Forget everything you know about canonical quantization

chill doggie

chill doggie

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The canonical quantization you're taught - replace {f,g} with [f, g]/iℏ - is, well, ill-defined. You can always add a classical 0 that becomes non-zero quantumly (thanks, non-commutation). But there's another way: geometric quantization.

It all starts with symplectic geometry. Think of a classical phase space, but forget coordinates for the moment. What you require is something called a symplectic form - a closed, non-degenerate 2-form, ω.

Now, if ω is "integral" (its cohomology class is integral), then you can construct a line bundle L over your phase space, with a connection ▽ whose curvature is -iω.
This, friends, is the so-called "prequantum line bundle."

But we're not quantizing yet. We require a
"polarization" — some clever way to slice up phase space into "position" and "momentum" subspaces (technically, a Lagrangian foliation). This is where the Kähler manifolds come in. They give this polarization.

Sections of L that are covariantly constant along your polarization form a vector space. If you've chosen a Kahler polarization, this space gets a natural inner product. Cool. Now you've got a Hilbert space, from pure geometry.

Classical observables (functions on phase space) can be "lifted" to operators on this Hilbert space. The problem is that this lift isn't unique. You need a "metaplectic correction" (a story for another time) to get it "just right.”

Geometric quantization is beautiful, and more sherdoggers should know this framework.
It even tells us that only certain classical states (Bohr-Sommerfeld leaves) have quantum counterparts.
IMG_8564.jpeg
 
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I know what some of you are thinking: “Why is the curvature imaginary? Seems like it's tacked on and then makes everything work out as a Hilbert space.”
But there’s more to it than just an arbitrary choice.

Wavefunctions and their inner product must respect the unitarity of evolution. The phase factor, eiϕ naturally arises in quantum mechanics due to the role of the complex exponential in describing phase evolution (think Schrödinger equation or interference).
By setting the curvature of the connection to −iω, the resulting geometric framework naturally accommodates these quantum phase factors.

The curvature being imaginary ensures that the resulting prequantum line bundle respects the symplectic form's scaling with ℏ, tying the classical geometry (ω) to the quantum one. This choice aligns the curvature with the Hermitian structure of the bundle, ensuring that the quantum Hilbert space has the correct inner product structure.

The condition that the symplectic form ω must have integral cohomology ensures that this imaginary curvature is well-defined and consistent with the topology of the symplectic manifold.

While canonical quantization simply replaces classical variables with operators in a somewhat ad hoc manner, geometric quantization ties quantum mechanics directly to the rich geometric structure of classical phase space. By using symplectic geometry and line bundles, it reveals how classical concepts, like phase space and observables, naturally lead to quantum states and operators.

This method not only preserves the integrity of the classical picture but also introduces quantum mechanics as a natural extension, showing a seamless connection between the two. In contrast, canonical quantization often feels like a disconnected set of rules, lacking this deeper geometric foundation. Geometric quantization, therefore, offers a more sophisticated and cohesive understanding of how quantum systems emerge from classical ones.

It’s like going from serving a succulent steak on a paper plate with a side of white rice to using a fine marble plate and pairing it with a vintage Cabernet.
 
chill doggie said:
I know what some of you are thinking: “Why is the curvature imaginary? Seems like it's tacked on and then makes everything work out as a Hilbert space.”
But there’s more to it than just an arbitrary choice.

Wavefunctions and their inner product must respect the unitarity of evolution. The phase factor, eiϕ naturally arises in quantum mechanics due to the role of the complex exponential in describing phase evolution (think Schrödinger equation or interference).
By setting the curvature of the connection to −iω, the resulting geometric framework naturally accommodates these quantum phase factors.

The curvature being imaginary ensures that the resulting prequantum line bundle respects the symplectic form's scaling with ℏ, tying the classical geometry (ω) to the quantum one. This choice aligns the curvature with the Hermitian structure of the bundle, ensuring that the quantum Hilbert space has the correct inner product structure.

The condition that the symplectic form ω must have integral cohomology ensures that this imaginary curvature is well-defined and consistent with the topology of the symplectic manifold.

While canonical quantization simply replaces classical variables with operators in a somewhat ad hoc manner, geometric quantization ties quantum mechanics directly to the rich geometric structure of classical phase space. By using symplectic geometry and line bundles, it reveals how classical concepts, like phase space and observables, naturally lead to quantum states and operators.

This method not only preserves the integrity of the classical picture but also introduces quantum mechanics as a natural extension, showing a seamless connection between the two. In contrast, canonical quantization often feels like a disconnected set of rules, lacking this deeper geometric foundation. Geometric quantization, therefore, offers a more sophisticated and cohesive understanding of how quantum systems emerge from classical ones.

It’s like going from serving a succulent steak on a paper plate with a side of white rice to using a fine marble plate and pairing it with a vintage Cabernet.
Click to expand...

After you defend and finish grad school, you can be a post doc for several years before you either become quant or a short order cook.
 
chill doggie said:
The canonical quantization you're taught - replace {f,g} with [f, g]/iℏ - is, well, ill-defined. You can always add a classical 0 that becomes non-zero quantumly (thanks, non-commutation). But there's another way: geometric quantization.

It all starts with symplectic geometry. Think of a classical phase space, but forget coordinates for the moment. What you require is something called a symplectic form - a closed, non-degenerate 2-form, ω.

Now, if ω is "integral" (its cohomology class is integral), then you can construct a line bundle L over your phase space, with a connection ▽ whose curvature is -iω.
This, friends, is the so-called "prequantum line bundle."

But we're not quantizing yet. We require a
"polarization" — some clever way to slice up phase space into "position" and "momentum" subspaces (technically, a Lagrangian foliation). This is where the Kähler manifolds come in. They give this polarization.

Sections of L that are covariantly constant along your polarization form a vector space. If you've chosen a Kahler polarization, this space gets a natural inner product. Cool. Now you've got a Hilbert space, from pure geometry.

Classical observables (functions on phase space) can be "lifted" to operators on this Hilbert space. The problem is that this lift isn't unique. You need a "metaplectic correction" (a story for another time) to get it "just right.”

Geometric quantization is beautiful, and more sherdoggers should know this framework.
It even tells us that only certain classical states (Bohr-Sommerfeld leaves) have quantum counterparts.
View attachment 1076825
Click to expand...
Gimme a sec... OK, done!
 
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