Correction: Spatial variability of medium dense sand deposit

Abbreviations

$A, B, H, \ldots$: operators; $D(A)$: domain of the operator $A$; $E, E_1, E_2, \ldots$: normed spaces; $\mathcal{X}, \mathcal{Y}, \ldots$: Banach spaces; $\mathcal{H}$: Hilbert space; $\mathscr{C}^0[0, 1]$: Banach space of continuous real-valued functions on $[0, 1]$ with the supremum norm; $\mathscr{C}^1[0, 1]$: Banach space of continuous and one-time differentiable real-valued functions on $[0, 1]$ with respect to its natural norm; $\mathfrak{D}_vf$: Gâteaux differential at $u\in U$ in the direction $v\in E$; $\mathcal{D}_uP$: operational tangent space at point $u$ to the manifold $P$; $\mathrm{T}_uP$: (kinematic) tangent space at $u$ to the manifold $P$; $H_n^p(x)$: Hermite probabilistic polynomial of order $n$; $D^\infty(A): = \bigcap_{n\in \bar{\mathbb{N}}}D(A^n)$: space of smooth vectors of an operator $B$; $D^a(A)$: space of analytic vectors of an operator $A$; $\dot\psi_{\psi_0}$: directional derivative on a normed space in the direction $\psi$ at the point $\psi_0$; $\nabla_{\dot\psi}$: partial derivative in the direction $\dot\psi$ on a normed space; $\mathcal{P}\mathcal{H}$: projective space of a Hilbert space $\mathcal{H}$; $\bar{\mathbb{N}}$: set of natural numbers including zero

1.   Introduction and state-of-the-art

The study of finite-dimensional symplectic manifolds has proven to be very fruitful in the description of mathematical and physical theories ^[1,2,3]. Among its applications, one finds the analysis of problems appearing in classical and quantum mechanics (see ^[1,3,4,5] and references therein).

The analysis of symplectic manifolds modeled on (possibly infinite-dimensional) Banach spaces, the hereafter called symplectic Banach manifolds (SBMs), is an appealing topic of research. This is illustrated by the mathematical works ^[6,7,8,9], or, more specficially, because SBMs are a particular case of other geometric structures on infinite-dimensional manifolds ^[10,11]. Moreover, SBMs are also interesting from a physical perspective ^{[12,13,14,15]}, and we will focus primarily on works such as ^[16,17,18]. Furthermore, SBMs appear in relevant physical topics such as Schrödinger equations (SEs) ^[17,18,19], non-linear SEs ^[7,12,13,15], and fluid mechanics ^{[19,20,21,22]}.

The theory of SBMs emerges as a generalization of finite-dimensional symplectic geometry obtained by addressing appropriate topological and analytical issues due to the infinite-dimensional nature of general Banach spaces. SBMs differ significantly from their finite-dimensional counterparts, e.g., there exists no immediate analogue of the Darboux theorem (see ^[9] and references therein). Relevantly, quantum and classical problems related to infinite-dimensional SBMs frequently require the study of non-smooth mathematical structures ^[18,19,23].

Schrödinger quantum mechanics represents a relevant field of application of symplectic geometry ^[5,14,17,24]. It is also worth noting that symplectic forms have been defined and used on manifolds modeled on different linear spaces: Banach ^[6,22,25], Hilbert ^[26,27], Fréchet ^{[15,25,28,29]}, convenient spaces ^{[19,23,25,26,30,31]}, etc. Despite this, many researchers study rigorously the differential geometric properties of $t$-dependent SEs on infinite-dimensional Hilbert manifolds. In this regard, ^[19,23] are probably the first geometric analyses, with other notable references including ^[26,32]. Most geometric works in the literature focus only on $t$-dependent or $t$-independent Schrödinger equations in finite-dimensional Hilbert spaces ^[4,17,24,27]. These works provide a clear geometric description of momentum maps in quantum mechanics, their relationships with other quantum mechanical formalisms, and more. However, few articles try to extend such results to an infinite-dimensional realm ^[17,18,28]. Let us review the literature on quantum mechanics approached through symplectic and other geometric methods in the infinite-dimensional context.

The first important pioneering works dealing with the problem of studying Hamiltonian symplectic systems on infinite-dimensional manifolds and quantum mechanics can be traced back to Marsden and his collaborators ^[19,23,26]. They developed methods to study Schrödinger equations on infinite-dimensional Hilbert spaces and other partial differential equations and problems mainly on manifolds modeled on normed spaces of different types. They also reviewed the differential calculus on locally convex topological vector spaces, introduced vector fields with domains, and other techniques to deal with quantum mechanical systems and partial differential equations. Moreover, Marsden and his collaborators also presented a generalized Hamiltonian approach based on the use of distributions and applied it to analyze the Bohr-Dirac correspondence principle in ^[23].

Later, other authors continued the study of quantum mechanical problems in a geometric manner. Some skipped most technical explanations and mathematical details, dealing mostly with bounded operators, finite-dimensional manifolds and/or not studying differentiability problems or domains (^[32] and references therein on the topic), or justifying results on physical grounds. It is worth stressing the relevance of ^[28], which is one of the fundamental references in the study of quantum mechanics via differential geometric methods. It should be noted that ^[28] uses, among other techniques, manifolds modeled on Fréchet spaces, which are quite difficult to deal with.

In the 1970s and 1980s, many authors were confident in the possibility of treating quantum mechanics on an infinite-dimensional manifold using differential geometry ^[12,13,33]. In particular, the authors in ^[33] treated the study of quantum mechanics in an infinite-dimensional projective space and analyzed the integrability of quantum mechanical systems. Moreover, ^[32] focuses on manifolds modeled on Hilbert spaces. It is worth stressing that ^[32] analyzes the literature on the geometric treatment of quantum mechanics available at that time. Other posterior works, like ^[29], presented certain quantum differentials and other approaches based on spaces of functions.

Furthermore, ^[18] provided a nice geometric approach to quantum mechanics modeled on Hilbert spaces, involving weak symplectic manifolds, and solving some problems in the literature. The work in ^[18] mostly focused on problems related to bounded operators, and partially skipped, in some interesting manners based on the use of relations, the study of the smoothness of geometric structures for self-adjoint operators. It is worth noting that ^[18] does not deal with $t$-dependent operators, which generally requires the study in detail of the common domain of several not necessarily self-adjoint operators on it, addressed in our paper. At moments, ^[18] assumes some domain of differentiability to apply its techniques; in others, relations are enough to prove their purposes.

Apart from ^[18], other researchers deal with potential approaches to the study of quantum mechanics via symplectic manifolds on special types of infinite-dimensional manifolds ^[15,30,31]. It is worth noting that there exists an analysis of Poisson manifolds and Hamiltonian systems in convenient spaces ^[30,31,34], which could be potentially used to study $t$-dependent Schrödinger equations, but it seems that no relevant advancements in this direction has been done so far. However, consider that convenient space methods are quite technical. Finally, there are several other works showing new potential approaches to $t$-dependent Schrödinger equations ^[35]. Of course, there are many works with applications in quantum mechanics using techniques from functional analysis ^[36,37,38] and many researchers who focus on the application of differential geometry in finite-dimensional Schrödinger equations or treat the problem at a physical level without dealing with technical problems. At this point, it is important to make some distinctions, as certain works can be very precise while omitting the study of specific aspects. For example, ^[18] provides a precise description by bypassing some smoothness considerations, for instance, by using an approach based on relations. The work ^[32] is supposed to be focusing on the infinite-dimensional case, but it does not analyze the mathematical challenges, such as smoothness, operator domains, etc. The work ^[17] examines key aspects of infinite-dimensional Schrödinger equations, like kinetic and operational tangent spaces, but does not thoroughly analyze operator domain definitions. Similar approaches are found in ^[4,5,39].

It should be stressed that several research groups may have their own conventions and assumptions in the study of symplectic geometry in infinite-dimensional manifolds. For instance, some reseachers admit symplectic forms on infinite-dimensional manifolds to have a non-trivial kernel (see the work [11, Remark 2.1]), which goes against the classical definition, which we follow here. In a more standard manner, these these structures are called presymplectic forms (see ^[40] for presymplectic forms on Banach spaces).

Summarizing, most researchers deal with the study via differential geometry of Schrödinger equations and other related equations on finite-dimensional manifolds ^[5,24,41,42]. In the infinite-dimensional realm, one assumes smoothness of structures in different manners or studies problems one can deal with without treating differentiability and other technical issues ^[4,17,43]. Some standard assumptions are the use of bounded or self-adjoint operators ^[18]. In general, most researchers mix some of the previous assumptions at different parts of their analsys. This significantly restricts the field of application of differential geometry in quantum mechanics. Several problems, like those related to families of operators that can be integrated to a Lie group action on an infinite-dimensional space, are related with operators that are not self-adjoint in a common domain to be analyzed. Hence, the development of an infinite-dimensional differential geometric approach allowing for the extension of previous ideas to general $t$-dependent unbounded Hamiltonian operators is extremely important, and it will be the main topic of this work.

It is worth stressing that the lack of a fully general geometric infinite-dimensional approach to $t$-dependent Schrödinger equations is due, in part, to the lack of smoothness of the structures used to describe unbounded operators and other related structures; the problems to define differential geometric objects, like the commutator of operators^[19], etc. Known methods to deal with these problems rely on using pure functional analysis techniques with no differential geometric methods or the justification of technical assumptions and results on physical grounds. Researchers using differential geometry techniques in quantum mechanical models in infinite-dimensional manifolds do not generally provide a detailed explanation of the differential geometric constructions they use. As far as we know, there is no rigorous differential geometric work dealing effectively with unbounded operators and their non-smoothness^[37].

We apply infinite-dimensional geometric and functional analysis techniques to the rigorous geometrical study of $t$-dependent SEs determined by (possibly not bounded) families of self-adjoint operators on separable Hilbert spaces. It is worth noting that since we deal with several self-adjoint operators at the same time, our techniques have to deal with non self-adjoint operators. Indeed, our methods can also be applied in a natural manner to operators on domains where they are not self-adjoint. For instance, analytic vectors naturally appear in problems related to a Lie group acting on an infinite-dimensional Hilbert space. Separable Hilbert spaces occur in most physical problems, and separability ensures the existence of a countable set of coordinates ^[27]. As a first new contribution, we have applied the theory of normed spaces, which admits a relatively simple differential geometric analysis and extensions to the Banach case, and analytic vectors ^[44,45,46] to improve the standard geometric approach given in ^[19] to deal with unbounded Hamiltonian operators. In particular, we mostly restrict ourselves to $t$-dependent Hamiltonian operators taking values in a Lie algebra of operators with an appropriate common domain of analytic vectors. It is worth noting that our approach can be applied to more general cases provided an adequate space of analytic vectors is known. Although the problems we are dealing with are not generally smooth, we will restrict ourselves to appropriate subspaces where problems are smooth enough to apply differential geometry. It is also worth noting that many of our practical applications restrict to the so-called strong SBMs, where a Darboux theorem exists ^[47] and other differential geometric techniques are applicable.

Our approach seems to be general enough to provide a rigorous theory while studying interesting physical systems, but it requires using some functional analysis techniques ^[19]. We have relied as much as possible on differential geometric techniques and simplifying technical assumptions. In particular, self-adjoint operators and their mean values are described as objects on manifolds modeled on normed spaces steaming from the manifold structure of Hilbert spaces. Due to the fact that functional analysis is not well-known for most researchers working on differential geometric methods in physics, we provide a brief introduction to the results on functional analysis and infinite-dimensional manifolds to be used.

