Full Fock Space Research Articles

Modular Structure and Inclusions of Twisted Araki-Woods Algebras

In the general setting of twisted second quantization (including Bose/Fermi second quantization, S-symmetric Fock spaces, and full Fock spaces from free probability as special cases), von Neumann algebras on twisted Fock spaces are analyzed. These twisted Araki-Woods algebras mathcal {L}_{T}(H) depend on the twist operator T and a standard subspace H in the one-particle space. Under a compatibility assumption on T and H, it is proven that the Fock vacuum is cyclic and separating for mathcal {L}_{T}(H) if and only if T satisfies a standard subspace version of crossing symmetry and the Yang-Baxter equation (braid equation). In this case, the Tomita-Takesaki modular data are explicitly determined. Inclusions mathcal {L}_{T}(K)subset mathcal {L}_{T}(H) of twisted Araki-Woods algebras are analyzed in two cases: If the inclusion is half-sided modular and the twist satisfies a norm bound, it is shown to be singular. If the inclusion of underlying standard subspaces Ksubset H satisfies an L^2-nuclearity condition, mathcal {L}_{T}(K)subset mathcal {L}_{T}(H) has type III relative commutant for suitable twists T. Applications of these results to localization of observables in algebraic quantum field theory are discussed.

On truncated t-free Fock spaces: Spectrum of position operators and shift-invariant states

The ergodic properties of the shift on the C⁎-algebras generated by creators and annihilators on both the full and m-truncated t-free Fock spaces are analyzed. In particular, the shift is shown to be uniquely ergodic with respect to the fixed-point algebra. In addition, for every m≥1, the invariant states of the shift acting on the m-truncated t-free C⁎-algebra are shown to yield a m+1-dimensional Choquet simplex, which collapses to a segment in the full case. Finally, the spectrum of the position operators on the m-truncated t-free Fock space is also determined.

Delayed thermalization in the mass-deformed Sachdev-Ye-Kitaev model

We study the thermalizing properties of the mass-deformed Sachdev-Ye-Kitaev model, in a regime of parameters where the eigenstates are ergodically extended over just portions of the full Fock space, as an all-to-all toy model of many-body localization (MBL). Our numerical results strongly support the hypothesis that, although considerably delayed, thermalization is still present in this regime. Our results add to recent studies indicating that MBL should be interpreted as a strict Fock-space localization.

Analysis of unitarity in conformal quantum gravity

We perform a canonical quantization of Weyl’s conformal gravity by means of the covariant operator formalism and investigate the unitarity of the resulting quantum theory. After reducing the originally fourth-order theory to second-order in time derivatives via the introduction of an auxiliary tensor field, we identify the full Fock space of quantum states under a Becchi–Rouet–Stora–Tyutin (BRST) construction that includes Faddeev–Popov ghost fields corresponding to Weyl transformations. This second-order formulation allows the formal tools of operator-based quantum field theory to be applied to quadratic gravity for the first time. Using the Kugo–Ojima quartet mechanism, we identify the physical subspace of quantum states and find that the subspace containing the transverse spin-2 states comes equipped with an indefinite inner product metric and a one-particle Hamiltonian that possesses only a single eigenstate. We construct the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula for the S-matrix in this spin-2 subspace and find that unitarity is violated in scattering events. The explicit way in which this violation occurs represents a new view on the ghost-problem in quadratic theories of quantum gravity.

Noncommutative rational Clark measures

Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

Note on bi-exactness for creation operators on Fock spaces

In this note, we introduce and study a notion of bi-exactness for creation operators acting on full, symmetric and anti-symmetric Fock spaces. This is a generalization of our previous work, in which we studied the case of anti-symmetric Fock spaces. As a result, we obtain new examples of solid actions as well as new proofs for some known solid actions. We also study free wreath product groups in the same context.

