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Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions

On a causal quantum stochastic double product integral related to Lévy area

R. L. Hudson, Yuchen Pei

Published Jul 25, 2018• 0 citations
Publisher

Abstract

We study the family of causal double product integrals $$ \prod\_{a < x < y < b}\Big(1 + i{\lambda \over 2}(dP\_x dQ\_y - dQ\_x dP\_y) + i {\mu \over 2}(dP\_x dP\_y + dQ\_x dQ\_y)\Big), $$ where $P$ and $Q$ are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in \[15]. The main problem solved in this paper is the explicit evaluation of the continuum limit $W$ of the latter, and showing that $W$ is a unitary operator. The kernel of $W - I$ is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.

DOI
10.4171/aihpd/60

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We study the family of causal double product integrals ∏_a<x<y<b(1+iλ2(dP_xdQ_y−dQ_xdP_y)+iμ2(dP_xdP_y+dQ_xdQ_y)),\prod\_{a < x < y < b}\Big(1 + i{\lambda \over 2}(dP\_x dQ\_y - dQ\_x dP\_y) + i {\mu \over 2}(dP\_x dP\_y + dQ\_x dQ\_y)\Big),∏_a<x<y<b(1+i2λ​(dP_xdQ_y−dQ_xdP_y)+i2μ​(dP_xdP_y+dQ_xdQ_y)), where PPP and QQQ are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in [15]. The main problem solved in this paper is the explicit evaluation of the continuum limit WWW of the latter, and showing that WWW is a unitary operator. The kernel of W−IW - IW−I is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.
We study the family of causal double product integrals \begin{equation*} \prod_{a < x < y < b}\left(1 + i{λ\over 2}(dP_x dQ_y - dQ_x dP_y) + i {μ\over 2}(dP_x dP_y + dQ_x dQ_y)\right) \end{equation*} where PPP and QQQ are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in [Hudson-Pei2015]. The main problem solved in this paper is the explicit evaluation of the continuum limit WWW of the latter, and showing that WWW is a unitary operator. The kernel of WWW is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.
This thesis is divided into two parts. In the first part (Chapters 1, 2, 3) various Robinson-Schensted (RS) algorithms are discussed. An introduction to the classical RS algorithm is presented, including the symmetry property, and the result of the algorithm Doob h-transforming the kernel from the Pieri rule of Schur functions h when taking a random word [O'C03a]. This is followed by the extension to a q-weighted version that has a branching structure, which can be alternatively viewed as a randomisation of the classical algorithm. The q-weighted RS algorithm is related to the q-Whittaker functions in the same way as the classical algorithm is to the Schur functions. That is, when taking a random word, the algorithm Doob h-transforms the Hamiltonian of the quantum Toda lattice where h are the q-Whittaker functions. Moreover, it can also be applied to model the q-totally asymmetric simple exclusion process introduced in [SW98]. Furthermore, the q-RS algorithm also enjoys a symmetry property analogous to that of the symmetry property of the classical algorithm. This is proved by extending Fomin's growth diagram technique [Fom79, Fom88, Fom94, Fom95], which covers a family of the so-called branching insertion algorithms, including the row insertion proposed in [BP13]. In the second part (Chapters 4, 5) we work with quantum stochastic analysis. First we introduce the basic elements in quantum stochastic analysis, including the quantum probability space, the momentum and position Brownian motions [CH77], and the relation between rotations and angular momenta via the second quantisation, which is generalised to a family of rotation-like operators [HP15a]. Then we discuss a family of unitary quantum causal stochastic double product integrals E, which are expected to be the second quantisation of the continuous limit W of a discrete double product of aforementioned rotation-like operators. In one special case, the operator E is related to the quantum Levy stochastic area, while in another case it is related to the quantum 2-d Bessel process. The explicit formula for the kernel of W is obtained by enumerating linear extensions of partial orderings related to a path model, and the combinatorial aspect is closely related to generalisations of the Catalan numbers and the Dyck paths. Furthermore W is shown to be unitary using integrals of the Bessel functions.
We replace the independent one-dimensional Brownian motions in the de…nition of classical Levy area by the mutually noncommuting momentum and position Brownian motions of non-Fock quantum stochastic calculus, which are independent in the sense of factorisation of joint characteristic functions. The corresponding quantum Levy area can then be constructed in a way similar to Levy’s original construction as a quantum martingale limit. Acknowledgement 1 The author thanks the referee for drawing his attention to a number of mistypes. 1 Introduction. Levy’s stochastic area [10], [11], as well as being of great intrinsic interest and possessing numerous connections with classical mathematical analysis and mathematical physics [7], has recently come to prominence in the theory of rough paths [12], [13], [5]. Here we present a quantum or noncommutative version of Levy area in which the two independent one-dimensional Brownian motions used to de…ne it are replaced by the pair of quantum Brownian motions (P;Q) ; forming the quantum Wiener process of [2]. Each of P and Q consists of a process of mutually commuting self-adjoint operators which can be simultaneously diagonalised as multiplication by a classical Brownian motion of appropriate variance in the Hilbert space of complex-valued functions square-integrable with respect to Wiener measure, But P and Q together satisfy the commutation relation P (s)Q(t) Q(t)P (s) = imin fs; tg I; (1)
Weconsider the analogueof Levy area, definedas an iterated stochastic integral, obtained by replacing two independent component one- dimensional Brownian motions by the mutually non-commuting momen- tum and position Brownian motions P and Q of either Fock or non-Fock quantum stochastic calculus, which are also stochastically independent in a certain sense. We show that the resulting quantum Levy area is trivially distributed in the Fock case, but has a non-trivial distribution in non-Fock quantum stochastic calculus which, after rescaling, interpolates between the trivial distribution and that of classical Levy area in the infinite tempera- ture limit. We also show that it behaves differently from the classical Levy area under a kind of time reversal, in both the Fock and non-Fock cases. 2000 AMS Mathematics Subject Classification: Primary: 81S25.