Fock Space Research Articles

Connes fusion of spinors on loop space

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

Essential positivity for Toeplitz operators on the Fock space

In this short note, we discuss essential positivity of Toeplitz operators on the Fock space, as motivated by a recent question of Perälä and Virtanen (Proc. Amer. Math. Soc. 151:4807–4815, 2023). We give a proper characterization of essential positivity in terms of limit operators. A conjectured characterization of essential positivity of Perälä and Virtanen is disproven when the assumption of radiality is dropped. Nevertheless, when the symbol of the Toeplitz operator is of vanishing mean oscillation, we show that the conjecture of Perälä and Virtanen holds true, even without radiality.

Effective action approach to the dynamical map

The dynamical map represents a fundamental concept in quantum field theory, providing a solution of the field equations in the Fock space of asymptotic fields. In this paper, we show how to express the dynamical map of a scalar field in the language of quantum effective action. This grants us new insights into the study of topological defects in quantum field theory by showing a connection between the usual least-action principle and Umezawa’s boson transformation method.

Dunkl-type Segal–Bargmann transform and its applications to some partial differential equations

Abstract In this paper, we give some applications of the Dunkl-type Segal–Bargmann transform ℬ α {\mathscr{B}_{\alpha}} in the field of partial differential equations, such as the time-dependent Dunkl–Dirac Laplacian equation and the time-dependent Dunkl–Schrödinger equation. The resolution of these types of problems is based on the techniques of the transmutation operators on the Dunkl-type Fock space ℱ α ⁢ ( ℂ d ) {\mathscr{F}_{\alpha}(\mathbb{C}^{d})} .

Superoscillations in high energy physics and gravity

We explore superoscillations within the context of classical and quantum field theories, presenting novel solutions to Klein–Gordon’s, Dirac’s, Maxwell’s and Einstein’s equations. In particular, we illustrate a procedure of second quantization of fields and how to construct a Fock space which encompasses Superoscillating states. Furthermore, we extend the application of superoscillations to quantum tunnelings, scatterings and mixings of particles, squeezed states and potential advancements in laser interferometry, which could open new avenues for experimental tests of quantum gravity effects. By delving into the relationship among superoscillations and phenomena such as Hawking radiation, the black hole (BH) information and the Firewall paradox, we propose an alternative mechanism for information transfer across the BH event horizon.

A note on invariant states of Gaussian quantum Markov semigroups

Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is characterized either by the property of their predual that preserves quantum Gaussian states. In this paper, show that gaussian QMSs can have only Gaussian invariant states if the spectrum of the drift matrix does not contain points on the imaginary axis. Moreover, in this case, any initial state converges in trace norm toward a unique gaussian invariant state.

Unconstrained Lagrangian formulation for bosonic continuous spin theory in flat spacetime of arbitrary dimension

We have discovered two unconstrained forms of free Lagrangian for continuous spin(CS) theory in arbitrary flat spacetime dimension for bosonic case. These Lagrangians, unlike that by Schuster and Toro, do not include delta functions and are conventional. The first form consists of five kinds of totally symmetric helicity fields and one kind of gauge parameter. By introducing auxiliary creation and annihilation operators, each is combined into a state vector in Fock space, including all ranks one by one. The Lagrangian imposes no constraints, such as trace conditions, on these fields or the gauge parameter field. Additionally, the Lagrangian does not contain higher-order derivative terms. In the limit as CS parameter μ approaches zero, it naturally reproduces a Lagrangian for helicity fields in higher spin(HS) theory, known as unconstrained quartet formulation. Permitting third-order derivatives, we also obtain the second unconstrained form of Lagrangian that can be written in terms of three kinds of fields, including μ, similar to the formulation by Francia and Sagnotti. Partial gauge fixing and partial use of equations of motion (EOM) on this Lagrangian yield a Fronsdal-like Lagrangian with a single double-traceless field, including μ. By imposing further gauge fixing on the field in the EOM with respect to divergence and trace, we confirm the reproduction of the modified Wigner equations already known in literature.

Schatten class Hankel operators on doubling Fock spaces and the Berger-Coburn phenomenon

Using the notion of integral distance to analytic functions, we give a characterization of Schatten class Hankel operators acting on doubling Fock spaces on the complex plane and use it to show that for f∈L∞, if Hf is Hilbert-Schmidt, then so is Hf¯. This property is known as the Berger-Coburn phenomenon. When 0<p≤1, we show that the Berger-Coburn phenomenon fails for a large class of doubling Fock spaces. Along the way, we illustrate our results for the canonical weights |z|m when m>0.

Finitely generated weakly monotone C∗-algebra

In this paper, we consider the [Formula: see text]-algebra generated by finitely many annihilation operators acting on the weakly monotone Fock space, and we call it weakly monotone [Formula: see text]-algebra. We give an abstract presentation for this algebra, showing that it is isomorphic to a suitable quotient of a Cuntz–Krieger [Formula: see text]-algebra 𝒪A corresponding to a suitable matrix [Formula: see text]. Furthermore, we show that the diagonal subalgebra of the weakly monotone [Formula: see text]-algebra is a MASA and we give a detailed description of its Gelfand spectrum.

