Local quantum information dynamics

Ryszard Paweł Kostecki

Perimeter Institute for Theoretical Physics
31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada0.15cmBasic Research Community for Physics
Mariannenstraße 89, 04315 Leipzig, Germany

June 10, 2016

This paper is intended to: 1) show how the local smooth geometry of spaces of normal quantum states over W$^*$-algebras (generalised spaces of density matrices) may be used to substantially enrich the description of quantum dynamics in the algebraic and path integral approaches; 2) provide a framework for construction of quantum information theories beyond quantum mechanics, such that quantum mechanical linearity holds only locally, while the nonlocal multi-user dynamics exhibits some similarity with general relativity. In the algebraic setting, we propose a method of incorporating nonlinear Poisson and relative entropic local dynamics, as well as local gauge and local source structures, into an effective description of local temporal evolution of quantum states by using fibrewise perturbations of liouvilleans in the fibre bundle of Hilbert spaces over the quantum state manifold. We apply this method to construct an algebraic generalisation of Savvidou's action operator. In the path integral setting, motivated by the Savvidou–Anastopoulous analysis of the role of Kähler space geometry in the Isham–Linden quantum histories, we propose to incorporate local geometry by means of a generalisation of the Daubechies–Klauder coherent state phase space propagator formula. Finally, we discuss the role of Brègman relative entropy in the Jaynes–Mitchell–Favretti renormalisation scheme. Using these tools we show that: 1) the propagation of quantum particles (in Wigner's sense) can be naturally explained as a free fall along the trajectories locally minimising the quantum relative entropy; 2) the contribution of particular trajectories to the global path integral is weighted by the local quantum entropic prior, measuring user's lack of information; 3) the presence of nonlinear quantum control variables results in the change of the curvature of the global quantum state space; 4) the behaviour of zero-point energy under renormalisation of local entropic dynamics is maintained by local redefinition of information mass (prior), which encodes the curvature change. We conclude this work with a proposal of a new framework for nonequilibrium quantum statistical mechanics based on quantum Orlicz spaces, quantum Brègman distances and Banach Lie algebras.

dedicated to Professor Stanisław Woronowicz on the occasion of  his 75$^{\,th}$ birthday

§ INTRODUCTION

Like a brush in hand

To paint a picture

Of what we would like to see

C.M. Schuldiner

In this Section we will motivate the approach of this paper by discussing how it is related to the structures and problems of Kähler geometric and C$^*$-algebraic approaches to quantum kinematics, as well as hamiltonian and path integral approaches to quantum dynamics. Our goal is to overview and explain the merits of the constructions that we pursue on the subsequent pages. The in-depth analysis of an approach to foundations of quantum theory[We distinguish between the quantum mechanics, understood as a framework defined in [361], and the quantum theory, understood as a (currently unknown) framework that should be capable of providing mathematically exact nonperturbative foundations for relativistic quantum field theory and nonequilibrium quantum statistical mechanics.] that we briefly postulate and use here is provided in another paper [214].

§.§ Quantum information geometric foundations: global postulates

The basic kinematic postulates of our framework are:

* Postulate 1: The underlying spaces of inquiry are W$^*$-algebras $\N$, instead of sample spaces and  Hilbert spaces.

* Postulate 2: The state spaces of quantified knowledge are sets $\M(\N)$ of positive normal states on W$^*$-algebras, instead of probabilistic models and spaces of density matrices.

* Postulate 3: The observables are arbitrary real valued functions $f:\M(\N)\ra\RR$ of normal states, instead of arbitrary real valued functions on sample spaces and self-adjoint operators.

* Postulate 4: Given an experimental configuration space $\Theta$, a method of model construction defining a mapping $\Theta\ni\theta\mapsto\phi(\theta)\in\M(\N)$, and a choice of the set of functions $\tilde{f}:\Theta\ra\RR$ that one is interested in, the set of observables that are relevant for a given problem is given by $\{f:\M(\N)\ra\RR\mid f\circ\theta=\tilde{f}\}$.

While C$^*$-algebras[A C$^*$-algebra is defined as an algebra $\C$ over $\CC$, equipped with: an operation $^*:\C\ra\C$ that algebraically abstracts the properties of a complex conjugation of complex numbers, and a norm $\n{\cdot}:\C\ra\RR^+$ that turns $\C$ into a Banach space satisfying $\n{x^*x}=\n{x}^2$ $\forall x\in\C$. W$^*$-algebras are characterised as such C$^*$-algebras for which there exists a Banach space, denoted $\C_\star$ and called a (Banach) predual, satisfying $(\C_\star)^\star\iso\C$. Given a Banach space $X$ over $\CC$, the operation $^\star:X\ra X^\star$ forms a Banach space of all continuous linear $\CC$-valued functionals, equipped with a supremum norm.] generalise algebras of complex continuous functions on compact topological spaces, W$^*$-algebras generalise $L_\infty$ spaces over localisable boolean algebras (or, equivalently, localisable measure spaces), so the problem of choice between them depends not only on the mathematical properties of a specific application but also on the general interpretation assigned to the quantum theoretic formalism. From the mathematical perspective of general integration theory (including integration on noncommutative W$^*$-algebras, on nonassociative Jordan algebras, and on spectral convex sets), it is completely natural to extend considerations from the sets of density matrices to the subsets of positive parts of Banach preduals of arbitrary W$^*$-algebras. The set of all (not necessarily normalised) density matrices is characterised as a positive part $\ell_1(\BH)^+$ of a noncommutative $\ell_1$ space associated to the W$^*$-algebra $\BH$ of bounded linear operators on a Hilbert space $\H$ (more precisely, $ℓ_1()_1()̋$, where $_1()̋$ is a Banach space of all trace class operators, equipped with a trace norm), while the Banach preduals $_⋆$ of arbitrary W$^*$-algebras $$ are characterised as a noncommutative $L_1$ spaces associated to these algebras, and this association is functorial with respect to W$^*$-isomorphisms [99]. Hence, if one considers quantum mechanics and probability theory as two instances of a more general class of information theories, then the use of W$^*$-algebras $$ and elements of $L_1()^+_⋆^+$ for the mathematical foundations of quantum mechanics is a \textit{natural} and \textit{exact} generalisation of mathematical formulation of probability theory in terms of a normalised measure theory (as proposed by Steinhaus [334] and developed by Kolmogorov [200]). This leads us: 1) to chose the framework of W$^*$-algebras $$ and normal positive states $ω∈^+_⋆$ instead the framework of C$^*$-algebras $$ and positive states $ω∈^⋆+$; 2) to consider subsets $()⊆^+_⋆$ as spaces of quantum states that are setting the arena for quantum kinematics. We view them as natural generalisation of probabilistic models $(,℧(),)⊆L_1(,℧(),)^+$. The observables in our framework are defined as \textit{arbitrary} real valued functions $f:()$. The observables in the standard sense of quantum mechanics are precisely determined as an affine subset of the observables in our sense:
\begin{equation}
f_x(\phi):=\phi(x)\;\;\forall x\in\N^\sa\;\forall\phi\in\N^+_\star.
\end{equation}
Postulates 1-4 do not yet determine how we are going to define specific kinematic and dynamic models. Following Chencov's geometric approach to foundations of statistical inference theory [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 244] (developed later by Amari and others [4, 5, 6, 7, 8, 9]), Jaynes information theoretic approach to foundations of statistical mechanics [166, 167, 168, 179, 169, 170, 172, 174, 175, 177] (developed later in the geometric direction by Ingarden and others [150, 156, 151, 152, 158, 153, 154, 155, 157]), and the program of smooth geometrisation of quantum mechanics [344, 236, 191, 69, 326, 68, 142, 264, 2, 10, 70, 111, 71, 147, 148, 299, 102, 103, 45, 21, 67, 46, 65, 29], we propose the following

\begin{itemize}
\item \textbf{Postulate 5:} \textit{The construction of specific models of kinematics and dynamics is based upon the geometric structures over state spaces, provided by quantum relative entropies and Banach Lie algebras, instead of scalar product of Hilbert space.}
\end{itemize}

In order to investigate the possible generalisations of quantum mechanical prescriptions of dynamics we want first to understand the geometric structures on state spaces. For a given Hilbert space $$̋ equipped with a scalar product $\s{\cdot,\cdot}_\H:\H\times\H\ra\CC$, the projection $\PP:\H\setminus\{0\}\ni\xi\mapsto\frac{\xi}{\n{\xi}_\H}\in\PP\H$ induces a manifold structure on $$̋, with tangent spaces given by $\H$, riemannian metric $^$̋ and symplectic form $\wbold^\H$ determined uniquely by a decomposition [1]
\begin{equation}
\s{\cdot,\cdot}_\H=\frac{1}{2}\gbold^\H(\cdot,\cdot)+\ii\frac{1}{2}\wbold^\H(\cdot,\cdot),
\end{equation}
and complex structure defined by $·,^(̋·)_=̋·,·_$̋. The tangent bundle of $\H$ over $$̋ can be viewed as a principal $U(1)$-bundle equipped with a $U(1)$ connection $1$-form $\nabla^\PP:\H\times\H\ni(\xi,\zeta)\mapsto\ii\s{\xi,\ddd\zeta}_\H\in\CC$. In our case, the lack of a unique global Hilbert space implies the lack of unique specification of riemannian metric and symplectic form derived from a scalar product. In order to facilitate a well-defined generalisation of riemannian and symplectic structure, the sets $\M(\N)$ can be equipped with two different smooth real Banach manifold structures. On one hand, an information manifold structure on the set $\N^+_{\star0}$ of all faithful (strictly positive) elements of $\N^+_\star$ is constructed by a choice of a quantum relative entropy functional on $\N^+_\star$, and has tangent spaces defined as noncommutative Orlicz spaces that provide suitable convergence behaviour of neighbourhoods of states as measured by constrained relative entropy minimisation (see Section <ref>). On the other hand, a Banach Lie–Poisson manifold structure is constructed by the choice of a Banach Lie algebra (such as the real Banach Lie algebra $\N^\sa$ of all self-adjoint elements of a W$^*$-algebra $\N$), and has tangent spaces defined as copies of a predual of this algebra (see Section <ref>). Given the information distance manifold structure on $\M(\N)$, the first two nonzero orders of Taylor expansion for a wide class of relative entropy functionals $D(\cdot,\cdot)$ on $\M(\N)$ give rise to the torsion-free smooth Norden–Sen geometries $(\M(\N),\gbold^D,\nabla^D,(\nabla^D)^\nsdual)$. The Norden–Sen geometry is defined by the condition (<ref>), which directly generalises the condition characterising the Levi–Civita affine connection $\nabla^\gbold$ in riemannian geometries $(\M,\gbold)$. The Fubini–Study riemannian metric $\gbold^\H$ becomes recovered as an extension to the boundary of pure states for a wide class of geometries $((),^D)$ [272]. On the other hand, given the choice of a Banach--Lie algebra $$ such that $()$ is a real Banach submanifold\footnote{For a discussion why an injective immersion of $\M(\N)$ into $\B_\star$ is not sufficient, see [39].} of $_⋆$ with $_ϕ_⋆_⋆$ $∀ϕ∈_⋆$ (or, more generally, $^⋆_x(_⋆)⊆_⋆$ $∀x∈$), the coadjoint action of $$ on $_⋆$ induces a Poisson structure on Fr\'{e}chet smooth real valued functions on $()$. If $$ is a Lie algebra of a group $G$, then the Banach Lie--Poisson manifolds $()$ are symplectic if they are coadjoint orbits of $G$. In particular, if $^$, then $G^$ (a group of all unitary elements of $$), and $_⋆^_⋆:={ϕ∈_⋆|ϕ(x^*)=ϕ(x)^*}(^)_⋆$. The example of such case is given by the orbit of a group of unitary operators of density matrices with finite fixed number $n∈$ of nonzero eigenvalues [38]. For $n=1$, one recovers precisely the symplectic structure $^$̋ on a projective Hilbert space $\PP\H$ that is induced from $·,·_$̋ on $\H$ [36].

The resulting description of quantum geometry can be summarised as follows:
\begin{enumerate}
\item[(a)] In the general setting of W$^*$-algebras $\N$ and quantum states defined as elements of $\N_\star^+$ the smooth manifold structure required to implement the infinite-dimensional quantum generalisation of Poisson geometry does not match with the smooth manifold structure required to implement the infinite-dimensional quantum generalisation of riemannian geometry (this issue is discussed in more details in [214]). As a result,  the geometry of a Hilbert space $\H$ (consisting of pure states), formulated in terms of  riemannian metric $\gbold^\H$ and symplectic structure $\wbold^\H$ defined over the same real Hilbert smooth manifold $\PP\H$, becomes generalised to the geometry of spaces $\M(\N)$ equipped with two different real Banach smooth manifold structures: of a Banach Lie--Poisson manifold and of a quantum information geometric (relative entropic) manifold. The former is determined by the choice of a Banach Lie algebra $\B$, and in the special cases reduces to a symplectic space. The latter is determined by the choice of a relative entropy functional $D:\M(\N)\times\M(\N)\ra[0,\infty]$, and in the special cases reduces to a torsion free Norden--Sen manifold, or just a riemannian space.
\item[(b)] Apart from the above two alternative systems of tangent, cotangent, and higher jet bundles, one can also introduce a bundle of Hilbert spaces over $\M(\N)\subseteq\N^+_\star$, that can serve as as an ambient framework to represent different geometrical objects. A natural candidate is a Gel'fand--Na\u{\i}mark--Segal bundle $\H\M(\N)$ of Hilbert spaces [256]. Because the bundle $\H\M(\N)$ is defined by states of an underlying manifold (as opposed to a projection $\PP$), there is no $\nabla^{\PP}$ connection $U(1)$ action in fibers. However, for $\M(\N)\subseteq\N^+_{\star0}$, each fibre of $\H\M(\N)$ bundle is equipped with a strongly continuous $U(1)$ action of a modular (Tomita--Takesaki) automorphism $\sigma^\omega:\RR\ni t\mapsto\Ad(\Delta_\omega^{\ii t})\in\Aut(\pi_\omega(\N))$. We will study the role of this automorphism in Section \ref{algebraic.action.operator.section}, showing that it resembles some interesting similarities with $\nabla^{\PP}$ when considered over a trajectory $\RR\ra\M(\N)\subseteq\N^+_{\star0}$. Yet, in Section \ref{quant.info.geom.section} we will show that, for any $\M(\N)\subseteq\N^+_{\star0}$, the bundle $\H\M(\N)$ carries a natural connection structure, with parallel transport given by the standard unitary transition operators $V_{\phi,\omega}:=J_{\phi,\phi}J_{\phi,\omega}$, where $J_{\phi,\omega}$ is a relative modular conjugation between two faithful normal GNS representations, while $J_{\phi,\phi}$ is a Tomita modular conjugation. We observe that this connection is Levi-Civita with respect to the Wigner--Yanase riemannian metric, and its local geodesic free fall corresponds to constrained minimisation of the Hilbert space norm (projective measurement).
\item[(c)] Because two manifold structures mentioned in (a) do not coincide, there seems to be no obvious candidate for a `complex structure' on a general quantum model $\M(\N)$. However, we notice that the use of complex Hilbert spaces of dimension $n$ instead of real Hilbert spaces of dimension $2n$ is crucially associated with the requirement that the generators of unitary transformations of these spaces should be represented by self-adjoint operators (identified with observables). Observing further that the standard quantum mechanical method of defining relevant observables proceeds by representations of Lie algebra $\glie$ of Lie group $G$ on the complex Hilbert space, we introduce the structure of a principal $G$-bundle $E$ over $\M(\N)$, equipped with a family of representations of an associated Lie algebra in the fibers of the GNS Hilbert space bundle $\H\M(\N)$. This leads to a rise of a $\glie$-valued connection form on $E$, and a fiberwise family of its representations on the fibers of $\H\M(\N)$. This structure exhibits some interesting features of the  relationship between Berry connection, complex structure of a Hilbert space, and construction of observables by means of representations of Lie algebras. The bundle structures of $E$ and $\H\M(\N)$ do not require $\M(\N)$ to be a Banach smooth manifold, but they require it to be a (Hausdorff and paracompact) topological space. We do not assume any \textit{a priori} relationship between the geometries of local causal dynamics, as provided by $\B$, and the transformations used to identify locally the relevant observables, as provided by $\glie$, because we consider it to be a model-dependent feature.
\item[(d)] Using the $\glie$-valued connection $\nabla^\glie$, we can define a kinematic propagator (in Prugove\v{c}ki sense) of the particles (in Wigner sense) as the holonomy of $\nabla^\glie$ along the geodesics of $\nabla^D$ (or $(\nabla^D)^\nsdual$ or $\nabla^{\gbold^D}$) affine connection.
\item[(e)] In Section \ref{MCP.bundle.section} we propose a construction of what we call the Morozova--Chencov--Petz Hilbert space bundle $\H^{\hhh}\M(\N)$. Its purpose is to use the riemannian metrics $\gbold^D$, associated with a class $D_\fff$ of quantum distances, in order to determine Hilbert spaces and corresponding representations that are different from the GNS construction, and include the information about the local riemannian geometry of a model (this is inspired by the ideas of [346, 292, 291]). In principle, it can be used as an alternative to $\H\M(\N)$ (especially for the purpose of the tasks (c)-(d)), however it is essentially harder to deal with mathematically. We consider this as an indication that the natural framework for a simultaneous implementation of geometric and algebraic tools used in this paper (entropic Norden--Sen geometries, Banach Lie--Poisson structure, perturbations of liouvilleans) are Banach dual pairs of noncommutative Orlicz spaces, used as tangent and cotangent spaces. However, the technical implementation of this idea requires one to develop a standard construction of Orlicz spaces for any (countably finite) W$^*$-algebra, and for a large class of quantum relative entropies, as well as to develop the theory of perturbations of ``Orlicz liouvilleans'' (by an analogy to $L_p$-liouvilleans of Jak\v{s}i\'{c} and Pillet [165]). While these tasks are beyond the scope of this paper, it can be understood as a testing ground for them, so we will discuss more precisely the above ideas in Sections \ref{local.infogeometry.histories.intro.section} and \ref{effective.local.quantum.dynamics.section}.
\end{enumerate}

