A philosophical analysis on Weinstein's works, Pt. 1

Introduction

Recently, I came across certain comments on Weinstein's work concerning the symplectic formalism of classical physical systems that not only adopted a naive realist (ontological) interpretation of it but also presented a completely simplistic and erroneous understanding of the underlying physical and mathematical concepts. In light of this, my aim here is to clarify his work by presenting and discussing it in a concise yet rigorous manner. Accordingly, this series will be divided in three parts. The first will focus on canonical geometric quantization; the second will explore Weinstein's contributions to geometric quantization from a categorical perspective; and the third and final part will address the philosophical implications and interpretations of his work. In particular, in this first part, I will introduce some basic concepts on sympletic geometry and a brief review on classical mechanics, in the context of sympletic geometry, and on quantum mechanics, in the context of ${C}^*-$Algebras.

A primer on Sympletic Geometric

Let $M$ be any topological space. We define a chart on $M$ as the pair $(U, \phi)$, where $U \subset M$ is open and $\phi: U \rightarrow M$ is such that $\phi(U) \subset \mathbb{R}^n$ is open. Given a collection $\{U_i, \phi_i\}$ of those charts, we say that it is an atlas $\mathcal{A}$ on $M$if they are a cover for $M$, i.e., if \[\bigcup_{i \in [1,n]} U_i = M\] and if all of it's charts are compatible with each other, i.e., if for any $i$, $j$ such that $U_i \cap U_j \neq \emptyset$, we have that $\phi_{i,j}: U_{i,j} \rightarrow \phi_{i,j}(U_{i,j}) \in C^\infty(M)$ and that $\phi_i \circ \phi_j^{-1}: \phi_i (U_i \cap U_j) \rightarrow \phi_j (\phi_i \cap \phi_j) \subset \mathbb{R}^n$ and $\phi_j \circ \phi_i^{-1}:\phi_j (U_i \cap U_j) \rightarrow \phi_i (U_i \cap U_j)\subset \mathbb{R}^n $, given $ \phi_{i,j} (U_i \cap U_j)$ open subsets of $\mathbb{R}^n$. We say that an atlas $\mathcal{A}$ is maximal for $M$ if, for any other atlas $\mathcal{A}'$, we have that if $\mathcal{A} \cup \mathcal{A}'$ is an atlas for $M$, then $\mathcal{A}' \subseteq \mathcal{A}$. Now, suppose that $\text{dim}(M)=n$. The map $\phi_i = (x^1, \dots, x^n)$ is smooth on $M$ and define the local chart $(x^1, \dots, x^n)$ on $U_i$. Furthermore, anynyan who has studied undegraduate level physics or linear algebra had, at some point, to change from one local chart to another (from, e.g., cylindrical to polar); to do so in the context of manifolds, we use the composite of two maps that compose two distinct charts, i.e. \[ \phi_i \circ \phi_j^{-1} (x) = (y^1(x^1,\dots,x^n), \dots, y^n (x^1,\dots, x^n)) \] Finally, $M$, together with it's maximal atlas, is what we define as a smooth manifold. Next, we discuss four important topics concerning differential geometry, namely smooth functions on manifolds, smooth maps between manifolds, tensors and differential forms. First of all, a function $f: M \rightarrow \mathbb{R} $ is said to be smooth if, for every $p \in M$ and for every $i \in [1,n]$, there is a chart $(U_i, \phi_i)$ with $x \in U_i$ such that $f_i \doteq f \circ \phi^{-1}_i: \phi_i (U_i) \rightarrow \mathbb{R}$ is a smooth function on the open set $\phi_i (U_i)$. Other than that, we can represent a smooth function in local coordinates via the function $\hat{f}: \phi(U) \rightarrow \mathbb{R}^n$, given by $\hat{f}(p) = f \circ \phi^{-1} (p)$; secondly, we can generalize that notion of smooth functions to smooth maps between manifolds. Let $M$ and $N$ be smooth manifolds. We say that the map $F: M \rightarrow N$ is smooth if, for every $p \in M$, there exist smooth charts $(U,\phi)$, containing $p$ and $(V, \psi)$ containing $F(p)$ such that $F(U) \subseteq V$ and the composite map $\psi \circ F \circ \phi^{-1}$ is smooth from $\phi(U)$ to $\psi(V)$. A primer example of a smooth maps between smooth manifolds are diffeomorphisms, that can be used to characterize sameness on a dimensional level of two smooth manifolds, via the following theorem:

A nonempty smooth manifold of dimension $m$ cannot be diffeomorphic to an $n$-dimensional smooth manifold unless $m = n$.

and on a boundary level via the theorem:

Suppose $M$ and $N$ are smooth manifolds with boundary and $F:M \rightarrow N$ is a Diffeomorphism. Then, $F(\partial M) = \partial N$ and $F$ restricts to a Diffeomorphismfrom $\text{Int} M $ to $\text{Int} N$.