On energy and particle production in cosmology: the particular case of the gravitino

It is well-known that the number of particles produced in cosmology, commonly defined in the literature from the Fock space of the instantaneous hamiltonian of the canonically normalized fields, is ambiguous. On the other hand, the energy computed from the energy-momentum tensor should be physical. We compare the corresponding Fock spaces and relate them through a Bogolyubov transformation. We find that for particles of spin 0, 1 and 3/2 the two Fock spaces are different, whereas they are the same for spin 1/2 fermions. For spin 0 and 1, for particles of high momenta the two Fock spaces align, as intuitively expected. For the spin 3/2, one finds two puzzles. The first one is that the two corresponding Fock spaces do not match even in the limit of high momenta. The second is that whereas we provide evidence for the equivalence theorem between longitudinal gravitinos and the goldstino in terms of an exact matching between the lagrangians and the instantaneous hamiltonians for the canonically normalized fields, the energy operator computed from the Rarita-Schwinger action does not seem to be captured in a simple way by the goldstino action. Our results suggest a re-analysis of non-thermal gravitino production in cosmology.

Iwahori‐metaplectic duality

AbstractWe construct a family of solvable lattice models whose partition functions include ‐adic Whittaker functions for general linear groups from two very different sources: from Iwahori‐fixed vectors and from metaplectic covers. Interpolating between them by Drinfeld twisting, we uncover unexpected relationships between Iwahori and metaplectic Whittaker functions. This leads to new Demazure operator recurrence relations for spherical metaplectic Whittaker functions. In prior work of the authors it was shown that the row transfer matrices of certain lattice models for spherical metaplectic Whittaker functions could be represented as ‘half‐vertex operators’ operating on the ‐Fock space of Kashiwara, Miwa and Stern. In this paper the same is shown for all the members of this more general family of lattice models including the one representing Iwahori Whittaker functions.

Boundedness of certain linear operators on twisted Fock spaces

Boundedness of certain linear operators on twisted Fock spaces

Block dual Toeplitz Operators on the Orthogonal Complements of Fock Spaces

Block dual Toeplitz Operators on the Orthogonal Complements of Fock Spaces

Boundedness and compactness of Hausdorff operators on Fock spaces

We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space F α p F^p_\alpha and taking its values into a larger one F α q , 0 &gt; p ≤ q ≤ ∞ F^q_\alpha ,\ 0 &gt; p \leq q \leq \infty , as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space F α ∞ F^\infty _\alpha is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become p p -summing are also included.

An Intruder-Free Fock Space Coupled-Cluster Study of the Potential Energy Curves of LiMg+ within the (2,0) Sector.

The potential energy curves (PECs) and spectroscopic constants of the ground and excited states of a LiMg+ molecular cation were investigated. We obtained accurate results for the fifteen lowest-lying states of the LiMg+ cation using the Intermediate Hamiltonian Fock Space Multireference Coupled Cluster (IH-FS-CC) method applied to the (2,0) sector. Relativistic corrections were accounted for using the third-order Douglas-Kroll method. In each instance, smooth PECs were successfully computed across the entire range of interatomic distances from equilibrium to the dissociation limit. The results are in good accordance with previous studies of this molecular cation. Notably, this study marks the first application of IH-FS-CC in investigating a mixed alkali and alkaline earth molecular cation, proving its usability in determining accurate PECs of such diatomics and their spectroscopic constants.

Weighted norm inequalities, embedding theorems and integration operators on vector-valued Fock spaces

Weighted norm inequalities, embedding theorems and integration operators on vector-valued Fock spaces

Complexity of fermionic states

How much information does a fermionic state contain? To address this fundamental question, we define the complexity of a particle-conserving many-fermion state as the entropy of its Fock space probability distribution, minimized over all Fock representations. The complexity characterizes the minimum computational and physical resources required to represent the state and store the information obtained from it by measurements. Alternatively, the complexity can be regarded as a Fock space entanglement measure describing the intrinsic many-particle entanglement in the state. We establish a universal lower bound for the complexity in terms of the single-particle correlation matrix eigenvalues and formulate a finite-size complexity scaling hypothesis. Remarkably, numerical studies on interacting lattice models suggest a general model-independent complexity hierarchy: ground states are exponentially less complex than average excited states, which, in turn, are exponentially less complex than generic states in the Fock space. Our work has fundamental implications on how much information is encoded in fermionic states. Published by the American Physical Society 2024

Characters of [formula omitted] and vertex operators

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group GLn(Fq). Green's theory of GLn(Fq) is recovered and enhanced under the realization of the Grothendieck ring of representations RG=⨁n≥0R(GLn(Fq)) as two isomorphic Fock spaces associated to two infinite-dimensional F-equivariant Heisenberg Lie algebras hˆF‾ˆq and hˆF‾q, where F is the Frobenius automorphism of the algebraically closed field F‾q. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space RG as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example.

Complex symmetry in the Fock space of several variables

In this paper we study the complex symmetry in the several variable Fock space by using the techniques of weighted composition operators and semigroups. We characterize unbounded weighted composition operators that are (real) complex symmetric with respect to a concrete conjugation. Using this characterization, we study complex symmetric semigroups and their generators. We realize such generators as first-order differential operators.

Extended Cesàro companion operators on generalized Fock spaces

In this paper we give some characterizations of the bounded and compact extended Cesàro companion operators acting between the generalized Fock spaces. We also obtain asymptotic estimates for the norm of these operators in terms of certain integral transforms, which are related to some function-theoretic properties of the symbols that induce the operators. As an application, we prove that there exist no nontrivial compact extended Cesàro companion integral and multiplication operators on generalized Fock spaces.