One-particle Hilbert Space Research Articles

Relative Entropy of Fermion Excitation States on the CAR Algebra

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

Asymptotic States and S-Matrix Operator in de Sitter Ambient Space Formalism

Within the de Sitter ambient space framework, the two different bases of the one-particle Hilbert space of the de Sitter group algebra are presented for the scalar case. Using field operator algebra and its Fock space construction in this formalism, we discuss the existence of asymptotic states in de Sitter QFT under an extension of the adiabatic hypothesis and prove the Fock space completeness theorem for the massive scalar field. We define the quantum state in the limit of future and past infinity on the de Sitter hyperboloid in an observer-independent way. These results allow us to examine the existence of the S-matrix operator for de Sitter QFT in ambient space formalism, a question which is usually obscure in spacetime with a cosmological event horizon for a specific observer. Some similarities and differences between QFT in Minkowski and de Sitter spaces are discussed.

Orthogonalization of fermionk-body operators and representability

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.

Quantum system partitioning at the single-particle level.

We discuss the partitioning of a quantum system through subsystem separation by unitary block-diagonalization (SSUB) applied to a Fock operator. For a one-particle Hilbert space, this separation can be formulated in a very general way. Therefore, it can be applied to very different partitionings ranging from those driven by features in the molecular structure (such as a solute surrounded by solvent molecules or an active site in an enzyme) to those that aim at an orbital separation (such as core-valence separation). Our framework embraces recent developments of Manby and Miller as well as the older ones of Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by SSUB. Moreover, it directly relates to decoupling approaches for relativistic four-component many-electron theory. For a Fock operator based on the Dirac one-electron Hamiltonian, one would like to separate the so-called positronic (negative-energy) states from the electronic bound and continuum states. The exact two-component (X2C) approach developed for this purpose becomes a special case of the general SSUB framework and may therefore be viewed as a system-environment decoupling approach. Moreover, for SSUB, there exists no restriction with respect to the number of subsystems that are generated-in the limit, decoupling of all single-particle states is recovered, which represents exact diagonalization of the problem. The fact that a Fock operator depends on its eigenvectors poses challenges to all system-environment decoupling approaches and is discussed in terms of the SSUB framework. Apart from improved conceptual understanding, these relations bring about technical advances as developments in different fields can immediately cross-fertilize one another. As an important example, we discuss the atomic decomposition of the unitary block-diagonalization matrix in X2C-type approaches that can inspire approaches for the efficient partitioning of large total systems based on SSUB.

Topological edge states for disordered bosonic systems

Quadratic bosonic Hamiltonians over a one-particle Hilbert space can be described by a Bogoliubov-de Gennes (BdG) Hamiltonian on a particle-hole Hilbert space. In general, the BdG Hamiltonian is not self-adjoint, but only J-self-adjoint on the particle-hole space viewed as a Krein space. Nevertheless, its energy bands can have non-trivial topological invariants like Chern numbers or winding numbers. By a thorough analysis for tight-binding models, it is proved that these invariants lead to bosonic edge modes which are robust to a large class of possibly disordered perturbations. Furthermore, general scenarios are presented for these edge states to be dynamically unstable even though the bulk modes are stable.

Erratum: Quantization of unstable linear scalar fields in static spacetimes [Phys. Rev. D 88 , 124005 (2013)

We discuss the quantization of an unstable field through the construction of a "one-particle Hilbert space." The system considered here is a neutral scalar field evolving over a globally hyperbolic static spacetime and subject to a stationary external scalar potential. In order to prove our results we assume spacetimes without horizons and that the theory possess a "mass gap." Our strategy consists in building a complex structure, which arises from a suitable positive bilinear form defined over the space of classical solutions of the field equation. Once the space of states of the quantum field has been set, it is possible to study the effect of the time translation symmetry on it. From the time translation operator we obtain an expression for the Hamiltonian operator associated with the unstable sector of the field. This last result coincides with findings from long ago showing that the unstable degrees of freedom of the field behave as non-relativistic particles in a parabolic potential barrier.

Finite energy quantization on a topology changing spacetime

The "trousers" spacetime is a pair of flat 2D cylinders ("legs") merging into into a single one ("trunk"). In spite of its simplicity this spacetime has a few features (including, in particular, a naked singularity in the "crotch") each of which is presumably unphysical, but for none of which a mechanism is known able to prevent its occurrence. Therefore it is interesting and important to study the behavior of the quantum fields in such a space. Anderson and DeWitt were the first to consider the free scalar field in the trousers spacetime. They argued that the crotch singularity produces an infinitely bright flash, which was interpreted as evidence that the topology of space is dynamically preserved. Similar divergencies were later discovered by Manogue, Copeland and Dray who used a more exotic quantization scheme. Later yet the same result obtained within a somewhat different approach led Sorkin to the conclusion that the topological transition in question is suppressed in quantum gravity. In this paper I show that the Anderson--DeWitt divergence is an artifact of their choice of the Fock space. By choosing a different one-particle Hilbert space one gets a quantum state in which the components of the stress-energy tensor (SET) are bounded in the frame of a free-falling observer.