With all these tools on the stage, we can approach the problem of construction of quantum dynamics.

\begin{itemize}
\item \textbf{Postulate 6:} \textit{The elementary form of causal dynamics is given by a Poisson flow generated by a smooth observable on a state space, as opposed to a unitary evolution on a Hilbert space. The elementary form of inferential dynamics is given by a (nonlinear) constrained quantum relative entropy minimisation, as opposed to a (linear) projection in a Hilbert space or L\"{u}ders' rule.}
\end{itemize}

Unitary evolution is a special case of a hamiltonian evolution on the self-adjoint part of a predual of a W$^*$-algebra, understood as a BLP space [36, 38, 253]. Derivation of L\"{u}ders' rules (selective and nonselective) as a special case of constrained quantum relative entropy minimisation\footnote{Technically, $\PPP^D_\Q(\phi)$ is a nonlinear projection in a positive cone of a noncommutative $L_1(\N)$ space, constrained by the data represented as a convex closed set $\Q_\Upsilon$ in a noncommutative Orlicz space $L_\Upsilon(\N)$, associated by means of a nonlinear bijective mapping $\ell:L_1(\N)\ra L_\Upsilon(\N)$, so that $\Q=\ell^{-1}(\Q_\Upsilon)$ forms a codomain of a projection. While in general we are interested in entropic projections for a wide class of quantum distance functionals, the results cited in this paragraph were proven for the most recognised example: the Umegaki--Araki noncommutative generalisation of the Kullback--Leibler distance.} (in short: entropic projection) for $D$ given by the Umegaki--Araki distance \eqref{Araki.std.rep.distance} was provided in [139] and [211].\footnote{Derivation of the nonselective L\"{u}ders' rule from minimisation of the Hilbert--Schmidt norm distance was provided much earlier in [141], while derivations of some special cases of the selective L\"{u}ders' rule using some symmetric quantum information distances were obtained in [131, 132, 81, 286]. See [211] for more discussion.} Hence, the basic dynamical setting of quantum mechanical evolution of quantum states, which is a unitary evolution followed by a projective measurement can be completely recovered as a special case of a \textit{causal inference instrument} given by the map\footnote{The composition in the reverse order can also be studied.}
\begin{equation}
\M_1(\N)\ni\phi\mapsto\PPP^D_\Q\circ w^{\B,h}_t(\phi)\in\M_2(\N),
\label{global.hamiltonian.entropic.map}
\end{equation}
\begin{equation}
\PPP^D_\Q(\psi):=\arginff{\psi\in\Q}{D(\omega,\psi)}
\end{equation}
is an entropic projection onto constrained set $$ for a quantum distance functional $D$, while $w^,h_t$ is a Poisson flow generated by a Banach Lie algebra $$ and a hamiltonian function $h$ for a time range $[0,t]$, corresponding uniquely to an integral line of a vector field
\begin{equation}
\XXX_h(\phi):=-\ad^\star_{\DF_\phi h}(\phi)\;\;\forall\phi\in\B_\star,
\end{equation}
where $_ϕh$ is a Fr\'{e}chet derivative of $h$ at $ϕ$, implementing the differential form $h(ϕ)$.

In [245] it was shown that partial trace is also a special case of entropic projection. Hence, all linear completely positive maps can be considered as a special case of the maps formed by composition of tensor products, Poisson flows, and entropic projections [215, 214]. This way the kinematic and dynamic setting of quantum mechanics and nonrelativistic quantum information theory becomes fully recovered as a special case of the framework specified by Postulates 1-6 above.

This leads us to:
\begin{itemize}
\item \textbf{Open problem:} \textit{Reconstruct (some aspects of) dynamics of quantum field theory and nonequilibrium quantum statistical mechanics using the framework specified by Postulates 1-6.}
\end{itemize}
Unlike in quantum mechanics, the dynamics of both these theories is sensitive to local geometric features of the kinematic structure of a quantum model. In consequence, we are lead to investigate how, and to what extent, the above geometric structures, and the corresponding nonlinear dynamical maps, can give account of the local structures in QFT and NQSM.

\subsection{Local quantum information dynamics in algebraic and path integral approaches\label{local.quantum.dynamics.intro.section}}

The main questions underlying the constructions carried on in this paper are: what if the correct setting for bridging the gap between algebraic and path integral setting is to use quantum state spaces and their geometry:
\begin{enumerate}
\item[1)] to define local evolution in the algebraic approach by means of \textit{locally} defined and perturbed liouvilleans?
\item[2)] \textit{instead} of using phase space geometry in the continuous time coherent state path integral ``quantisation''?
\item[3)] to describe renormalisation as purely information theoretic procedure?
\end{enumerate}
The discussion below is intended to show that the proposal of the geometric framework for locally quantum information theories that we provide in Section \ref{locally.quantum.information.relativity.section} is remarkably grounded in the insights coming from \textit{three} very distinct theoretical frameworks: 1) a geometric extension of algebraic hamiltonian dynamics with local gauge and local sources by means of local perturbation of liouvilleans; 2) a generalisation of the Daubechies--Klauder path integration to an algebraic setting; 3) a geometric Jaynes--Mitchell--Favretti renormalisation, applicable in nonequilibrium quantum statistical mechanics.

The standard Haag--Kastler [128, 127] setting of an algebraic approach to quantum field theory is widely considered as being unable to incorporate the local gauge principle\footnote{E.g. \cytat{The Lagrangean and the Feynman path integral are at present indispensable tools in the characterization and study of a specific theory. Together with the local gauge principle they pose questions which in the algebraic approach are not understood and should be tackled.} [127].} (the global gauge principle has been partially incorporated to an algebraic approach by means of the Doplicher--Haag--Roberts theory [85, 86]). Apart from renormalisation techniques, this principle is a fundamental tool in the construction of the predictively sound models in quantum field theory. Its maintenance by the lagrangean/path-integral approach leads to an abandonment of the algebraic approach by most of the practitioners of QFT, but this is provided at the price of replacing mathematically well-defined objects by symbolic (and usually perturbative) techniques of calculations. This makes QFT very different from quantum mechanics, because the latter facilitates construction of predictive models without the expense of mathematical precision. In this paper we intend to show that the consideration of geometric structures on the spaces $^+_⋆$ of normal states over W$^*$-algebras $$, as well as construction of effective local dynamics by means of local perturbations of liouvilleans, may provide an extension of an algebraic approach capable of dealing with the local gauge principle and the use of `external sources', typical in the path integral formalism. Our point of departure from the Haag--Kastler perspective is to consider locality in the Prugove\v{c}ki sense [281], being associated with the fiber at a given point of an underlying space (so the Lorentz or Poincar\'{e} covariance condition is to be applied fiberwisely), as opposed to a neighbourhood of this point (e.g. as given by the special relativistic diamond). This allows to think of local GNS Hilbert space associated to the manifold of quantum states as (a model of, or as a container of) local tangent space, corresponding to a local quantum mechanical description provided by a single (quantum bayesian) user. Further extension from the bundle of (self-dual) Hilbert spaces to the  bundle of dual pairs of noncommutative Orlicz spaces is necessary to allow the geometry of local quantum inference to be governed by a wide class of conventions of estimation, beyond the Wigner--Yanase riemannian metric (so that the quantum nonequilibrium thermodynamic Kubo--Mori--Bogolyubov metric, as well as the quantum estimation theoretic Helstrom--Uhlmann--Bures metric can be included on the equivalent mathematical footing). The construction of an underlying manifold structure commits to the principle of equivalence between local inference by means of constrained maximisation of relative entropy, and the free fall along the geodesics of the dually flat local geometry, derived from this entropy.

On the path integral side, our approach is directly inspired by the Daubechies--Klauder [77, 194, 195, 199, 35, 198] continuous-time regularised coherent state phase space approach to path integration, and the closely related Ana\-sto\-pou\-los--Savvidou [11, 12, 13] analysis of decoherence functional in the Isham--Linden quantum histories approach. Both have shown that one can think of the underlying dynamical objects of respective theories (path integrals and decoherence functionals) as consisting of the hamiltonian evolution perturbed by the geometric structures on the space of quantum states. Our goal here is to follow Klauder's remark \cytat{If there is ever any hope to define path integrals rigorously as path integrals over a set of paths (functions of time), then it is \textit{essential} to give up the notion that the paths involved are sharp value paths and replace that with another interpretation of which the expectation value paths is a completely satisfactory example} [197] by extending these approaches to the state spaces over W$^*$-algebras, and relating them with the local liouvillean approach to algebraic dynamics. The virtue of the Daubechies--Klauder approach is that it provides a mathematically rigourous continuous time regularisation of the functional integral in a way that gives the same results under arbitrary canonical transformations of the underlying phase space. This is not true for most of other approaches to quantisation, not only path integral based. The restrictions on the class of hamiltonians that are allowed in order to maintain this procedure to be well defined are quite mild. The key ingredient of this approach is introducing a regulariser that represents a riemannian metric on the phase space (corresponding to a Fubini--Study metric on coherent quantum states), and determines a pinned Wiener measure of the Brownian process on the phase space.

%Our work was originally intended as  a generalisation of these results beyond the framework of Hilbert spaces and pure states.
%this paper is: how the above geometric structures on the spaces of states on W$^*$-algebras can be used to define the kinematics and dynamics of quantum theory beyond quantum mechanics which would be potentially capable of providing the exact mathematical formalism for (a suitable formulation of) quantum field theory?
%(and, more generally, the series of related papers \cite{Kostecki:2010:AIP,Kostecki:2011:OSID,Kostecki:2011:principles,Duch:Kostecki:2011,Kostecki:2011:Vaxjo,Kostecki:2014,Hellmann:Kaminski:Kostecki:2016,Kostecki:2016:microhauptwerk})
%\subsection{Bridging the gap between hamiltonian and lagrangean quantum dynamics}
%As a result, it seems reasonable to split the investigations of the state-space-based construction of quantum lagrangeans into two different works. By this reason, we avoid discussion of riemannian (and more generally, relative entropic) aspects of the quantum state space geometry and dynamics here, leaving it for another paper.

The heuristic ideas underlying our treatment of quantum dynamics are: 1) Quantum kinematics and dynamics should be defined without recourse to classical models and their quantisation; 2) Classical (phase space, but also space-time) geometry should be considered as a locally emergent feature describing particular properties of the multi-agent information processing systems and not as a fundamental structure (background); 3) Local spatial (phase space or space-time) variables should arise as epistemic (e.g. operational) parameters of information states (see e.g. [289, 90]). This heuristics is in a disagreement with the perspectives of the orthodox algebraic and path integral approaches (yet, there are some exceptions\footnote{In particular, \cytat{the interpretation of the formal path integral (...) in terms of paths $\mathrm{p}(t)$ and $\mathrm{q}(t)$ for which the meaning of the variables is that of expectation values is far more acceptable than the one in which the meaning is that of both sharp position and sharp momentum (eigen)values. (...) One is almost tempted to assert that the usual interpretation in terms of sharp eigenvalues is ``wrong'', because it cannot be consistently maintained, while the interpretation in terms of [expectation] values is ``right'', because it can be consistently maintained} [196].}), however we consider this disagreement as a virtue, because it allows us to learn something new.\footnote{See [205, 206, 207, 208] for some wi(l)der heuristic ideas that have lead to the current work.}

The main mathematical tools used in what follows are: the Hilbert space bundle over $^+_⋆$ arising from the Gel'fand--Na\u{\i}mark--Segal representation, introduced recently by Odzijewicz and Sli\.zewska [256], the Derezi\'{n}ski--Jak\v{s}i\'{c}--Pillet theory [80] of unbounded perturbation of standard liouvilleans, the B\'{o}na--Odzijewicz--Ratiu construction [36, 37, 38, 253] of the Banach--Lie--Poisson manifold structure on the self-adjoint part $^_⋆$ of the Banach predual of $$, Jen\v{c}ov\'{a}'s construction [184, 185] of real Banach smooth manifold over the spaces $^+_⋆01$ of normalised faithful (strictly positive) elements of $^+_⋆$, based on quantum relative entropic perturbations of states and noncommutative Orlicz spaces, the Daubechies--Klauder continuous time regularised coherent state path integrals [77, 194, 196, 365, 198], and Favretti's geometrisation [100] of the Jaynes--Mitchell source theory [241, 173, 177, 118, 120]. Section \ref{geometric.structures.section} provides an introduction to most of these mathematical tools. More specifically, Section \ref{W.star.BLP.section} is intended as an introduction to the Banach--Lie--Poisson structure of preduals of W$^*$-algebras. This material is mostly based on papers [38, 253]. In Section \ref{standard.liouvilleans.section} we review the construction of a standard liouvillean (including standard representation and Haagerup's theorem), some relative modular theory, as well as the Odzijewicz--Sli\.zewska construction of the GNS bundle of Hilbert spaces. In Section \ref{relat.entr.MCP.bundle} we discuss some results from quantum information geometry which we will use in the subsequent sections, including Jen\v{c}ov\'{a}'s construction of a smooth quantum manifold structure. The Daubechies--Klauder approach is discussed in Section \ref{Hilbert.space.geometry.path.integrals}, while Favretti's approach is discussed in Section \ref{Favretti.section}.
\subsubsection{Locally perturbed liouvilleans\label{locally.perturbed.liouvilleans.intro.section}}
Section \ref{algebraic.hamiltonian} provides an elementary analysis of the relationship between W$^*$-dynamical systems, hamiltonian flows on BLP spaces, and standard liouvilleans of the W$^*$-dynamical systems $(,,α)$. In order to study the relationship of the BLP structure with the usual usage of standard liouvilleans\footnote{By the Haagerup theorem [130] for standard representations of W$^*$-algebras, for every pair of a W$^*$-dynamical system $(\N,\RR,\alpha)$ and a standard representation $(\H,\pi,J,\stdcone)$ of a W$^*$-algebra $\N$, there exists a unique unitary evolution on $\H$ that represents $\alpha$ leaving $\stdcone$ unchanged. Its generator is an unbounded operator, called the \textit{standard liouvillean}.}, we begin with characterisation of the class of weakly-$\star$ continuous representations $\alpha:\RR\ra\Aut(\N)$ whose predualised actions on $\N_\star^{\sa}$ can be described as Poisson flows of some hamiltonian vector field.