Our work solves some technical difficulties to ensure the standard differential geometric approach of the examples under study and other similar ones. For instance, we provide several technical details in the theory of analytic vectors related to the metaplectic representation, angular momentum operators, and other physical examples, which seem to be not developed elsewhere. Once these details are given, the further application of differential geometric techniques is simpler and operational. In particular, our approach shows that the case of physical unbounded observables and other operators admitting a basis of eigenfunctions (without being necessarily symmetric), may be studied throughout our differential geometric techniques. It is worth noting that our approach is much simpler than the one in ^[19], which requires the use of topological vector spaces and deals with complicated issues.

As an application of our techniques, the above-mentioned $t$-dependent SEs are projected onto (generally infinite-dimensional) projective spaces using a Marsden-Weinstein (MW) reduction. Our methods represent an appropriate framework, for instance, for the generalization of the standard differential geometric works by G. Marmo and collaborators on quantum mechanics (see, e.g., ^[4,5,48]) or other authors ^[27,49] to the analysis of quantum mechanical problems on infinite-dimensional Hilbert spaces not determined by bounded operators.

Let us detail the structure of this work. At the end of this introduction, we present a table summarizing the major symbols used throughout this work. Section 2 is devoted to the basic functional analysis techniques and notations to be used hereafter. Differential geometry modeled on normed spaces ^[19,50] and Banach spaces ^[1,16] are discussed in Section 3. In Section 4, the theory of Banach and normed symplectic manifolds is briefly addressed. After studying weak and strong SBMs, we survey their related Darboux theorems, the canonical one- and two-forms on cotangent bundles to Banach manifolds, and introduce the MW reduction theorem. In Section 5, we provide a symplectic approach to infinite-dimensional separable Hilbert spaces. In Section 6, we focus on vector fields induced by unbounded self-adjoint, observable, and diagonalizable operators, while we analyze Hamiltonian functions for quantum models in Section 7. In Section 8, we study a class of $t$-dependent Hamiltonian systems related to the $t$-dependent SEs appearing in this work. In Section 9, a MW reduction procedure is used to project the above-mentioned $t$-dependent SEs onto a, generally, infinite-dimensional projective space. We also consider $t$-dependent SEs related to unbounded $t$-dependent Hamiltonian operators as a special type of Hamiltonian systems on Hilbert manifolds. Finally, we summarize our results and survey prospective lines of research in Section 10.

2.   Fundamentals on operators

Let us discuss several basic facts on functional analysis to be applied hereafter, e.g., in the mathematical formulation of Schrödinger quantum mechanics with unbounded operators (see ^[16,27,51] for details).

We call operator a linear map $A$ defined on a dense subspace $D(A)$ of a normed space $E_1$ taking values on a normed space $E_2$ of the form

Operators are sometimes called unbounded operators to highlight that they do not need to be continuous. It is worth stressing that defining an operator amounts to giving both its domain and its value on it. If two operators

with $i = 1, 2$, satisfy that $D(A_1)\subset D(A_2)$ while $A_1$ and $A_2$ take the same values in $D(A_1)$, then we say that $A_2$ is an extension of $A_1$. Two operators are equal, and we write $A_1 = A_2$, when $A_1$ and $A_2$ have the same domain and they take the same values on it.

To define the sum and composition of operators with possibly different domains, we proceed as follows. Let

with $i = 1, 2$, be two operators. If $D(A_1)\cap D(A_2)$ is dense in $E_1$, then $A_1+A_2$ is the operator on $D(A_1)\cap D(A_2)$ whose value at

is $A_1\psi+A_2\psi.$ Similarly, given an operator

such that

is dense in $E_1$, then we can define the composition

Let

be an operator on a Hilbert space $\mathcal{H}$. Consider the associated operator

Whenever $H_{\psi}$ is continuous on $D(H)$, the Riesz representation theorem ^[27] ensures that there exists $\psi_0 \in \mathcal{H}$ such that

for every $\phi\in D(H)$. If $H_{\psi}$ is continuous for every $\psi$ in a dense subset $D(H^\dagger)$ of $\mathcal{H}$, then one can define the adjoint of $H$, denoted by $H^{\dagger}$, as the operator

In other words,

for every $\psi\in D(H^\dagger)$ and $\phi\in D(H)$.

Let us now detail certain relevant types of operators.

Definition 2.1. An operator

is called:

a) Symmetric: if $\langle \psi \vert H \phi \rangle = \langle H \psi \vert \phi \rangle$ for all $\phi, \psi \in D(H)$;

b) Skew-symmetric: if $\langle \psi \vert H \phi \rangle = -\langle H \psi \vert \phi \rangle$ for all $\phi, \psi \in D(H)$;

c) Self-adjoint: if $H^{\dagger} = H$;

d) Skew-self-adjoint: if $H^{\dagger} = -H$;

e) Essentially self-adjoint: if $H$ admits a unique self-adjoint extension;

f) Observable: if $H$ is symmetric and admits a basis of eigenvectors.

Let us briefly comment several facts. If $H$ is symmetric, then the equality

for $\psi, \phi\in D(H)$ ensures that $H_\psi$ is continuous for every $\psi\in D(H)$ and

Hence, $D(H^\dagger)$ is dense in $\mathcal{H}$ and, consequently, $H^\dagger$ is a well-defined operator. Moreover,

where $H^{\dagger}\big\vert_{D(H)}$ is the restriction of $H^\dagger$ to $D(H)$, and $H^{\dagger}$ is therefore an extension of $H$. On the other hand, if $H$ is symmetric and

then $H^{\dagger} = H$ and vice versa. An analogue discussion can be applied to the case of $H$ being a skew-symmetric operator.

It is relevant to stress that any self-adjoint operator ${A}$ on a separable Hilbert space is unitarily equivalent to a multiplication operator ^[27]. Moreover, by Stone's theorem on one-parameter unitary groups, self-adjoint operators correspond to infinitesimal generators of $t$-parametric unitary groups.

Definition 2.2. Let

be an operator between normed spaces. We say that $A$ is open if the image of every open set in $E_1$ by $A$ is open. The operator $A$ is said to be closed if its graph is closed in $E_1\times E_2$ relative to the product topology.

Sometimes in the literature, a map whose graph is closed is said to be graph-closed, and the term closed is applied to operators so that $A(C)$ is closed whenever $C\subset D(A)$ is closed. However, we will stick to the terminology established in Definition 2.2.

Theorem 2.3. (Closed graph theorem ^[27]) Let

be an operator between Banach spaces. The operator $A$ is continuous if and only if its graph is closed in $\mathcal{X}\times \mathcal{Y}$.

If the

in Theorem 2.3 is continuous, then the closed graph theorem yields that $A$ is closed. Nevertheless, an operator defined on a non-closed domain may be non-continuous and closed simultaneously. For instance, the derivative

where $\mathscr{C}^1[0, 1]$ and $\mathscr{C}^0[0, 1]$ stand for the spaces of differentiable with continuous derivative and continuous functions on $[0, 1]$ with their standard norms, respectively, can be proved to be closed but not continuous. More generally, every self-adjoint operator is closed ^[27]. Then, if $H$ is a symmetric operator such that $D(H) = \mathcal{H}$, then $H$ is bounded.

Theorem 2.4. (Open mapping theorem [51, Theorem 4.10]) If

is a surjective continuous operator between Banach spaces, then $A$ is an open map.

3.   Geometry on normed and Banach manifolds

Our geometric study of quantum models will entail the analysis of classes of normed spaces ^[19]. Since the Banach fixed point theorem does not hold on normed spaces, first-order differential equations in normal form do not need to admit unique integral curves ^[25,34]. There are no straightforward analogues of the Implicit function and Inverse function theorems either (see ^[25] for details). This section recalls the major results on differential calculus on normed spaces to be used hereafter. For details, we refer to ^[19,25,50].

We hereafter assume that every normed vector space $E$ over a field $\mathbb{K}$ admits an unconditional Schauder basis, namely a topological basis $\{e_i\}$ so that every $v\in E$ admits a unique decomposition

for certain unique constants $\{\lambda^i\}\subset\mathbb{K}$ such that the sum does not depend on the order of summation of its terms. Hereafter, all basis are considered to be unconditional Schauder bases. The field $\mathbb{K}$ is hereafter assumed to be the field of real or complex numbers.

Let $E$ and $F$ be normed spaces and let $B(E, F)$ denote the space of continuous linear maps from $E$ to $F$. The space $B(E, F)$ admits an induced topology. It is worth noting that, if $E, F$ are normed, the space $B(E, F)$ does not admit a topology with good properties (for details see ^[19]). Similarly, $B^r(E, F)$ stands for the space of continuous $r$-linear maps on $E$ taking values in $F$.

Partial derivatives on normed vector spaces can be defined using the Gâteaux differential. In particular, a function

on an open $U$ of the normed space $E$ admits a Gâteaux differential at $u\in U$ in the direction $v\in E$ if the limit

exists. We call the function $u\mapsto (\mathfrak{D}_vf)(u)$ the partial derivative of $f$ in the direction $v\in E$. If $\mathfrak{D}_vf(u)$ exists for every $v\in E$ at $u\in U$, we say that $f$ is Gâteaux differentiable at $u$ and we define

This function satisfies that

for every $v\in E$ and $\lambda \in \mathbb{K}$. Nevertheless, $(\mathrm{d} f)_u$ does not need to be linear and, even if it is linear, it does not need to be continuous. If $(\mathrm{d} f)_u$ is linear and continuous, then we say that $f$ is Fréchet differentiable at $u$.

The function

is said to be differentiable if it is Fréchet differentiable at every point of $U$ and the mapping $u\mapsto \mathrm{d} f_u$ is continuous [50, p.38]. The function

is said to be differentiable of order $r+1$ at $a\in U$ if it is differentiable of order $r$ and the differential of order $r$ of $f$, let us say $d^{\, r}f$, exists at $a$ [50, Section 4.4] and [52, Chapter 9]. We say that $f$ is smooth or of class $\mathscr{C}^\infty$ if it is differentiable for every $r$.

The composite mapping theorem and the Leibniz rule hold on sufficiently differentiable functions on normed spaces ^[19]. Although there is no mean value theorem in topological vector spaces, a mean value theorem holds on normed spaces. In any case, if

for every $u\in U$ and $U$ is connected, then $f$ is constant on $U$ (see ^[19,50]). This result will be employed for studying the existence of Hamiltonian functions related to self-adjoint operators in quantum mechanics (see Section 6).

Once a notion of differentiability has been stated on normed vector spaces, the definition of a manifold modeled on a normed vector space is similar to the one on a standard manifold modeled on a finite-dimensional vector space (see ^[19,50,52]).

If not otherwise stated, manifolds are assumed to be smooth and modeled on a separable Banach space $\mathcal{X}$, i.e., the manifold $P$ admits a smooth differentiable structure whose charts take values in $\mathcal{X}$. Exceptions to this general rule appear in our description of quantum mechanical problems related to unbounded operators, which require the use of smooth manifolds modeled on normed spaces that do not need to be complete. It is worth noting that avoiding spaces that are not normed is a key to obtain a description that is technically appropriate for physical problems without adding intricate technical problems.

It can be proved that there exists a natural one-to-one correspondence between the space $\mathrm{T}_uP$ of tangent vectors at $u\in P$, which are understood as equivalence classes of curves

passing through $u$ at $t = 0$ and having the same, well-defined, Gâteaux differential at $t = 0$ in the direction $1\in \mathbb{R}$ ^[16,17,19], and the space $\mathcal{D}_uP$ of derivations

where $\mathscr{C}^\infty_u(P)$ is the space of germs at $u\in P$ of functions on $P$, of the form

for a curve $\gamma$ with $\gamma(0) = u$ and any $f\in \mathscr{C}^\infty_u(P)$. Let us prove the analogue of this result on normed spaces under our given general assumptions. The case for general manifolds modeled on normed spaces follows then by using local charts and their compatibility relations.