Large-scale behaviour of local and entanglement entropy of the free Fermi gas at any temperature**In memory of Enrico Fermi (1901–1954) on the occasion of the birthday of the ideal Fermi gas.

The leading asymptotic large-scale behaviour of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensional Euclidean space at zero absolute temperature, T = 0, is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at . The leading large-scale term of this thermal EE turns out to be twice the first-order finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the interface of the bipartition. However, it is given by a rather complicated integral derived from a semiclassical trace formula for a certain operator on the underlying one-particle Hilbert space. But in the zero-temperature limit , the leading large-scale term of the thermal EE considerably simplifies and displays a -singularity which one may identify with the known logarithmic enhancement at T = 0 of the so-called area-law scaling.

Entanglement Dynamics of Second Quantized Quantum Fields

We study the entanglement dynamics in the system of coupled boson fields. We demonstrate that there are different natural notions of locality in this context leading to inequivalent notions of entanglement. We concentrate on the particle picture, when entanglement of one particle is determined by one-particle density matrix. We study, in detail, the effect of interaction preserving populations of individual one-particle states. We show that if the system is initially in a disentangled state with the definite total number of particles and the dimension of the one-particle Hilbert space is more than two, then only potentials of the special form admit complete entanglement, which is shown to be reached at NOON states. If the system is initially in Glauber’s coherent state, complete entanglement is not reached despite the presence of two entangling channels in this case. We conclude with studying the time evolution of entanglement of photons in a cavity with multiple quantum dots in the limit of large number of photons. We show that in a relatively short time scale the completely entangled states belong to the class of graph states and are formed due to the interaction with dots in resonance with the cavity modes.

Quantization of unstable linear scalar fields in static spacetimes

We discuss the quantization of an unstable field through the construction of a ``one-particle Hilbert space.'' The system considered here is a neutral scalar field evolving over a globally hyperbolic static spacetime and subject to a stationary external scalar potential. In order to prove our results we assume spacetimes without horizons and that the theory possess a ``mass gap.'' Our strategy consists in building a complex structure, which arises from a suitable positive bilinear form defined over the space of classical solutions of the field equation. Once the space of states of the quantum field has been set, it is possible to study the effect of the time translation symmetry on it. From the time translation operator we obtain an expression for the Hamiltonian operator associated with the unstable sector of the field. This last result coincides with findings from long ago showing that the unstable degrees of freedom of the field behave as nonrelativistic particles in a parabolic potential barrier.

QUANTUM FIELDS ON CURVED MOMENTUM SPACE

Relativistic particles with momentum space described by a group manifold provide a very interesting link between gravity, quantum group symmetries and non-commutative field theories. We discuss how group valued momenta emerge in the context of three-dimensional Einstein gravity and describe the related non-commutative field theory. As an application we introduce a non-commutative heat-kernel, calculate the associated spectral dimension and comment on its non-trivial behavior. In four spacetime dimensions the only known example of momenta living on a group manifold is encountered in the context of the κ-Poincaré algebra introduced by Lukierski et al. 20 years ago. I will discuss the construction of a one-particle Hilbert space from the classical κ-deformed phase space and show how the group manifold structure of momentum space leads to an ambiguity in the quantization procedure. The tools introduced in the discussion of field quantization lead to a natural definition of deformed two-point function.

Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes

We investigate whether a symmetric, second order Wick polynomial T of a free scalar field, including derivatives, is essentially self-adjoint on the natural (Wightman) domain in a quasi-free (i.e., Fock space) Hadamard representation. Our results apply to arbitrary spacetime dimensions d ≥ 2, but we do restrict our attention to the case where T is smeared with a test-function from a particular class \documentclass[12pt]{minimal}\begin{document}${\cal S}$\end{document}S, namely the class of sums of squares of test-functions. (This class of smearing functions is smaller than the class of all non-negative test-functions – a fact which follows Hilbert's theorem.) Combining techniques from microlocal and functional analysis we prove that T is essentially self-adjoint if it is a Wick square (without derivatives). In the presence of derivatives we prove the weaker result that T is essentially self-adjoint if its compression to the one-particle Hilbert space is essentially self-adjoint. For the latter result, we use Wüst's theorem and an application of Konrady's trick in Fock space. In the presence of derivatives, we also prove that one has some control over the spectral projections of T, by describing it as the strong graph limit of a sequence of essentially self-adjoint operators.