A characterization of rationality in free semicircular operators

For free semicircular elements realized on the full Fock space, we prove an equivalence between rationality of operators obtained from them and finiteness of the rank of their commutators with right annihilation operators. This is an analogue of the result for the reduced C⁎-algebra of the free group by G. Duchamp and C. Reutenauer which was extended by P. A. Linnell to densely defined unbounded operators affiliated with the free group factor. While their result was motivated by quantized calculus in noncommutative geometry, we state our results in terms of free probability theory.

Fatou's theorem for non-commutative measures

A classical theorem of Fatou asserts that the Radon–Nikodym derivative of any finite and positive Borel measure, μ, with respect to Lebesgue measure on the complex unit circle, is recovered as the non-tangential limits of its Poisson transform in the complex unit disk. This positive harmonic Poisson transform is the real part of an analytic function whose Taylor coefficients are in fixed proportion to the conjugate moments of μ.Replacing Taylor series in one variable by power series in several non-commuting variables, we show that Fatou's Theorem and related results have natural extensions to the setting of positive harmonic functions in an open unit ball of several non-commuting matrix-variables, and a corresponding class of positive non-commutative (NC) measures. Here, an NC measure is any positive linear functional on a certain self-adjoint unital subspace of the Cuntz–Toeplitz algebra, the C⁎−algebra generated by the left creation operators on the full Fock space.

On the Thermodynamics of the q-Particles.

Since the grand partition function for the so-called q-particles (i.e., quons), , cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for , and its q-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to . We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., ) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor in the n term of the power series expansion with respect to the fugacity z. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons .

On the polyanalytic short-time Fourier transform in the quaternionic setting

<p style='text-indent:20px;'>In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb's uncertainty principle. These results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window.</p>

Completely positive maps for imprimitive complex reflection groups

In 1994, M. Bożejko and R. Speicher proved the existence of completely positive quasimultiplicative maps from the group algebra of Coxeter groups to the set of bounded operators. They used some of them to define an inner product associated to creation and annihilation operators on a direct sum of Hilbert space tensor powers called full Fock space. Afterwards, A. Mathas and R. Orellana defined in 2008 a length function on imprimitive complex reflection groups that allowed them to introduce an analogue to the descent algebra of Coxeter groups. In this article, we use the length function defined by A. Mathas and R. Orellana to extend the result of M. Bożejko and R. Speicher to imprimitive complex reflection groups, in other words to prove the existence of completely positive quasimultiplicative maps from the group algebra of imprimitive complex reflection groups to the set of bounded operators. Some of those maps are then used to define a more general inner product associated to creation and annihilation operators on the full Fock space. Recall that in quantum mechanics, the state of a physical system is represented by a vector in a Hilbert space, and the creation and annihilation operators act on a Fock state by respectively adding and removing a particle in the ascribed quantum state.

Quantum Ergodicity in the Many-Body Localization Problem.

We generalize Page's result on the entanglement entropy of random pure states to the many-body eigenstates of realistic disordered many-body systems subject to long-range interactions. This extension leads to two principal conclusions: first, for increasing disorder the "shells" of constant energy supporting a system's eigenstates fill only a fraction of its full Fock space and are subject to intrinsic correlations absent in synthetic high-dimensional random lattice systems. Second, in all regimes preceding the many-body localization transition individual eigenstates are thermally distributed over these shells. These results, corroborated by comparison to exact diagonalization for an SYK model, are at variance with the concept of "nonergodic extended states" in many-body systems discussed in the recent literature.

Non-commutative rational functions in the full Fock space

A rational function belongs to the Hardy space, H 2 H^2 , of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function r ∈ H 2 \mathfrak {r} \in H^2 is particularly simple: The inner factor of r \mathfrak {r} is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational. We extend these and other basic facts on rational functions in H 2 H^2 to the full Fock space over C d \mathbb {C} ^d , identified as the non-commutative (NC) Hardy space of square-summable power series in several NC variables. In particular, we characterize when an NC rational function belongs to the Fock space, we prove analogues of classical results for inner-outer factorizations of NC rational functions and NC polynomials, and we obtain spectral results for NC rational multipliers.