Let $(E, \Vert \cdot \Vert)$ be a real normed space. By our general assumption on normed spaces, $E$ admits an unconditional Schauder basis $\{\psi_n\}$. Let $\{f^n\}$ be the corresponding coordinate functions in $E^*$, which are continuous. Thus, if $\psi\in E$, then there exist unique escalars $c^n = f^n(\psi)$ such that

The elements of $\{f^n\}$ do not need to be a basis of the dual space $E^*$ in the sense that certain elements of $E^*$ may not be the limit of a sequence of finite linear combinations of $\{f^n\}$ in the strong topology of $E^*$ (consider for instance $\ell^1$ with the basis $e_n = (0, \ldots, 1(n-{\rm th\, \, position}), 0, \ldots)$ with $n\in \mathbb{N}$, and its dual $(\ell^1)^* = \ell^\infty$ with the elements $\sum_{n\in \mathbb{N}}f^n\in \ell^\infty$). If the functions $\{f^n\}$ are a basis of $E^*$, then $\{\psi_n\}$ is called a shrinking basis. Anyhow, the value of any element of $E^*$ at an element $v\in E$ can be obtained as a linear combination $\sum_{n}\lambda_nf^n(v)$, where the sum is understood as a limit in the field $\mathbb{K}$. In general, the $\{\lambda_n\}$ satisfy relations ensuring that the sum exists and is continuous. These relations can be different from the ones satisfied by the $\{c^n\}$, e.g., the space $\ell^p$ of sequences ${\bf c} = (c^1, c^2, \ldots)$ of numbers with

with respect to the norm

satisfies that $\ell^{p*}\simeq \ell^{q}$ for $1/p+1/q = 1$.

Every equivalence class of curves passing through $\psi_0\in E$ is determined by the Gâteaux differential of the curves at $t = 0$ in the direction $1\in \mathbb{R}$ and vice versa. Then, there is a one-to-one correspondence between $\mathrm{T}_{\psi_0}E$ and $E$, and we can write $\mathrm{T}_{\psi_0}E\simeq E$.

In turn, $\psi\in E$ determines a derivation

Obviously $\dot\psi_{\psi_0}\in \mathcal{D}_{\psi_0}E$. The subindex $\psi_0$ will be omitted if it is clear from context, or irrelevant, so as to simplify the notation.

If

then it follows from (3.1) that

for each $n$ and $\psi = \phi$. Then, every element of $\mathcal{D}_{\psi_0}E$ determines a unique element of $E$, which defines a unique tangent vector at $\psi_0$. Then, we can identify $\mathrm{T}_{\psi_0}E$ and $\mathcal{D}_{\psi_0}E$ and write

Moreover, each $\psi\in E$ is associated with a derivation $\dot{\psi}\in \mathrm{T}_{\psi_0}E$ via the isomorphism

The previous construction allows for building the tangent and cotangent bundles of a normed manifold $P$ along with other geometric structures, e.g., generalized tangent bundles, following the same ideas as in the finite-dimensional case, provided the infinite-dimensional nature of these structures is taken into account. In particular, a vector field is a section of the tangent bundle $\mathrm{T} P$, more precisely, it is a map

such that

where

is the projection onto the base space $P$ and $\mathrm{Id}_P$ is the identity map on $P$ (see ^[19]). Similarly, a $k$-form is a section of the bundle

where $(\bigwedge^k \mathrm{T}^\ast P)_u$ is the space of continuous $\mathbb{R}$-valued $k$-linear skew-symmetric mappings on $\mathrm{T}_uP$. We hereafter write $\Omega^k(P)$ for the space of $k$-forms on $P$ and we denote by $\mathscr{C}^\infty(P)$ the space of smooth functions on $P$. Similarly to the finite-dimensional manifold case, one can define the exterior derivative

which exists on an appropriate subset of $\Omega^k(P)$ of, differentiable enough, $k$-forms.

If $P$ is not modeled on a Banach space, then a vector field on $P$ may not have certain integral curves ^[34]. Hence, additional conditions must be required to ensure the existence of integral curves. Moreover, our study of quantum mechanical problems will demand the definition of vector fields with a domain (see [19, p.366]) for details).

Definition 3.1. A vector field on a domain $D\subset P$ is a map

such that $D$ is a manifold modeled on a normed space that, as a subset of $P$, is dense in $P$, the inclusion $\iota:D\hookrightarrow P$ is smooth, and $\tau\circ X = \mathrm{Id}_D$. Moreover, $X$ is called smooth if

is smooth.

Example 3.2. Let us provide an example of a vector field on a domain that will be generalized and studied in detail in Section 6. Consider the complex Hilbert space $L^2(\mathbb{R})$ of equivalence classes of square integrable complex functions on $\mathbb{R}$ that coincide almost everywhere. This space is naturally a real manifold modeled on a real Hilbert space given by the differential structure induced by the global map

where $L^2_{\mathbb{R}}(\mathbb{R})$ is $L^2(\mathbb{R})$ considered, in the natural way, as a real Hilbert space. The isomorphisms

for every $\psi \in L^2(\mathbb{R})$, allow us to write

and to consider every operator

as a map

In particular, consider

such that $D(\partial_x)$ is spanned by linear combinations of a finite family of elements of the basis of functions

with

and $H_n(x)$ being the Hermite polynomial of order $n$. Then, $D(\partial_x)$ is dense in $L^2(\mathbb{R})$. Moreover, $D(\partial_x)$ is a normed vector space relative to the restriction of the topology of $L^2(\mathbb{R})$ to $D(\partial_x)$, but it is not complete. The restriction of the global atlas of $L^2(\mathbb{R})$ to $D(\partial_x)$ induces a second one on $D(\partial_x)$ that is modeled only on a normed, not complete, space because $L^2(\mathbb{R})$ is the topological closure of $D(\partial_x)$, which is therefore different from $D(\partial_x)$. Note that

where one assigns every element of $D(\partial_x)$ its coordinates in the given basis. Then, the inclusion of $D(\partial_x)$ into $L^2(\mathbb{R})$ reads in these coordinates as

It is therefore a linear bounded mapping and it becomes smooth. Moreover, the coordinates of an element in $D(\partial_x)$ may be any finite sequence of non-zero constants, while any element of $L^2(\mathbb{R})$ is a square summable sequence. Evidently, every smooth function on $L^2(\mathbb{R})$ is smooth on $D(\partial_x)$. Since

we can write that

In coordinates adapted to the basis $H_n^p(x)e^{-x^2/2}$ of $D(\partial_x)$ and $L^2(\mathbb{R})$, where $H_n^p(x)$ is the Hermite probabilistic polynomial of order $n$, and using that

for any non-negative integer $n$, one has

Hence, $X_{\partial_x}$ becomes a vector field on a domain.

Example 3.2 can be generalized to vector fields on domains induced by observables, which will be the standard case in our study of quantum mechanics. This case will also turn out to be very frequent in the literature ^[19].

Similarly to vector fields on a domain, one can define differential $k$-forms on a domain. We write $\Omega^k(D)$ for the space of differential $k$-forms on a domain $D$. Moreover, it is straightforward to define $t$-dependent vector fields on a domain ^[19].

Definition 3.3. A (local) flow for

is a map of the form

where $\varepsilon > 0$ and $U\subset P$ is open and contains $D$, such that

for all $u\in D$ and $t\in (-\epsilon, \epsilon)$.

At first glance, it may seem strange that the flow of a vector field $X$ on a domain $D$ may be defined on an open set $U$ strictly containing $D$, but this will be useful and natural in this and other works (see ^[19]). More precisely, skew-self-adjoint operators on a Hilbert space will give rise to vector fields on domains whose flows are defined on the full $\mathbb{R}\times \mathcal{H}$ but they are differentiable only on a subspace of it. It is worth noting that an extension of the above for the case of $t$-dependent vector fields can be accomplished quite straightforwardly ^[19].

Example 3.4. Recall the vector field $X_{\partial_x}$ from Example 3.2. For an element $\psi\in D(\partial_x)$, since $\partial_x \psi\in D(\partial_x)$ and $\psi$ is an analytic function, one has that

and $\exp(t\partial_x)\psi$ may not belong to $D(\partial_x)$ ^[27]. Since ${{{\rm{i}}}} \partial_x$ admits a self-adjoint extension, the Stone-von Neumann theorem ensures that $\exp(t\partial_x)$ can be defined as a unitary map on $L^2(\mathbb{R})$ and $X_{\partial_x}$ admits a flow given by

For elements $\psi\in D(\partial_x)$, one observes that

but $F$ may not be differentiable for $\psi \notin D(\partial_x)$.

Note that $D(\partial_x)$ can be chosen so as to give rise to a flow $F$ of $X_{\partial_x}$ on a domain that is invariant under the maps $F_t$ for $t\in \mathbb{R}$. For instance, consider $\partial_x$ to be defined on the Schwartz space $\mathcal{S}(\mathbb{R})$. It can be proved that

for every $\psi\in \mathcal{S}(\mathbb{R})$ and $x\in \mathbb{R}$. Hence, the new $D(\partial_x)$ is invariant relative to $\exp(t\partial_x)$. Moreover,

on elements of $\mathcal{S}(\mathbb{R})$. Note that we are using the Schwarz space with the norm induced by $L^2(\mathbb{R})$.

Since the sum and composition of vector fields related to operators with different domains do not need to be defined at a new common domain, the Lie bracket of such vector fields may not be defined. In this work, conditions will be given to ensure that the Lie bracket of studied vector fields is defined on a dense subspace of $P$. This happens, for instance, for observables having a common basis of eigenvalues on the space they are defined on. Then, one can consider the direct sum of the subspaces generated by the elements of such a basis. This is a normed, but generally not complete, dense subspace.

It is interesting to our purposes to consider the case when a subset $D$ of a manifold $P$ modeled on a Banach space is a manifold modeled on a normed space. In this case, the exterior differential operator $\mathrm{d}$ can be similarly defined on differential forms on $D$. Moreover, $\mathrm{T}_p D\subset \mathrm{T}_pP$ and one can consider whether a function $f$: $D\rightarrow \mathbb{R}$ is such that its differential, namely

can be extended at points $u\in D$ to elements of $(\mathrm{d} f)_u\in \mathrm{T}^\ast_pP$. This will be important in our quantum mechanical applications and it becomes the main reason to provide the following definition.

Definition 3.5. Let $D$ be a subset of a manifold $P$ that is, again, a manifold modeled on a normed space. We say that $f$: $D \rightarrow \mathbb{R}$ is admissible when

can be extended continuously to $\mathrm{T}_{u}P$.

For a vector field $X$ on a domain $D_X \subset P$, which is assumed to be a manifold modeled on a normed space such that $X(D_X)\subset \mathrm{T} D_X$, the inner contraction, $\iota_{X}$, on differential forms with $D_X\supset D_\alpha$ is defined by restricting the contraction with $X$ of differential forms at points of $P$ and its evaluation on vector fields on $D_X$.

Similarly, the Lie derivative

is defined using Cartan's formula

Theorem 3.6. Let $X$ be a vector field on a domain $D_X\subset P$ and let

be the flow of $X$. Then,

for any $\alpha \in \Omega^k(P)$, values $t, s\in (-\epsilon, \epsilon)$, and arbitrary $u\in D_X$.

