classical state (changes) in nLab

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Contents

Idea

A classical state is a state of a system of classical mechanics.

In principle, a pure state in classical mechanics specifies completely all information about the state of the system, while a mixed state is a probability measure on the space of pure states. This space of pure states my may be identified with thestate space? in Lagrangian mechanics or with the phase space in Hamiltonian mechanics.

Definition

We give a definition in a very general context.

For AA a commutative unital associative algebra that encodes a system of classical mechanics (say, the associative algebra underlying a Poisson algebra), a classical state is an ℝ\mathbb{R}-linear function

ρ:A→ℝ \rho\colon A \to \mathbb{R}

that satisfies

• normalization ρ(1)=1\rho(1) = 1;

• positivity for all a∈Aa \in A we have ρ(a 2)≥0\rho(a^2) \geq 0.

This is essentially the definition of quantum state, but formulated for commutative algebras and over the real numbers.

If we take AA to be a **-algebra over the complex numbers, then we may take ρ\rho to be a ℂ\mathbb{C}-linear function from AA to ℂ\mathbb{C} instead.

	classical mechanics	semiclassical approximation	…	formal deformation quantization	quantum mechanics
order of Planck's constant ℏ\hbar	𝒪(ℏ 0)\mathcal{O}(\hbar^0)	𝒪(ℏ 1)\mathcal{O}(\hbar^1)	𝒪(ℏ n)\mathcal{O}(\hbar^n)	𝒪(ℏ ∞)\mathcal{O}(\hbar^\infty)
states	classical state	semiclassical state			quantum state
observables	classical observable				quantum observable

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

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