Feynman-Diagrammatic Description of the Asymptotics of the Time Evolution Operator in Quantum Mechanics

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on \({\mathbb{R}^n}\), with magnetic and potential terms. In particular, for each classical path γ connecting points q ₀ and q ₁ in time t, we define a formal power series V _γ (t, q ₀, q ₁) in \({\hbar}\), given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V _γ ) satisfies Schrödinger’s equation, and explain in what sense the \({t \to 0}\) limit approaches the δ distribution. As such, our construction gives explicitly the full \({\hbar\to 0}\) asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Infinitesimal Deformations of a Formal Symplectic Groupoid

Given a formal symplectic groupoid G over a Poisson manifold (M, π ₀), we define a new object, an infinitesimal deformation of G, which can be thought of as a formal symplectic groupoid over the manifold M equipped with an infinitesimal deformation \({\pi_0 + \varepsilon \pi_1}\) of the Poisson bivector field π ₀. To any pair of natural star products \({(\ast,\tilde\ast)}\) having the same formal symplectic groupoid G we relate an infinitesimal deformation of G. We call it the deformation groupoid of the pair \({(\ast,\tilde\ast)}\) . To each star product with separation of variables \({\ast}\) on a Kähler–Poisson manifold M we relate another star product with separation of variables \({\hat\ast}\) on M. We build an algorithm for calculating the principal symbols of the components of the logarithm of the formal Berezin transform of a star product with separation of variables \({\ast}\) . This algorithm is based upon the deformation groupoid of the pair \({(\ast,\hat\ast)}\) .     

Low Density Phases in a Uniformly Charged Liquid

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form \({\gamma=(\langle u_i,v_j\rangle)_{i,j=1}^n}\), where the vectors u_i and v_j are in the unit ball of a real Hilbert space. In this work we study the probability of the nonlocal nature of these correlations as a function of \({\alpha=\frac{m}{n}}\), where the previous vectors are sampled according to the Haar measure in the unit sphere of \({\mathbb R^m}\). In particular, we prove the existence of an \({\alpha_0 > 0}\) such that if \({\alpha\leq \alpha_0}\), \({\gamma}\) is nonlocal with probability tending to 1 as \({n\rightarrow \infty}\), while for \({\alpha > 2}\), \({\gamma}\) is local with probability tending to 1 as \({n\rightarrow \infty}\).     

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

Let \({T=\mathbb R^d}\) . Let a function \({QT^2\to\mathbb C}\) satisfy \({Q(s,t)=\overline{Q(t,s)}}\) and \({|Q(s,t)|=1}\). A generalized statistics is described by creation operators \({\partial_t^\dagger}\) and annihilation operators ? _t , \({t\in T}\), which satisfy the Q-commutation relations: \({\partial_s\partial^\dagger_t = Q(s, t)\partial^\dagger_t\partial_s+\delta(s, t)}\) , \({\partial_s\partial_t = Q(t, s)\partial_t\partial_s}\), \({\partial^\dagger_s\partial^\dagger_t = Q(t, s)\partial^\dagger_t\partial^\dagger_s}\). From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which Q(s, t) is equal to q if s < t, and to \({\bar q}\) if s > t. Here \({q\in\mathbb C}\) , |q| = 1. We start the paper with a detailed discussion of a Q-Fock space and operators \({(\partial_t^\dagger,\partial_t)_{t\in T}}\) in it, which satisfy the Q-commutation relations. Next, we consider a noncommutative stochastic process (white noise) \({\omega(t)=\partial_t^\dagger+\partial_t+\lambda\partial_t^\dagger\partial_t}\) , \({t\in T}\) . Here \({\lambda\in\mathbb R}\) is a fixed parameter. The case λ = 0 corresponds to a Q-analog of Brownian motion, while λ ≠ 0 corresponds to a (centered) Q-Poisson process. We study Q-Hermite (Q-Charlier respectively) polynomials of infinitely many noncommutatative variables \({(\omega(t))_{t\in T}}\) . The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define Q-cumulants of a field \({(\xi(t))_{t\in T}}\). This allows us to define a Q-Lévy process as a field \({(\xi(t))_{t\in T}}\) whose values at different points of T are Q-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a Q-Lévy process, and derive a Nualart–Schoutens-type chaotic decomposition for such a process.     

