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Given an arbitrary countably generated rigid C*-tensor category, we construct a fully faithful bi-involutive strong monoidal functor onto a subcategory of finitely generated projective bimodules over a simple, exact, separable, unital C*-algebra with a unique trace. The C*-algebras involved are built from the category using the Guionnet–Jones–Shlyakhtenko construction. Out of this category of Hilbert C*-bimodules, we construct a fully faithful bi-involutive strong monoidal functor into the category of bifinite spherical bimodules over an interpolated free group factor. The composite of these two functors recovers the functor constructed by Brothier, Hartglass, and Penneys.

We study the optimality of the remainder term in the two-term Weyl law for the Dirichlet Laplacian within the class of Lipschitz regular subsets of $\mathbb{R}^d$. In particular, for the short-time asymptotics of the trace of the heat kernel we prove that the error term cannot be made quantitatively better than little-$o$ of the second term., Comment: 19 pages

We consider the variety of Filippov (n-Lie) algebra structures on an (n + 1)-dimensional vector space. The group GLn(K) acts on it, and we study the orbit closures with respect to the Zariski topology. This leads to the definition of Filippov algebra degenerations. We present some fundamental results on such degenerations, including trace invariants and necessary degeneration criteria. Finally, we classify all orbit closures in the variety of complex (n + 1)-dimensional Filippov n-ary algebras.We consider the variety of Filippov (n-Lie) algebra structures on an (n + 1)-dimensional vector space. The group GLn(K) acts on it, and we study the orbit closures with respect to the Zariski topology. This leads to the definition of Filippov algebra degenerations. We present some fundamental results on such degenerations, including trace invariants and necessary degeneration criteria. Finally, we classify all orbit closures in the variety of complex (n + 1)-dimensional Filippov n-ary algebras.

The group of area preserving diffeomorphisms of the two sphere, SDiff(S2), is one of the simplest examples of an infinite dimensional Lie group. It plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related, newly defined symmetry groups. We investigate its representation theory using the method of coadjoint orbits. We describe the Casimir functions and the Cartan algebra. Then we evaluate the trace of a simple SDiff(S2) operator using the Atiyah-Bott fixed point formula. The trace is divergent but we show that it has well-defined truncations related to the structure of SDiff(S2). Finally, we relate our results back to the recent appearances of SDiff(S2) in black hole physics.The group of area preserving diffeomorphisms of the two sphere, SDiff(S2), is one of the simplest examples of an infinite dimensional Lie group. It plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related, newly defined symmetry groups. We investigate its representation theory using the method of coadjoint orbits. We describe the Casimir functions and the Cartan algebra. Then we evaluate the trace of a simple SDiff(S2) operator using the Atiyah-Bott fixed point formula. The trace is divergent but we show that it has well-defined truncations related to the structure of SDiff(S2). Finally, we relate our results back to the recent appearances of SDiff(S2) in black hole physics.

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators $\mathcal{A}(A_1,\ldots A_K)$. By applying the Shemesh and Amitsur - Levitzki theorems to analyse the structure of the algebra $\mathcal{A}(A_1,\ldots A_K)$ one can predict the asymptotic dynamics for a class of operations.

We locate the set of pairs $(\rho_{1},\rho_{2})$ of Gaussian states of a single mode electromagnetic field that exhibit maximal trace distance subject to the energy constraint $\langle a^{\dagger}a \rangle_{\rho_{1}}=\langle a^{\dagger}a \rangle_{\rho_{2}} = E$. Any such pair allows to achieve the minimum possible error in the task of binary distinguishability of two single mode, isoenergetic Gaussian quantum signals. In particular, we show that the logarithm of the minimal error probability for distinguishing two maximally trace distant, isoenergetic Gaussian states scales as $-E^{2}$, less than the achievable scaling of the minimal error probability for distinguishing, e.g., a pair of isoenergetic Heisenberg-Weyl coherent states with energy $E$ or a pair of isoenergetic quadrature squeezed states with energy $E$. For the case of a field consisting of $M>1$ modes, we locate the set of pairs of maximally trace distant isoenergetic, isocovariant Gaussian states. These results have basic applications in the theory of continuous variable quantum communications with Gaussian states of light., Comment: 11 pages, 2 figures

This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein's inequality and one of Lieb's less well-known concave trace functions, allows one to obtain conditions for equality. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein's elegant proof of the relevant concavity theorem of Lieb., Comment: 28 pages, latex Added reference to M.J.W. Hall, "Quantum Information and Correlation Bounds" Phys. Rev. A, 55, pp 100--112 (1997)

