Schrodinger vs Hamilton: A Structural Analogy

Seeking similarities and differences when trying to understand quantum mechanics

I’m trying to learn some Quantum Field Theory, and part of the process involves thinking harder about regular old quantum mechanics from different viewpoints. Here’s an extended analogy between the ‘Schrodinger Picture’ of quantum mechanics and the ‘Hamiltonian Formalism’ of classical mechanics.

Basic Terms of the Theories

States

• Classical mechanics The potential states of a system form a manifold called phase space (often the cotangent bundle to a configuration space, but this is not necessary). A point in phase space specifies the entirety of a specific state of the physical system at a given moment in time.
• Quantum mechanics: The potential states of a system form a vector space we’ll call state space (often $L^2$ of a configuration space, but this is not necessary). A vector in this space specifies the entirety of a specific state of a physical system at a given moment in time.

Physics

• Classical mechanics: The general idea of ‘physics’ is encoded by an additional structure; the symplectic form $\omega$. (This lets us talk about lagrangian submanifolds, making distinctions between things like ‘position’ and ‘momentum’).
• Quantum mechanics: The general idea of physics is encoded by additional structure: an inner product turning this into a Hilbert space (this lets us talk about things like the Fourier Transform at least in $L^2$ examples, making distinctions between “position” and “momentum”).

Dynamics

• Classical mechanics: The dynamics are given by a flow on phase space. This flow preserves the physics, so must be by symplectomorphisms. In general we require a slightly stronger condition, that it is Hamiltonian (loosely, that its derivative is exact not just closed).
• Quantum mechanics: The dynamics are given by a flow on state space. This flow preserves the physics, so must be by unitary transformations. (And, a small technical point: because of difficulties in infinite dimensions, we require that it is strongly continuous.)

Measurement

• Classical mechanics: Observable quantities about a system are modeled by real valued functions on phase space, taking a state (point on the manifold) to a real number (the value of that quantity: eg energy, or momentum in a given direction).
• Quantum mecahnics: Observable quantities about a system are represented by hermitian operators on state space. The possible values of a quantity are the eigenvalues of the given operator. These operators take a state to a probability distribution on the eigenvalues, giving the probability of observing that given value upon measurement.

Quantitative Dynamics

This description seems rather bare-boned; particularly it seems we don’t know nearly enough to understand what time evolution looks like. But in fact we do!

Classical Mechanics

• Denote the time evolution flow of symplectomorphisms by $S(t)$. Differentiating the flow $S$ gives a vector field $X=\frac{d}{dt}S$ on phase space, and a corresponding dynamical law $\frac{d}{dt} = X$.
• Any symplectic action has $\iota_\omega X$ closed; the further constraint that our flow be Hamiltonian is just the requirement that $\iota_\omega X$ is exact.
• Thus, there exists some function $H$ where $\iota_\omega X = dH$. We will call this function the Hamiltonian (this only defines up to an additive constant)
• This tells us that dynamics are fully specified by a choice of observable on $H$ phase space: making such a choice, we have Hamilton’s equations:

$\frac{d}{dt}= X_H\hspace{1cm}\textrm{for}\hspace{1cm}\iota_\omega X_H = dH$

Quantum Mechanics

• Denote the time evolution flow of unitary transformations by $U(t)$. This is a strongly continuous 1-parameter unitary subgroup of $\mathrm{Isom}(H)$.
• By the Stone theorem, there exists a self-adjoint operator $H$ (defined on a subspace of state space) such that we may write $U(t)=\exp(i t H)$
• Differentiating along a solution to the flow shows $H$ controls infinitesimal time translations: $\frac{d}{dt}U(t)\psi = \frac{d}{dt}\exp(itH)\psi = iH\exp(itH)\psi = iH U(t)\psi$
• This tells us that dynamics are fully specified by a choice of observable on $H$ state space: making such a choice, we have Schrodinger’s equation: $\frac{d}{dt}=iH$

Symmetries

Classical Mechanics

A symplectomorphism $g$ of the phase space $M$ is a symmetry of the classical system if it preserves time evolution. Precisely, if $S_t\colon M\to M$ is the time evolution operator, we requre $g.S_t(\sigma)= S_t(g.\sigma)$ This places some strong constraints on any Lie group $G$ of symmetries: if $g_s$ is a 1-parameter flow by symmetries (whose differential is the vector field $\Gamma$).

Quantum Mechanics

A unitary transformation $g$ of the state space $H$ is a symmetry of the quantum system if it preserves time evolution. Precisely, if $U_t\colon H\to H$ is the time evolution operator, we require $g.U_t(\psi)=U_t(g.\psi)$ This places some strong constraints on any Lie group $G$ of symmetries: if $g_s$ is a strongly continuous 1-parameter flow by symmetries (generated by a self adjoint operator $\Gamma$ as $g_s = \exp(is\Gamma)$ by Stone’s theorem).

Questions about this perspective

• Why do we need to require extra properties of time evolution (or behavior under symmetries) for classical mechanics? When the phase space is simply connected we don’t need any extra assumptions as the first homology is trivial, so closed forms and exact forms coincide. But in general this is necessary: what is the geometrical motivation for requring that time evolution be by a 1-parameter group whose derivative is exact? Is the reason we can get away with only requiring unitarity in quantum mechanics that the Hilbert space is contractible, and thus has trivial first homology? Or are these unrelated considerations?
• Does quantum mechanics really live in a Hilbert space, or in a projective space? The observable quantities (probability distributions) depend not on the specific vector but only on its span. The flow in hilbert space is unitary, meaning it preserves norm (but does modify phase).
• Is this the right perspective on quantum symmetries? Again it seems we have a phase issue going on: we need to recover projective representations, not unitary representations of the symmetry groups.
• The two underlying geometric worlds are not the same (symplectic manifolds, vs Hilbert spaces). But Hilbert spaces are (very simple, though infinite dimensional) symplectic manifolds if thought of as a real vector space with a complex structure. Is there a way to make sense more rigorously of the analogy between Schrodinger’s equation and Hamilton’s equation?
• Is part of the difficulty here that usual classical systems are finite dimensional whereas usual quantum systems are infinite dimensional? Of course there are exceptions (spin), but would the analogy be closer if we looked at classical field theory vs quantum mechanics, where the classical phase space is a vector space of field profiles, instead of a nonlinear configuration space?