Universal Components of Random Nodal Sets

We give, as L grows to infinity, an explicit lower bound of order \({L^{\frac{n}{m}}}\) for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of P with eigenvalues below L. Here, P denotes an elliptic self-adjoint pseudo-differential operator of order \({m > 0}\), bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed n-dimensional manifold M equipped with some Lebesgue measure. In fact, for every closed hypersurface \({\Sigma}\) of \({\mathbb{R}^n}\), we prove that there exists a positive constant \({p_\Sigma}\) depending only on \({\Sigma}\), such that for every large enough L and every \({x \in M}\), a component diffeomorphic to \({\Sigma}\) appears with probability at least \({p_\Sigma}\) in the vanishing locus of a random section and in the ball of radius \({L^{-\frac{1}{m}}}\) centered at x. These results apply in particular to Laplace–Beltrami and Dirichlet-to-Neumann operators.     

A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥T _c,k =J/arctan?(1/k) the limiting Gibbs measure is unique, and for T<T _c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if \(T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})\), with \(\sqrt{k} then there is a new uncountable set \({\mathcal{G}}_{k,k_{0}}\) of Gibbs measures. Moreover \({\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}\), for k ₀≠k′₀. Therefore if \(T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})\), \(T_{c,\sqrt{k}+1} then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}\(\cup(\bigcup_{k_{0}:\sqrt{k}.     

$${\mathbb{Z}_N}$$ -Curves Possessing No Thomae Formulae of Bershadsky–Radul Type

A \({\mathbb{Z}_N}\) -curve is one of the form \({y^{N}=(x-\lambda_{1})^{m_{1}}\cdots(x-\lambda_{s})^{m_{s}}}\) . When N = 2 these curves are called hyperelliptic and for them Thomae proved his classical formulae linking the theta functions corresponding to their period matrices to the branching values λ₁, . . . , λ _s . In his work on Fermionic fields on \({\mathbb{Z}_N}\) -curves with arbitrary N, Bershadsky and Radul discovered the existence of generalized Thomae’s formulae for these curves which they wrote down explicitly in the case in which all rotation numbers m _i equal 1. This work was continued by several authors and new Thomae’s type formulae for \({\mathbb{Z}_N}\) -curves with other rotation numbers m _i were found. In this article we prove that for some choices of the rotation numbers the corresponding \({\mathbb{Z}_N}\) -curves do not admit such generalized Thomae’s formulae.     

Limit of Fluctuations of Solutions of Wigner Equation

We consider fluctuations of the solution W _ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W _ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the initial data is singular in the x variable, that is, W _ε (0, x, k) = δ(x)f(k) and \({f\in {\mathcal{S}}(\mathbb{R}^d)}\), then the laws of the rescaled fluctuation \({Z_\varepsilon(t):=\varepsilon^{-1/2}[W_\varepsilon(t,x,k)-\bar{W}(t,x,k)]}\) converge, as ε → 0+, to the solution of the same radiative transport equation but with a random initial data. This complements the result of [6], where the limit of the covariance function has been considered.     

Existence and Uniqueness of SRB Measure on <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Generic Hyperbolic Attractors

Let M be a smooth Riemannian manifold. We show that for C ¹ generic \({f\in {\rm Diff}^1(M)}\), if f has a hyperbolic attractor Λ _f , then there exists a unique SRB measure supported on Λ _f . Moreover, the SRB measure happens to be the unique equilibrium state of potential function \({\psi_f\in C^0(\Lambda_f)}\) defined by \({\psi_f(x)=-\log|\det(Df|E^u_x)|, x\in \Lambda_f}\), where \({E^u_x}\) is the unstable space of T _x M.     

