated and used to define the “type” of a phase. The KMS phase diagrams are worked out explicitly and compared with the thermodynamic phase structures obtained in the preceding works.

We continue the discussion of a bipolaronic superconductor (resp. an anisotropic antiferromagnet in quasispin formulation) as formulated in a previous work, based on a quantum-statistical, microscopic mean-field model. The grand-canonical thermodynamic limit is compared with the canonical thermodynamic limit in terms of a net of perturbations, becoming singular in the infinite lattice limit. A generalized thermostatistical framework is elaborated which covers model potentials with infinite parts. The function of the limiting free energy density in selecting the (stable) phases with broken symmetry is graphically illustrated. The phase diagrams for the two types of ensembles are shown to differ in the region where both the gauge symmetry and the invariance under sublattice exchange are broken. In particular, the type of the phase transitions, the order of the critical points, and the shape of some phase boundaries are found to depend on the ensemble, which clarifies certain controversial topics for these models. The uniqueness of the limiting Gibbs states with free boundary conditions in all thermodynamic phase regions is proved, and their decomposition into pure phase states in terms of a symmetric measure is evaluated. The field operators of the condensed particles are determined in the representations over the limiting Gibbs states.

For the case of infinitely many photon (Boson) modes we investigate the unitary implementability of a class of symplectic one-parameter groups (more exactly, of the associated groups of Bogoliubov automorphisms on the CCR algebra) in the Fock representation and in representations of the CCR algebra, which are symplectically related and inequivalent to Fock. Furthermore, the existence of the associated (squeezing) quadratic Hamiltonians is discussed. Finally, applications in the theory of quantization in QED and in the Luttinger model are pointed out.

Taking into account some philosophical notionson realism a reformulation of“macro-realism” according to Leggett-Garg isput forward. A macroscopic phase dynamics based on amicroscopic SQUID-model is discussed in this context.

Within the algebraic frame of smeared photon fields, the coherence properties of squeezed multi-photon states are elaborated. The squeezing Bogoliubov automorphisms are constructed by means of suitable one-mode symplectic transformations in the infinite-dimensional test-function space. Transposed into the state space, these transformations are shown to preserve first-order coherence when applied to coherent, classical, Fock-normal states in spite of rendering them nonclassical. The cases, where first-order coherence is destroyed, are also classified. Very special conditions on the initial state modes and on the operation parameters are worked out, under which even second-order coherence is still valid for the transformed classical coherent states, in spite of exhibiting squeezed fluctuations.

In the frame of operator-algebraic quantum statistical mechanics we calculate the grand canonical equilibrium states of a bipartite, microscopic mean-field model for bipolaronic superconductors (or anisotropic antiferromagnetic materials in the quasispin formulation). Depending on temperature and chemical potential, the sets of statistical equilibrium states exhibit four qualitatively different regions, describing the normal, superconducting (spin-flopped), charge ordered (antiferromagnetic), and coexistence phases. Besides phase transitions of the second kind, the model also shows phase transitions of the first kind between the superconducting and the charge ordered phases. A unique limiting Gibbs state is found in its central decomposition for all temperatures, even in the coexistence region, if the thermodynamic limit is performed at fixed particle density (magnetization).

We compare the weakly inhomogeneous BCS-model of a previous investigation with a weak perturbation of a symmetrized Hubbard model in the pair formalism on the homogeneous lattice. In terms of our operator algebraic discussion we prove the existence of unique limiting Gibbs states at fixed particle density (and temperature) for both models (which guarantees the physical equivalence of the large finite models with their thermodynamic limit). A close similarity of the condensate structure between the two models is made explicit, especially the appearance of two critical densities. This likeness manifests itself also in the derived self-consistency equations for the momentum resp. position dependent, complex gap parameters. The results on a stable gap and macroscopic phase are thus transferable from the BCS- to this Hubbard model. Only the localization of the condensate of the Hubbard model at zero momentum bears some resemblance to the Bose-Einstein condensation.

Combining the signal and the idler modes we show that each nondegenerate squeezing quadratic Hamiltonian may be transformed into the form of a degenerate squeezing Hamiltonian, if one uses the smeared field formalism. For the case of infinitely many photon modes we discuss the existence of squeezing quadratic Hamiltonians in Fock space. This gives a limitation on the squeezing parameters, which guarantees that all squeezed vacua are representable as vectors in Fock space. If this condition is not satisfied (the case of strong squeezing) then all squeezed vacua are outside the Fock space and mutually inequivalent.