I study various properties of the critical limits of correlators containing insertions of conserved and anomalous currents. In particular, I show that the improvement term of the stress tensor can be fixed unambiguously, studying the RG interpolation between the UV and IR limits. The removal of the improvement ambiguity is encoded in a variational principle, which makes use of sum rules for the trace anomalies a and a'. Compatible results follow from the analysis of the RG equations. I perform a number of self-consistency checks and discuss the issues in a large set of theories., 15 pages

Characteristic classes in space-time manifolds are discussed for both even- and odd-dimensional spacetimes. In particular, it is shown that the Einstein--Hilbert action is equivalent to a second Chern-class on a modified Poincare bundle in four dimensions. Consequently, the cosmological constant and the trace of an energy-momentum tensor become divisible modulo R/Z., 12 pages, no figure

Based on an exact trace formula for a one-dimensional ray-splitting system, we derive novel combinatorial identities for cyclic binary sequences (P\'olya necklaces)., Comment: 15 pages

We propose an approach to obtaining new trace formulas of the Gel’fand–Levitan–Buslaev–Faddeev type, valid for Hilbert–Schmidt perturbations. In this way we obtain a new trace formula for Schrodinger operators on the half-line with long-range potentials.

It is a well-known fact that the first and last non-trivial coefficients of the characteristic polynomial of a linear operator are respectively its trace and its determinant. This work shows how to compute recursively all the coefficients as polynomial functions in the traces of successive powers of the operator. With the aid of Cayley-Hamilton's theorem the trace formulas provide a rational formula for the resolvent kernel and an operator-valued null identity for each finite dimension of the underlying vector space. The 4-dimensional resolvent formula allows an algebraic solution of the inverse metric problem in general relativity., 11 pages, no figures, LaTeX. The title has been changed in this new version

In this paper we consider a weighted trace formula for Schrodinger operators. More precisely, let ψjℏ and Ejℏ denote the eigenfunctions and eigenvalues of a Schrodinger-type operator Hℏ with a discrete spectrum. Let ψ(x,ξ) be a coherent state centered at a point (x,ξ) of a hyperbolic closed orbit γ. We show that, as ℏ→0, the leading term of ∑jφ{[Ej(ℏ)−E]/ℏ}|(ψ(x,ξ),ψjℏ)|2 can be expressed in terms of the analytic continuation on the upper and lower half-planes of the positive and negative frequencies part of φ. The result is also related to complex trajectories surrounding γ.

We show that the WKB approximation gives the exact result in the trace formula of ``$CQ^N$'', which is the non-compact counterpart of $CP^N$, in terms of the ``multi-periodic'' coherent state. We revisit the symplectic 2-forms on $CP^N$ and $CQ^N$ and, especially, construct that on $CQ^N$ with the unitary form. We also revisit the exact calculation of the classical patition functions of them.

We construct the path integral formula in terms of ``multi-periodic'' coherent state as an extension of the Nielsen-Rohrlich formula for spin. We make an exact calculation of the formula and show that, when a parameter corresponding to the magnitude of spin becomes large, the leading order term of the expansion coincides with the exact result. We also give an explicit correspondence between the trace formula in the multi-periodic coherent state and the one in the ``generalized'' coherent state., 28 pages, latex

Josza's definition of fidelity for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C$^*$-algebras $A$ that possess a faithful trace functional $\tau$. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements $\rho\in A$ for which $\tau(\rho)=1$. The second of our two settings is more operator theoretic: by fixing a faithful normal semifinite trace $\tau$ on a semifinite von Neumann algebra $M$, we define and consider the fidelity of pairs of positive operators in $M$ of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of $A$ or of the predual $M_*$. Our results also yield a new proof of a theorem of Moln\'ar on the structure of quantum channels on the trace-class operators that preserve fidelity., Comment: To appear in Journal of Mathematical Physics

We discuss a non--commutative integration calculus arising in the mathematical description of anomalies in fermion--Yang--Mills systems. We consider the differential complex of forms $u_0\ccr{\eps}{u_1}\cdots\ccr{\eps}{u_n}$ with $\eps$ a grading operator on a Hilbert space $\cH$ and $u_i$ bounded operators on $\cH$ which naturally contains the compactly supported de Rham forms on $\R^d$ (i.e.\ $\eps$ is the sign of the free Dirac operator on $\R^d$ and $\cH$ a $L^2$--space on $\R^d$). We present an elementary proof that the integral of $d$--forms $\int_{\R^d}\trac{X_0\dd X_1\cdots \dd X_d}$ for $X_i\in\Map(\R^d;\gl_N)$, is equal, up to a constant, to the conditional Hilbert space trace of $\Gamma X_0\ccr{\eps}{X_1}\cdots\ccr{\eps}{X_d}$ where $\Gamma=1$ for $d$ odd and $\Gamma=\gamma_{d+1}$ (`$\gamma_5$--matrix') a spin matrix anticommuting with $\eps$ for $d$ even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes' non--commutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace., Comment: 16 pages, latex