Our main conclusion from this analysis is that the relationship between Poisson flows and standard liouvilleans should be \textit{localised}: instead of requiring a Poisson flow to globally agree with a family of norm continuous isometries arising from a predetermined W$^*$-dynamical system, we can start from a \textit{quantum Poisson system} (defined as a set $\M(\N)\subseteq\N^+_\star$ equipped with \textit{some} Banach Lie--Poisson manifold structure, not necessarily determined by the coadjoint action of the Lie algebra of self-adjoint elements of $\N$), and determine a fiberwise family of  local W$^*$-dynamical systems generated by a $1$-form corresponding to a hamiltonian vector field on the state space. This way we consider the fiberwise family of local standard liouvilleans as a Hilbert space/algebraic counterpart of the smooth manifold/geometric hamiltonian vector field of the Poisson flow.

The main technique used in Section \ref{from.local.gauge.to.lagrangeans.section} is: 1) to represent a (possibly, nonlinear) local Poisson flow \textit{on the state space manifold} in each fibre of the GNS Hilbert space bundle by constructing a \textit{local} standard liouvillean, generating unitary evolution uniquely corresponding to the hamiltonian vector field of this flow, and then: 2) to perturb it using objects that represent additional geometric structures on the state space. The resulting structure is shown to determine a nonlinear instrument on $\N^+_\star$ (in the sense of [78]), which we call a \textit{local liouvillean instrument}. It describes the temporal evolution of quantum states determined by the postulated `internal' dynamics (a W$^*$-dynamical system, a Poisson flow, or a globally defined vector field) perturbed by the \textit{geometric} structures on the quantum model. In other words, the local liouvillean instrument encodes the effective dynamics, that takes into account a nontrivial geometry of the space of quantum states. In addition, we discuss the possible expressions for time dependent $n$-point correlation functions that can be constructed using the above structures. Both local liouvillean instruments and correlation functions are understood as tools as quantification of the effective dynamics.

%In consequence, in Section \ref{from.local.gauge.to.lagrangeans.section} we systematically use the DJP approach to perturbation of standard liouvilleans to construct local GNS bundle representations of nonlinear Poisson dynamics perturbed by the action of local gauge connection as well as additional source terms. We discuss the possible quantification of this dynamics in two ways: by means of liouvillean instruments and by means of correlation functions.%, and by means of quantum lagrangeans.
%The uniqueness of this unitary evolution is facilitated
%and we consider it

Noticing that both the GNS bundle and the tangent bundle of the manifold of quantum states can allow in principle for introduction of a nontrivial action of some Lie group $G$ on fibres, we propose to consider a specific relationship between local gauge (principal $G$-bundle connection) structure and the GNS bundle. We begin with incorporation of the (fiberwise representation of the) action of the nontrivial gauge connection $\mathbf{A}$ (one-form valued in the Lie algebra $\glie$ of $G$) into the perturbation of the local liouvillean, discussing also the possible relationship between affine connection on the tangent bundle of the quantum manifold and propagation of quantum particles (in Wigner sense) in the GNS fibre bundle.

Apart from the gauge connection, we study also the class of objects that, from the perspective of the BLP structure, could be considered as nonlinear quantum fields. These are introduced as the additional source/sink terms, representing the Lie algebra valued differential one-forms on the base quantum manifold. (The idea of using source-based approach is inspired by Schwinger's [309, 310] and the Mitchell--Jaynes [241, 177] approaches.) %The perturbed standard liouvillean, which we call a \textit{local liouvillean}, generates the unitary evolution in a fibre that takes into account the effects introduced by the local gauge structure, as well as by any other \textit{local} ``quantum field'' structure that is defined by means of the self-adjoint (not necessarily bounded) operators acting fibrewise on the fibres of the GNS bundle.

%some , and  %We close the paper with a construction of a corresponding class of quantum lagrangeans, and determining under which conditions this construction is possible.

%The research carried in this paper is intended as a stimulation for further investigation of the relationship between the GNS bundle and smooth geometric structures on the quantum state spaces. In particular, the setting of the quantum information geometric manifolds by Jen\v{c}ov\'{a}  provides a basis for alternative constructions, with different tangent and cotangent spaces, which seem to be more accessible for the information geometric approach to renormalisation. We will return to this issue in [214].

As a result, we construct a setting that allows to define various nonlinear quantum models equipped with smooth geometric structures that can be represented directly in terms of the families of operators acting locally on the fibres of the GNS bundle of Hilbert spaces over the model. It seems that this framework covers quite well some of the components of the lagrangean framework (under nonorthodox assumption that space-time/phase space geometry is emergent from the geometry of quantum state spaces). The investigated correspondences between geometric and algebraic structures can be briefly summarised as:
\begin{center}
\begin{tabular}{c|c}
\textit{$C^\infty$-geometric} & \textit{GNS-bundle-algebraic}\\\hline
principal $G$-bundle sections & gauge propagators\\
one-forms & local quantum field source operators\\
Lie algebra valued one-forms & local gauge quantum fields\\
global charges & global source strengths\\\hline
\end{tabular}
\end{center}
Under some additional conditions we can establish also some relationships between structures of tangent and the GNS bundles:
\begin{center}
\begin{tabular}{c|c|c}
\textit{$C^\infty$-geometric} & \textit{GNS-bundle-algebraic} & \textit{extra condition}\\\hline
hamiltonian vector fields & standard liouvilleans & (PC$_2$)\\
geodesic trajectories & gauge geodesic propagations & (QP$_1$)\\\hline
\end{tabular}
\end{center}
There is also a correspondence between the local liouvillean instruments acting on $\M(\N)$ and local liouvillean operators acting on the fibres of the GNS bundle. These instruments might be nonsmooth. Two main ideas regarding the local quantum dynamics contained in Section \ref{from.local.gauge.to.lagrangeans.section} can be summarised as:
\begin{align}
\mbox{local gauge dynamics}&=\mbox{local Poisson dynamics}+\mathbf{A}\mbox{-propagation},\\
\mbox{local liouvillean dynamics}&=\mbox{local Poisson dynamics}+\mathbf{A}\mbox{-propagation}+\mbox{action of sources}.
\end{align}

An especially interesting possibility for introducing an affine connection $\nabla$ on a tangent bundle $\T\M(\N)$ is a third order Taylor expansion of a quantum relative entropy functional $D$. In such case the gauge geodesic propagation of quantum particles can be carried precisely along the lines of local information flow, defined by a constrained maximisation of a relative entropy, and equivalent to the local $\nabla^D$-geodesic free fall. This particular application shows a virtue of using the Hilbert space bundle combined with the technique of local perturbation of standard liouvilleans: it allows to accomodate different smooth manifold structures on the space of quantum states into a single fiber-wise operator formulation. See [214] for further discussion.

In Section \ref{algebraic.action.operator.section} we provide another example of application of this technique, constructing an algebraic generalisation of Savvidou's action operator. It is specified by perturbation of a standard liouvillean $L_\alpha$ of a weak-$\star$ continuous $*$-automorphism $\alpha:\RR\ra\Aut(\N)$ of a W$^*$-algebra $\N$ by the generator $K_\omega:=-\log\Delta_\omega$ of the Tomita--Takesaki modular automorphism $\sigma^\omega:\RR\ra\Aut(\N)$. This can be tentatively interpreted as incorporation of an action of a $U(1)$-connection on a fibre bundle of Hilbert spaces over a real line of a trajectory of $\alpha_\star$ on $\N^+_\star$. For any W$^*$-algebra $\N$, the Falcone--Takesaki theory [99] functorially associates a `core' von Neumann algebra $\core$. If $\N$ is equipped with a faithful normal algebraic state $\omega$, then there exists a canonical unitary isomorphism $\core\iso\N\rtimes_{\sigma^\omega}\RR$ with the crossed product corresponding to a W$^*$-dynamical system $(\N,\RR,\sigma^\omega)$ formed by a modular $*$-automorphism $\sigma^\omega$ of $\N$. This crossed product is a von Neumann algebra generated by the operators $\pi_{\sigma^\omega}(x)$ and $u_\RR(t)=\ee^{-\ii t\tilde{V}}$ acting on the space $\H_\omega\otimes L_2(\RR,\dd t)$ by means of the equations \eqref{pi.sigma} and \eqref{lambda.tau}. The covariance equation  \eqref{modular.histories.covariance}, where a self-adjoint linear operator $K_\omega$ is equal to the Tomita--Takesaki modular hamiltonian, turns the `Liouville' (in Savvidou's sense) action of $\ee^{-\ii t\tilde{V}}$ on $L_2(\RR,\dd t)$ into the action of $\ee^{-\ii tK_\omega}$ on the space $\H_\omega$. Hence, one can say that it `internalises the description of external unitary kinematics'. The perturbed operator $L_\alpha+K_\omega=L_\alpha-\log\Delta_\omega$ provides an algebraic replacement of the quantum histories description of action operator given by equations \eqref{s.h.auto} and \eqref{s.h.auto2}.
\subsubsection{Local information geometry in quantum histories\label{local.infogeometry.histories.intro.section}}
By analysis of the virtues and drawbacks of the above formulation of an algebraic action operator, we come to a conclusion that the proper candidate for a description of the geometric perturbation of a dynamics due to the local change of state in a projective measurement is not $K_\omega$, but a standard unitary transition operator $V_{\phi,\omega}$ (which is not easily incorporable into the local liouvillean framework). It is a parallel transport of the Levi-Civita connection $\nabla^{1/2}$ of the Wigner--Yanase metric $\gbold^{1/2}$, and projections along its ``free fall'' geodesics are equal to the linear projections in a (standard representation) Hilbert space. This observation leads us to revisit the use of a Fubini--Study metric $\gbold^{\mathrm{FS}}$ on the space of coherent states (which coincides, up to a multiplicative scalar factor $4$, with $\gbold^{1/2}$, when the latter is extended to the boundary of the pure states) for the purposes of regulation of the propagator of a quantum dynamics defined by means of functional integration. As a result, we propose a suitable generalisation of the Daubechies--Klauder expression \eqref{DK.Wiener.expression} for regularised continous time path integration, replacing the coherent states over phase space by all states in the given quantum model $\M(\N)\subseteq\N^+_{\star0}$. In what follows, we will discuss the conceptual aspects of the mathematical formulation that we provide.

Motivated by the Anastopoulos--Savvidou analysis of the term $\ee^{\ii\int p\dot{q}}$ in the Daubechies--Klauder formula \eqref{DK.Wiener.expression} as a holonomy of the Berry connection, we replace it by
\begin{equation}
\exp\left(\ii\int\dd t\s{\Omega_{\phi(t)},\ddd_{\nabla^{1/2}}(\phi(t))\Omega_{\phi(t)}}_{\H_{\phi(t)}}\right),
\label{one.form.nabla.onehalf.holonomy}
\end{equation}
where $\H_{\phi(t)}$ is the GNS Hilbert space associated with $\phi(t)\in\M(\N)$, $\Omega_{\phi(t)}$ is its representing vector, while $\ddd_{\nabla^{1/2}}$ is a $\nabla^{1/2}$ connection $1$-form. This is equivalent to a local integral of an infinitesimal entropic projection generated by a quantum Br\`{e}gman distance $D_{1/2}$ on $\M(\N)$. While the necessary mathematical background describing the relationship between entropic projections and geodesic free falls is discussed in Section \ref{distances.NS.geom.section}, let us briefly explain the conceptual perspective behind using it to define the dynamics in quantum theory.

In general, the entropic projections $\PPP^D_{\Q}$ can be used to generate the global temporal evolution of quantum models following the ideas of Jaynes [179, 241, 173, 177], promoted from an absolute to relative entropy by Schl\"{o}gl [300, 302, 301, 304, 303, 305, 306] and Hobson [145, 146] (see [118, 120, 341, 50] for the recent account on further developments of these approaches).\footnote{While the definition of $\int\mu p\log\frac{p}{q}$ as well as its conceptualisation as a measure of relative information gain is due to Kullback [219, 218], the use of this object for defining information dynamics can be credited to the above authors.} Given a time dependent set of constraints $\Q(s)$, the map
\begin{equation}
\phi_0\mapsto\PPP^D_{\Q(s)}(\phi_0)
\label{entropic.time.evolution}
\end{equation}
selects a unique trajectory of quantum states, if for each $s$ the set $\Q(s)$ is such that it gives a unique solution to the corresponding minimisation problem (in order to recover the typical formulation of dynamical problems, one may additionally require the map $s\mapsto\PPP^D_{\Q(s)}(\phi_0)$ to be continuous, and $\PPP^D_{\Q(0)}(\phi_0)=\phi_0$).

However, this construction is not the same as local re-updating of the state in time $s$ to the state in time $s+\delta s$ by the new data. While in principle it is nothing wrong with it (after all, the classical action principle $\delta S=0$ is an inherently nonlocal construction), it is interesting to see whether a \textit{local} entropic dynamics can be proposed. The equivalence of entropic projections with geodesic projections for the class of \textit{Br\`{e}gman distances} $D_\Psi$ provides such a possibility. In such case, instead of consideration of subsequent stages of a relative entropy driven evolution that is nonlocally determined by an initial state $\phi_0$, one can just follow the $\nabla^{D_\Psi}$-geodesics of the $\nabla^{D_\Psi}$-connection (derived from $D_\Psi$, as a third order  Taylor expansion, by means of the Eguchi equations, see Section \ref{distances.NS.geom.section}), maintaining the condition of the $(\nabla^{D_\Psi})^\nsdual$-convexity and $(\nabla^{D_\Psi})^\nsdual$-affinity of the local constraints, as well as their $(\gbold^{D_\Psi},\nabla^{D_\Psi},(\nabla^{D_\Psi})^\nsdual)$-orthogonality with respect to the local $\nabla^{D_\Psi}$-geodesic trajectory. This way the local user's inference, based on smooth reminimisation of an information distance (locally $D_\Psi$-optimal learning process) becomes equivalent with a free fall along $\nabla^{D_\Psi}$-geodesics. One can call it a \textit{local equivalence principle} of an ``information gravity''. The restriction of an arbitrary $D_\Psi$ and $\nabla^{D_\Psi}$ to $D_{1/2}$ and $\nabla^{1/2}$, as expressed by \eqref{one.form.nabla.onehalf.holonomy}, is caused by two reasons: requirement of showing explicit backwards compatibility with the Daubechies--Klauder path integrals, and also the structural restrictions of the GNS bundle.\footnote{Using Hasegawa [137] representation of a tangent space in terms of functions of density operator, we could generalise the use of the GNS bundle at least to the case of $\nabla^\gamma$ connections derived from $D_\gamma$. However, this would be restricted to the finite dimensional case. Moreover, we are interested here more in the search of an appropriate analytic setting for the general theory than in the explicit calculations of special finite dimensional cases.} In order to use other $\nabla^{D_\Psi}$ connections, we would have to systematically apply noncommutative Orlicz spaces, which is beyond the scope of this paper.

In addition, we replace an affine function $h(z(t))$ in the Daubechies--Klauder formula \eqref{Klauder.path.integral}, which is corresponding to a Killing hamiltonian vector field and is generated by a coherent state expectation value of a self-adjoint hamiltonian operator, by any smooth function on $\M(\N)$, understood as a hamiltonian function on a BLP manifold. This leads us to ask how one can relate local entropic and local hamiltonian dynamics in the histories context.