For a diffeomorphism $\phi$: $P_1\rightarrow P_2$ and a vector field $X$ on a domain $D_X\subseteq P_1$, the push-forward of $X$ by $\phi$, if it exists, has domain $\phi(D_X)\subseteq P_2$ and is defined by

If the flow of $X$ is given by $F_t$, then the flow of $\phi_* X$ is

for every $t\in (-\epsilon, \epsilon)$. In particular, one has to assume that $\phi(D_X)$ is dense in $P_2$ for the vector field $\phi_*X$ to exist.

Definition 3.7. A diffeomorphism $\phi$: $D\rightarrow D$ of a domain $D\subset P$ is said to be admissible if and only if for every $u\in D$, the map

extends to a continuous linear map

The following results are the generalization to the infinite-dimensional context of the quotient manifold theorem ^[16].

Theorem 3.8. Let

be a free and proper action on a Banach manifold. Then, $P/G$ is a manifold and

is a submersion.

Proposition 3.9. If $G$ is a compact Lie group, then every Lie group action

is proper.

4.   Symplectic geometry in the infinite-dimensional context

In this section, we introduce and study symplectic and related structures on manifolds modeled on normed and Banach spaces. As shown next, the fact that a symplectic form is defined on such, generally infinite-dimensional, manifolds yields several relevant differences with respect to the case of symplectic forms on finite-dimensional manifolds. It is worth noting that our presentation is focused on Banach manifolds and related cases, which are enough for our purposes. We skip the theory of symplectic forms on convenient spaces, given for instance in ^[30,31,34], which is much more technical.

Definition 4.1. A symplectic form is a closed differential two-form $\omega$ on a manifold $P$ modeled on a normed space that is non-degenerate, namely

is injective for every $u\in P$. If $\omega_u^{\flat}$ is an isomorphism for every $u\in P$, then we call $\omega$ a strong symplectic form. A (resp. strong) symplectic manifold is a pair $(P, \omega)$, where $\omega$ is a (resp. strong) symplectic form on $P$.

Every strong symplectic form is always a symplectic form, whereas a symplectic form $\omega$ on $P$ is strong provided $\omega^\flat_u$ is also surjective for every $u\in P$. The latter can be warranted, for instance, when $\omega$ is a symplectic form and $P$ is finite-dimensional. If the symplectic form $\omega$ of a symplectic manifold $(P, \omega)$ is known from context, then one will simply say that $P$ is a symplectic manifold.

The Darboux theorem is a well-known result from standard finite-dimensional symplectic geometry that states that every symplectic manifold admits a local coordinate system in which the associated symplectic form takes a canonical form ^[1]. This result cannot be extended to symplectic forms on manifolds modeled on general Banach spaces, as shown by Marsden for the case of SBMs ^[19], which in turn led to different generalizations of the Darboux theorem for SBMs ^[1,49].

The extension of the standard Darboux theorem to SBMs is accomplished through constant differential forms. Let us describe this notion. Every skew-symmetric continuous bilinear form on a normed space $E$, let us say

gives rise to a unique differential two-form $\omega_0$ on $E$ by considering the natural isomorphism

and defining $\omega_0$ to be the only differential form satisfying that

for every $u\in E$. Such a type of differential two-form on a normed space $E$ defined from of a skew-symmetric bilinear form on $E$ is called a constant differential two-form.

Theorem 4.2. (Strong Darboux theorem ^[1]) If $(P, \omega)$ is a strong Banach symplectic manifold, then there exists an open neighbourhood around every $u\in P$ where $\omega$ is the pull-back of a constant differential two-form.

The cotangent bundle of any finite-dimensional manifold admits a so-called tautological differential one-form whose differential gives rise to a strong symplectic form ^[16]. Similarly, the cotangent bundle of a Banach manifold $P$ is endowed with a canonical differential one-form and a related symplectic form, which is not strong in general [19, Theorem 2.4].

Definition 4.3. Let $P$ be a manifold modeled on a Banach space $\mathcal{X}$ and let

be the cotangent bundle projection. The tautological differential one-form on $\mathrm{T}^\ast P$ is the differential one-form $\theta$ on $\mathrm{T}^\ast P$ of the form

and the canonical two-form $\omega$ on $P$ is defined as $\omega = - \mathrm{d} \theta$.

Let us give a coordinate expression for $\theta$ and $\omega$ on a separable Banach manifold $P$ modeled on a Banach space $\mathcal{X}$ with an unconditional Schauder basis. It is worth recalling that the separability does not imply the existence of an unconditional Schauder basis, as follows from the very famous counterexample found by Enflo. On a sufficiently small open set $U\subset P$, one can consider a coordinate system $\{x^j\}$. The coordinate system $\{x^j, p_j\}$ adapted to $\mathrm{T}^\ast P$ allows us to establish a local diffeomorphism

In turn, the coordinate system $\{x^j, p_j\}$ induces another adapted coordinate system to $\mathrm{T} \mathrm{T}^\ast U$, namely $\{x^j, p_j, \dot x^j, \dot p_j\}$, allowing us to define a diffeomorphism

In these coordinates,

Then,

More generally, one has the following result for Banach spaces (see [19, Proof of Theorem 2.4] for details). In essence, the reflexivity of $\mathcal{X}$ amounts to the fact that $\omega$ gives rise to an isomorphism between $\mathrm{T} P$ and $\mathrm{T}^\ast P$.

Theorem 4.4. The canonical two-form $\omega$ on $\mathrm{T}^\ast P$ is a symplectic form. In addition, $\omega$ is a strong symplectic form if and only if $P$ is modeled on a reflexive Banach space.

Definition 4.5. Let $P$ be a Banach manifold and let

denote a bilinear map. If $(\mathscr{C}^\infty(P), \{\cdot, \cdot\})$ is a Lie algebra and $\{\cdot, \cdot\}$ satisfies the Leibniz rule, namely

for all $f, g, h\in \mathscr{C}^\infty(P)$, then $\{ \cdot, \cdot\}$ is called a Poisson bracket or a Poisson structure on $P$, and in this case $(P, \{\cdot, \cdot \})$ is said to be a Poisson manifold.

Note that the Poincare Lemma works also for infinite-dimensional manifolds (see ^[53]).

Definition 4.6. Let $(P, \omega)$ be a symplectic manifold. A vector field $X$: $P \rightarrow \mathrm{T} P$ is called Hamiltonian if there exists a function $h\in \mathscr{C}^\infty(P)$ such that

The function $h$ is called a Hamiltonian function of $X$. Whenever $\iota_{X}\omega$ is closed, and thus locally exact (see ^[19,53]), we say that $X$ is a locally Hamiltonian vector field.

The set of (locally) Hamiltonian vector fields on $(P, \omega)$ will be denoted by (resp. ${\rm{Ham}}_{\rm{loc}}(P, \omega)$) ${\rm{Ham}}(P, \omega)$ or (resp. ${\rm{Ham}}_{\rm{loc}}(P)$) ${\rm{Ham}}(P)$ when $\omega$ is understood from context.

If $\omega$ is a strong symplectic form, then every $h\in \mathscr{C}^\infty(P)$ is the Hamiltonian function of a Hamiltonian vector field as in the case of symplectic forms on finite-dimensional manifolds. For general SBMs, each $\omega_u^{\flat}$, with $u\in P$, is only in general injective and some $h\in \mathscr{C}^\infty(P)$ may not give rise to any Hamiltonian vector field. If a function $h\in \mathscr{C}^\infty(P)$ has a Hamiltonian vector field, it is unique. This motivates the following definition.

Definition 4.7. Let $(P, \omega)$ be a Banach symplectic manifold. We call a function $h\in \mathscr{C}^\infty(P)$ admissible if it is the Hamiltonian function of a Hamiltonian vector field.

We will denote by $X_h$ the unique Hamiltonian vector field associated with an admissible function $h$. The following result is immediate.

Proposition 4.8. Let $(P, \omega)$ be a symplectic manifold. The set ${\rm{Ham}}_{\rm{loc}}(P)$ is a Lie algebra, while ${\rm{Ham}}(P)$ is an ideal of ${\rm{Ham}}_{\rm{loc}}(P)$ and

In particular, if $X_1, X_2\in {\rm{Ham}}_{\rm{loc}}(P)$, then

Every symplectic form on a finite-dimensional manifold gives rise to a Poisson structure. This is no longer true for a general symplectic form, as not every function is associated with a Hamiltonian vector field. A way to ensure this is given by the following proposition.

Proposition 4.9. A strong symplectic manifold $(P, \omega)$ induces a Poisson structure on $P$ given by

Definition 4.10. Let $(P_1, \omega_1)$ and $(P_2, \omega_2)$ be two symplectic manifolds. A diffeomorphism $\phi$: $P_1 \rightarrow P_2$ is called a symplectic map, or a symplectomorphism, if $\phi^*\omega_2 = \omega_1$.

We hereafter focus on physical systems determined by the so-called Hamiltonian systems.

Definition 4.11. A Hamiltonian system is a triple $(P, \omega, h)$, where $(P, \omega)$ is a symplectic manifold and

is a $t$-dependent function on $P$ such that for every $t\in \mathbb{R}$, the function $h_t\in \mathscr{C}^\infty(P)$ is an admissible function for $(P, \omega)$. We call $h$ the Hamiltonian function of the system.

Every Hamiltonian system $(P, \omega, h)$ gives rise to a unique $t$-dependent vector field $X$, called the Hamiltonian vector field of the system, satisfying that

The system of differential equations describing the integral curves

of $X$ reads

This system of differential equations is called the Hamilton's equations for the system $(P, \omega, h)$. To emphasise the role of the Hamiltonian function $h$ to determine its associated vector field, we will sometimes write $X_h = X$. In the MW reduction procedure, one is interested in a special family of Hamiltonian systems admitting a particular type of symmetries described next.

Definition 4.12. A $G$-invariant Hamiltonian system relative to the Lie group action

is a Hamiltonian system $(P, \omega, h)$ such that

If

is understood from context, we will simply talk about a $G$-invariant Hamiltonian systems, omitting the Lie group action.

Definition 4.13. A Lie group action

on a symplectic manifold $(P, \omega)$ is called symplectic if, for every $g\in G$, the map

is a symplectomorphism. Moreover, $\Phi$ is called Hamiltonian if the fundamental vector fields of $\Phi$ are Hamiltonian relative to the symplectic form $\omega$.

Example 4.14. Let $\mathcal{H}$ be a complex Hilbert space and consider the constant symplectic form $\omega_B$ on $\mathcal{H}$ induced by the continuous skew-symmetric bilinear form

Given the skew-adjoint operator ${{{\rm{i}}}} \mathrm{Id}_\mathcal{H}$, the Stone-von Neumann theorem ^[27] states that each $\exp({{{\rm{i}}}} t \mathrm{Id}_{\mathcal{H}})$ is a continuous mapping on $\mathcal{H}$ and, since it is linear, it is also smooth. Since $t{{{\rm{i}}}} \mathrm{Id}_{\mathcal{H}}$ is continuous

and the operators $\exp({{{\rm{i}}}} t \mathrm{Id}_{\mathcal{H}})$, for $t\in \mathbb{R}$, span a group isomorphic to $U(1)$, namely the unitary group of transformations on a one-dimensional Hilbert space. Define the Lie group action

Then,

for every $\psi_1, \psi_2\in \mathcal{H}$ and $g\in U(1)$. Consequently,

for every $g\in G$ and $\Phi$ is a symplectic action.

Definition 4.15. Let $(P, \omega)$ be a symplectic manifold. A momentum map for a Lie group action

is a mapping $J$: $P \rightarrow \mathfrak{g}^*$ such that

where $\xi_P$ is the fundamental vector field of $\Phi$ related to $\xi\in\mathfrak{g}$, namely

Thus,

is a Hamiltonian function of $\xi_P$ for each $\xi\in \mathfrak{g}$.