Submodules in Polydomains and Noncommutative Varieties

Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of n-fold tensor products of Drury–Arveson spaces.

Blaschke–singular–outer factorization of free non-commutative functions

By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, a bounded analytic function is called inner or outer if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, H2, of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function θ decomposes uniquely as the product of a Blaschke inner function and a singular inner function, where the Blaschke inner contains all the vanishing information of θ, and the singular inner factor has no zeroes in the unit disk.We prove an exact analogue of this factorization in the context of the full Fock space, identified as the Non-commutative Hardy Space of analytic functions defined in a certain multi-variable non-commutative open unit ball.

A de Branges-Beurling theorem for the full Fock space

We extend the de Branges-Beurling theorem characterizing the shift-invariant spaces boundedly contained in the Hardy space of square-summable power series to the full Fock space over Cd. Here, the full Fock space is identified as the Non-commutative (NC) Hardy Space of square-summable Taylor series in several non-commuting variables. We then proceed to study lattice operations on NC kernels and operator-valued multipliers between vector-valued Fock spaces. In particular, we demonstrate that the operator-valued Fock space multipliers with common coefficient range space form a bounded general lattice modulo a natural equivalence relation.

Lebesgue Decomposition of Non-Commutative Measures

AbstractWe extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

An operator that relates to semi-meander polynomials via a two-sided $q$-Wick formula

We consider the sequence $( Q_n )_{n=1}^{\infty}$ of semi-meander polynomials which are used in the enumeration of semi-meandric systems (a family of diagrams related to the classical stamp-folding problem). We show that for a fixed natural number $d$, the sequence $( Q_n (d) )_{n=1}^{\infty}$ appears as sequence of moments for a compactly supported probability measure $\nu_d$ on the real line. More generally, we consider a two-variable generalization $Q_n (t,u)$ of $Q_n(t)$, which is related to a natural concept of self-intersecting meandric system; the second variable of $Q_n (t,u)$ keeps track of the crossings of such a system (and one has, in particular, that $Q_n (t,0)$ is the original semi-meander polynomial $Q_n (t)$). We prove that for a fixed natural number $d$ and a fixed real number $q$ with $|q| < 1$, the sequence $( Q_n (d,q) )_{n=1}^{\infty}$ appears as sequence of moments for a compactly supported probability measure $\nu_{d:q}$ on the real line. The measure $\nu_{d;q}$ is found as scalar spectral measure for an operator $T_{d;q}$ constructed by using left and right creation/annihilation operators on a $q$-deformation of the full Fock space introduced by Bozejko and Speicher. The relevant calculations of moments for $T_{d;q}$ are made by using a two-sided version of a (previously studied in the one-sided case) $q$-Wick formula, which involves the number of crossings of a pair-partition.

Operators affiliated to the free shift on the Free Hardy Space

The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions in the open complex unit disk with outer denominators. This definition extends naturally to the commutative and non-commutative multi-variable settings of the Drury-Arveson space and the full Fock space over Cd. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions in a non-commutative open unit ball in several matrix-variables, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space acts as right multiplication by a non-commutative function in the right free Smirnov class (and analogously, replacing ‘right’ with ‘left’).

Cyclic row contractions and rigidity of invariant subspaces

It is known that pure row contractions with one-dimensional defect spaces can be classified up to unitary equivalence by compressions of the standard d-shift acting on the full Fock space. Upon settling for a softer relation than unitary equivalence, we relax the defect condition and simply require the row contraction to admit a cyclic vector. We show that cyclic pure row contractions can be “transformed” (in a precise technical sense) into compressions of the standard d-shift. Cyclic decompositions of the underlying Hilbert spaces are the natural tool to extend this fact to higher multiplicities. We show that such decompositions face multivariate obstacles of an algebraic nature. Nevertheless, decompositions are obtained for nilpotent commuting row contractions by analyzing function theoretic rigidity properties of their invariant subspaces.