The trace of the heat kernel is expanded in a basis of nonlocal curvature invariants of $N$th order. The coefficients of this expansion (the nonlocal form factors) are calculated to third order in the curvature inclusive. The early-time and late-time asymptotic behaviours of the trace of the heat kernel are presented with this accuracy. The late-time behaviour gives the criterion of analyticity of the effective action in quantum field theory. The latter point is exemplified by deriving the effective action in two dimensions., Comment: 22 pages, REVTeX, Alberta Thy 45-93

The purpose of this article is to study the unbounded Tomita–Takesaki theory by trace functionals on O*‐algebras, and to apply the results to the algebras associated with the canonical commutation relation, pq−qp=−i1 (CCR‐algebras).

A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list of its solutions. The set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the Grobner‐basis methods. Each solution was then transformed into a canonical form (based on the various trace matrices of R) for final elimination of duplicates and subcases. If we use homogeneous parametrization the solutions can be combined into 23 distinct cases, modulo the well‐known C, P, and T reflections, and rotations and scalings R=κ(Q⊗Q)R(Q⊗Q)−1.

Quantum matrices $A(R)$ are known for every $R$ matrix obeying the Quantum Yang-Baxter Equations. It is also known that these act on `vectors' given by the corresponding Zamalodchikov algebra. We develop this interpretation in detail, distinguishing between two forms of this algebra, $V(R)$ (vectors) and $V^*(R)$ (covectors). $A(R)\to V(R_{21})\tens V^*(R)$ is an algebra homomorphism (i.e. quantum matrices are realized by the tensor product of a quantum vector with a quantum covector), while the inner product of a quantum covector with a quantum vector transforms as a scaler. We show that if $V(R)$ and $V^*(R)$ are endowed with the necessary braid statistics $\Psi$ then their braided tensor-product $V(R)\und\tens V^*(R)$ is a realization of the braided matrices $B(R)$ introduced previously, while their inner product leads to an invariant quantum trace. Introducing braid statistics in this way leads to a fully covariant quantum (braided) linear algebra. The braided groups obtained from $B(R)$ act on themselves by conjugation in a way impossible for the quantum groups obtained from $A(R)$., Comment: 27 pages

In the present paper, we present an integrable hierarchy for the Zakharov-Ito system. We first construct the Lenard recursion sequence and zero curvature representation for the Zakharov-Ito system, following Cao’s method as significantly generalized by other authors. We then construct the bi-Hamiltonian structures employing variational trace identities but woven together with the Lenard recursion sequences. From this, we are in a position to construct an integrable hierarchy of equations from the Zakharov-Ito system, and we obtain the recursion operator and Poisson brackets for constructing this hierarchy. Finally, we demonstrate that the obtained hierarchy is indeed Liouville integrable.

In this article, we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator ∑i=1d(−∂i2)s,s∈(0,1] on an open and bounded subdomain Ω⊂Rd and predict bounds on the sum of the first N eigenvalues, the counting function, the Riesz means, and the trace of the heat kernel. Moreover, utilizing the connection of coherent states to the semi-classical approach of quantum mechanics, we determine the sum for moments of eigenvalues of the associated Schrodinger operator.

In this paper we classify a certain family of solutions of Lovelock gravity in the Chern-Simons (CS) case, in arbitrary (odd) dimension, d ⩾ 5. The spacetime is characterized by admitting a metric that is a warped product of a two-dimensional spacetime M2 and an (a priori) arbitrary Euclidean manifold Σd−2 of dimension d − 2. We show that the solutions are naturally classified in terms of the equations that restrict Σd−2. According to the strength of such constraints we found the following branches in which Σd−2 has to fulfill: a Lovelock equation with a single vacuum (Euclidean Lovelock Chern-Simons in dimension d − 2), a single scalar equation that is the trace of an Euclidean Lovelock CS equation in dimension d − 2, or finally a degenerate case in which Σd−2 is not restricted at all. We show that all the cases have some degeneracy in the sense that the metric functions are not completely fixed by the field equations. This result extends the static five-dimensional case previously discussed in Dotti et ...

In this note we correct a mistake made in my paper [J. Math. Phys. 51, 033506 (2010)]. Using the contour integration method we shall obtain a new regularized trace formula for Schrodinger operator with energy-dependent potential.