Theorem 4.16. (Hamiltonian Noether's theorem ^[54]) Consider a Hamiltonian system $(P, \omega, h)$. Let

be a Lie group action on a symplectic manifold $(P, \omega)$ with a momentum map

If the $t$-dependent function $h$ on $P$ is $G$-invariant, namely

for all $g\in G$ and $t\in \mathbb{R}$, then $J$ is conserved on the trajectories of $X_h$, that is

for all $t\in \mathbb{R}$, where $F$ is the flow of $X_h$.

To introduce the symplectic MW reduction, one may consider a quite general property of momentum maps given by the following definition^[54], which makes use of the coadjoint action

on the dual to the Lie algebra $\mathfrak{g}^*$ of $G$.

Definition 4.17. A momentum map

of a Lie group action

is said to be Ad$^*$-equivariant when

for all $g\in G$, where ${\rm{Ad}}^*$ is the coadjoint action of $G$, namely

and $({\rm{Ad}}^*)_{g}$ is the transpose of the morphism

In other words, $J$ is ${\rm{Ad}}^*$-equivariant if the diagram aside commutes for every $g\in G$

The ${\rm{Ad}}^*$-equivariance of momentum maps is not really necessary in what follows: It is only commonly used. Instead, one can use that a momentum map related to a Lie group action

is always equivariant related to a new Lie group action

defined by

for every $g\in G$ and $\vartheta\in \mathfrak{g}^*$. Moreover,

is defined to be the well defined constant element in $\mathfrak{g}^*$ of the form

The Lie group action $\Delta$ is frequently called the affine Lie group action related to $J$ and $\Phi$. We hereafter write $G_\mu$ for the isotropy subgroup of $G$ leaving invariant an element $\mu\in \mathfrak{g}^*$ relative to its associated affine Lie group action.

When $(P, \omega, h)$ is a $G$-invariant Hamiltonian system, one expects that the $G$-invariance of the system will enable the simplification of the system, e.g., giving rise to a new symplectic manifold of "smaller" dimension. This is accomplished through the MW reduction theorem enabling one to go from a symplectic manifold having a Hamiltonian Lie group action of symmetries of $h$ to another "reduced" symplectic manifold. Then, the initial Hamiltonian function $h$ will also give rise to a new $t$-dependent function on the 'reduced' manifold, which finishes the reduction procedure. To do so, we first need the give the following definition.

Definition 4.18. Given a map

between two manifolds, we say that $n\in N$ is a weakly regular value of $f$ if $f^{-1}(n)\subset P$ is a submanifold,

for every $u\in f^{-1}(n)$, and $\ker \mathrm{T}_{u}f$ admits a closed supplementary in $\mathrm{T}_uP$.

Note that the condition on the closed supplementary in $\mathrm{T}_uP$ is immediately satisfied when $P$ is a finite-dimensional manifold. Nevertheless, this property is not always satisfied in the infinite-dimensional case. In the infinite-dimensional realm, immersions, submersions and other types of mappings are required to satisfy similar conditions to extend to the infinite-dimensional case the properties they have on finite-dimensional manifolds.

Theorem 4.19. (MW reduction theorem) Consider a symplectic manifold $(P, \omega)$. Let $G$ be a Lie group acting on $P$ and let

be an associated momentum map. Let $\mu$ be a weakly regular value of $J$ and suppose that $G_{\mu}$ acts freely and properly on the submanifold $J^{-1}(\mu)$. If

is the natural embedding of $J^{-1}(\mu)$ within $P$ and

is the canonical projection, then there exists a unique symplectic form $\omega_{\mu}$ on

such that

It is important to point out that Theorem 4.19 does not require the Banach symplectic manifold $P$ to be strongly symplectic ^[54]. The following theorem uses the MW reduction theorem to reduce the dynamics.

Theorem 4.20. Let the conditions of Theorem 4.19 hold. Let $(P, \omega, h_t)$ be a $G$-invariant Hamiltonian system relative to the Lie group action

Then,

● The flow of $X_h$, namely $F$, leaves $J^{-1}(\mu)$ invariant;

● The diffeomorphisms $F_t$ commute with every $\Phi_{g}$, for $g\in G_\mu$, giving rise to a flow $K$ on $P_\mu$ such that $\pi_\mu\circ F_t = K_t\circ \pi_\mu$;

● The flow $K$ corresponds to a vector field with a Hamiltonian function $h_\mu\in \mathscr{C}^\infty(P_\mu)$ determined by $h_{\mu}\circ \pi_{\mu} = h\circ \jmath_{\mu}$.

It is remarkable that the proper description of the theorem in the infinite-dimensional case requires Theorem 3.8 so as to ensure that $J^{-1}(\mu)/G_\mu$ is a submanifold.

5.   Symplectic forms on separable Hilbert spaces

Most quantum mechanical theories assume Hilbert spaces to be separable, i.e., to admit a countable dense subset of vectors ^[55], and so is done hereafter. Let us now endow a separable Hilbert space with a strong symplectic form. This will provide a rigorous mathematical extension to the infinite-dimensional setting of the standard approach to finite-dimensional Hilbert spaces ^[4,17,39,48]. We provide certain details that are frequently absent in the literature ^[32] and focus on problems on infinite-dimensional Hilbert spaces, but the finite-dimensional case is immediate from our considerations.

A complex Hilbert space $(\mathcal{H}, \langle \cdot\vert \cdot \rangle)$ is separable if and only if it admits a countable orthonormal basis $\{\psi_j\}_{j\in\mathbb{N}}$ ^[27]. This is quite a ubiquitous condition in quantum problems, but it may fail to hold, for instance, in scattering processes ^[56]. When $\mathcal{H}$ is separable, it admits an orthonormal countable basis that enables us to endow $\mathcal{H}$ with a real differentiable structure modeled on a Hilbert space.

Choose real-valued coordinates $\{q^j, p_j\}_{j\in \mathbb{N}}$ on $\mathcal{H}$ given by

Let $\ell_2$ be the Hilbert space of complex square-summable series considered in the natural way as a real vector space. One can then construct a global chart

In fact, every $\psi\in\mathcal{H}$ admits a unique family of complex constants $c_j\in\mathbb{C}$ such that

It is worth recalling that since $\mathcal{H}$ is a Hilbert space

and this holds independently of the order of the elements of the basis. This is not true for a general Schauder basis of a Banach space. Consequently,

and

Thus,

Conversely, every sequence

is the image of a unique element

In view of (5.1), one has that

and $\varphi$ is an isometry. Since $\varphi$ is an isometry and it has an inverse, its inverse is also an isometry and it is therefore continuous. Consequently, $\mathcal{H}$ and $\ell^2$ are homeomorphic. This gives rise to a real differentiable structure on $\mathcal{H}$.

Recall that the tangent space to $\mathcal{H}$ at $\phi$ is the space of equivalence classes of curves

such that

We will denote by $\dot\psi_\phi$ the equivalence class of the curve $\gamma_\psi$. Moreover,

and there exists an isomorphism

Moreover, we represent $\lambda_\phi(\psi_j)$ and $\lambda_\phi ({\rm i}\psi_j)$ by $(\partial_{q^j})_{\phi}$ and $(\partial_{p_j})_{\phi}$, respectively. Hence, $\{(\partial_{q^j})_{\phi}, (\partial_{p_j})_{\phi}\}_{j\in\mathbb{N}}$ becomes a basis of $\mathrm{T}_{\phi}\mathcal{H}$ and

It is worth noting that each $\mathrm{T}_\phi\mathcal{H}$ is then isomorphic to $\ell^2$ and the chosen basis is shrinkable.

The cotangent space of $\mathcal{H}$ at $\phi$, i.e.,

admits a basis $\{(\mathrm{d} q^j)_{\phi}, (\mathrm{d} p_j)_{\phi}\}_{j\in\mathbb{N}}$ dual to $\{(\partial_{q^j})_{\phi}, (\partial_{p_j})_{\phi}\}_{j\in\mathbb{N}}$. Then,

At each $\phi$, one has then that

In other words, the elements of $\mathrm{T}^*_\phi\mathcal{H}$ are those of the form $\lambda_j({\mathrm{d}} q^j)_\phi+\mu^j({\mathrm{d}} p_j)$ with $(\lambda_j, \mu^j)_{j\in \mathbb{N}}$ forming a sequence in $\ell^2$.

Observe that

is a non-degenerate bilinear (over $\mathbb{R}$) form. Indeed, $\mathcal{B}$ is well-defined because if ${ \bf c}_1, {\bf c}_2\in \ell_2$, then

and the map $\mathcal{B}$ is bilinear and continuous. Since

we obtain that $\mathcal{B}$ is skew-symmetric. Then, $\mathcal{B}$ allows us to endow $\mathcal{H}$ with a unique constant differential two-form $\omega$ such that

on every $\phi\in \mathcal{H}$. By using the isomorphism $\lambda_\phi$, one has that

takes the local form

Then, $\mathrm{d}\omega = 0$ and $\omega$ is a non-degenerate form.

To see that $\omega$ is a strong symplectic form, note that $\mathcal{H}$ admits a submanifold $\mathcal{Q}$ given by the zeros of the functions $\{p^j\}_{j\in \mathbb{N}}$. In fact, the $\{q^j\}_{j\in \mathbb{N}}$ form a coordinate system on $\mathcal{Q}$ taking values in the real vector space of square-summable sequences $(q_j)_{j\in \mathbb{N}}$, making $\mathcal{Q}$ into a manifold modeled on a Hilbert space. Then, the variables $\{q^j, p_j\}_{j\in \mathbb{N}}$ can be understood as the adapted variables to

In view of (5.2), we get that $\omega$ is the canonical two-form of $\mathrm{T}^\ast\mathcal{Q}$. Since $\mathrm{T}^\ast\mathcal{Q}$ is modeled on a Hilbert space, which is always reflexive ^[16], Theorem 4.4 shows that $\omega$ is a strong symplectic form.

6.   Vector fields related to operators

In geometric finite-dimensional quantum mechanics, operators can easily be described via the so-called linear vector fields, which are smooth and even analytic functions ^[17,19]. Let us extend this idea to bounded, first, and to general (possibly unbounded) operators on infinite-dimensional Hilbert spaces. This will involve addressing several issues concerning the lack of differentiability of the linear vector fields related to unbounded operators.

We showed in Section 5 that $\mathcal{H}$ is a smooth manifold modeled on a real Hilbert space through a global chart given by the coordinates $\{q^j, p_j\}_{j\in\mathbb{N}}$. This gives rise to a global coordinate system $\{q^j, p_j, \dot q^j, \dot p_j\}_{j\in\mathbb{N}}$ adapted to $\mathrm{T}\mathcal{H}$. This coordinate system gives rise to a diffeomorphism between $\mathrm{T}\mathcal{H}$ and $\mathcal{H}\oplus\mathcal{H}$. Then, every bounded operator

gives rise to a vector field

Since $B$ is bounded, it is continuous and

is also a smooth mapping. It follows immediately that $X_{B}$, which amounts to a bounded operator

where $\mathcal{H}\times \mathcal{H}$ is endowed with its natural Hilbert structure induced by the one on $\mathcal{H}$, is a smooth vector field.

In infinite-dimensional quantum mechanics, operators are standardly unbounded, they have a domain, and it is non-trivial to provide a procedure to ensure that they can be described somehow through linear vector fields on a domain that will be smooth enough for their differential geometric treatment. In particular, one knows that an unbounded operator

on a Hilbert space $\mathcal{H}$ is related to a vector field with a domain (3.2). However, it is important to discover what else can be said. Let us study this process by means of the theory of analytic vectors (see ^{[27,44,46,57]} for details).