Combined with the causal inference Ansatz \eqref{global.hamiltonian.entropic.map}, the principle \eqref{entropic.time.evolution} leads to a \textit{global} evolution
\begin{equation}
\phi(t,s)=\PPP^D_{\Q(s)}\circ w^{\B,h}_t(\phi_0).
\label{two.times.global.causal.inference.instrument}
\end{equation}
While one can chose $t\in[r_0,r_1]$ and $s\in[r_1,r_2]$, $r_0,r_1,r_2\in\RR$, there is no obligation to do so. In general, $w^{\B,h}_t$ represents a causal evolution governed by the principle of a \textit{local conservation of absolute energy} (undestood as an element of the local space of quantitative effects, such as the self-adjoint observables in quantum mechanics), while $\PPP^D_{\Q(s)}$ represents an inferential evolution governed by the principle of a \textit{global growth of relative entropy} (understood as a function on the global space of states). It was first observed by K\k{e}pi\'{n}ski [192, 193] that each of those dynamical processes carries its own notion of time. Our work grew out from consideration of this duality, and \textit{a priori} independence of two associated notions of time, as the fundamental principle of physical dynamics. In the context of the present paper, as we discuss it below, we postulate that the `energetic' time of causality and the `entropic' time of inference are equal, but only \textit{infinitesimally}.\footnote{Due to incoherence between the standard use of the word `local' in physics and in mathematics, it is hard to propose any universally optimal terminology for distinguishing between different regimes. In this paper we use the terms: \textit{global} to refer to objects acting on all space $\M$; \textit{local} and (equivalently) \textit{infinitesimal} to refer to objects acting at $\phi\in\M$; \textit{nonlocal} to refer to objects acting in some neighbourhood of $\phi\in\M$ (maybe \textit{quasi-local} would be a better term). Within our setting, the local regime corresponds to a single user system, defined by the states-and-effects kinematics equipped with the causal-inferential dynamics, global regime corresponds to a multi-user framework, while nonlocality corresponds to the issues of construction of effective multi-user kinematics and dynamics, based on the choice of specific criteria of synchronisation between individual user's systems, at the expense of some individual properties being no longer maintained at the effective level. See Sections \ref{curvature.intro.section} and \ref{locally.quantum.information.relativity.section} for more discussion.} This forms the first principle of local information dynamics. More precisely, we consider the local causal dynamics, governed by $\ddd h_\B$,\footnote{The notation $\ddd h_{\{\cdot,\cdot\}_\B}$ would be completely precise, and symmetric with the notation $\ddd_{\nabla^{D_\Psi}}$, but also quite expensive visually.} and the local inferential dynamics, governed by $\ddd_{\nabla^{D_{\Psi}}}$, as \textit{two} independent fundamental dynamical processes that should be treated on the equal footing as components generating jointly an infinitesimal temporal evolution in a \textit{single} time. In other words, we postulate that the complete description of the local dynamics should be governed by a differential $1$-form
\begin{equation}
{\cal F}_{D_\Psi,h_\B}:=\ddd h_\B+\ddd_{\nabla^{D_\Psi}}.
\label{local.causal.inferential.form}
\end{equation}

However, the existing constructions of a quantum information manifold use different tangent-cotangent space structure than the quantum Poisson spaces, so, as a result, the addition operation in \eqref{local.causal.inferential.form} is precisely as meaningful, as is adding the element of the noncommutative Orlicz space to the element of a Banach Lie subalgebra $\B$ of $\N^\sa$ (or of $\N$, if one wants to use some nonstandard possibilities). The best solution to this problem would be to represent the action of $\B$ on the cotangent space, given by the noncommutative Orlicz space $L_\Upsilon(\N)$. Yet, even in the simple case, when $\B$ is a Lie algebra of a group of unitaries $\N^\uni$, a corresponding functional analytic study requires to prove the analogue of the Haagerup theorem [130] for the noncommutative Orlicz spaces, resulting in the $L_\Upsilon$-liouvillean (a generalisation of the Jak\v{s}i\'{c}--Pillet $L_{1/\gamma}$-liouvilleans [165]). This is beyond the scope of this paper. As a result, the natural functional analytic representation of the above differential geometric framework is not yet available, and we need to carry our investigations by means of a bit less canonical tools, as reviewed in Section \ref{qig.foundations.intro.section}. The analysis of local perturbations of standard (i.e., $L_{1/2}$-) liouvilleans associated with the GNS Hilbert bundle, carried in Section \ref{from.local.gauge.to.lagrangeans.section} and briefly reviewed in Section \ref{locally.perturbed.liouvilleans.intro.section}, is intended to be a warm-up investigation preceding the (currently unavailable) theory of local perturbations of $L_\Upsilon$-liouvilleans, associated with the tangent and cotangent spaces of quantum information manifolds based on quantum Br\`{e}gman distances. In this sense, the current paper can be considered as an investigation of the mathematical framework for the local Ansatz \eqref{local.causal.inferential.form}, as well as a family of related dynamical problems.\footnote{In future work, we will also consider a replacement of $\ddd h_\B$ by the connection $1$-form of the quantum relative free energy, as generated by the Fenchel--Legendre conjugate of a relative entropy.}

\subsection{Effective local quantum dynamics\label{effective.local.quantum.dynamics.section}}
Comparing the path integral propagator \eqref{RPK.propagator} with the local liouvillean propagator \eqref{liouv.prop}, we can see that both consist essentially of the subtraction of the local free fall along $\nabla^{1/2}$ geodesics from the local hamiltonian flow. This free fall corresponds to the local $D_{1/2}$-projection, which can be in turn interpreted as a continuous projective measurement. This leads us to revisit the problem of analytical implementation of the postulate that the \textit{local} geometric dynamics should be generated by the $1$-form \eqref{two.times.global.causal.inference.instrument}. In particular, applying the infinitesimal approximation $\phi=\omega+\delta\omega$ in the local liouvillean propagator \eqref{liouv.prop} results in the generator $\pi_\omega(\DF_\omega h)-\log(J_{\omega+\delta\omega}J_{\omega+\delta\omega,\omega})$ , which can be considered as an implementation of \eqref{local.causal.inferential.form}.

As a warm-up, consider the case $\N\iso\BH$, $\B\iso\BH^\sa$, $\M(\N)=\schatten_1(\H)^+_{1}$. Combining the insights of B\'{o}na [37, 38] and Grandy [119, 120], one can propose the following generalisation of the von Neumann equation,
\begin{equation}
\ii\frac{\dd}{\dd t}\rho(t)=
[\ddd h(\rho(t)),\rho(t)]-\frac{\partial}{\partial t}\rho(t),
\label{dissipative.generalisation.of.vonNeumann.eq}
\end{equation}
\begin{equation}
\frac{\partial}{\partial t}\rho(t)=\sum_{i\in I}P_i\frac{\partial}{\partial t}p_i(t)
\end{equation}
is determined by the nonhamiltonian change of probabilities $\{p_1(t),\ldots,p_n(t),\ldots\}$ that determine $\rho(t)$ by means of $\rho(t)=\sum_{i\in I}P_ip_i(t)$, given $\sum_{i\in I}P_i=\II$, $P_iP_j=\dirac_{ij}P_i=\dirac_{ij}P_j$, and $P_i\in\Proj(\BH)$ $\forall i\in I$. Grandy, following Jaynes [179, 172, 175], proposes to use maximum absolute relative entropy to construct the evolution $p(t)$. As compared to Jaynes' approach, we propose to replace the use of absolute entropy on probability densities by the use of relative quantum entropies on quantum states. The resulting geometrisation of a contribution $\frac{\partial}{\partial t}\rho(t)$ by means of a connection $1$-form $\ddd_{\nabla^{1/2}}(\phi(t))$ can be provided by a choice of a frame (ordered list of vector fields) $\xi(t):=(\xi_1(t),\ldots,\xi_n(t),\ldots)\in\H_{\rho(t)}$, such that $\ab{\xi_i(t)}^2=p_i(t)$ $\forall i\in I$, and evaluation
\begin{equation}
\label{d.gamma.two.sum.christo}
\end{equation}
Contracting the missing indices with $\xi^i(t)$, we derive the explicit representation of \eqref{one.form.nabla.onehalf.holonomy} as
\begin{equation}
\exp\left(\ii\int\dd t\,\xi_j(t)\sum_k({\Gamma^{\nabla^{1/2}}})^j_{\;\;ki}(\xi(t))P^k\xi^i(t)\right).
\end{equation}
Hence, \textit{if} one implements the principle \eqref{two.times.global.causal.inference.instrument} as a formal equation
\begin{equation}
\dot{\rho}(t)=-\ii\left[{\cal F}_{h,\nabla^{1/2}}(\rho(t)),\rho(t)\right],
\label{formal.effective.form.perturbation}
\end{equation}
\textit{then} the latter can be represented in the above situation as
\begin{equation}
\ii\frac{\dd}{\dd t}\rho=[\ddd h(\rho(t)),\rho(t)]-\sum_{k,i}({\Gamma^{\nabla^{1/2}}})^j_{\;\;ki}(\xi(t))P^k(P^i-J_{\rho(t)}P^iJ_{\rho(t)}),
\label{RPK.local.equation}
\end{equation}
where the modular conjugations $J_{\rho(t)}$ arise from a commutator of $\ddd_{\nabla^{1/2}}$ with $\rho(t)$.

Alternatively, taking into account our earlier observation that the standard transition unitary $V_{\omega,\phi}=J_{\phi}J_{\phi,\omega}$ is \textit{exactly} a parallel transport of $\nabla^{1/2}$, we can begin from the $\nabla^{1/2}$-parallel transport equation of a vector $v^a$ along the trajectory $\rho(t)$,
\begin{equation}
\frac{\dd}{\dd t}v^a(t)=-\sum_{b,c}({\Gamma^{\nabla^{1/2}}})^a_{\;\;bc}(\rho(t))v^b(t)\left(\frac{\dd}{\dd t}\rho(t)\right)^c.
\end{equation}
Substituting $v=\dot{\rho}(t)$, and integrating out, we get
\begin{equation}
\ii\frac{\dd}{\dd t}\rho(t)=
-\int_{-\infty}^t\dd t \sum_{b,c}({\Gamma^{\nabla^{1/2}}})^a_{\;\;bc}(\rho(t))\left(\frac{\dd}{\dd t}\rho(t)\right)^b\left(\frac{\dd}{\dd t}\rho(t)\right)^c.
\label{geodesic.free.fall}
\end{equation}
This equation, describes the equation of motion of the free fall along the $\nabla^{1/2}$-geodesic trajectory $\rho(t)$, when represented in the GNS Hilbert space bundle by means of $\H_{\rho(t)}\iso\schatten_2(\H)$.

An infinitesimal transformation $\rho\mapsto\rho+\der\rho$ can be decomposed as [137]
\begin{equation}
\der\rho:=\widetilde{\der}\rho+[\rho,W]=\sum_{i=1}^n\left(\frac{\partial\rho(\theta)}{\partial\theta^i}+[\rho,W_i]\right)\ddd\theta^i,
\label{orthogonal.decomp.of.rho}
\end{equation}
where $\widetilde{\der}\rho=\sum_{i=1}^n\frac{\partial\rho(\theta)}{\partial\theta^i}\ddd\theta^i$ is defined by $[\widetilde{\der}\rho,\rho]=0$, and $W=\sum_{i=1}^nW_i\ddd\theta^i$ is an antiself-adjoint operator (hence, $k_i^*=k_i:=\ii W_i$). The mappings $\der$, $\widetilde{\der}$ and $[\,\cdot\,,W]$ are derivations on $\BH$. This determines a decomposition of tangent space at $\rho$ into the direct product of the corresponding subspaces. An explicit representation of the tangent space in terms of $\schatten_2(\H)$ space by means of finite dimensional coordinate parametrisation $\RR^n\supsetneq\Theta\ni\theta\mapsto\rho(\theta)\in\schatten_1(\H)^+$ reads [137]
\begin{equation}
\T_\rho\ell_{1/2}(u)=\sum_{i=1}^nu^i\left(\sqrt{\rho}\frac{\partial\rho}{\partial\theta^i}+2[\sqrt{\rho},W_i]\right).
\end{equation}
As a result of these considerations, if we interpret the principle \eqref{two.times.global.causal.inference.instrument} as a statement that the effective local dynamics is generated by the sum of vectors $\dot{\rho}(t)$ arising independently from the hamiltonian flow and the geodesic free fall \eqref{geodesic.free.fall}, then we should use the equation
\begin{align}
\ii\frac{\dd}{\dd t}\rho(t)&=[\ddd h(\rho(t)),\rho(t)]\label{RPK.effective}\\
&-\int_{-\infty}^t\dd t \sum_{b,c}({\Gamma^{\nabla^{1/2}}})^a_{\;\;bc}(\rho(t))\left(\sum_{i}u^i\left(\sqrt{\rho}\frac{\partial\rho}{\partial\theta^i}+2\left[\sqrt{\rho},k_i\right]\right)\right)^b\left(\sum_{i}u^i\left(\sqrt{\rho}\frac{\partial\rho}{\partial\theta^i}+2\left[\sqrt{\rho},k_i\right]\right)\right)^c.\nonumber
\end{align}
This equation describes the \textit{effective local dynamics}, including causality and inference effects on the equal footing (thus, \textit{paralelly processing} them). Comparing the nonhamiltonian parts of the equations  \eqref{RPK.local.equation} and \eqref{RPK.effective}, we see that the equation \eqref{RPK.local.equation} can be at best some sort of approximation of \eqref{RPK.effective}. We interpret it as an indication of the weakness of the implementation \eqref{formal.effective.form.perturbation}, as compared with \eqref{RPK.effective}.

The interpretation of \eqref{local.causal.inferential.form} an infinitesimal analogue (but not a generator) of the ``entropic inference following causal Poisson evolution'' global W$^*$-geometric dynamics \eqref{global.hamiltonian.entropic.map}, and the fact that the latter allows to reconstruct CPTP maps as a special case [215, 214], suggests to intepret \eqref{RPK.effective} as a nonlinear geometric analogue of the Lindblad--Gorini--Kossakowski--Sudarshan equation [230, 117].
\subsection{Curvature measures desynchronisation in the multi-user inference\label{curvature.intro.section}}
In this Section we will consider the problem of the effective \textit{nonlocal} quantum information dynamics. As opposed to effective local dynamics, which provides the infinitesimal description of causality and inference from the perspective of a single user (thus, allowing for an immediate subjective bayesian interpretation), nonlocality is intended to describe the multi-user (intersubjective) conventions of relating causal and inferential dynamics of individual users. In our opinion the geometric space is an emergent property of the specific intersubjective conventions of information (causal and inferential) dynamics, which is, in turn, relative to a specific choice of (class of) users. In this text we are focused on an analysis of a specific example, rooted in quantum information geometry. More foundational discussion will be postponed to another paper [214].

In principle, given two or more different users, each with his/her own method of providing inferences and evaluating causal evolution, nothing can be said about how their information dynamics is related. If only the inferential aspect of information dynamics is taken into account, then such situation can be understood as incommensurability of inferences provided by different subjective bayesians with their arbitrary choices of respective priors and of methods of updating. In order to relate these different instances of local information dynamics, one needs to introduce some method of translation between the respective evolutions, as well as their initial assumptions. Each such method represents a specific intersubjective convention, which allows to translate between individual instances of information dynamics at the expense of constraining its possible forms to such that are subjectible to a given convention. In particular, for inferential part of the information dynamics, there should be a way of identification of a given state of information as `the same' state for all users under consideration. Note that there is no need for a such identification being made globally for all \textit{possible} users: it is sufficient if one can do it for different sets of users that are under the scope of interest.