The propagation of guided waves in photonic crystal fibers (PCFs) is studied. A PCF can be regarded as a perfectly two dimensional photonic crystal with a line defect along the axial direction. This problem can be treated as an eigenvalue problem for a family of noncompact self-adjoint operators. Under the assumption that the background spectrum has a gap, we prove that a line defect can create an eigenvalue of any given fixed value in the gap, provided that the defect is strong enough. Based on a decoupling of regions in R2 by means of Dirichlet and Neumann boundaries, then using the trace ideal estimates, we study asymptotic distribution of eigenvalues and bounds on the number of eigenvalue branches. In particular, we show that if the defect is weak enough, no eigenvalues can be created inside the gap.

A method for computing the trace of the kernel of the homogeneous Fredholm’s equation for resonant states arising from nonlocal potentials is proposed. We show that this integral formulation is convergent.

We describe a new approach to the hard rod model and construct a Ruelle-Mayer-type transfer operator which satisfies a dynamical trace formula. This yields a meromorphic continuation of the dynamical zeta function in two variables among them the inverse temperature.

Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such $N^2\times N^2$ matrices for odd $N$ are studied here. Presence of $\frac 12(N+3)(N-1)$ free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are $N$ possible states at each site. The trace of the transfer matrix is shown to depend on $\frac 12(N-1)$ parameters. For order $r$, $N$ eigenvalues consitute the trace and the remaining $(N^r-N)$ eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the $r$-th roots of unity, or lower dimensional multiplets corresponding to factors of the order $r$ when $r$ is not a prime number. The modulus of any eigenvalue is of the form $e^{\mu\theta}$, where $\mu$ is a linear combination of the free parameters, $\theta$ being the spectral parameter. For $r$ a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable $S$-matrices are constructed starting from our braid matrices. Perspectives are discussed.

We study trace functions on the form $ t\to\tr f(A+tB) $ where $ f $ is a real function defined on the positive half-line, and $ A $ and $ B $ are matrices such that $ A $ is positive definite and $ B $ is positive semi-definite. If $ f $ is non-negative and operator monotone decreasing, then such a trace function can be written as the Laplace transform of a positive measure. The question is related to the Bessis-Moussa-Villani conjecture. Key words: Trace functions, BMV-conjecture., Minor change of style, update of reference

We deal with trace formulas for half-line Schrodinger operators with long-range potentials. We generalize the Buslaev–Faddeev trace formulas to the case of square integrable potentials. The exact relation between the number of the trace formulas and the number of integrable derivatives of the potential is also given. The main results are optimal.

A nonrelativistic superparticle moving freely on the super‐Riemann surface of genus g≥2 is investigated. The classical motion is characterized by the super‐Fuchsian group. The periodic orbits are unstable and the system is classically chaotic. Quantization is performed in the path integral method. The kernel function is explicitly given in the semiclassical approximation. The quantized energy spectrum is related to the length spectrum of the classical periodic orbits, which is a superanalog of Selberg’s trace formula.

The Regge pole spectrum of the ABFST multiperipheral model with a resonance kernel is studied, using both the forward and nonforward partial‐wave integral equations. Insight into the complicated pattern of the trajectory spectrum is provided by an analysis of the singularity structure of the basic kernel using appropriate representations for it, and by an elucidation of the process of generation of trajectories in the weak‐coupling limit. This also establishes a framework for understanding the characteristics of the approximate solutions to the problem that are taken up subsequently. The trace approximations for both the nonforward and forward equations are worked out in detail. The traces involved are evaluated in convenient closed forms from which all the necessary information can be extracted easily. It is found that the approximation preserves, to a fair degree of accuracy, the trajectory‐generating singularity structures of the relevant kernels, and that good `effective' trajectory positions for the leading and secondary poles are obtained in both the forward and nonforward cases. It is also shown that, in this approximation, other phenomena such as complex Regge poles, the threshold behavior of the trajectories, and the intercept and slope of the leading trajectory can be investigated in a close simulation of the actual situation. Recent factorizable approximations are then examined from the point of view of the pole spectra they lead to, and it is concluded that, by and large, they oversimplify the problem and that their shortcomings are thus more extensive than those of the trace approximation.

We investigate the form Levinson’s theorem takes when the two‐body scattering amplitude is not decomposed into partial waves. It is found that the theorem changes its structure in this case and is not merely the sum over angular momentum of the well‐known partial wave results. The energy dependent quantity that replaces the partial wave phase shift turns out to be the trace of the two‐body time delay operator. This extended version of the theorem remains valid for scattering by nonspherically symmetric potentials.