Definition 6.1. An analytic vector for an operator

is an element $\phi\in \mathcal{H}$ such that

and

for some $C_\phi > 0$ and every $n\in \bar{\mathbb{N}}$^*. The elements of $\in D^\infty(B)$ are called smooth vectors (of $B$) ^[58]. We write $D^a(B)$ for the space of analytic vectors of $B$.

^*Note that there is a typo in [57, Definition 7.1].

Both smooth and analytic vectors for a given operator $B$ are vector spaces. The fact that $D^\infty(B)$ is a vector space is clear from the linearity of $B$. Moreover if $\phi, \psi\in D^a(B)$, then a simple calculation, namely

shows that $D^a(B)$ is a vector space.

The term "smooth" vector comes from the fact that, if $B$ is skew-self-adjoint, the mapping

is smooth if and only if $\psi\in D^\infty(B)$ (see [57, p. 145]). For a general operator $B$, one has that

The existence of analytic vectors for operators can be illustrated by Nelson's analytic vectors theorem, which states that a symmetric operator $A$ on a Hilbert space $\mathcal{H}$ is essentially self-adjoint if and only if $A$ admits a dense domain $D^a(A)$ of analytic vectors ^[46]. We hereafter assume $B$ to be skew-self adjoint on some domain, if not otherwise stated. Analytic vectors allow for many other interesting simplifications in the theory of skew-adjoint operators. They are also useful in the differential geometric study of quantum mechanical problems. Let us also consider the following interesting result.

Proposition 6.2. ([57, Proposition 7.8]) Let $A$ be a self-adjoint operator and consider an analytic vector $\psi\in D^a(A)$. Let $C_{\psi}\in\mathbb{C}$ be such that

for every $n\in \bar{\mathbb{\mathbb{N}}}$. If $z\in \mathbb{C}$ and $|z| < C_{\psi}^{-1}$, then $\psi\in D(e^{z {A}})$ and

Let $A$ be a skew-self-adjoint operator. Since $e^{tA}$ is a continuous operator defined on $\mathcal{H}$, it stems from the definitions of smooth and analytic vectors for $A$ that

for every $t\in \mathbb{R}$ [57, p.149]. In fact, if $\psi\in D^\infty(A)$, then $s\mapsto \exp(sA)\psi$ is smooth and

is again smooth.

Assume that $V$ is a finite-dimensional Lie algebra of skew-self-adjoint operators on $\mathcal{H}$. One may wonder when the exponential of the elements of $V$, which are continuous automorphisms on $\mathcal{H}$ by the Stone-von Neumann Theorem ^[27], can be understood as automorphisms on $\mathcal{H}$ related to a continuous Lie group representation on $\mathcal{H}$. The following definition, based on the one given in ^[7] establishes the family of Lie algebras $V$ admitting such a property and defines rigorously what we mean.

Definition 6.3. A finite-dimensional Lie algebra $V$ of operators with a common domain $\mathcal{D}$ within a Hilbert space $\mathcal{H}$ is integrable if there exists an injective Lie algebra representation

and a continuous Lie group action

(relative to the natural topologies on $G\times \mathcal{H}$ and $\mathcal{H}$), where $G$ is the connected and simply connected Lie group of $\mathfrak{g}$, such that

We call $\rho$ the infinitesimal Lie group action associated with $\Phi$.

Geometrically, the above definition tells us that the fundamental vector fields of the Lie group action $\Phi$ of $G$ coincide on $\mathcal{D}$ with the vector fields associated with the elements of $V$. Moreover, the fundamental vector fields of this action satisfy that

where $D$ is an invariant common domain of smooth functions where the operators exponentials of the elements of $V$ are well defined and smooth enough. In other words, despite the lack of differentiability of the vector fields related to operators, the geometric expression of the Lie derivative of vector fields still holds.

Example 6.4. Consider the operators ${{{\rm{i}}}} \hat p$, ${{{\rm{i}}}} \hat x$ and ${{{\rm{i}}}} \widehat{{\rm Id}}$ on $L^2(\mathbb{R})$ spanning a three-dimensional Lie algebra of skew-self-adjoint operators. Consider the subspace of analytic functions of the Schwarz space $\mathcal{S}(\mathbb{R})$. This Lie algebra of operators admit a dense subspace of analytic vectors given by the functions $H_n(x)e^{-x^2/2}$ and it can be integrated. Moreover, these functions belong to $\mathcal{S}(\mathbb{R})$, which is invariant relative to the action of $\exp(\lambda {{{\rm{i}}}} \widehat p), \exp(\lambda {{{\rm{i}}}} \widehat x)$ and $\exp(\lambda {{{\rm{i}}}} \widehat {\rm Id})$ for $\lambda \in \mathbb{R}$. For instance,

Hence, expression (6.1) gives

It is a consequence of an argument given by Goodman ^[45] and Nelson ^[46] that if a finite-dimensional real Lie algebra of skew-symmetric operators is integrable, then the next criterium holds ^[44].

Theorem 6.5. (FS$^3$ criterium ^[44]) Every finite-dimensional Lie algebra of skew-symmetric operators on a Hilbert space $\mathcal{H}$ is integrable if it admits a basis with a common dense invariant subspace of analytic vectors.

There are other criteria of integrability of finite-dimensional Lie algebras of skew-symmetric operators on Hilbert spaces ^[46,58], but they are, in general, more complicated to use. On the other hand, the FS$^3$ criterion can be relatively easily proved to hold in many physically relevant problems appearing in physics ^[17,59]. In particular, if $A_1, \ldots, A_n$ are observables admitting a common basis of eigenvectors $\{e_n\}_{n\in \mathbb{N}}$, then ${{{\rm{i}}}} A_1, \ldots, {{{\rm{i}}}} A_n$ admit a basis in $\mathcal{H}$ of common eigenvectors $\{e_n\}_{n\in \mathbb{N}}$ and all their elements are analytic relative to ${{{\rm{i}}}} A_1, \ldots, {{{\rm{i}}}} A_n$. Moreover, the direct sum of subspaces

is a normed not complete space (in general) of analytic vectors relative to ${{{\rm{i}}}} A_1, \ldots, {{{\rm{i}}}} A_n$ that is invariant relative to the operators $\exp({{{\rm{i}}}} \lambda_\alpha A_\alpha)$ with $\lambda_\alpha\in \mathbb{R}$ and $\alpha = 1, \ldots, r$.

Assume then that the skew-symmetric operators of the Lie algebra $V$ have a common invariant domain $D_V$ of smooth vectors on $\mathcal{H}$. In practical cases, each element of $V$ is skew-self-adjoint in an appropriate domain and its exponential exists on $\mathcal{H}$. Note that $D_V$ can be extended by the exponentials of the elements of $V$ to give a domain that is invariant relative to them. The related continuous unitary action

satisfies that

is smooth on $D_V^\infty$ ^[60].

Let us now prove the following useful result:

Theorem 6.6. Every observable $B$ on a Hilbert space $\mathcal{H}$ gives rise to an integrable vector field on a domain

and

Proof. We know that $\mathrm{T}\mathcal{H}$ is diffeomorphic to $\mathcal{H}\oplus\mathcal{H}$. Hence, $B$ can be associated with the vector field on a domain given by $X_{B}$. Note that the direct sum of the subspaces generated by the eigenvalues of $B$ gives a dense subspace

of smooth vectors for $B$ in $\mathcal{H}$. Hence, $D^\infty(B)$ is not empty and becomes dense in $\mathcal{H}$. Moreover, $D^a(B)\supset E_{B}$ is also dense in $\mathcal{H}$ and the Nelson's analytic vectors theorem shows that $B$ is integrable. The norm of $\mathcal{H}$ can be restricted to $D^{\infty}(B)$, which becomes a normed space. The space $E_{B}$ is dense in $D^{\infty}(B)$ and we obtain a basis for $D^{\infty}(B)$. The maps of the global atlas on $\mathcal{H}$ can be restricted to $D^{\infty}(B)$, which makes $D^\infty({B})$ into a manifold modeled on a normed space. The inclusion

is smooth relative to the previous differential structures because every smooth function on $\mathcal{H}$ remains smooth when restricted to $D^{\infty}(B)$. In fact, the variations of a smooth function on $\mathcal{H}$ remain smooth the directions tangent to $D^{\infty}(B)$ at points of $D^{\infty}(B)$.

On an element $\psi\in D^\infty(B)$, one has that $F_\psi$ is smooth. In view of this, one can see that $X_{B}$ is a vector field on the domain $D^{\infty}(B)$ and

see ^[60] for further details. Hence, $X_{B}$ is an integrable vector field on the domain $D^\infty(B)$. Since

one has that $X_{B}$ can be considered also as a vector field on a domain $D^\infty(B)$ taking values in $\mathrm{T} D^\infty(B)$. □

It is worth studying non-symmetric operators for which a vector field approach is possible. Let us give an example.

Example 6.7. Consider the space of square integrable functions on $[-l, l]\subset \mathbb{R}$ for $l > 0$, which is a Hilbert space, and the operator

In particular, consider $D(-{{{\rm{i}}}}\widehat p^2)$ to be the space spanned by finite linear combinations of functions

Hence, $D(-{{{\rm{i}}}}\widehat{p}^2)$ becomes a normed space relative to the norm induced by $L^2([-l, l])$ that is isomorphic to $\mathbb{R}^\mathbb{N}$ and dense in $L^2([-l, l])$. Then, one obtains an operator

It is worth stressing that $X_{-{{{\rm{i}}}} \widehat{p}^2}$ is not smooth and $-{{{\rm{i}}}}\widehat{p}^2$ is not symmetric. However, one may define a flow

Note that this map is well defined as

Moreover, $D(-{{{\rm{i}}}} \widehat{p}^2)$ consists of analytic vectors for $-{{{\rm{i}}}}\widehat{p}^2$ and $\exp(-{{{\rm{i}}}} t\widehat{p}^2)$ is continuous on $D(-{{{\rm{i}}}} t\widehat{p}^2)$. Indeed, $D(-{{{\rm{i}}}} t\widehat{p}^2)$ is invariant relative to $\exp(-{{{\rm{i}}}} t\widehat{p}^2)$. Moreover,

7.   Hamiltonian functions induced by self-adjoint operators

In the previous section, we showed that vector fields generated by skew-self adjoint operators may be well-defined on an appropriate domain. We will show that these vector fields can be considered as Hamiltonian vector fields in a domain using functions on domains. For instance, we have to deal with functions on Hilbert spaces of the type

where $H$ is a symmetric operator. The function $f_H$ is called the average value of the operator $H$ ^[27] and it is defined only on the domain of $H$, which is dense in $\mathcal{H}$.

In general, the whole symplectic formalism can be modified to deal with quantum mechanical systems by considering that structures are defined on a dense subset of $\mathcal{H}$ having the structure of a normed space and using the fact that the operators of the theory are self- or skew-self-adjoint. Most structures to be used are not smooth, but they are differentiable enough to apply differential geometric techniques. Many results can be extended to symmetric operators and other more general setups.

Let us start by defining how to write in terms of quantum operators the differentials of the real and imaginary parts of certain average values of operators. This is done in the following lemma:

Lemma 7.1. If $f_A$ is the average function of a symmetric operator

then

Therefore, $(\mathrm{d} f_A)_{\psi}$ can be extended to a unique element of $\mathrm{T}^\ast_\psi\mathcal{H}$.