The general setting for these considerations can be defined as follows. Given an abstract set $\M$ of users, with a single user represented as a point $\phi\in\M$ equipped with a vector space of \textit{local states} $V(\phi)$ and a Banach dual vector space $V^\dual(\phi)$ of \textit{local effects} (one can think of them in terms somewhat similar to [232], but the duality does not have to be carried by Banach space structure, but e.g. by convenient vector space structure).\footnote{By the reasons discussed above and below, we postulate that \textit{if} $\M$ is given by $\M(\N)$, \textit{then} $V(\phi)$ should be specified as $L_{\Upsilon(\phi)}(\N)$. However, in order to distinguish the conceptual and the representational aspects of our approach, we state it in more general terms.} In order to model causality and inference, each user can chose locally his/her own `causal' Banach Lie algebra $\B$ acting on $V^\dual(\phi)$, as well as its own `inferential' \textit{Br\`{e}gman functional} $\tilde{D}_\Psi$ on $V(\phi)$, the latter determined by the duality between $V(\phi)$ and $V^\dual(\phi)$ and the choice of a function $\Psi:V(\phi)\ra\RR$. Given a set $U(\phi)\subseteq\M$ of users, such that $\phi\in U(\phi)$, a choice of a function $\ell_\phi:U(\phi)\ra V(\phi)$ allows to use the Br\`{e}gman \textit{functional} $\tilde{D}_\Psi:V(\phi)\times V(\phi)\ra[0,\infty]$ in order to construct the Br\`{e}gman \textit{distance} $D_\Psi:U(\phi)\times U(\phi)\ra[0,\infty]$ by means of
\begin{equation}
\end{equation}
Thus, the choice of the function $\ell_\phi$ defines how the local user interprets `inferentially' the subset $U(\phi)$ of the set $\M$, while the choice of a hamiltonian function $h(\phi)$ defines how he/she interprets it `causally'. The forms $\ddd h_\B(\phi)$ and $\ddd_{\nabla^{D_\Psi}}(\phi)$, constructed as elements of $V^\dual(\phi)$, are encoding the corresponding infinitesimal dynamics.\footnote{On the level of implementation, it is sufficient to model $V(\phi)$ and $V^\dual(\phi)$ as convenient vector spaces in order to guarantee that the infinitesimal calculus is well defined.} The translation between different users in the set $\Q\subseteq\M$ requires, within this model, to specify relationship between $\ell_\phi$ and $\ell_\psi$, as well as $h(\phi)$ and $h(\psi)$, for all elements $\phi,\psi\in\Q$.

The \textit{local} state-effect kinematics is completely described using the pair $(V(\phi),V^\dual(\phi))$ at a given point $\phi\in\M$, while the resulting \textit{local} causal-inferential dynamics is generated by \eqref{local.causal.inferential.form}. This leads to a question to what extent, and at what expense, this dynamics can be extended to a larger \textit{nonlocal} area of $\M$, for example allowing to interconnect the dual pairs of vector spaces of different users by means of the (not necessarily global) sheaf of tangent and cotangent spaces. On the conceptual level, this corresponds to the question how the local (individual) state-and-effect dual pairs of different users $\phi\in\M$ can be mutually related, how their respective causal and inferential local dynamics can be synchronised, and what is the price to pay for it? For example, even if all users $(\phi,V(\phi),V^\dual(\phi))$ agree to use the same generating objects for their respective local system of causality (e.g., a Banach Lie algebra $\B$) and local system of inference (e.g., a discrimination function $\Psi$), their individual choices of functions $h_\B(\phi)$ and $\ell_\phi$ may be not extendible to a set $\M$, resulting in sheaves of different effective local dynamics \eqref{local.causal.inferential.form}. The problem of conditions for integrability of \eqref{local.causal.inferential.form} is thus directly related to the issue of  nonlocal (multi-user) synchronisation of local systems of causality and inference about local states and effects.

%\footnote{Despite using the suggestive notation $\ddd_{\nabla^{D_\Psi}}(\phi)$, the intrinsic consistency of the above interpretation requires that this object should be determined completely by $\tilde{D}_\Psi$ and the linear structure of $V(\phi)$ and $V(\phi)^\star$, so it should carry no information about the embeddings $(\ell_{V},\ell_{V^\star})$. This follows from the equivalence of the condition $x=\PPP^{\tilde{D}_\Psi}_C(y)$ with the conditions \cite{Bauschke:Borwein:Combettes:2001, Bauschke:2003,Alber:2007}
%\duality{z-x,\DG_y\Psi-\DG_x\Psi}_{V(\phi)\times V(\phi)^\star}\leq0\;\;\forall z\in C
%\widetilde{D}_\Psi(z,x)+\widetilde{D}_\Psi(x,y)&\leq \widetilde{D}_\Psi(z,y)\;\;\forall z\in C.
%Because every Banach space can be equipped with a tangent space modeled on itself, we can consider the infinitesimal change $y+\ddd y$ and ask how this influences the change of the solution $\PPP^{\tilde{D}_\Psi}_C(y)$. In \cite{Resmerita:2004} it was shown that this change is continuous under some additional assumptions on $\Psi$. We avoid the lack of more refined characterisation by requiring Br\`{e}gman functionals to be}

In general, the convention defining emergent multi-user inference can be arbitrarily different from the convention defining the emergent multi-user causality, and they both can differ from any of the local instances of inference and causality that are amalgamated into the emergent structure. As a result, the emergence of a nonlocal spatio-temporal causal-inferential dynamics is be provided at the expense of its departure from the local causal-inferential dynamics. In the context of the structures analysed in this paper, it is represented by the appearance of the system of local entropic priors associated with a specific integral line of a vector field on a model $\M$. (From the closely related point of view, one can notice that the postulate of Section \ref{local.infogeometry.histories.intro.section} identifies the local inferential time with local causal time for each individual user \textit{separately}, but it does not say anything about mutual relationships of those time parameters for different users. In principle, one could also study theories for which the local causal and inferential time of each user would \textit{not} be identified, yet the multi-user synchronisation of those two temporal structures would be considered. We find this perspective very attractive, but it is beyond the scope of the current paper.)

On the technical level, we observe that the Fubini--Study riemannian metric, used in the regularising term in the Daubechies--Klauder and the Anastopoulos--Savvidou approaches, can be replaced by the second order Taylor expansion $\gbold^{1/2}$ of a quantum relative entropy $D_{1/2}$. This leads us to postulate to use a quantum entropic prior \textit{localised} to a neighbourhood of a given state as a general geometric form of the regulariser (see Section \ref{entropic.priors.section} for a discussion of the notion of an entropic prior in the commutative case). Integration of a local entropic prior along a given trajectory on a state manifold constructs a regularised weight for this trajectory.

On the conceptual level, we interpret the appearance of local entropic prior as a measure of (impossibility of) synchronisation of the causal-inferential dynamics of the subsequent local users at a given spatio-temporal trajectory. More specifically, the choice of a \textit{single} nonlocal (multi-agent) time trajectory sets up the nonlocal vector field, corresponding to a specific system of synchronisation (=~nonlocal/noninertial observation frame) for the local forms ${\cal F}_{D_\Psi,h_\B}$. This choice is arbitrary, but with each such choice, the passage from point to point on the corresponding nonlocal trajectory of a \textit{single spatialised time} adds an additional term to an effective dynamics \textit{along this trajectory}. Only some specific conventions of the multi-user inference avoid the path-dependence of the synchronisation of the inference: in the example that we study here they are given by such models $(\M(\N),D_\Psi)$ for which the scalar curvature $\kappa(\nabla^{\gbold^\Psi})$ is constant over $\M(\N)$.

One can wonder why this phenomenon of ``breaking of symmetry'' between causality and inference results in an entropic (hence inferential), as opposed to a hamiltonian (hence causal) contribution. If all users along the trajectory (as well as in the neighbourhood of this trajectory, in order to have a situation that is more canonical than that of a $1$-dimensional manifold) agree to have the same choice of $\B$ and $\Psi$, then in principle they should share the same nontrivial inferential and causal geometry, so there is yet no reason for ``breaking of the symmetry''. However, the Carath\'{e}odory--Jacobi--Lie theorem (which generalises the Darboux theorem) implies that for any symplectic manifold with a hamiltonian function $h$, the function $h$ is a conserved quantity in an open neighbourhood for any $\phi$ for which $\ddd h(\phi)\neq0$. An analogous statement is not true in riemannian geometry, as measured by the curvature. The generalisation from symplectic to Poisson geometry and from riemannian to Norden--Sen geometry does not change qualitatively this difference. Thus, the passage from infinitesimal/local (jets) to neighbourhood/nonlocal (germs) is trivial for causality, but not for inference. So, while it is possible to nonlocally synchronise the local causal structure along a trajectory of users (whenever it is modelled by the Poisson geometry), it is impossible to do this with the local inferential structure (whenever it is modelled by the Norden--Sen geometry). It is a very interesting phenomenon, which may be interpreted by saying that while the global synchronisation of local systems of causality is a trivial consequence of choice of the same system of locally conserved quantities for each user, the global synchronisation of local systems of inference is impossible, with the system of local entropic priors measuring the scale of this impossibility for each of the local users along the trajectory. The local entropic priors could be in principle replaced by any other local priors. Each of such choices represents a different \textit{initial system of assumptions} that are used by local users for the purposes to measure the desynchronisation between their local/jet inferences, as provided by means of $\tilde{D}_\Psi$ on $V(\phi)$, and their nonlocal/germ inferences, as provided by means of $D_\Psi$ on $\M$. The local priors are just measures on the local neighbourhoods in the model $\M$, and they have to be \textit{postulated independently} from the choice of ${\cal F}_{D_\Psi,h_\B}$.\footnote{While a detailed discussion of this phenomenon is beyond the scope of the current paper, we want to note that a somewhat similar situation (roughly speaking, a necessity of using additional geometric structures on the pre-state spaces in order to select a specific sheaf of states) was discovered by Bostelmann [42] in the Haag--Ojima approach [129] to construction of germs of algebraic states corresponding to a choice of a specific predictive dynamical theory in the algebraic approach.}

Thus, while the equation ${\cal F}_{D_\Psi,h_\B}=0$ (whenever valid) can be understood as a causal-inferential analogue of Einstein's version of Newton's first law of motion (\textit{for a single user}, from his/her own perspective), the local priors are somewhat similar to the second law of thermodynamics or to the universality of the gravitation: the globalisation of inferences is provided at the expense of inevitability of making those inferences dependent on additional arbitrary assumptions, that are in principle different for each user, and \textit{are nonobservable in the infinitesimal causal-inferential reference frame}. The shadow of dependence of the multi-userr inferences, as well as of the effective \textit{nonlocal} causal-inferential dynamics, on arbitrary additional assumptions (\textit{attributed} to other users by the user residing at the end point of the integrated trajectory) is a price paid for a requirement of global spatialisation of the inferential dynamics. The lack of the similar effect in the case of causal dynamics is in essence a result of a local linearity of the Banach Lie--Poisson spaces. If other mathematical structure would be chosen to model causality, the similar dialectics may occur. For example, one could model causal dynamics by means of extremum of quantum relative free energies, defined as the Legendre--Fenchel conjugates of quantum information distances. The nonlocal trivialisation of the causal structure by means of the CJL theorem would not be applicable in such case. As a result, the dual, local relative free energy priors, should be also included, as an additional regularising term, into integration giving rise to an effective dynamics. In general, we propose to consider the priors on the model $\M$ as an information theoretic analogue of the notion of mass, so the local prior at the neighbourhood $U(\phi)$ of $\phi\in\M$ can be interpreted as an information theoretic analogue of the mass of the user $\phi$ distributed over the set $U(\phi)$ of users. The special case in which the integration against the local relative free energy priors would cancel the contribution arising from the local entropic priors can be then considered to be an analogue of Einstein's principle of equivalence of gravitational (inferential) and inertial (causal) mass.

As observed in [235, 3, 365], if the riemannian geometries of the phase space used for the Wiener measure regularisation have nonconstant scalar curvature, then the weighting of the phase space paths is nonuniform, corresponding to the phase space point dependency of the zero-point energy. On the other hand, in a completely different context, Jaynes has argued [166, 171, 176] that the zero-point energy should be interpreted not as an ontic feature of the system, but as observer's measure of uncertainty regarding his/her own prediction of the value of energy, as based on his/her own \textit{prior information}. Quite independently from these considerations, Rodr\'{\i}guez has developed the theory of entropic priors [288] interpreting them as the \cytat{statistical representation of the vacuum of information in a given hypothesis space} [289]. Our formulation recombines the above insights, integrating them into a single statement: zero-point energy's point-dependence is a manifestation of a dependence of nonlocal integrability of local inferential structure on the local geometry of user's prior knowledge (or, equivalently, prior ignorance).\footnote{This gives also some justice to an otherwise quite cryptic remark of Jaynes on the book [277] on functional integration: \cytat{A useful start on understanding of these phenomena, but still lacking any coherent theoretical basis -- which we think is supplied only by the principle of maximum entropy as a method of reasoning.} [178].}

However, while our proposal is well founded on the geometric side, and has also an interesting information theoretic interpretation, it has to be regarded as heuristic from the perspective of stochastic process based foundations for path integration. More specifically, the Daubechies--Klauder formulation relies on the interpretation of $\int\dd t p\dot{q}=\int p \dd q$ as the Stratonovich integral (so $\dd q$ is considered as a Stratonovich differential $\dd_{\mathrm{S}}q$), and on the interpretation of (roughly) $\int\mathcal{D}p\mathcal{D}q\ee^{-\frac{1}{2\upsilon}\int\dd t(\dot{p}^2+\dot{q}^2)}$ as the pinned Wiener measure (see [35] for a systematic mathematical treatment of these objects in terms of the Berezin--Toeplitz operators). The generalisation to a wide class of connections $\nabla^{D_\Psi}$ and local entropic priors (even if kept at the second order riemannian level of $\gbold^D$) asks for a systematic development of a technique of stochastic integration of random walks $\mathbf{X}$ on $\RR^n$ associated with the Brownian motions on smooth manifolds $\M$, $\dim\M=n$, that could systematically address the  functional integration of the above geometric structures beyond the level of heuristic treatment that is standard for physicists. In particular, following [149, 190], let us consider the system of stochastic differential equations
\begin{align}
\dd_\mathrm{S}\mathbf{X}&=\mathbf{e}\,\dd_\mathrm{S}\mathbf{B},\\
\dd_\mathrm{S}\mathbf{e}&=H_\mathbf{e}\dd_\mathrm{S}\mathbf{X},
\end{align}
where $\dd_\mathrm{S}$ are Stratonovich differentials, $\mathbf{B}$ is an euclidean brownian motion on $\RR^n$, frame $\mathbf{e}$ is a map from $\RR^n$ to the tangent space of $\M$, and $H$ is a horizontal lift of a tangent space at $\mathbf{X}$ to the tangent space at $\mathbf{e}$, dependent on the choice of an affine connection $\nabla$ on $\M$. If this connection is not Levi-Civita, then the process $\mathbf{X}$ will be not markovian. Furthermore, a suitable riemannian metric reproducing the Laplace--Beltrami operator can be uniquely constructed by an appropriate choice of an elliptic diffusion. Hence, the exact mathematical foundation for our generalisation of the Daubechies--Klauder formula is possible at least for the second order Taylor expansion of the entropic prior and for different connections $\nabla^{D_\Psi}$. While we consider the task of a systematic treatment of this topic to be of high importance, it will be left beyond the scope of this paper.

The issue of renormalisation cannot be omitted in any foundational discussion of local quantum dynamics. Section \ref{information.theoretic.renormalisation} is dedicated to the study how the tools of quantum information geometry can be used in order to deal with the tasks of renormalisation. In Sections \ref{section.JM.source.theory} and \ref{Favretti.section} we briefly review the Jaynes--Mitchell source theory [241, 173, 177, 118, 120] and its geometric generalisation by Favretti [100], respectively. This approach provides a geometric implementation of the idea of renormalisation of dynamics by reduction of dimensionality of the model by fixing the control parameter, which is specified as a constraint on the space of information states (as opposed to the space of functions or operators). We observe that the Jaynes--Mitchell--Favretti approach is canonically related to the use of Br\`{e}gman distances, and that it can be used to provide a \textit{local} description of entropic information dynamics and multiparameter nonlinear quantum control problems on an arbitrary quantum manifold. The key insight of this approach can be summarised as: renormalisation of the action of control parameters (sources) leads to departure of the geometry of a model $\M$ from a dually flat one. Hence, the appearance of nonzero curvature is an indicator of a nontrivial constraints for information dynamics. Moreover, the nonconstancy of this curvature indicated local dependence of these constraints. From the perspective of our approach to the Daubechies--Klauder path integrals, we postulate that the renormalised description of dynamics should use renormalised riemannian metric $\tilde{\gbold}$ instead of $\gbold^{D}$ in the regulariser. This corresponds to replacement of a ``vacuum of information'' by the ``vacuum of information storing the shadow of the knowledge about the sources that were renormalised out''.