Being honest with you, we will not use any of these results in what follows, but I found it interesting to state them; thirdly, a tensor of rank $(p, q)$ is an algebraic objetic defined as a multilinear map

\[T: \underbrace{V^* \times \dots \times V^*}_{p \; \text{times}} \times \underbrace{V \times \dots \times V}_{q \; \text{times}} \rightarrow \mathbb{R}\]

given $V$ a vector space $-$ for intance, the soon to be defined $T_pM$ and his big brother $TM$ are vector spaces $-$ and $V^*$ it's dual, i.e. the space of linear maps $\beta: V \rightarrow \mathbb{R}$, which is himself a vector space; lastly, we define a differential form as an special type of tensor, in the sense that a differential $p$-form $\omega$ is a completely antisymmetric $(0,p)$-tensor, i.e., a tensor that for any $i$ and $j$ satisfies $\omega (x^1,\dots, x^i, \dots, x^j, \dots, x^p) = - \omega (x^1,\dots, x^j, \dots, x^i, \dots, x^p) $. The space of these forms is often denoted by $\Omega^p(M)$.

Thereafter, consider $p \in M$. A linear map $\gamma: C^\infty(M) \rightarrow \mathbb{R}$ is called a derivation at $p$ if it satisfies $\gamma(fg) = f(p)\gamma g + g(p)\gamma f$, for all $f,g \in C^\infty(M)$. The set of all smooth derivations at $p$, denoted by $T_p M $, is what we define as the tangent space to $M$ on $p$. With that, we can now define a tangent bundle of $M$ as

\[ TM \doteq \bigsqcup_{p \in M} T_p M \]
The dual of the tangent bundle, in the sense we defined above, is the cotangent space $(T_pM)^*$ at $p$ and, similarly, the bundle $(TM)^*$ is the disjoint union of all of these spaces. One important feature of $(TP)^*$ is the natural projection map $\pi: (TM)^* \rightarrow M$, that send $\gamma \in (T_pM)^*$ to $p \in M$. The canonical basis for $(TM)^*$ are given with respect to the canonical basis of $M$, as $\{\partial/\partial x^i \} =(\partial/\partial x^1, \dots, \partial/\partial x^n)$. In order to bring the abstract tangent spaces discussed above into the practical world, we will discuss the differential of smooth maps on manifolds and some adjacent results. That being said, consider two smooth manifolds $M$ and $N$ with $\text{dim}(M, N) = n$ and a map $F: M \rightarrow N$. We then define the map

\[ dF_p: T_pM \rightarrow T_{F(p)}N \]

as the differential of $F$ at $p$. This means that, given $\gamma \in T_p M$ and $f \in C^\infty(M)$, we have that $dF_p(\gamma)(f) = \gamma (f \circ F)$. Noting that

\[ dF_p(\gamma)(fg) = \gamma ((fg) \circ F) = \gamma((f \circ F)(g \circ F)) = f \circ F(p) \gamma (g \circ F) + g \circ F(p) \gamma (f \circ F) = f(F(p))dF_p(\gamma)(g) + g(F(p))dF_p (\gamma)(f) \]

nyan can conclude that $dF_p (\gamma): C^\infty(N) \rightarrow \mathbb{R}$ is a derivation at $F(p)$. The differential have the following properties: In what follows, I will state two important propositions that will help us bring that abstract concept of tangent and cotangent spaces down to Earth:

The basics on Quantum and Classical Mechanics

In today's physics, a classical physical system is modelled as a sympletic (or multisympletic) manifold $ (M, \omega)$ equipped with an smooth map $H: M \rightarrow \mathbb{R}$ that encapsulates the dynamic of the system

Geometric Quantization

In geometric quantization, the goal to be achieved is to find a quantization map $Q: C^\infty(M) \rightarrow \mathcal{L}(\mathcal{H})$ such that it satisfies the Dirac's quantization conditions, namely

$Q_{f_1 + f_2} = Q_{f_1} + Q_{f_2} \; \text{and} \; Q_{cf_1} = c Q_{f_1}, \; \text{given} \; c \in \mathbb{R}$;
$ Q_1 = \text{id}_\mathcal{H} $;
$Q_{\bar{f}} = (Q_f)^*$;
$ [Q_f, Q_g] = -i \hbar Q_{\{f,g\}} $;
$f_1,...,f_n \; \text{complete} \Rightarrow Q_{f_1}...Q_{f_n} \; \text{complete}$.