Decay Behaviors of the <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript> Hadronic Molecules

In this proceeding, we present our recent work on decay behaviors of the P_c hadronic molecules, which can help to disentangle the nature of the two P_c pentaquark-like structures. The results turn out that the relative ratio of the decays of P _c ⁺ (4380) to \({\bar D *}{\Lambda _c}\) and J/ψp is very different for P_c being a \({\bar D *}{\Sigma _c}\) or \(\bar D\Sigma _c *\) bound state with \({J^P} = \frac{{{3 - }}}{2}\) And from the total decay width, we find that P_c(4380) being a \(\bar D\Sigma _c *\) molecule state with \({J^P} = \frac{{{3 - }}}{2}\) and P_c(4450) being a \({\bar D *}{\Sigma _c}\) molecule state with \({J^P} = \frac{{{5 + }}}{2}\) is more favorable to the experimental data.     

Meixner Class of Non-Commutative Generalized Stochastic Processes with Freely Independent Values I. A Characterization

Let T be an underlying space with a non-atomic measure σ on it (e.g. \({T=\mathbb R^d}\) and σ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of T, with freely independent values. Such a process (field), ω = ω(t), \({t\in T}\) , is given a rigorous meaning through smearing out with test functions on T, with \({\int_T \sigma(dt)f(t)\omega(t)}\) being a (bounded) linear operator in a full Fock space. We define a set CP of all continuous polynomials of ω, and then define a non-commutative L ²-space L ²(τ) by taking the closure of CP in the norm \({\|P\|_{L^2(\tau)}:=\|P\Omega\|}\) , where Ω is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between L ²(τ) and a (Fock-space-type) Hilbert space \({\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,\gamma_n)}\) , with explicitly given measures γ _n . We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set CP invariant. (Note that, in the general case, the projection of a continuous monomial of order n onto the n ^th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions λ and η ≥ 0 on T, such that, in the \({\mathbb F}\) space, ω has representation \({\omega(t)=\partial_t^\dagger+\lambda(t)\partial_t^\dagger\partial_t+\partial_t+\eta(t)\partial_t^\dagger\partial^2_t}\) , where \({\partial_t^\dagger}\) and ? _t are the usual creation and annihilation operators at point t.     

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements H _xy , indexed by \({x,y \in \Lambda \subset \mathbb{Z}^d}\), are independent, uniformly distributed random variables if \({\lvert{x-y}\rvert}\) is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales \({t\ll W^{d/3}}\). We also show that the localization length of the eigenvectors of H is larger than a factor W ^d/6 times the band width. All results are uniform in the size \({\lvert{\Lambda}\rvert}\) of the matrix.     

Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials

For an infinitely renormalizable quadratic map \({f_c: z\mapsto z^2+c}\) with the sequence of renormalization periods {k _m } and rotation numbers {t _m  = p _m /q _m }, we prove that if \({\limsup k_m^{-1} \log |p_m| >0 }\), then the Mandelbrot set is locally connected at c. We prove also that if \({\limsup |t_{m+1}|^{1/q_m} <1 }\) and q _m → ∞, then the Julia set of f _c is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor.     

Influence of Molecular Effects on the Emission of Sound in a Low-Velocity Impact of a Drop on Water Surface

The dynamics of an impact pressure pulse and the evolution of the coalescence bridge between a drop and the surface of water are investigated in experiments on the impact of the drop on water surface in the range of low impact velocities U. Experimental sequences of radii r _i of the bridge, which are approximated by a function of the form t^1/2, are extrapolated to the instant of contact and are compared with radii r _k of the outer contour of the cross section formed by the bottom part of the drop with the surface. The impact pressure pulse exhibits the critical dependence on ratio \({\varepsilon _{\text{i}}}(U) = {\dot r_i}/{\dot r_k}\) of the velocities of spreading. The value of ε = 1 determines the velocity threshold below (above) which the pressure is hydrodynamic (hydroacoustic) by nature.     

A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite-range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime \({\beta < \beta_c}\), and the mean-field lower bound \({\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}\) for \({\beta > \beta_c}\). For finite-range models, we also prove that for any \({\beta < \beta_c}\), the probability of an open path from the origin to distance n decays exponentially fast in n. For the Ising model, we prove finiteness of the susceptibility for \({\beta < \beta_c}\), and the mean-field lower bound \({\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}\) for \({\beta > \beta_c}\). For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for \({\beta < \beta_c}\).     