We also introduce another type of information geometric renormalisation of inferential dynamics of quantum states, which describes situations where none of  specific control (covariate) parameter is fixed, but the quantum model is subjected to the action of completely positive maps. This procedure is based on the use of $D_\fff$ distances as well as associated \textit{contraction coefficients}, introduced by Ruskai et al [72, 64, 293, 227]. For an alternative (and essentially more developed) approach to renormalisation based on $D_\fff$ on $\N_\star^+$, see [30, 31, 32] (c.f. [79] for a pedagogical introduction).
\subsection{Locally quantum multi-user information relativity\label{locally.quantum.information.relativity.section}}
The discussion in this paper is aimed at the construction of the framework of the multi-user (intersubjective) information relativity with emergent spaces. While most of discussion is kept in the framework build upon the W$^*$-algebras, it is more a useful testing ground (and a verifying constraint for backwards compatibility), then a desired property of the framework. In other words, our intention is to get rid of the \textit{Postulate 1} of Section \ref{qig.foundations.intro.section} by a suitable reformulation of the \textit{Postulate 5} and \textit{6}. We propose to consider the following tentative structure:
\begin{enumerate}
\item[1)] Consider arbitrary set $\M$ of users.
\item[2)] each user $\phi\in\M$ is equipped with his/her own vector space $V(\phi)$ containing his/her descripton of `states' (`configurations', `preparations', `inputs') and a dual vector space $V^\dual(\phi)$, containing his/her description of `effects' (`registrations', `results', `outputs').
\item[3)] User's notion of inference (respectively, causality) is modelled by a choice of set of of endomorphisms of $V(\phi)$ (respectively, $V^\dual(\phi)$).
\item[4)] In particular, inferences on $V(\phi)$ can be provided by means of the Br\`{e}gman functional $\tilde{D}_{\Psi,\phi}:V(\phi)\times V(\phi)\ra[0,\infty]$, while the causality on $V^\dual(\phi)$ can be provided either by the Legendre--Fenchel conjugate of $\tilde{D}_{\Psi,\phi}$, or by a representation of some Lie algebra.
\item[5)] More specifically, if one wants to do statistical inference, it is necessary to make clear how the behaviour of finite data sets is associated with the specific idealisations used in the theoretic framework, as represented by the `ideal' theoretical states and effects. In order to assert such relation, it is necessary to admit some statistical tools that are representing the control over a `convergence to an ideal form of a data set', for any given nonideal form of a real data set. Large number estimation and asymptotic estimation are two typical tools (on the geometric level, they correspond, respectively to relative entropy maximisation, and the local linearity). If a user introduces a discrimination function (information potential, absolute entropy) $\Psi_\phi:V(\phi)\ra\,]-\infty,+\infty]$, then he/she is able to quantify the rates of convergence of sequences of data. The key property of the Br\`{e}gman functional is that it allows for a generalised pythagorean theorem \eqref{generalised.pythagore.Bregman}, which is a nonlinear generalisation of the fundamental property of euclidean and Hilbert spaces. Yet, it is doing it without necessity of assuming that $V(\phi)$ is normed, or even metrisable. As a result, user's local inferences on $V(\phi)$ based on entropic projections $\PPP^{\tilde{D}_{\Psi,\phi}}$ allow to decompose the information distance to an `ideal inference' ($\tilde{D}_{\Psi,\phi}(x,y)$) into a sum of a distance to an effective solution (satisfying given constraints and minimising the distance) and an uncertainty within the constrained space. (By the Legendre--Fenchel duality, completely parallel considerations are applicable to causality on $V^\dual(\phi)$ determined by the relative free energy $\tilde{D}_{\Psi^\lfdual,\phi}:V^\dual(\phi)\times V^\dual(\phi)\ra[0,\infty]$.)
\item[6)] The geometric structures on the set $\M$ are introduced as quantitative means of relating (synchronising) inferences and causality of different users.
\item[7)] Each user $\phi$ provides his/her own mappings from $\M$ into $V(\phi)$, given by bijective embeddings $\ell_\phi:U(\phi)\ra V(\phi)$, where $U(\phi)\subseteq\M$ is the subset of users in $\M$ that a user $\phi$ considers as representable in terms of his/her `configuration' space $V(\phi)$.
\item[8)] Using the embeddings $\ell_\phi$, each user can relate his/her individual inferences on $V(\phi)$ with other users using (a part of) the same set $U$. In particular, the relative Br\`{e}gman entropy $D_\Psi$ on $\M$ is induced by
\begin{equation}
\tilde{D}_{\Psi,\phi}(\ell_\phi(\omega_1),\ell_\phi(\omega_2))=D_{\Psi,U(\phi)\cap U(\psi)}(\omega_1,\omega_2)=\tilde{D}_{\Psi,\psi}(\ell_\psi(\omega_1),\ell_\psi(\omega_2))\;\;\forall \omega_1,\omega_2\in U(\phi)\cap U(\psi).
\end{equation}
This provides the means to relate the \textit{large number} inferential dynamics of different users (such as in Sanov's theorem).
\item[8)] The local smooth manifold structure induced by $D_\Psi$ on $\M$ provides the means to relate \textit{asymptotic} inferential dynamics of different users. In particular, the equivalence of local geodesic projection and local $D_\Psi$ projections can be understood as a method of locally linear synchronisation of inferences of different users. In such case the curvature of the manifold $\M$ measures impossibility of ideal synchronisation of inferences between different users. The relationships with the notions of causality, effective dynamics, and renormalisation were discussed in the previous Section. They are dependent only on the notion of Br\`{e}gman distance on an arbitrary set $\M$, and Eguchi equations on an arbirary smooth manifold $\M$, hence they hold in general, without assuming that $\M$ is a subset of $\N^+_\star$ for some W$^*$-algebra $\N$.
\item[9)] The reconstruction of the special case, when $\M$ is equal to the set $\M(\N)$ of states over a globally defined W$^*$-algebra $\N$ is an open problem. Our conjecture is that quantum mechanics can be characterised as a set $\M$ equipped with an induced structure of the riemannian manifold $(\M,\gbold^{D_\Psi})$ and a Poisson manifold such that the set of extremal points of the convex hull of $\M$ admits an induced metric and induced symplectic structure that determine a K\"{a}hler manifold satisfying the standard properties of the quantum mechanical K\"{a}hler manifolds.
\end{enumerate}
In principle, the above scheme is applicable to a wide class of postquantum information theoretical settings, such as general probabilistic theories. In what follows, having in mind the possible applications in nonequilibrium quantum statistical mechanics, we will focus on its implementation in the context of W$^*$-algebras and the associated functional analytic spaces.

While \eqref{SS} considers only the case of local $\nabla^{1/2}$/$D_{1/2}$-projections, [139, 211, 215] consider only the case of global $D_0$-projections. Yet, we think that the natural geometric objects for construction of local  and nonlocal quantum kinematics and dynamics are arbitrary Banach Lie algebras (to describe causality) and arbitrary quantum Br\`{e}gman distances (to describe inference), connected together via dual pairs of noncommutative Orlicz spaces. More specifically, following a discussion in Section \ref{local.infogeometry.histories.intro.section} we think that the problems considered in this paper should be readdressed in a more general  foundational framework, based on the following principles:
\begin{enumerate}
\item[1)] The use of GNS Hilbert bundle should be replaced by a suitable bundle of noncommutative Orlicz spaces $L_{\Upsilon(\phi)}(\N)$ playing the role of tangent spaces $\T_\phi\M(\N)$, understood as the spaces of local `configurations' (e.g. $\phi(\theta)\mapsto\theta\mapsto\left(\frac{\partial}{\partial\theta_i}\right)$), with their Banach duals $L_{\Upsilon(\phi)}(\N)^\star$ playing the role of cotangent spaces $\T^\ct_\phi\M(\N)$, understood as the spaces of local `effects' ($f(\phi)\mapsto\ddd f\mapsto(\ddd f_i)$). Somewhat similar ideas were considered earlier in [341, 233], but only in a global context, restricted to a single `tangent' and `cotangent' space.
\item[2)] There should be provided a canonical construction of a quantum Br\`{e}gman distance $D_\Psi$ associated with a Banach dual pair $(L_{\Upsilon(\phi)}(\N),L_{\Upsilon(\phi)}(\N)^\star)$, a function $\Psi:L_{\Upsilon(\phi)}(\N)\ra\RR$, and a family of embeddings $\ell_\phi:\N^+_\star\ra L_{\Upsilon(\phi)}(\N)$, such that the second order G\^{a}teaux derivative of $D_\Psi$ determines a map
\begin{equation}
\gbold_\phi^{D_\Psi}:L_{\Upsilon(\phi)}(\N)\times L_{\Upsilon(\phi)}(\N)^\star\ra[0,\infty],
\end{equation}
while the third order G\^{a}teaux derivatives determine the connections $\nabla^{D_\Psi}$ and $(\nabla^{D_\Psi})^\nsdual$, with the respective parallel transports equal to the isometric transition operators
\begin{align}
\transport^{\nabla^{D_\Psi}}_{\phi,\omega}&:\T_\phi\M(\N)\ra\T_\omega\M(\N),\\
\transport^{(\nabla^{D_\Psi})^\nsdual}_{\phi,\omega}&:\T^\ct_\phi\M(\N)\ra\T^\ct_\omega\M(\N),
\end{align}
and satisfying the generalised Norden--Sen duality
\begin{equation}
\gbold^{D_\Psi}_\phi(\transport^{\nabla^{D_\Psi}}_{\phi,\omega}(x),\transport^{(\nabla^{D_\Psi})^\nsdual}_{\phi,\omega}(y))=\gbold^{D_\Psi}_\omega(x,y).
\end{equation}
This is intended to implement the principle discussed in Section \ref{Orlicz.Br\`{e}gman.section}: the local structure of an information manifold should implement the local equivalence of an entropic projection (user's inference) and a  geodesic free fall (an absence of geometric `gravity').
\item[3)] Lie algebras $\glie$ and Banach Lie algebras $\B$ should be represented on $L_{\Upsilon(\phi)}(\N)^\star$, the latter giving rise to a Banach Lie--Poisson manifold structure on $\M(\N)$ determined by the action of $\B_\star$ on $L_{\Upsilon(\phi)}(\N)$.
\item[4)] As a result, both $\ddd_{\nabla^{D_\Psi}}(\phi)$ and $\ddd h_\B(\phi)$ become the elements of the same operator space $L_{\Upsilon(\phi)}(\N)^\star$, so the local effective dynamics can be formulated in terms of a $1$-form ${\cal F}=\ddd h_\B+\ddd_{\nabla^{D_\Psi}}$, treating local causality and local inference on an equal footing. Note that, in face of the discussion in Sections \ref{locally.perturbed.liouvilleans.intro.section}, \ref{local.infogeometry.histories.intro.section}, and \ref{Orlicz.Br\`{e}gman.section}, the condition ${\cal F}=0$ can be understood as an information-theoretic analogue of Einstein's version of Newton's first law of motion: a rest in a local causal-inference frame (provided by user's choice of generators $\B$ for causal dynamics in a cotangent space of effects, and a discrimination function $\Psi$ for inferential dynamics in a tangent space of states).
\item[5)] The local tangent BLP hamiltonian vector field, generating a local W$^*$-dynamical system associated with a local fiber of a GNS Hilbert bundle, should be represented in terms of $L_\Upsilon$-liouvillean, acting on the $L_{\Upsilon(\phi)}(\N)$ tangent space. The incorporation of the contribution of the local free fall along $\nabla^{D_\Psi}$-connection geodesics should be provided by the perturbation of this $L_\Upsilon$-liouvillean, defined in an analogy with the Jak\v{s}i\'{c}--Pillet $L_{1/\gamma}$-liouvillean [165].
\item[6)] The local entropic prior should be constructed using $D$ and $\gbold^D$ representing the \textit{effective geometry} of a model $\M(\N)$, after the renormalisation of all contributions from control sources. One can interpret the local entropic prior at a point $\phi\in\M$ as an information theoretic analogue of the (inferential) mass of the user at $\phi$. This interpretation gives a particularly neat meaning to the observation [235, 3], discussed in Section \ref{local.quantum.dynamics.intro.section}, that the regularising term in the Daubechies--Klauder formula leads to a point-dependence of a zero-point energy if{}f the curvature of the Fubini--Study metric is nonconstant. From our point of view, it means: the curvature $\kappa^{\nabla^{D}}$ of a quantum model is a measure of desynchronisation of the ideal multi-user inference (as given locally by $\nabla^{D_\Psi}$-geodesics, or, equivalently, $\ddd_{\nabla^{D_\Psi}}$), reflected in the influence of a local mass on the local zero-point energy. The case of information model with a constant curvature is an inferential analogue of the Carath\'{e}odory--Jacobi--Lie theorem for symplectic manifold, allowing for a global trivial synchronisation of local causality systems.
\item[7)] Given these constructions, the dynamical Ansatz \eqref{SS} should be generalised by means of the replacement of the GNS Hilbert bundle by a corresponding dual bundle of noncommutative Orlicz spaces.
\item[8)] In principle, every quantum Br\`{e}gman functional $\tilde{D}_\Psi$ on $L_{\Upsilon(\phi)}(\N)$ determines the Legendre--Fenchel conjugate functional $\tilde{D}^\lfdual_{\Psi}$ on $L_{\Upsilon(\phi)}(\N)$, which can be naturally interpreted as a relative free energy. If one would replace the use of the (nonlocally trivially synchronisable) $1$-form $\ddd h_\B$ by the form $\ddd_{\nabla^{D^\lfdual_\Psi}}$, then it would be necessary to introduce a corresponding local entropic prior (causal mass), measuring the influence of desynchronisation of local systems of causality, along the spatial trajectory on $\M$, on the effective quantitative nonlocal evolution. The postulate of the local cancellation of effects of those two (inferential and causal) priors, when integrated together, would be an information theoretic analogue of the postulate of equality of gravitational and inertial mass.
\item[9)] In the JMF source theoretic approach the local departure of riemannian metric from the hessian geometry is described by the equation \eqref{JMF.renormalisation.of.metric}. Applying the change of geometry $\gbold^{D_\Psi}\mapsto\tilde{\gbold}$ to a definition of a prior $\exp\left(-k\frac{\epsilon^2}{2}\int_\pathgamma\dd t\,\gbold_{ab}(\phi)\dot{\phi}^a\dot{\phi}^b\right)\sqrt{\det(\gbold(\phi))}$, we can see that the change of the scalar curvature $\kappa(\phi)$ of the model is reflected in the redefinition of the local prior (information mass) $P$. This leads to a more general problem. Given any neighbourhood $U\subseteq\M$ of a user $\phi\in\M$, one can ask how to determine the departure of a prior $P$ on $U$ (from the originally postulated one) that is caused by the changes of curvature $\kappa$ of a model $\M$ associated with a given system of inference, when the transformation of geometry of a model $\M$ (e.g., due to the Jaynes--Mitchell renormalisation) is considered. The analogy with general relativity that we were pursuing in this paper, as well as an observation [365] that the uniform weighting of the trajectories requires constant scalar curvature, leads us to conjecture that
\begin{equation}
\delta\int_UP=\delta\int_U\kappa.
\end{equation}
\end{enumerate}
We hope that this mathematical framework will allow for a unified treatment of the foundations of nonequilibrium quantum statistical mechanics, as discussed from different perspectives in [120], [341], and [165] (see also [50]). More generally, we consider it to be a testing ground for a construction of a predictive dynamical theory that would unify several concepts of general relativity with quantum information theory into a single framework of multi-agent (post)quantum information relativity. Maybe this sounds a bit like a fountain of conjectures and high hopes, but we believe that the science fiction of today is just an advanced propagator of a scientific research of tomorrow.
\ifvargaugecompile %%% IF OF IF...ELSE GAUGE COMPILE %%%%%
\section{Quantum geometry \& global dynamics\label{geometric.structures.section}}
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\subsection{Quantum geometry\label{geometric.structures.section}}
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Sections \ref{W.star.BLP.section}, \ref{standard.liouvilleans.section}, \ref{distances.NS.geom.section}, \ref{quant.info.geom.section} do not contain new results or constructions, except of the definition of the quantum Poisson system.
\subsection{Quantum Banach Lie--Poisson spaces\label{W.star.BLP.section}}
\else %%% ELSE OF IF...ELSE GAUGE COMPILE %%%%%

\fi %%% END OF IF...ELSE GAUGE COMPILE %%%%%%%%

\subsubsection{Banach--Lie--Poisson spaces}

Let $\KK\in\{\RR,\CC\}$. A vector space $X$ over $\KK$ is called a \df{Lie algebra} if{}f it is equipped with a function $[\cdot,\cdot]:X\times X\ra X$ such that for all $f_1,f_2,f_3\in X$ and for all $\lambda\in\KK$
\begin{enumerate}
\item[1)] $[f_1,f_2]=-[f_2,f_1]$ (antisymmetry),
\item[2)] $[f_1,f_2+\lambda f_3]=[f_1,f_2]+\lambda[f_1,f_3]$ (linearity),
\item[3)] $[f_1,[f_2,f_3]]+[f_3,[f_1,f_2]]+[f_2,[f_3,f_1]]=0$ (Jacobi identity).
\end{enumerate}
The function $[\cdot,\cdot]$ is called the \df{Lie bracket}. If $(X,[\cdot,\cdot])$ satisfies also the Leibniz's rule
\begin{enumerate}
\item[4)] $\{f_1,f_2f_3\}=\{f_1,f_2\}f_3+f_2\{f_1,f_3\}\;\;\forall f_1,f_2,f_3\in X$,
\end{enumerate}
then $[\cdot,\cdot]$ is called the \df{Lie--Poisson bracket} [229], while $(X,[\cdot,\cdot])$ is called a \df{Lie--Poisson algebra}. If $G$ is a Lie group, then a Lie algebra of its generators will be denoted $\Lie(G)$. A vector space $X$ over $\KK$ is called a \df{Banach Lie algebra} if{}f it is a Banach space with a norm $\n{\cdot}$, a Lie algebra, and its Lie bracket $[\cdot,\cdot]:X\times X\ra X$ is bilinear and continuous in the topology of $\n{\cdot}$. If $\glie$ is a Banach Lie algebra, then the \df{adjoint map} $\rpktarget{ADMAP}\ad_x:\glie\ni y\mapsto[x,y]\in\glie$ and \df{coadjoint map} $\ad_x^\banach:\glie^\banach\ra\glie^\banach$,
\begin{equation}
\duality{y,\ad^\banach_x(z)}_{\glie\times\glie^\banach}:=\duality{\ad_x(y),z}_{\glie\times\glie^\banach}\;\;\forall z\in\glie^\banach,
\end{equation}
are norm continuous for each $x\in\glie$.