However, it is not possible, at least not in this context; the best we can do is to define Q such that it satisfies the first four conditions. What his means is that in what follows, $Q$ will never satisfy such condition, but the attempt to do so will return some interesting results. That being said, given a sympletic manifold $(M, \omega)$, it's quantization starts by the step known as prequantization. To do so, we start by defining a complex line bundle over $M$, as follows:

Let $(M, \omega)$ be a sympletic manifold. The pair $(L, \nabla)$, given $\sigma_\alpha: U_\alpha \rightarrow M$ a section of $L$ and \[ (\nabla_X \sigma)_\alpha = X(\sigma_\alpha) - \frac{i}{\hbar} \theta_\alpha(X)\sigma_\alpha \] $\forall X \in \mathfrak{X}_\mathbb{C}(M)$, is defined as a complex line bundle over $M$.

Next, we define the condition on which a manifold can possess a prequantum line bundle, in other words, the condition for a manifold M to be quantizable. This conditions is called Weil integrality condition, given by:

Let $(M, \omega)$$ be a symplectic manifold. Then there exists a prequantum line bundle if and only if for every closed 2-dimensional submanifold $\Sigma$ of $M$ we have \begin{equation} \int_\Sigma \omega \in 2\pi\hbar\mathbb{Z} \end{equation} i.e. $[\omega] \in H^2(M, 2\pi\hbar\mathbb{Z})$.

If that conditions holds for a manifold M, then it can be characterized as the Hilbert space $\mathcal{H}^{\text{pre}} = L^2(L)$ and we can define the first ingredient on the recipe for construct $Q$ — the prequantization map —, as: \[ P_f \doteq - i \hbar \nabla_{X_f} + f \] As it is, $P_{f}$ satifies the first four Dirac's quantization conditions. In a attempt for it to satisfy the fifth one, we introduct two concepts: the polarization and the half-form. A polarization' goal is to select one half of the states in our Hilbert space, by selecting at every point $p \in M$ half of the directions in the tangent space $T_p M$ and asking the sections of the line bundle that is comprising our Hilbert space to be constant along these directions. The collection of these directions for a point $p$ is the space $\mathcal{P}_p \subset T_p M$, such that $\mathcal{P} = \bigcup_{p \in M} \mathcal{P}_p$. $\mathcal{P}$ must satisfy the polarization condition $\nabla_X \sigma = 0$, for all $X \in \Gamma(P)$, which implies that:

For all $X, Y \in \Gamma(P)$, the Lie bracket $[X,Y] \in \Gamma(P)$ - we say that $\mathcal{P}$ is involutive;
For all $p \in M$, $\mathcal{P}_p \subset T_p M$ is lagrangian. In other words, $\mathcal{P}$ is lagrangian.

So, in the end, our polarization $\mathcal{P}$ is just a involutive lagrangian subbundle of TM. However, it is usual to extend $\mathcal{P}$ by adding complex directions to it, such that it is now a subbundle of $T_p M \otimes \mathbb{C} = (TM)_\mathbb{C}$, instead of $TM$. The half-form $]delta_p$ is introducted to compensate for a problem with real polarizations, where the space of these polarizations, $\Gamma(\mathcal{P})$, was often empty. So, using $\delta_p$ we define a new line bundle $L_p = L \otimes \delta_p$ such that the product of two sections is naturally a volume form on the quotient. A mathematical rigorously discussion about the half-forms (and their theoretical construction in this context) would be too extensive, so I will leave it as it is. Take all about it for granted. That said, the sections of these bundles are called P-wave functions if they satisfy:

Suppose we have a symplectic manifold $(M,\omega)$ with prequantum line bundle $(L,\nabla,\langle\cdot,\cdot\rangle)$, a polarization $P$ and a half-form bundle $\delta_{P}$. Let $L_{P}=L\otimes\delta_{P}$. A section $\tilde{\sigma}=\sigma\otimes\psi\in\Gamma(L_{P})$ is called a $\textit{\(P$-wave function}\) if for all $X\in\Gamma(P)$ \[ (\nabla_{X} + \nabla^{\delta_{P}}_{X})(\tilde{s}) = \nabla_{X}\sigma \otimes \psi + \sigma \otimes \nabla^{\delta_{P}}_{X}\psi = 0. \]