Anatomy of a deformed symmetry: Field quantization on curved momentum space

In certain scenarios of deformed relativistic symmetries relevant for noncommutative field theories particles exhibit a momentum space described by a non-Abelian group manifold. Starting with a formulation of phase space for such particles which allows for a generalization to include group-valued momenta we discuss quantization of the corresponding field theory. Focusing on the particular case of $\ensuremath{\kappa}$-deformed phase space we construct the one-particle Hilbert space and show how curvature in momentum space leads to an ambiguity in the quantization procedure reminiscent of the ambiguities one finds when quantizing fields in curved space-times. The tools gathered in the discussion on quantization allow for a clear definition of the basic deformed field mode operators and two-point function for $\ensuremath{\kappa}$-quantum fields.

An Algebraic Construction of Boundary Quantum Field Theory

We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras A_V on the Minkowski half-plane M_+ starting with a local conformal net A of von Neumann algebras on the real line and an element V of a unitary semigroup E(A) associated with A. The case V=1 reduces to the net A_+ considered by Rehren and one of the authors; if the vacuum character of A is summable A_V is locally isomorphic to A_+. We discuss the structure of the semigroup E(A). By using a one-particle version of Borchers theorem and standard subspace analysis, we provide an abstract analog of the Beurling-Lax theorem that allows us to describe, in particular, all unitaries on the one-particle Hilbert space whose second quantization promotion belongs to of E(A^(0)) with A^(0) the U(1)-current net. Each such unitary is attached to a scattering function or, more generally, to a symmetric inner function. We then obtain families of models via any Buchholz-Mach-Todorov extension of A^(0). A further family of models comes from the Ising model.

Fredholm Determinants and the Statistics of Charge Transport

Using operator algebraic methods we show that the moment generating function of charge transport in a system with infinitely many non-interacting Fermions is given by a determinant of a certain operator in the one-particle Hilbert space. The formula is equivalent to a formula of Levitov and Lesovik in the finite dimensional case and may be viewed as its regularized form in general. Our result embodies two tenets often realized in mesoscopic physics, namely, that the transport properties are essentially independent of the length of the leads and of the depth of the Fermi sea.

Non-Volkov solutions for a charge in a plane wave

We continue the study of the supersymmetric vector multiplet in a purely quantum framework. We obtain some new results which make the connection with the standard literature. First we construct the one-particle physical Hilbert space taking into account the (quantum) gauge structure of the model. Then we impose the condition of positivity for the scalar product only on the physical Hilbert space. Finally we obtain a full supersymmetric coupling which is gauge invariant in the supersymmetric sense in the first order of perturbation theory. By integrating out the Grassmann variables we get an interacting Lagrangian for a massive Yang-Mills theory related to ordinary gauge theory; however the number of ghost fields is doubled so we do not obtain the same ghost couplings as in the standard model Lagrangian.

Massive gravity as a quantum gauge theory

We present a new point of view on the quantization of the massive gravitational field, namely we use exclusively the quantum framework of the second quantization. The Hilbert space of the many-gravitons system is a Fock space F+ (Hgraviton) where the one-particle Hilbert space Hgraviton carries the direct sum of two unitary irreducible representations of the Poincaré group corresponding to two particles of mass m > 0 and spins 2 and 0, respectively. This Hilbert space is canonically isomorphic to a space of the type Ker(Q)/Im(Q) where Q is a gauge charge defined in an extension of the Hilbert space Hgraviton generated by the gravitational field hμν and some ghosts fields uμ, ũμ (which are vector Fermi fields) and vμ (which is a vector Bose field).Then we study the self interaction of massive gravity in the causal framework. We obtain a solution which goes smoothly to the zero-mass solution of linear quantum gravity up to a term depending on the bosonic ghost field. This solution depends on two real constants as it should be; these constants are related to the gravitational constant and the cosmological constant. In the second order of the perturbation theory we do not need a Higgs field, in sharp contrast to Yang-Mills theory.

Massless Relativistic Wave Equations and Quantum Field Theory

We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincare group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fiber degrees of freedom of the covariant representation. They are characterized by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between \( {\mathcal E}(2) \) (the two-fold cover of the 2-dimensional Euclidean group) and \( \overline{{\mathcal E}(2)}. \)

MODULAR LOCALIZATION AND WIGNER PARTICLES

We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano–Wichmann relations and a representation of the Poincaré group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita–Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh–Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and of de Sitter spacetime.

On the quantization of the linearized gravitational field

We present a new point of view on the quantization of the gravitational field, namely we use exclusively the quantum framework of the second quantization. More explicitly, we take as one-particle Hilbert space, the unitary irreducible representation of the Poincarégroup corresponding to a massless particle of helicity 2 and apply the second quantization procedure with Einstein-Bose statistics. The resulting Hilbert space +() is, by definition, the Hilbert space of the gravitational field. Then we prove that this Hilbert space is canonically isomorphic to a space of the type Ker(Q) / Im(Q) where Q is a supercharge defined in an extension of the Hilbert space +() by the inclusion of ghosts: some fermion ghosts uµ, µ which are vector fields and a bosonic ghost which is a scalar field. This has to be contrasted with the usual approaches where only the fermion ghosts are considered. However, a rigorous proof that this is, indeed, possible seems to be lacking in the literature.