Let $M$ be a real Banach smooth manifold, and $\rpktarget{CIF}\CIF(M;\RR)$ denotes the space of all infinitely Fr\'{e}chet differentiable $\RR$-valued functions on $M$. Then a real \df{Poisson structure} on $M$ is defined as a function $\rpktarget{POISSON}\{\cdot,\cdot\}:\CIF(M;\RR)\times\CIF(M;\RR)\ra\CIF(M;\RR)$ such that $(\CIF(M;\RR),\{\cdot,\cdot\})$ is a Lie algebra [229]. %
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\footnote{Note that, on an algebraic level, the conditions i.2) and 4) on $\{f,\cdot\}$ are the same as the conditions 1. and 2. on $\der(\cdot)$ in Section \ref{derivations.section}.}
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%\item[i)] ,
%\item[ii)] $\{f_1,\cdot\}$ satisfies Leibniz's rule: .
If $M$ above is \textit{finite dimensional}, then $(M,\{\cdot,\cdot\})$ is called a real \df{Poisson manifold} [228, 367, 354, 368]. If $M$ is a real Banach smooth manifold, then the cotangent space at $x$ can be defined by $\T_x^\ct M:=(\T_xM)^\banach$, and each element of $\T_x^\ct M$ has a form
\begin{equation}\rpktarget{DDDf}
\ddd f(x)\equiv\ddd_x f:\T_xM\ni v\mapsto\ddd_x f(v):=v(f)\in\RR
\end{equation}
for some $f\in\CIF(M;\RR)$. Let $\T^\ct{}^\ct M:=\bigcup_{x\in M}\T_x^\ct{}^\ct M$, where $\T_x^\ct{}^\ct M:=(\T^\ct_x M)^\banach=(\T_x M)^\banach{}^\banach$. If $(M,\{\cdot,\cdot\})$ is a real Poisson manifold, then every $k\in\CIF(M;\RR)$ determines a unique vector field $\XXX_k\in\T M$ by\rpktarget{HAMILVF}
\begin{equation}
\XXX_k(f):=\bigcup_{x\in M}\left\{x\mapsto\duality{\XXX_k(x),\ddd f(x)}_{\T_xM\times\T_x^\ct M}\right\}=\{f,k\}\;\forall f\in\CIF(M;\RR).
\label{hamiltonian.vector.field}
\end{equation}
Such $\XXX_k$ is called a \df{hamiltonian vector field}, while the corresponding $k$ is called the \df{Hamilton function}. If $M$ is infinite dimensional, then the value of $\{\cdot,\cdot\}$ at $x\in M$ depends only on the differentials $\ddd f(x),\ddd k(x)\in\T_x^\ct M$. Hence, there exists a section $\rpktarget{VARPI}\varpi$ of the vector bundle $\bigwedge^2\T^\ct{}^\ct M$ such that
\begin{equation}
\{f,k\}=\varpi(\ddd f,\ddd k).
\end{equation}
Let $\varpi$ be a \textit{smooth section}, meaning that for each $x\in M$ there exists a continuous bilinear antisymmetric function $\varpi_x:\T_x^\ct M\times\T_x^\ct M\ra\RR$ such that $x\mapsto\varpi_x$ is smooth. Let the function $\natural:\T^\ct M\ra\T^\ct{}^\ct M$ be the bundle map covering identity, isometric on fibres, and satisfying
\begin{equation}
\natural_x(\ddd f(x)):=\varpi_x(\cdot,\ddd f),
\end{equation}
which means that
\begin{equation}
(\natural_x(\ddd f(x)))(\ddd k(x))=\{k,f\}(x)\;\;\forall f,k\in\CIF(M;\RR).
\end{equation}
\begin{equation}
\XXX_k:=\varpi(\cdot,\ddd k)=\natural(\ddd k)=\{\cdot,k\}
\label{smooth.section.ttwom}
\end{equation}
is a smooth section of $\T^\ct{}^\ct M$, but may not be a vector field on $M$, because $\T M\subseteq\T^\ct{}^\ct M$ and $\T M\not\iso\T^\ct{}^\ct M$ in general. In order to solve this problem, Odzijewicz and Ratiu [253] proposed (see also further discussion and results in [254, 27, 28, 255, 351, 287, 252]) to define a real \df{Banach Poisson manifold} as a pair $(M,\{\cdot,\cdot\})$ of a real Banach smooth manifold $M$ and a Poisson structure $\rpktarget{POISSON.TWO}\{\cdot,\cdot\}$ on it such that the function $\natural:\T^\ct M\ra\T^\ct{}^\ct M$ defined above satisfies $\natural(\T^\ct M)\subseteq\T M$.

If $(M,\{\cdot,\cdot\})$ is a real Banach Poisson manifold, then every $k\in\CIF(M;\RR)$ determines a unique vector field $\XXX_k\in\T M$ defined by \eqref{smooth.section.ttwom}, and called a \df{hamiltonian vector field}. Such $k$ is then called the \df{Hamilton function}. Every real Poisson manifold is a real Banach Poisson manifold, so this terminology is consistent. Odzijewicz and Ratiu [253] define also a \textit{holomorphic} Banach Poisson manifold which provides an analogous setting for $\KK=\CC$, but we will not use its specific properties here, so we omit its definition. If a Banach space $X$ over $\KK$ is equipped with a Poisson structure $\{\cdot,\cdot\}$ that turns it into a (real or holomorphic) Banach Poisson manifold, then $X^\banach\subseteq\CIF(X;\KK)$. Moreover, $\{\cdot,\cdot\}$ is linear on $\CIF(X;\KK)$ if $\{X^\star,X^\star\}\subseteq X^\star$.

A \df{Banach--Lie--Poisson space} is defined [253] as a pair $(X,\{\cdot,\cdot\})$ such that
\begin{enumerate}
\item[1)] $X$ is a Banach space over $\KK$,
\item[2)] $(X,\{\cdot,\cdot\})$ is a real (if $\KK=\RR$) or holomorphic (if $\KK=\CC$) Banach Poisson manifold,
\item[3)] $X^\banach\subseteq\CIF(X;\KK)$ is a Banach Lie algebra with respect to $\{\cdot,\cdot\}$.
\end{enumerate}
In such case, the restriction of $\{\cdot,\cdot\}$ on $\CIF(X;\KK)$ to $X^\banach$ will be denoted by $[\cdot,\cdot]$. As proved in [253], the Banach space $X$ is a BLP space $(X,\{\cdot,\cdot\})$ if{}f $X^\banach$ is a Banach Lie algebra $(X^\banach,[\cdot,\cdot])$ satisfying
\begin{equation}
\ad_x^\banach(X)\subseteq X\subseteq X^\banach{}^\banach\;\;\forall x\in X^\banach,
\end{equation}
and in such case $\{\cdot,\cdot\}$ is given by
\begin{equation}
\{f,k\}(z)=\duality{z,[\DF_zf,\DF_zk]}_{X\times X^\banach}\;\;\forall f,k\in\CIF(X;\KK)\;\forall z\in X.
\label{BLP.three}
\end{equation}
Moreover, under those conditions the hamiltonian vector field associated to \textit{any} $k\in\CIF(X;\KK)$ reads\rpktarget{HAMILT.ZWEI}
\begin{equation}
\XXX_k(z)=-\ad^\banach_{\DF_zk}(z)\;\;\forall z\in X.
\label{BLP.four}
\end{equation}

If $(X,\{\cdot,\cdot\})$ is a BLP space, and if $X\iso\T_xX$ $\forall x\in X$, then $\T_x^\ct X\iso(\T_x X)^\banach\iso X^\banach$, so if $f\in X^\banach$, then one can identify $\ddd_x f\in\T_x^\ct X$ with $\DF_xf\in X^\banach$. As a result, for any $z\in X$ and $y\in X^\banach$ the linearity of $y$ gives $\DF_zy=y$, and for every $x\in X$ one has [38, 253]
\begin{align}
\duality{y,\ad^\banach_x(z)}_{X^\banach\times X^\banach{}^\banach}&=\duality{z,[x,y]}_{X\times X^\banach}=\{x,y\}(z)=-\{y,x\}(z)\\
&=(\XXX_x(y))(z)=\duality{\DF_zy,\XXX_x(z)}_{X^\banach\times X^\banach{}^\banach}=-\duality{y,\XXX_x(z)}_{X^\banach\times X^\banach{}^\banach}.
\end{align}
\begin{equation}
\XXX_x(z)=-\ad^\banach_x(z)\;\;\;\forall z\in X\;\forall x\in X^\banach.
\end{equation}

The notion of the BLP space can be viewed as a suitable generalisation of the important properties of \textit{strong} symplectic manifold to infinite dimensional situation which need not admit decomposition into symplectic leaves. If $(M_1,\{\cdot,\cdot\}_1)$ and $(M_2,\{\cdot,\cdot\}_2)$ are BLP spaces, then a smooth function $w:M_1\ra M_2$ is called a \df{Poisson map} if{}f
\begin{equation}
\{f\circ w,k\circ w\}_1=\{f,k\}_2\circ w\;\;\forall f,k\in\CIF(M_2;\KK).
\label{BLP.five}
\end{equation}
As shown in [237], the condition iii) above makes \eqref{BLP.five} equivalent to
\begin{equation}
\XXX_k\circ w=\T w\circ\XXX_{k\circ w}\;\;\forall k\in\CIF(M_2;\KK).
\end{equation}
If $(M,\{\cdot,\cdot\})$ is a BLP space and $h\in\CIF(M;\KK)$, then the \df{Hamilton equation}
\begin{equation}
\frac{\dd}{\dd t}f(w^h_t(x))=\{h,f(w^h_t)\}(x)\;\;\forall f\in\CIF(M;\KK)\;\;\forall t\in\RR\;\;\forall x\in M
\label{hamilton.eqn}
\end{equation}
determines a unique \textit{local} map $\rpktarget{WHTM}w^h_t:M\ra M$, called a \df{hamiltonian flow} of $h$, which is a Poisson map. The solutions of the equation $x(t)=w^h_t(x)$ with $x(0)=x$ need not exist \textit{globally}, that is, for all $t\in\RR$ and all $x\in M$. If they exist globally, then the hamiltonian vector field $\{\cdot,h\}$ is called \df{complete}.