Squeezing processes are commonly described in terms of quadratic Hamiltonians, which generate unitary implementations of Bogoliubov transformations of the quantized electromagnetic field. Here the behaviour of the quasifree, the classical, and the coherent photon states under general squeezing Bogoliubov transformations is investigated. It is found that there is a great variety of mixed classical states, which remain classical under the squeezing operation, whereas each pure classical state becomes non-classical. Especially, some classical, microscopic first order coherent states remain classical and coherent of first order under one-mode squeezing. This contrasts squeezing of macroscopic coherent states.

A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre‐Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self‐contained fashion.

By interpreting infinitely many two-level atoms as a mean field quantum lattice system in a recent paper the time evolution of the Dicke Maser model has been elaborated in terms of operator algebraic methods. Using these results, here the emitted radiation of the infinite Dicke model is investigated. It is shown how the collective behaviour of the atoms influences the quantized radiation, which for large times becomes classically coherent (in the sense of Glauber). The field modes which are (approximately) resonant with the level-splitting energy of the atoms are found to be the essential part of the generated coherent light, and thereby determine its macroscopic nature. Furthermore, the destruction and revival of coherence, the mean number of the emitted photons during the time evolution, as well as their spatial distribution are discussed.

By means of operator algebraic methods the dynamics of the Dicke model is investigated in the limit where the number of the two-level atoms goes to infinity and the interaction strength remains on a finite level. The infinite atomic system is treated as a mean field quantum lattice system. It is shown that the limiting dynamics is essentially determined by the collective behaviour of the atoms. With Trotter's product formula and perturbation theoretical methods we obtain explicit expressions for the unitary time evolution operators in the uncoupled representation.

Bardeen–Cooper–Schrieffer (BCS)‐like models for permutation invariant electronic interactions are investigated in terms of recent rigorous mean‐field methods. Their limiting dynamics is constructed in the Heisenberg picture on (extensions of) the full canonical commutation relation (CAR) algebra and also restricted to (extensions of) the pair algebra. The classical part of the proper BCS dynamics on the CAR algebra is shown to exhibit 15 macroscopic collective variables, the time dependence of which is explicitly integrated and discussed.

$C^*$-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-$*$-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-$*$-algebras instead of $C^*$-algebras. Fourier transformation and representation theory of the measure Banach-$*$-algebras are combined with the theory of continuous projective group representations to arrive at the genuine $C^*$-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general) infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter $\hbar$. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation., This is a contribution to the Special Issue on Deformation Quantization, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

The quantum statistical frame for infinite multi-lattice spin systems is introduced. The thermodynamic functionals specific internal energy, entropy and free energy are shown to exist on the set of permutation invariant states for polynomial mean field interactions by direct estimation methods. Their dependence on the relative sizes of the sublattice systems is made explicit. The set of homogeneous minimal free energy states is shown to be a Bauer simplex which contains all limiting Gibbs states. For the extremal minimal free energy states the self-consistency equations are derived.

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.

The possibility of a coherent superposition of two macroscopically different states is discussed, e.g., in connection with the most popular Schrodinger “cat paradox” (Schrodinger 1935, Audretsch and Mainzer 1990). The usual point of view takes it for granted that the attributes “living” and “dead” of the system “cat” are so different that their combination into a state, in which none of them is actualized, seems paradoxical. Thus from this point of view the fundamental problem arises how to deal with quantum theory in a manner avoiding this apparent paradox.