Proof. By definition of the contraction of the differential of a function with a vector field

for all $\phi, \psi\in D(A)$. Now, observe that

so that $(\mathrm{d} f_H)_{\psi}$ is bounded on $D(A)$. Since $D(A)$ is a dense subspace of $\mathcal{H}$ and $(\mathrm{d} f_A)_\psi(\dot{\phi})$ is bounded on $D(A)$, then, in view of the Hahn-Banach theorem, $(\mathrm{d} f_A)_{\psi}$ may be continuously extended to a linear continuous form on $\mathrm{T}_{\psi}\mathcal{H}$. Since $D(A)$ is dense in $\mathcal{H}$, this extension is unique. □

The above lemma shows that $f_A$ is Fréchet differentiable at every point of $D(A)$ and it allows us to introduce the following definition:

Definition 7.2. Let $f_H$ be the average value of a symmetric operator

Its differential is the mapping

where

Definitions from symplectic geometry can be adapted to the context of quantum mechanics by assuming that they must be restricted appropriately to domains.

Definition 7.3. We say that a vector field on a domain

is Hamiltonian relative to a symplectic manifold $(\mathcal{H}, \omega)$ if

where the right-hand side is understood as the Gâteaux differential of a Fréchet differentiable function

which is defined at points of $D$.

It is worth noting that the mapping $\mathrm{d} h_\psi$ does not need to exists for $\psi\notin D(H)$, while the contraction is in points of the domain of $X$.

Proposition 7.4. The vector field with a domain $X_{-{{{\rm{i}}}} H}$ on $\mathcal{H}$ induced by a symmetric operator

where $D(H)$ is a domain of $H$, is Hamiltonian relative to the Hamiltonian function

Proof. Using the expression for the canonical symplectic form on $\mathcal{H}$, we obtain, in view of Lemma 7.1, that

□

8.   On $t$–dependent Hamiltonian operators

To make a differential geometric approach to $t$-dependent SEs possible, we present a specific type of $t$-dependent SEs to be studied. In particular, we are concerned with the ones of the form

where the $t$-dependent operator $H(t)$ are determined by the following proposition:

Definition 8.1. A $t$-dependent Hamiltonian operator is a $t$-parametrised family of self-adjoint operators $H(t)$ on a Hilbert space $\mathcal{H}$ such that all the $H(t)$ admit a common invariant domain $\mathcal{D}$ of analytic vectors in $\mathcal{H}$.

Note that we do not impose each $H(t)$ to be self-adjoint on $\mathcal{D}$. Since $\mathcal{D}$ will be a subset of the domains $D(H(t))$, with $t\in \mathbb{R}$, where each $H(t)$, for a fixed $t$, is self-adjoint for a certain domain $D(H(t))$, the operators $H(t)$ are only symmetric on $\mathcal{D}$. Definition 8.1 is satisfied by relevant physical systems. The following example shows one of them.

Example 8.2. The quantum operators

are self-adjoint operators on $L^2(\mathbb{R})$ with respect to appropriate domains ^[27,38,57]. The operators $H_1, \ldots, H_6$ span a Lie algebra of symmetric operators on a common domain $\mathcal{D}$ of functions of the form $e^{-x^2/2}P(x)$, where $P(x)$ is any polynomial ^[57]. The space $\mathcal{D}$ admits an algebraic basis $e^{-x^2/2}H^p_n(x)$ with $n = 0, 1, 2, \ldots$, where the $H^p_n(x)$ are the so-called Hermite probabilistic polynomials. The functions $e^{-x^2/2}H_n^p(x)$ constitute a basis of $L^2(\mathbb{R})$ and, then, $\mathcal{D}$ is dense in $L^2(\mathbb{R})$. The operators $H_1, \ldots, H_6$ are symmetric on $\mathcal{D}$. Moreover, $\mathcal{D}$ is also invariant relative to the action of $H_1, \ldots, H_6$. Hence, every $t$-dependent Hamiltonian operator on $L^2(\mathbb{R})$ of the form

where the functions $b_\alpha(t)$ are any real $t$-dependent functions, is a $t$-dependent Hamiltonian operator with a domain $\mathcal{D}$.

The $t$-dependent Hamiltonian (8.2) embraces $t$-dependent harmonic oscillators and other related systems ^[17]. Note that

is a Lie algebra of skew-symmetric operators on $\mathcal{D}$. Moreover, if one restricts to the functions

where $m = 0, 1, 2, 3, \ldots$, one finds after a simple calculation that these elements are analytic for $H_4, H_5, H_6$. The constant bound for a vector $\psi_m$ according to the Definition 6.1 of analytic vector could be, for instance

for all the operators ${{{\rm{i}}}} H_4$–${{{\rm{i}}}} H_6$. Then, $\mathcal{D}$ consists of analytic vectors for ${{{\rm{i}}}} H_4$–${{{\rm{i}}}} H_6$. Moreover, it can be proved that $H^p_m(x)e^{-x^2/2}$ are also analytic for ${{{\rm{i}}}} H_1$–${{{\rm{i}}}} H_3$. Hence, the elements of $\mathcal{D}$ are analytic for ${{{\rm{i}}}} H_1, \ldots, {{{\rm{i}}}} H_6$. Therefore, ${{{\rm{i}}}} H_1, \ldots, {{{\rm{i}}}} H_6$ can be integrated giving rise to the so-called metaplectic representation of the metaplectic group.

It is worth noting that $\mathcal{D}$ is not invariant relative to the operators of the form $\exp({{{\rm{i}}}} \lambda H_\alpha)$ for any real $\lambda$. For instance,

and the latter does not belong to $\mathcal{D}$. Recall also that the operators $\exp({{{\rm{i}}}} \lambda H_4)$ exist because the given operators are self-adjoint for an appropriate domain.

Example 8.3. Consider the space

and the operators

Using an appropriate basis of polynomials in $x, y, z$, one can prove that $\mathcal{D}$ is a space of analytic vectors for the above operators.

The $t$-dependent SEs on $\mathcal{H}$ describe the dynamics of quantum systems and the elements of $\mathcal{H}$ represent its quantum states. Under the conditions given in Definition 8.1 on the associated $H(t)$, we can prove that the $t$-dependent SE can be considered as a Hamiltonian system.

Using the diffeomorphism

we can understand the solutions to (8.1) as curves in $\mathcal{H}$ whose tangent vectors at $\psi(t)$ are given by

Note that this equality makes sense only in the space $\mathcal{D}$ which is included in the space of smooth vectors for all the operators $-{{{\rm{i}}}} H(t)$ for $t\in \mathbb{R}$.

Finally, Proposition 7.4 ensures that the $t$-dependent vector field $X$ is, for every $t\in \mathbb{R}$, a Hamiltonian vector field with Hamiltonian function $\frac12\langle\psi|H(t)\psi\rangle$. As a consequence, we obtain the following theorem:

Theorem 8.4. A $t$-dependent SE related to a $t$-dependent Hamiltonian $H(t)$ is the differential equation for the integral curves of the $t$-dependent vector field on a domain associated with the $t$-dependent Hamiltonian system $(\mathcal{H}, \omega, \frac12\langle\psi|H(t)\psi\rangle)$.

The domain $\mathcal{D}$ of analytic vectors is not invariant relative to the exponential of the operators $H(t)$ for $t\in \mathbb{R}$. This happens, for instance, in Example 8.2. There are other many case where this can be assumed.

Definition 8.5. A quantum $t$-dependent Hamiltonian system is a Hamiltonian system of the form $(\mathcal{H}, \omega, \frac12\langle \psi| H(t)\psi\rangle)$, where $H(t)$ is a $t$-dependent Hamiltonian operator on $\mathcal{H}$.

Hence, the above states that the $t$-dependent SEs defined by a $t$-dependent Hamiltonian are the Hamilton equations of the $t$-dependent Hamiltonian.

9.   Reduction process for t-dependent SEs

In this section, we describe the projection of a $t$-dependent SE determined by a $t$-dependent Hamiltonian operator, which satisfies appropriate technical conditions, onto a projective space via a MW reduction.

Recall that the projective space, ${\mathcal{PH}}$, of a Hilbert space $\mathcal{H}$ is the space of equivalence classes of proportional non-zero elements of $\mathcal{H}$. Every equivalence class in ${\mathcal{PH}}$ is called a ray of $\mathcal{H}/\{0\}$. Physically, the projective space ${\mathcal{PH}}$ is what is really relevant, as different elements of $\mathcal{H}$ belonging to the same ray have the same physical meaning ^[56].

More particularly, let us apply the MW reduction to a $t$-dependent SE determined by a $t$-dependent Hamiltonian operator $H(t)$ on a Hilbert space $\mathcal{H}$ to describe its projection onto the projective space $\mathcal{PH}$ as a $t$-dependent Hamiltonian system relative to a reduced symplectic form. To accomplish the process for $t$-dependent SEs related to unbounded operators, we first define what an invariant function is. As in previous sections, to define it, one considers an appropriate restriction of the infinite-dimensional Hilbert space to an appropriate normed domain, as it happens in quantum mechanics. Thus, it is convenient to recall what we mean by a symmetry of a function relative to a Lie group action.

Definition 9.1. Let

be a Lie group action and let $D$ be a dense complex subspace of $\mathcal{H}$. A $t$-dependent function

is said to be invariant under $\Phi$ (or $G$-invariant when $G$ is known from context) if

for all $g\in G$ and every $t\in\mathbb{R}$.

Implicitly, Definition 9.1 implies that $\Phi_g(D)\subset D$ for every $g\in G$. Otherwise, if $\Phi_g(\psi)\notin D$ for certain $g\in G$ and $\psi\in D$, it would follow that

since the left-hand side of the equality would not be defined. Naturally, one says that a $t$-independent function

is $G$-invariant relative to a Lie group action

if $\jmath^*f$, where

is $G$-invariant in the sense given in Definition 9.1.

Example 9.2. Consider the multiplicative group

and the Lie group action

Let us prove that if $H$ is a symmetric operator with domain $D(H)$, then its average function $f_H$ is $U(1)$-invariant. In fact, the domain $D(H)$ is a $\mathbb{C}$-linear subspace of $\mathcal{H}$. Moreover,

Therefore, $f_H$ is invariant relative to $\Phi_{e^{{\rm i}\theta}}$ for every $e^{{\rm i}\theta} \in U(1).$ Recall that since $H$ is symmetric, $f_H$ is a real function. Consequently, $f_H$ is a $U(1)$-invariant function. It is worth noting that, if $\psi\in D(H)$, then $e^{{{{\rm{i}}}} \theta}\psi\in D(H)$ for every $\theta\in \mathbb{R}$ and $\Phi_g(\psi)\in D(H)$ for every

Hence, $D(H)$ is invariant under $\Phi$.

Let us now reduce a $t$-dependent SE (8.1) associated with a $t$-dependent Hamiltonian operator $H(t)$ on a separable Hilbert space $\mathcal{H}$ onto its projective Hilbert space, $\mathcal{P}\mathcal{H}$, via a MW symplectic reduction.

Consider global real-valued coordinates $\{q^j, p_j\}_{j\in\mathbb{N}}$ on $\mathcal{H}$ as mentioned in Section 5. The initial data for the reduction is given by

where the $b_1(t), \ldots, b_r(t)$ are arbitrary $t$-dependent real-valued functions and $H_{\alpha}$, with $\alpha = 1, \dotsc, r$, are self-adjoint operators such that all elements of $\langle H_1, \ldots, H_r \rangle$ share a common domain of analytic vectors $D\subset\mathcal{H}$. We stress that we do not require $H_1, \ldots, H_r$ to be self-adjoint on $D$. Moreover, note that $\{{\mathrm{d}} q^j, {\mathrm{d}} p_j\}_{j\in\mathbb{N}}$ is a basis of $\mathrm{T}^*\mathcal{H}$ and every element of each $\mathrm{T}^*_{(q, p)}\mathcal{H}$ is the limit of a sum of finite linear combinations of elements of $\{{\mathrm{d}} q^j, {\mathrm{d}} p_j\}_{j\in\mathbb{N}}$.