\ifvargaugecompile %%% IF OF IF...ELSE GAUGE COMPILE %%%%%
\subsubsection{W$^*$-algebra predual as a BLP space}
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\subsubsection{W$^*$-algebra predual as a BLP space}
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If $M$ is a Banach space, and a Banach smooth manifold modelled on itself by means of an identity mapping then for each $x\in M$ there is a Banach space isomorphism $\T_xM\iso M$. If $\N$ is a W$^*$-algebra, then the Banach Lie algebra structure of $(\N_\star)^\banach\iso\N$ is given by its commutator $[x,y]:=xy-yx$, while $\rpktarget{AD.SECOND}\ad_x:=[x,\,\cdot\,]=\LLL_x-\RRR_x$ and $\ad_x^\banach=\LLL^\banach_x-\RRR^\banach_x$ are defined by weakly-$\star$ continuous maps $\LLL_x:\N\ni y\mapsto xy\in\N$, $\RRR_x:\N\ni y\mapsto yx\in\N$. The condition $\ad_x^\banach(\N_\star)\subseteq\N_\star$ holds for all $x\in\N$, so $\N_\star$ is a BLP space that is a holomorphic Banach Poisson manifold (modelled on itself by the atlas consisting of one chart, an identity mapping $\id_{\N_\star}$) with the Poisson structure given by \eqref{BLP.three},\rpktarget{POISSON.DREI}
\begin{equation}
\{f,k\}(\phi)=\phi([\DF_\phi f,\DF_\phi k])\;\;\forall f,k\in\CIF(\N_\star;\CC)\;\forall\phi\in\N_\star.
\label{BLP.six}
\end{equation}
As a result, the hamiltonian vector field associated to every $k\in\CIF(\N_\star;\CC)$ by means of \eqref{BLP.four} takes a form\rpktarget{HAMILT.DREI}
\begin{equation}
\XXX_f(\phi)=-\ad^\banach_{\DF_\phi f}(\phi)=\LLL^\banach_{\DF_\phi f}(\phi)-\RRR^\banach_{\DF_\phi f}(\phi)\;\;\forall\phi\in\N_\star.
\label{BLP.seven}
\end{equation}
These results, including the BLP space structure of $\N_\star$, were discovered by B\'{o}na [36, 37, 38] in the $\N_\star=\schatten_1(\H)=\BH_\star$ case, and were generalised to arbitrary W$^*$-algebras by Odzijewicz and Ratiu [253]. We will call \eqref{BLP.seven} the \df{B\'{o}na--Odzijewicz--Ratiu equation}. %, and we will speak of the \df{B\'{o}na--Odzijewicz--Ratiu structure} to refer to the BLP space structure of the preduals of W$^*$-algebras (or of their self-adjoint parts, see below).
If $\N=\BH$ then $\N_\star\iso\schatten_1(\H)$ and for every $\rho\in\schatten_1(\H)$ and every $x,y\in\BH$
\begin{equation}
\duality{y,-\ad^\banach_x(\rho)}_{\BH\times\BH^\banach}=-\duality{[x,y],\rho}_{\BH\times\BH^\banach}=-\tr_\H([x,y]\rho)=\duality{[x,\rho],y}_{\schatten_1(\H)\times\BH},%=\tr(y[x,\rho])
\label{BLP.eight}
\end{equation}
which follows from the fact that $\schatten_1(\H)$ is an ideal in $\BH$. As a result,
\begin{equation}
\end{equation}
and the BOR equation \eqref{BLP.seven} turns to the Lax equation [224]
\begin{equation}
\XXX_f(\rho)=[\DF_\rho f,\rho]\;\;\forall\rho\in\schatten_1(\H).
\end{equation}
In particular, a choice of the Hamilton function $h(\rho):=\tr_\H(H\rho)$, where $H\in\BH$ but is not necessarily self-adjoint, turns \eqref{hamilton.eqn} to
\begin{equation}
\frac{\dd}{\dd t}\rho(t)=-\ad^\banach_{\DF_\rho h}(\rho)=[H,\rho].
\end{equation}
\subsubsection{Quantum Poisson systems}
The above equation is derived for a Poisson structure on $\N_\star=\schatten_1(\H)$ viewed as a holomorphic Banach Poisson manifold. However, the standard construction of unitary dynamics in quantum mechanics makes us to be more interested in the real Banach Poisson manifold $\N_\star^\sa$ (and submanifolds of it that are also subsets of $\N^+_\star$), equipped with the Poisson structure coinduced by the action of the Lie algebra $\N^\asa$ of anti-selfadjoint elements of $\N$. More precisely, the set $\N^\uni$ of all unitary elements of a W$^*$-algebra $\N$ is a real Banach Lie group and has a real Lie Banach algebra $\Lie(\N^\uni)=\N^\asa:=\{x\in\N\mid x=-x^*\}$ with $[x,y]:=xy-yx$ [43]. The elements of the Banach Lie algebra $\N^\asa$ can be represented by $x\in\N^\sa=\ii\N^\asa$, using the Lie bracket $\N^\sa\times\N^\sa\ni(x,y)\mapsto\ii[x,y]\in\N^\sa$, which corresponds to the commutator $[\ii x,\ii y]=\ii z$ in $\N^\asa$. This algebra has a unique Banach predual, given by $\N_\star^\sa\iso L_1(\N)^\sa:=\{\phi\in L_1(\N)\mid\phi=\phi^*\}$, with an isomorphism $(\N_\star^\sa)^\banach\iso\N^\sa$ defined by duality
\begin{equation}
\N_\star^\sa\times\N^\sa\ni(\phi,x)\mapsto\duality{\phi,x}_{\N_\star^\sa\times\N^\sa}:=\phi(x)\in\RR.
\label{duality.sa.sa}
\end{equation}
The adjoint representation $\Ad(\N^\uni)$ of a Banach Lie group $\N^\uni$ on $\Lie(\N^\uni)=\ii\N^\sa=\N^\asa$,
\begin{equation}\rpktarget{AD.BIG}
\Ad(u)x:=uxu^*\;\;\forall u\in\N^\uni\;\forall x\in\N^\sa
\end{equation}
determine the coadjoint representation $\Ad^\banach(\N^\uni)$ on $(\N^\banach)^\sa$,
\begin{equation}
\duality{x,\Ad^\banach(u)\phi}_{\N^\sa\times(\N^\banach)^\sa}:=\duality{\Ad(u^{-1})x,\phi}_{\N^\sa\times(\N^\banach)^\sa}\;\;\forall x\in\N^\sa\;\forall\phi\in(\N^\banach)^\sa.
\end{equation}
Using these properties, one can show that [38, 253]
\begin{equation}
\ad^\banach_x(\N_\star^\sa)\subseteq\N_\star^\sa\subseteq(\N^\asa)^\banach\;\;\forall x\in\N^\asa.
\end{equation}
The space $\N_\star^\sa$ can be equipped with a real Banach smooth manifold structure modelled on itself by the atlas consisting of one chart, which is determined by the identity mapping on $\N_\star^\sa$. As a result, $\T_\phi(\N_\star^\sa)\iso\N_\star^\sa$ $\forall \phi\in\N_\star^\sa$. So, it is possible to use \eqref{BLP.three} and \eqref{duality.sa.sa} to \textit{define} the BLP structure on $\N_\star^\sa$ by
\begin{equation}
\{f,k\}(\phi):=\ii\phi([\DF_\phi f,\DF_\phi k])\;\;\forall f,k\in\CIF(\N_\star^\sa;\RR)\;\forall\phi\in\N_\star^\sa.
\label{BLP.on.N.sa}
\end{equation}
As a result, the Hamilton equation \eqref{hamilton.eqn} for $h\in\CIF(\N_\star^\sa;\RR)$ reads
\begin{equation}
\frac{\dd}{\dd t}f(\phi(t))=\{h,f\}(\phi(t))=\ii(\phi(t))\left([\DF_{\phi(t)}h,\DF_{\phi(t)}f]\right).
\label{BLP.nine}
\end{equation}
The spaces $\N^+_{\star1}$, $\N^+_\star$ and $(\N_\star^\sa,\{\cdot,\cdot\}_{\N_\star^\sa})$ are subsets of $(\N^\banach)^\sa$ that are invariant with respect to $\Ad^\banach(\N^\uni)$. As shown in [27], they decompose into union of orbits of $\Ad^\banach(\N^\uni)$, which in turn are weak symplectic manifolds, which provides the symplectic foliation of the BLP space $(\N_\star^\sa,\{\cdot,\cdot\}_{\N_\star^\sa})$. Similarly, $(\N_\star,\{\cdot,\cdot\}_{\N_\star})$ is invariant with respect to the action of the Banach Lie group $\N^\inv$ of all invertible elements of $\N$. If $\N=\BH$ and $\rho\in\schatten_1(\H)^\sa$, then the calculation analogous to \eqref{BLP.eight} gives
\begin{equation}
\ad_x^\banach(\rho)=[\rho,x]\;\;\forall x\in\BH^\asa\;\forall \rho\in\schatten_1(\H)^\sa.
\end{equation}
As a result, the BOR equation \eqref{BLP.seven} on $\N_\star^\sa=\schatten_1(\H)^\sa$ takes the form
\begin{equation}
\XXX_f(\rho)=[\rho,\DF_\rho f]\;\;\forall\rho\in\schatten_1(\H),
\end{equation}
while the Hamilton equation \eqref{BLP.nine} becomes [38]
\begin{equation}
\frac{\dd}{\dd t}f(\rho(t))=\ii\,\tr_\H\left([\rho(t),\DF_{\rho(t)}h]\DF_{\rho(t)}f\right).
\label{BLP.ten}
\end{equation}
Because of the identity
\begin{equation}
\frac{\dd}{\dd t}f(\rho(t))=\tr_\H\left((\DF_{\rho(t)}f)\frac{\dd}{\dd t}\rho(t)\right),
\end{equation}
the equation \eqref{BLP.ten} is equivalent to the \df{B\'{o}na equation} [36, 187, 38],
\begin{equation}
\ii\frac{\dd}{\dd t}\rho(t)=[\DF_{\rho(t)}h,\rho(t)].
\label{gen.vN}
\end{equation}
The solutions of \eqref{gen.vN} are state-dependent unitary operators $U(\rho,t)$. They do not form a group, but satisfy a cocycle relationship:
\begin{equation}
U(\rho,t+s)=U((\Ad(U(\rho,t)))(\rho),s)U(\rho,t)\;\;\forall t,s\in\RR.
\end{equation}
In the special case, when $h(\rho)=\tr_\H(\rho H)$ for $H\in\BH^\sa=\ii\BH^\asa$, \eqref{gen.vN} turns to the \df{von Neumann equation}
\begin{equation}
\ii\frac{\dd}{\dd t}\rho(t)=[H,\rho(t)].
\label{vN.equation.from.BLP}
\end{equation}
%\textbf{\texttt{[add about solutions and examples]}}

So far we have followed the B\'{o}na--Odzijewicz--Ratiu approach, hence our main object of interest was the real Banach manifold $\N^\sa_\star$, equipped with the BLP space structure coinduced by the Banach--Lie algebra $\N^\asa$ via $\N^\sa$, corresponding to the group $\N^\uni$ of unitary elements of $\N$. However, in principle, a geometric setting for nonlinear dynamics of quantum models can be generated by an arbitrary Banach Lie algebra $\B$ over $\RR$ such that:
\begin{enumerate}
\item[(i)] its Banach predual space $\B_\star$ exists, is unique, and is a real Banach Poisson manifold,
\item[(ii)] $\ad^\star_x(\B_\star)\subseteq\B_\star\subseteq\B^\star$ $\forall x\in\B$,
\item[(iii)] there exists a nonempty set $\M(\N,\B)\subseteq\N^+_\star$ that is a real BLP submanifold of $\B_\star$.
\end{enumerate}
In what follows, we will call such Banach Lie algebras $\B$ to be \df{well-adapted}. For any choice of $h\in\CIF(\M(\N,\B);\RR)$, the pair $(\M(\N,\B),h)$ will be called a \df{quantum Poisson system} whenever
\begin{equation}
w^h_t(\phi)\in\M(\N,\B)\;\;\forall \phi\in\M(\N,\B)\;\;\forall t\in\RR.
\label{quantum.Poisson.system.condition}
\end{equation}
Hence, each quantum Poisson system is determined by a choice of: a space of quantum states $\M(\N)\subseteq\N^+_\star$, a Banach Lie algebra $\B$, a tangent bundle (real Banach manifold) structure on $\M(\N)$, and a real Fr\'{e}chet smooth function on $\M(\N)$, satisfying the conditions (i), (ii), (iii), and \eqref{quantum.Poisson.system.condition}. The assumptions $\B\iso\N^\sa$ and $\T_\phi\N^\sa_\star:=\N_\star^\sa$ $\forall\phi\in\N^\sa_\star$ recover the BOR setting completely. Note that a general quantum Poisson system $(\M(\N,\B),h)$ does not have to be related to any group, so in particular to a group of unitary operators $\N^\uni$. The only shared property (securing the backwards compatibility with quantum mechanical setting) is implementation of the Poisson flow on the predual by means of a coinduced action of a Banach--Lie algebra.

\ifvargaugecompile %%% IF OF IF...ELSE GAUGE COMPILE %%%%%
%\subsection{Symplectic submanifolds}
\else %%% ELSE OF IF...ELSE GAUGE COMPILE %%%%%
\subsubsection{Symplectic submanifolds}
[[write about symplectic orbits (and maybe also lagrangeans!)]]
[[write also about the solutions to B\'{o}na equation etc!]]
\fi %%% END OF IF...ELSE GAUGE COMPILE %%%%%%%%

\ifvargaugecompile %%% IF OF IF...ELSE GAUGE COMPILE %%%%%
\subsection{Relative modular operators, standard liouvilleans and the GNS bundle\label{standard.liouvilleans.section}}
A \df{weight} on a W$^*$-algebra $\N$ is defined as a function $\omega:\N^+\ra[0,+\infty]$ such that $\omega(0)=0$, $\omega(x+y)=\omega(x)+\omega(y)$, and $\lambda\geq0\limp\omega(\lambda x)=\lambda\omega(x)$, with the convention $0\cdot(+\infty)=0$. A weight is called: \df{faithful} if{}f $\omega(x)=0\limp x=0$; \df{finite} if{}f $\omega(\II)<\infty$; \df{semi-finite} if{}f a left ideal in $\N$ given by $\nnn_\phi:=\{x\in\N\mid\phi(x^*x)<\infty\}$ is weakly-$\star$ dense in $\N$; \df{normal} if{}f $\omega(\sup\{x_\iota\})=\sup\{\omega(x_\iota)\}$ for any uniformly bounded increasing net $\{x_\iota\}\subseteq\N^+$. A space of all normal semi-finite weights on a W$^*$-algebra $\N$ is denoted $\W(\N)$, while the subset of all faithful elements of $\W(\N)$ is denoted $\W_0(\N)$. Hence, $\N_\star^+\subset\W(\N)$ and $\N_{\star0}^+\subset\W_0(\N)$.  For $\psi\in\W(\N)$, $\supp(\psi)=\II-\sup\{P\in\Proj(\N)\mid\psi(P)=0\}$. An element $\omega\in\N^+_\star$ is faithful if{}f $\supp(\omega)=\II$.

A \df{representation} of a W$^*$-algebra $\N$ is defined as a pair $(\H,\pi)$ of a Hilbert space $\H$ and a $*$-homomorphism $\rpktarget{pi}\pi:\N\ra\BH$. A representation $\pi:\N\ra\BH$ is called: \df{nondegenerate} if{}f $\{\pi(x)\xi\mid (x,\xi)\in\N\times\H\}$ is dense in $\H$; \df{normal} if{}f it is continuous with respect to the weak-$\star$ topologies of $\N$ and $\BH$; \df{faithful} if{}f $\ker(\pi)=\{0\}$. An element $\xi\in\H$ is called \df{cyclic} for a W$^*$-algebra $\N\subseteq\BH$ if{}f $\N\xi:=\bigcup_{x\in\N}\{x\xi\}$ is norm dense in $\BH$. A representation $\pi:\N\ra\BH$ is called \df{cyclic} if{}f there exists $\Omega\in\H$ that is cyclic for $\pi(\N)$. According to the Gel'fand--Na\u{\i}mark--Segal theorem [106, 311] for every pair $(\N,\omega)$ of a W$^*$-algebra $\N$ and $\omega\in\N^{\banach+}$ there exists a triple $(\H_\omega,\pi_\omega,\Omega_\omega)$ of a Hilbert space $\H_\omega$ and a cyclic representation $\pi_\omega:\N\ra\BH$ with a cyclic vector $\Omega_\omega\in\H_\omega$\rpktarget{h.omega}, and this triple is unique up to unitary equivalence. An analogue of this theorem for weights follows the similar construction, but lacks cyclicity. If $\omega$ is a weight on a W$^*$-algebra $\N$, then there exists the Hilbert space $\H_\omega$, defined as the completion of $\nnn_\omega/\ker(\omega)$ in the topology of a norm generated by the scalar product $\s{\cdot,\cdot}_{\omega}:\nnn_\omega\times\nnn_\omega\ni(x,y)\mapsto\omega(x^*y)\in\CC$,
\begin{equation}\rpktarget{h.omega.zwei}
\H_\omega:=\overline{\nnn_\omega/\ker(\omega)}=\overline{\{x\in\N\mid\omega(x^*x)<\infty\}/\{x\in\N\mid\omega(x^*x)=0\}}=\overline{\nnn_\omega/\I_\omega},
\end{equation}
and there exist the maps\rpktarget{rep.omega.zwei}\rpktarget{pi.omega.zwei}
\begin{align}
[\cdot]_\omega:\nnn_\omega\ni x&\mapsto [x]_\omega\in\H_\omega,
\label{GNS.class.weight}\\
\pi_\omega:\N\ni x&\mapsto([y]_\omega\mapsto[xy]_\omega)\in\BBB(\H_\omega),
\label{GNS.rep.weight}
\end{align}
such that $[\cdot]_\omega$ is linear, $\ran([\cdot]_\omega)$ is dense in $\H_\omega$, and $(\H_\omega,\pi_\omega)$ is a representation of $\N$. If $\omega\in\W(\N)$ then $(\H_\omega,\pi_\omega)$ is nondegenerate and normal. It is also faithful if $\omega\in\W_0(\N)$.

A standard representation [130] of a W$^*$-algebra $\N$ is defined as a quadruple $(\H,\pi,J,\stdcone)$ of a Hilbert space $\H$, a nondegenerate faithful weakly-$\star$ continuous representation $\pi:\N\ra\BH$, a conjugation $J:\H\ra\H$, and a self-polar cone\footnote{A subspace $\D$ of a Hilbert space $\H$ is called a self-polar cone if{}f $\lambda\xi\in\D$ $\forall\xi\in\D$ $\forall\lambda\geq0$ and $\D=\{\zeta\in\H\mid\s{\xi,\zeta}_\H\geq0\;\forall\xi\in\D\}$.}  $\stdcone\subseteq\H$, satisfying the conditions
\begin{equation}
J\pi(\N) J=\pi(\N)^\comm,\;\;
\xi\in\stdcone\;\limp\;J\xi=\xi,\;\;
\pi(x)J\pi(x)\stdcone\subseteq\stdcone,\;\;
\end{equation}
For any standard representation
\begin{equation}
\forall\phi\in\N^+_\star\;\;\exists!\stdembed_\pi(\phi)\in\stdcone\;\;\forall x\in\N\;\;\phi(x)=\s{\stdembed_\pi(\phi),\pi(x)\stdembed_\pi(\phi)}_\H.
\label{std.vector.representative}
\end{equation}