Making Almost Commuting Matrices Commute

Suppose two Hermitian matrices A, B almost commute (\({\Vert [A,B] \Vert \leq \delta}\)). Are they close to a commuting pair of Hermitian matrices, A′, B′, with \({\Vert A-A' \Vert,\Vert B-B'\Vert \leq \epsilon}\) ? A theorem of H. Lin [3] shows that this is uniformly true, in that for every \({\epsilon > 0}\) there exists a δ > 0, independent of the size N of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how δ depends on \({\epsilon}\) . We give uniform bounds relating δ and \({\epsilon}\) . The proof is constructive, giving an explicit algorithm to construct A′ and B′. We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices. Within the context of quantum measurement, this implies an algorithm to construct a basis in which we can make a projective measurement that approximately measures two approximately commuting operators simultaneously. Finally, we comment briefly on the case of approximately measuring three or more approximately commuting operators using POVMs (positive operator-valued measures) instead of projective measurements.     

Harmonic Pinnacles in the Discrete Gaussian Model

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.     

Meixner Class of Non-commutative Generalized Stochastic Processes with Freely Independent Values II. The Generating Function

Let T be an underlying space with a non-atomic measure σ on it. In [Comm. Math. Phys. 292, 99–129 (2009)] the Meixner class of non-commutative generalized stochastic processes with freely independent values, \({\omega=(\omega(t))_{t\in T}}\) , was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions \({Z=(Z(t))_{t\in T}}\) such that Z(t) commutes with ω(s) for any \({s,t\in T}\). Then a generating function can be understood as \({G(Z,\omega)=\sum_{n=0}^\infty \int_{T^n}P^{(n)}(\omega(t_1),\dots,\omega(t_n))Z(t_1)\dots Z(t_n)}\) \({\sigma(dt_1)\,\dots\,\sigma(dt_n)}\) , where \({P^{(n)}(\omega(t_1),\dots,\omega(t_n))}\) is (the kernel of the) n ^th orthogonal polynomial. We derive an explicit form of G(Z, ω), which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators \({\partial_t,t \in T}\) . In contrast to the classical case, we prove that the operators ? _t related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality.     

Conformal Mappings and Dispersionless Toda Hierarchy

Let \({\mathfrak{D}}\) be the space consists of pairs (f, g), where f is a univalent function on the unit disc with f(0) = 0, g is a univalent function on the exterior of the unit disc with g(∞) = ∞ and f′(0)g′(∞) = 1. In this article, we define the time variables \({t_n, n\in \mathbb{Z}}\), on \({\mathfrak{D}}\) which are holomorphic with respect to the natural complex structure on \({\mathfrak{D}}\) and can serve as local complex coordinates for \({\mathfrak{D}}\) . We show that the evolutions of the pair (f, g) with respect to these time coordinates are governed by the dispersionless Toda hierarchy flows. An explicit tau function is constructed for the dispersionless Toda hierarchy. By restricting \({\mathfrak{D}}\) to the subspace Σ consists of pairs where \({f(w)=1/\overline{g(1/\bar{w})}}\), we obtain the integrable hierarchy of conformal mappings considered by Wiegmann and Zabrodin [31]. Since every C ¹ homeomorphism γ of the unit circle corresponds uniquely to an element (f, g) of \({\mathfrak{D}}\) under the conformal welding \({\gamma=g^{-1}\circ f}\), the space Homeo _C (S ¹) can be naturally identified as a subspace of \({\mathfrak{D}}\) characterized by f(S ¹) = g(S ¹). We show that we can naturally define complexified vector fields \({\partial_n, n\in \mathbb{Z}}\) on Homeo _C (S ¹) so that the evolutions of (f, g) on Homeo _C (S ¹) with respect to ? _n satisfy the dispersionless Toda hierarchy. Finally, we show that there is a similar integrable structure for the Riemann mappings (f ^?1, g ^?1). Moreover, in the latter case, the time variables are Fourier coefficients of γ and 1/γ ^?1.