The $t$-dependent SE (8.1) induced by $H(t)$ defines a $t$-dependent Hamiltonian vector field on $\mathcal{H}$ with a domain $D\subset \mathcal{H}$ of the form

such that

for all $\alpha = 1, \dotsc, r$ and $\psi\in D$. We will hereafter consider the reduction of $\omega$ to the projective space, and then the reduction of the $t$-dependent SE associated with $H(t)$.

9.1.   The MW reduction of the symplectic form on $\mathcal{H}$

Consider the Lie group action

This Lie group action is symplectic relative to the symplectic form (5.2), as proved in Example 4.14.

To obtain the momentum map associated with $\Phi$, we must relate every fundamental vector field of $\Phi$ to a Hamiltonian function. The Lie algebra $\mathfrak{u}(1)$ of $U(1)$ consists of pure imaginary numbers, that is

The fundamental vector field of $\Phi$ generated by ${{{\rm{i}}}}\in\mathfrak{u}(1)$ reads

This is the vector field associated with the skew-self-adjoint and bounded operator

In view of these results and Proposition 7.4, an appropriate momentum map $J$ corresponding to the action $\Phi$ is given by

so that the values of $J$ at every $\psi\in\mathcal{H}$ are proportional to the square of the norm of $\psi$. Let us prove that $J$ is smooth. The directional derivative of $J$ in the direction ${\psi}$ at $\psi_0$, namely $\dot\psi_{\psi_0}J$, becomes

This gives rise to a linear and bounded Gateaux differential of $J$ on $\mathcal{H}$ for each $\psi_0\in\mathcal{H}$. Hence, $J$ is differentiable (see ^[19,25,50] for details). Let us write $\nabla_{\dot\psi}J$ for the partial derivative of $J$ in the direction $\dot\psi$. The second-order directional derivative of $J$ along $\phi$ and $\psi$ is constant:

Hence, all second-order directional derivatives are constant and continuous relative to $\dot\psi, \dot\phi$. Higher-order derivatives are zero. Thus, $J$ is smooth.

Let us show that $\Phi$ is ${\rm{Ad}}^*$-equivariant. Since $U(1)$ is an Abelian Lie group, its adjoint action ${\rm{Ad}}$ is trivial, that is

for every $g\in U(1)$. Then, the coadjoint action is also trivial. Moreover, $J$ is invariant with respect to the action of $U(1)$ on $\mathcal{H}$, since unitary transformations do not change the norm of elements of the Hilbert space $\mathcal{H}$. Therefore,

Thus, the momentum map $J$ is ${\rm{Ad}}^*$-equivariant.

Let us prove that every $\mu < 0$ is a regular value of $J$. Since $J$ is smooth, it is enough to prove that

is surjective with a split kernel for all non-zero elements $\psi\in\mathcal{H}$ (see ^[16]). The tangent map of $J$ at $\psi\in\mathcal{H}$ is

Note that $\mathrm{T}_{\psi}J\neq 0$ if and only if $\psi\neq 0$. Therefore,

for $\psi\neq 0$ and thus $\mathrm{T}_{\psi}J$ is surjective if and only if $\psi\in \mathcal{H}\backslash\{0\}$. Since $J$ is smooth, $\mathrm{T}_{\psi}J$ is a continuous map, $\ker \mathrm{T}_{\psi}J$ is a closed subspace of $\mathrm{T}_{\psi}\mathcal{H}$, which becomes a Hilbert space relative to $\mathfrak{Re}\langle \cdot\vert\cdot\rangle$. Moreover, $\mathrm{T}_{\psi}J$ has a split kernel, since

where $(\ker \mathrm{T}_{\psi}J)^{\perp}$ is a closed subspace. Hence, all $\mu < 0$ are regular values of $J$, and, for these values, we have the submanifolds

As mentioned before,

for every $g\in U(1)$. Therefore, the isotropy group at $\mu$ is

Let us prove that $U(1)$ acts freely and properly on $J^{-1}(\mu)$. The condition

for some $\theta\in [0, 2\pi)$, implies that

so that $e^{{{{\rm{i}}}}\theta} = 1$ is the identity element of $U(1)$ and, as a result, $\Phi$ is a free action. Moreover, $U(1)$ is homeomorphic to the group

Therefore, $U(1)$ is a compact Lie group. In view of Proposition 3.9, the Lie group action $\Phi$ is also proper. Consequently, Theorem 3.8 ensures that

is a smooth manifold for all $\mu < 0$. Note that if

is such that the images of two elements of $J^{-1}(\mu)$ under $\pi_\mu$ are the same, both elements are the same up to a proportional complex constant with module one. This amounts to saying that the images of two elements of $J^{-1}(\mu)$ are the same if their images under

are the same. As

one can indeed identify

and see that

Since the differentiable structure on $P_{\mu}$ is one making

into a differentiable submersion and

the smooth structure on $P_\mu$ is the one of

Hence,

is a smooth manifold for all $\mu < 0$.

Note that all assumptions of the MW reduction theorem are satisfied for $\mu < 0$. We may now implement the reduction procedure for the symplectic form. Let us rewrite $\pi_{\mu}$ as

and consider the natural embedding

The symplectic form on the projective space $\mathcal{PH}$ is the only symplectic form on this latter manifold, let us say $\omega_\mu$, such that

In other words,

9.2.   MW reduction of SEs

Consider the $t$-dependent Schrödinger equation associated with (9.1). The vector fields

with $\alpha = 1, \ldots, r$, are tangent to $J^{-1}(\mu)$ at the points where they are defined. In fact, the self-adjointness of $H_\alpha$ for each $\alpha = 1, \ldots, r$, implies that, on points of $D$, we have

Then, the restriction of $X_\alpha$ to $J^{-1}(\mu)$ is a well-defined vector field: it is not only tangent where defined, but it is defined on a subset that is dense in $J^{-1}(\mu)$. Let us prove this.

Since $D$ is dense in $\mathcal{H}$, for every $\psi\in J^{-1}(\mu)$ with $\mu\neq 0$, there exists a sequence $(\psi^D_n)_{n\in\mathbb{N}}$ of non-zero elements of $\mathcal{H}\backslash\{0\}$ that belong to $D$, such that $\psi^D_n \rightarrow \psi$. Note that

If $\psi^D_n \rightarrow \psi$, then $(\sqrt{k}\psi_n^D / \Vert \psi^D_n \Vert)_{n\in\mathbb{N}}$ is a sequence in $J^{-1}(\mu)\cap D$, such that

Consequently,

is dense in $J^{-1}(\mu)$ and

The flow of $X_\alpha$, namely

leaves invariant $J^{-1}(\mu)$. Additionally, it commutes with every $\Phi_g$, since each $\Phi_g$ is a multiplication by a phase and the $\exp(-t{{{\rm{i}}}} H_\alpha)$ are linear mappings for every $t\in \mathbb{R}$. This gives rise to an induced flow on $P_\mu$ of the form

Let us write $X_ \alpha^\mu$ for the restriction of $X_\alpha$ to $J^{-1}(\mu)$, which is tangent to this submanifold. Since the projection $\pi_{\mu}$ amounts to the restriction of $\pi$ to $J^{-1}(\mu)$, and $X^\mu_\alpha$ is the restriction of $X_\alpha$ to $J^{-1}(\mu)$, one obtains that the projection of $X^\mu_\alpha$ onto $J^{-1}(\mu)/U(1)_\mu$ is the projection of $X_\alpha$ on $\mathcal{H}_0$ onto $\mathcal{PH}$.

To prove that the projection of $X^\mu_\alpha$ on $\mathcal{H}_0$ onto $P_\mu$ is a vector field $K_\alpha^\mu$, one has to prove that such a projection is well defined on a dense subset of $P_\mu$. In fact, Theorem 3.8 shows that

is a surjective submersion. Therefore,

and $\pi_{\mu}(D_{\mu})$ is dense in $\mathcal{PH}$. We conclude that $K^\mu_{\alpha}$ is well-defined with a domain being a dense subset of $\mathcal{PH}$.

Let us prove that $K^\mu_\alpha$ is a Hamiltonian vector field on $P_\mu$ with a Hamiltonian function given by

for $\alpha = 1, \dotsc, r.$ In fact,

Then, $f_{{\alpha}}^{\mu}$ is the Hamiltonian function of $K^\mu_\alpha$.

The above shows how the $t$-dependent vector field $X$ on the domain $D$ gives rise to the $t$-dependent vector field

on $J^{-1}(\mu)\cap D$ projecting onto a $t$-dependent vector field

on

whose integral curves are determined by the $t$-dependent projective SE.

Since $K^{\mu}$ is defined on a manifold $D_{\mu}$ modeled on a normed space, we cannot ensure straightforwardly the existence of its integral curves. Nevertheless, their existence is inferred from the existence of the integral curves of the $t$-dependent vector field $X$.

10.   Conclusions and outlook

We present a careful symplectic geometric approach to the SEs on separable Hilbert spaces determined through $t$-dependent unbounded self-adjoint Hamiltonian operators satisfying a quite general condition. The projection of such equations onto the projective space has been described as a MW reduction. We describe carefully the necessary technical geometric and functional analysis details to accomplish the description of the results given. This has been accomplished in a mainly differential geometric way and by avoiding complicated functional analysis techniques.

In particular, the most relevant novelty of our paper is to propose the theory of analytic vectors to deal with the problem of non-smoothness of geometric structures in the infinite-dimensional setting, which is intrinsic to $t$-dependent Schrödinger equations. We have also shown that analytic vectors can be used in a practical manner to study rigorously particular relevant physical examples. Moreover, our work provides a brief introduction and state-of-the-art to the study of symplectic manifolds modeled on infinite-dimensional topological spaces of different types and their application to quantum mechanics.

There is much room for additional developments. In the future, we aim to study how to extend to the infinite-dimensional context using several geometric techniques to deal with SEs (see ^[4,60]). Moreover, other types of MW reductions will be contemplated. We expect to analyze the Hamilton-Jacobi theory in infinite-dimensional symplectic manifolds.

Author contributions

Lucas suggested most initial ideas, techniques, and examples for the work. He also collaborated in writing the work. Rivas contributed to improving all the ideas, techniques, and examples provided by Lucas. He also collaborated in writing the work. Lange contributed to improving all the ideas, techniques, and examples provided by Lucas. She also carefully verified the calculations. Additionally, she collaborated in writing all parts of the work. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

Lucas acknowledges financial support by the project number 1082 of the University of Warsaw. Lange was partially financed by the project MTM2015-69124-REDT entitled "Geometry, Mechanics and Control" financed by the Ministerio de Ciencia, Innovación y Universidades (Spain). Lucas, Lange and Rivas were partially financed by project IDUB with number PSP: 501-D111-20-2004310. Lucas acknowledges a Simons-CRM professorship. Rivas acknowledges funding from Lucas's CRM-Simons professorship, the Spanish Ministry of Science and Innovation, grants PID2021-125515NB-C21, and RED2022-134301-T of AEI, and Ministry of Research and Universities of the Catalan Government, project 2021 SGR 00603, to accomplish a stay at the CRM which allowed him to complete some of the main parts of the present work. We would like to thank the anonymous referees for reading carefully our work, giving constructive critics that allowed us to improve our manuscript, and point out potential directions of further research.

Conflict of interest

The authors have no competing interests relevant to the content of this article.