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Register a new userFrom n+1-level atom chains to n-dimensional noises
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In quantum physics, the state space of a countable chain of (n+1)-level atoms becomes, in the continuous field limit, a Fock space with multiplicity n. In a more functional analytic language, the continuous tensor product space over R of copies of the space C^{n+1} is the symmetric Fock space Gamma_s(L^2(R;C^n)). In this article we focus on the probabilistic interpretations of these facts. We show that they correspond to the approximation of the n-dimensional normal martingales by means of obtuse random walks, that is, extremal random walks in R^n whose jumps take exactly n+1 different values. We show that these probabilistic approximations are carried by the convergence of the basic matrix basis a^i_j(p) of $otimes_N CC^{n+1}$ to the usual creation, annihilation and gauge processes on the Fock space.
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For n even, we prove Pozhidaevs conjecture on the existence of associative enveloping algebras for simple n-Lie algebras. More generally, for n even and any (n+1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map from L to U(L) is injective. We use noncommutative Grobner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.
We consider the general $mathcal{N}{=},4,$ $d{=},3$ Galilean superalgebra with arbitrary central charges and study its dynamical realizations. Using the nonlinear realization techniques, we introduce a class of actions for $mathcal{N}{=},4$ three-dimensional non-relativistic superparticle, such that they are linear in the central charge Maurer-Cartan one-forms. As a prerequisite to the quantization, we analyze the phase space constraints structure of our model for various choices of the central charges. The first class constraints generate gauge transformations, involving fermionic $kappa$-gauge transformations. The quantization of the model gives rise to the collection of free $mathcal{N}{=},4$, $d{=},3$ Galilean superfields, which can be further employed, e.g., for description of three-dimensional non-relativistic $mathcal{N}{=},4$ supersymmetric theories.
We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $ C^{ast} $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{ast} $-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence-uniqueness results for KMS states associated to the graphs.
The 1971 Fortuin-Kasteleyn-Ginibre (FKG) inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008 one of us (Sahi) conjectured an extended version of this inequality for all $n>2$ monotone functions on a distributive lattice. Here we prove the conjecture for two special cases: for monotone functions on the unit square in ${mathbb R}^k$ whose upper level sets are $k$-dimensional rectangles, and, more significantly, for arbitrary monotone functions on the unit square in ${mathbb R}^2$. The general case for ${mathbb R}^k, k>2$ remains open.
A transition matrix $[U_{i,j}]_{i,jgeq 0}$ on $mathbb{N}$ is said to be almost upper triangular if $U_{i,j}geq 0Rightarrow jgeq i-1$, so that the increments of the corresponding Markov chains are at least $-1$; a transition matrix $[L_{i,j}]_{i,jgeq 0}$ is said to be almost lower triangular if $L_{i,j}geq 0Rightarrow jleq i+1$, and then, the increments of the corresponding Markov chains are at most $+1$. In the present paper, we characterize the recurrence, positive recurrence and invariant distribution for the class of almost triangular transition matrices. The upper case appears to be the simplest in many ways, with existence and uniqueness of invariant measures, when in the lower case, existence as well as uniqueness are not guaranteed. We present the time-reversal connection between upper and lower almost triangular transition matrices, which provides classes of integrable lower triangular transition matrices. These results encompass the case of birth and death processes (BDP) that are famous Markov chains (or processes) taking their values in $mathbb{N}$, which are simultaneously almost upper and almost lower triangular, and whose study has been initiated by Karlin & McGregor in the 1950s. They found invariant measures, criteria for recurrence, null recurrence, among others; their approach relies on some profound connections they discovered between the theory of BDP, the spectral properties of their transition matrices, the moment problem, and the theory of orthogonal polynomials. Our approach is mainly combinatorial and uses elementary algebraic methods; it is somehow more direct and does not use the same tools.
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