nLab geometry of physics -- quantum mechanics

This is a sub-entry of geometry of physics.

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So far we have discussed extended prequantum field theory: Lagrangians and their induced action functionals and prequantum n-bundles. Now we turn to actual quantum field theory. A prequantum field theory is supposed to induce a quantum field theory under the last step of higher geometric quantization: a choice of polarization (or equivalent) and the passage to the corresponding space of states of polarized sections of the prequantum n-bundles. This step that connects prequantum field theory with quantum field theory we discuss below in Geometric Quantization.

Here we discuss the structure of the outcome of this process.

Quantum mechanics

Model Layer

• FQFT – Exposition and Introduction – Quantum mechanics in Schrödinger picture

• quantum mechanics

Worldvolumes and cobordisms

• cobordism

Spectral triple

• spectral triple

Semantic Layer

Internal categories in an $\infty$-topos

Simplicial objects in an $\infty$-topos

• simplicial object in an (∞,1)-category

Category objects in an $\infty$-topos

• Segal condition

• internal category in an (∞,1)-category

Syntactic Layer

Directed homotopy type theory

• directed homotopy type theory