In other terms, a P-wave functions is a polarized section of $L_p$ and, in a sense, they compose a refinement of the concept of well known square-integrable wave functions $\psi \in L^2(\mathbb{C})$, i.e., the traditional wave functions as exposed in quantum mechanics courses. To see how the product of two sections defines a volume you have to note (or verify) that $\langle \tilde{\sigma}_1, \tilde{\sigma}_2 \rangle_{L_p} = \langle \sigma_1, \sigma_2 \rangle (\varphi_1, \varphi_2) $ is invariant under the flow of any vector field $X \in \Gamma(P)$, and therefore defines a volume form on $Q = \frac{M}{D}$. Despite that efforts, $P$ still don't satisfy the fifth Dirac's quantization condition, but that allow us to define a new and more appropriate Hilbert space for simply connected leaves, as $\mathcal{H} = L^2(\Gamma_p(L_p))$, i.e., as the completion of $\Gamma_p(L_p)$. When treating non-simply connected leaves, we need to introduced yet another new concept: the cohomological wave functions. In general, given $\nabla$ a flat connection on a vector bundle $E$, we can define a complex form as $\Omega^*(M,E) = \Gamma(\Lambda^*T^*M \otimes E)$ and it's differential as $d^{\nabla} (\omega \otimes \sigma) = d\omega \otimes \sigma + \omega \wedge \nabla \sigma$ To extend these concepts to line bundles $L_0$, with a connection given by $\nabla_{L_p} = \nabla^L _+ \nabla^{\delta_p}$, we have to restrict differential forms to $P$. This is equivalent to say that $\omega_p \in \Omega_k (M , L_p)$ is $P$-closed if

\[ (d \nabla^{L_{P}} \omega_{P}) \bigg|_{p} = 0 \]

and that it is $P$-exact if there exists an $\alpha_p$ such that

\[ \left(\omega_{P} - d^{\nabla^{L}_{P}}\alpha\right)|_{p} = 0 \]

This is needed, basically, due to the fact that $\nabla^{L}$ is not flat and that $\nabla^{\delta_p}$ is ill defined. As P-exact forms are P-closed forms, we can define the cohomology group:

\[ H^k(M, P, L_P) := \frac{\Omega_p^{k_{P\text{-closed}}}(M, P, L_P)}{\Omega_p^{k_{P\text{-exact}}}(M, P, L_P)} \]

These groups defines the cohomological wave functions, an even more refined idea of wave functions, as they compose a generalization of P-wave functions as discussed above. Lastly, we enter the step of quantization. More precisely, the step in which we construct the quantization map and operator $Q$ and $\hat{Q}$, respectively, using all the other results obtained so far. That being said, given $f \in C^{\infty}(M)$ we define the quantization map using $P$ as:

Given $\tilde{\sigma} = \sigma \otimes \psi$ and $L_{p} = L \otimes \delta_{P}$, we define the map $Q_{f}: C^{\infty}(M) \to H^{0}(M, P, L_{P})$, as \[ Q_{f}(\tilde{\sigma}) := P_{f}(\sigma) \otimes \psi + \sigma \otimes (-i\hbar L_{X_{f}} \psi) \]

Which obeys the first four Dirac's quantization conditions, by construction. As a side note, we say that f is a quantizable function if $ [X, X_{f}] \in \Gamma(P)$, i.e., if the hamiltonian vector fields of $f$ are polarization-preserving. Then we define a quantization operator as Finally, we define (summarize would be a better term, tbh) a canonical geometric quantization — and, consequently, all the results obtained so far — in the following theorem:

Suppose we have a sympletic manifold $(M, \omega)$, together with a prequantum line bundle $(L, \nabla)$, a polarization $P$ and a bundle of half-forms $\delta_P$. Denote $\hat{\mathcal{H}} = H^{\bullet}(M,P,L_P)$ and $C^\infty(M)_P$ the collection of functions whose hamiltonian vector fields are polarization-preserving. Then, the map $\hat{Q}_f: C^\infty(M) \rightarrow H^{\bullet}(M,P,L_P)$, as defined above, satisfies the first, second and fourth Dirac's quantization conditions. Moreover, the space of smooth polarized sections of $L_p$, denoted by $H^{0}(M,P,L_P)$, has a pairing that defines our volume form, i.e., by $\langle \tilde{\sigma}_1, \tilde{\sigma}_2\rangle_{L_P} = \langle \sigma_1, \sigma_2(\psi_1, \psi_2)$, and the map $Q_f: C^\infty(M) \rightarrow H^{0}(M,P,L_P)$ as defined above satisfies the first four Dirac's quantization conditions with respect to this pairing.

The second part will be all about category theory and Weinstein's works.