For an arbitrary multiply connected cavity a solution theory for the Maxwell equations is developed under ideal conductor boundary conditions. The trajectories of the electromagnetic fields are decomposed into their Helmholtz–Hodge components, including so–called cohomological fields which correspond to the harmonic differential forms. A canonical formalism is formulated in the Coulomb gauge, which comprises only the transversal and cohomological components. The canonical vector potentials and their conjugate fields are smeared in terms of smooth test functions. The Poisson bracket for the smeared fields is based on a symplectic form in the test function space and the corresponding infinite dimensional symplectic Lie algebra is introduced. The conservation quantities of Maxwell theory arise in this manner as co–momentum images of the Lie algebra. By means of a symplectic transformation the transversal dynamical generator is diagonalized and defines a special complexification of the test function space. A diagonalization of the cohomological dynamical generator is proved impossible. Since the Weyl quantization provides a symplectic equivariant map of the classical fields onto the quantum fields (affiliated with our C*–Weyl algebra), we have in quantum electrodynamics the same freedom of symplectic transformations and complexifications in the test function space (and not in the representing Hilbert space). Using the diagonalizing complexification we introduce the (smeared) transversal creation and annihilation operators and arrive at a unique particle structure for the transversal fields in a Fock representation. The symplectic Lie algebra elements, multiplied by i, constitute the one–photon observables, where especially the diagonal dynamical Maxwell generator generalizes Einstein's photon energy to our general set up. The quantum co–momentum map acquires the form of a second quantization of the one-photon observables and leads to particle conserving multi–photon Hamiltonians for the complex linear Lie algebra elements, whereas the anti-linear Lie algebra elements generate particle non–conserving squeezing Hamiltonians. Since there is no particle structure for the quantized cohomological fields their collective nature is confirmed, illustrating the difference between quantization and particle discretization.

We give in the first section an introduction into the basic notions of operator algebraic quantum theory with the emphasis on the state space of the quasi-local algebra. Representation theory and decomposition theory are related with the macroscopic distinguishability (disjointness) of states. In section 2 we use this formalism to discuss superconductivity in terms of a gauge covariant BCS-model in which the operators of the macroscopic phase and of the condensed Cooper pairs play a decisive role. Two weakly coupled BCS-superconductors constitute a model for the Josephson junction, in which the condensed Cooper pairs tunnel across the isolation barrier of the junction and provide one component of the two-fluid model. The Josephson relations are microscopically derived as operator equations.

As a microscopic description of the Josephson junction, two BCS models, are studied in the strict pair formulation with quite an arbitrary weak coupling potential. The modular formalism, the separate gauge transformations, and the limiting dynamics are analyzed for the interacting system in terms of the GNS representation of the uncoupled limiting Gibbs state. By means of the Connes theory the condensed Cooper pair and the quasiparticle spectrum is shown to be stable against weak perturbations. The modular formalism is used to construct a local approximation to the renormalized particle number operator and, by this, its time dependence, in spite of this observable not being affiliated with the von Neumann algebra of the temperature representation. The time derivation from this unbounded operator‐valued function coincides with the limit of the local currents and splits under a natural assumption into a sum of the Josephson and the quasiparticle current operator extending the two‐fluid picture also to the coupled model.

In the framework of generalized quantum theory using aW*-algebraic formalism, we introduce a completely symmetric coherence relation for states which is also applicable to nonpure states. Making use of lattice theoretic results the properties of this relation, especially its connection with incompatibility, are investigated. By means of algebraic decomposition theory the investigation is reduced to the case of factors where a complete classification of the coherence classes is given.

It is shown that the local existence of certain thermodynamic potentials allows for the complete diagonalization of thermodynamic stability conditions, so that only one-dimensional extrema have to be investigated.

For a class of discrete mean field models the limiting dynamics is investigated in the representations of generalized grand canonical states. It is demonstrated that for a certain form of spontaneous symmetry breakdown the W*‐automorphism dynamics exhibits a uniquely determined nontrivial classical part, which is essential for the explanation of macroscopic quantum phenomena.

Infinite series of Wick powers of the free, massive Bose field are analysed in terms of test function spaces of typeS for arbitrary space dimension. By direct estimates of the smeared phase space integrals sufficiency conditions for the existence of the vacuum expectation values are derived. These conditions are shown to be precise. The field-operators are defined on a dense invariant domain in Fock space, where they satisfy the Wightman axioms with the possible exception of locality. Localisable and nonlocalisable fields are dealt within the same frame. The behaviour of spectral functions and the strength of singularities are discussed.

In the temperature dependent reconstructed quantum mechanics over the limiting Gibbs state for two weakly coupled superconductors the effective dynamics is worked out and the current operator is expressed in terms of gauge covariant quasi-particle and condensed Cooper pair fields, where only the latter exhibit macroscopic quantum features.