A local spectral sum rule for nonrelativistic scattering in two dimensions is derived for the potential class v∈L4/3 (R2). The sum rule relates the integral over all scattering energies of the trace of the time‐delay operator for a finite region Σ⊆R2 to the contributions in Σ of the pure point and singularly continuous spectra.

The spectral sequences method is employed to study the cohomology space of the Becchi–Rouet–Stora (BRS) operator, which describes the general coordinate transformations in a two‐dimensional polynomial Lagrangian field theory. A pure external gravitational model is considered. In the Fadeev–Popov charge‐one sector, two classes of elements are found: the first represents the ordinary trace anomalies, in the second the presence of the anomalies recently calculated by Bardeen and Zumino are pointed out.

The expressions for the Gel’fand invariants (Casimir operators) of U(n), O(n), and Sp(n) in terms of the IR labels are derived by relating them to the trace of a suitably defined operator Pk. The method has unity and simplicity, since trace (Pk) can be related directly to the IR labels using the Weyl dimension function—without having to determine the eigenvectors of P.

A reduction of the Duffin‐Kemmer‐Petiau algebra to a direct sum of irreducible subalgebras for spin‐0 and spin‐1 bosons is presented. The subalgebras are defined by multiplication rules for the linearly independent basis elements. In the representations discussed the spin projection operators are independent basis elements of the subalgebras. The formal utility of these representations is demonstrated by obtaining the reduction of arbitrary operator products and trace theorems. The practical utility is demonstrated by application to the analysis of free and interacting boson field currents. Most importantly, one can understand the differences between DKP nonconserved currents and those obtained from second‐order wave equations.

We derive the Weyl, the normal, and the antinormal ordered forms of the exponential of a multimode quadratic expression in boson operators. The trace of this exponential operator is also evaluated.

The completeness relation for the system of ‘‘squared’’ solutions of the discrete analog of the Zakharov–Shabat problem is derived. It allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results. These include: (i) the expansion of the potential and its variations over the system of ‘‘squared’’ solutions, the expansion coefficients being the scattering data and their variations, respectively; thus the interpretation of the inverse scattering transform (IST) as a generalized Fourier transform becomes obvious; (ii) compact expressions for the trace identities through the operator Λ, for which the ‘‘squared’’ solutions are eigenfunctions; (iii) brief exposition of the spectral theory of the operator Λ; (iv) direct calculation of the action‐angle variables based on the symplectic form of the completeness relation; (v) the generating functional of the M operators in the Lax representation; (vi) the quantum ve...

A matrix product that generalizes the Kronecker matrix product is introduced and its properties are documented. This product is applied to classical statistical mechanics where its trace properties lead to both the transfer matrix and graphical expansion methods of evaluating the partition function.

It is proved that every density matrix is the limit, in the sense of weak operator convergence, of a sequence of operators each of which may be represented as an integral over projection operators onto coherent states (in the sense of Glauber) with a square‐integrable weight function. This result is a special case of one that holds for all operators with trace and for overcomplete families of states other than just the coherent states. We prove our more general result, at no cost of complexity, within the more general framework of continuous‐representation theory. The significance of our results for representing traces of operators is indicated.

Continuing the development of a previous paper on generalized isoperimetric inequalities (i.e., rearrangement inequalities for Green's functions), we extend the theory to the case of Green's functions for a potential which approaches zero at infinity. Specialization to domain potentials and long times gives Polya and Szego's isoperimetric inequality for the electrostatic capacity. Long times and a more general potential give a new isoperimetric inequality (for the ``scattering length'' of a potential). We also obtain from another specialization a curious isoperimetric inequality for the trace of the phase shift operator of scattering theory (for a given energy).

The partition function of a one‐dimensional system of particles with all interactions active is formulated in terms of Ising ferromagnetic spin states. It is shown how the partition function for two‐ and three‐dimensional systems can be obtained from the one‐dimensional one by restricting the number of bonds per particle. After writing the partition function in matrix form, the operator of interest is diagonalized and its trace expressed in the form of a convenient infinite series. The series is shown to be absolutely convergent and its analytic properties are briefly investigated. It is then applied to one‐, two‐, and three‐dimensional systems with nearest‐neighbor interactions. The validity of the model is established by summing the one‐dimensional series and comparing it with the known solution obtainable by other methods. Two‐ and three‐dimensional series are determined by algebraic techniques identical to the one‐dimensional series. These are seen to agree with the low‐temperature expansions obtained by other authors. The method of this article is seen to have the advantage of simplicity and uniformity regardless of the dimension of the system. A subsequent article is devoted to the thermodynamics of the model.