nLab quantum observable

See also at algebra of observables.

Idea

An observable in quantum physics.

Definition in geometric quantization

We consider the notion of quantum observables in the the context of geometric quantization. See also quantum operator (in geometric quantization).

On a symplectic manifold

Let $(X, \omega)$ be a (pre-)symplectic manifold, thought of as the phase space of a physical system.

Assume that $\omega$ is prequantizable (integral) and let $\nabla : X \to \mathbf{B} U(1)_{conn}$ be a prequantum bundle $E \to X$ with connection for $\omega$, hence with curvature $F_\nabla = \omega$. Write $\Gamma_X(E)$ for the space of smooth sections of the associated complex line bundle. This is the prequantum space of states.

Definition

For $f \in C^\infty(X, \mathbb{C})$ a function on phase space, the corresponding pre-quantum operator is the linear map on prequantum states

$$\hat f : \Gamma_X(E) \to \Gamma_X(E)$$

given by

$$\psi \mapsto -i \nabla_{v_f} \psi + f \cdot \psi \,,$$

where

• $v_f$ is the Hamiltonian vector field corresponding to $f$;

• $\nabla_{v_f} : \Gamma_X(E) \to \Gamma_X(E)$ is the covariant derivative of sections along $v_f$ for the given choice of prequantum connection;

• $f \cdot (-) : \Gamma_X(E) \to \Gamma_X(E)$ is the operation of degreewise multiplication pf sections.

Remark

In terms of Higher geometric prequantum theory we may, as discussed there, identify the Poisson bracket Lie algebra $\mathfrak{Poisson}(X,\omega)$ with the Lie algebra of the group of automorphism $\exp(O) \colon \nabla \stackrel{\simeq}{\to} \nabla$ regarded in the slice over $\mathbf{B}U(1)_{conn}$. Moreover, the space of sections is equivalently the space of maps $\Psi \colon \nabla \to \mathbb{C}//U(1)_{conn}$ in the slice from $\nabla$ into the differential refinement of the smooth universal line bundle $\mathbb{C}//U(1) \to \mathbf{B}U(1)$. In this formulation the action of prequantum operators is just the precomposition action

$$\widehat{\exp(O)} \colon (\nabla \stackrel{\Psi}{\to} \mathbb{C}//U(1)_{conn}) \mapsto (\nabla \stackrel{exp(O)}{\to} \nabla \stackrel{\Psi}{\to} \mathbb{C}//U(1)_{conn}) \,.$$

Now after a choice of polarization a quantum state is a prequantum wave function which is covariantly constant along the Lagrangian submanifolds of the foliation. Not all prequantum operators will respect the space of such quantum states inside all quantum states. Those that do become genuine quantum operators.

Definition

Let $\mathcal{P}$ be a polarization of the symplectic manifold $(X,\omega)$. then a quantum state or wavefunction is a prequantum state $\psi$ such that $\nabla \Psi$ vanishes along the leaves of the polarization.

A quantum operator is a prequantum operator which preserves quantum states among all prequantum states.

Proposition

A prequantum operator given by a Hamiltonian function $f$ with Hamiltonian vector field $v_f$ is a quantum operator, def. , with respect to a given polarization $\mathcal{P}$ precisely if its flow preserves $\mathcal{P}$, hence precisely if

$$[v_f, \mathcal{P}] \subset \mathcal{P} \,.$$

Remark

If $\mathcal{P}$ is a Kähler polarization then its underlying almost complex structure it induces a spin^c structure, as discussed there. If $\rho \colon G \to QuantMorph(X,\nabla)$ is a Hamiltonian action (a homomorphism to the quantomorphism group) such that each prequantum operator $\rho(g)$ is a quantum operator in that it preserves the polarization, by prop. , then the corresponding spin^c structure is $G$-invariant. Accordingly the index of the spin^c Dirac operator which gives the geometric quantization by cohomological quantization exists not just in K-theory, where it yields the space of quantum states, but even in $G$-equivariant K-theory, exhibiting a representation of $G$ on the Hilbert space. This is the action of the quantum observables given by $\rho$ from the point of view of cohomological quantization.

Example

Over a phase space which is a cotangent bundle and with respect to the corresponding canonical vertical polarization, a Hamiltonian function is a quantum operator precisely if it is at most linear in the canonical momenta.

See for instance (Blau, around p. 35)

On an $n$-plectic smooth $\infty$-groupoid

(…)

Irreducible representations and superselection sectors

The space of quantum states forms a linear representation of a given algebra of observables. The decomposition of that into irreducible representations is physically the decomposition into superselection sectors.

Related concepts

product in perturbative QFT	$\,\,$ induces
normal-ordered product	Wick algebra (free field quantum observables)
time-ordered product	S-matrix (scattering amplitudes)
retarded product	interacting quantum observables

• Hamiltonian operator

• self-adjoint operator, daseinisation

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

References

• Roland Omnès, §3 of: The Interpretation of Quantum Mechanics, Princeton University Press (1994) [ISBN:9780691036694]

In algebraic quantum theory

In traditional AQFT:

• Nikolay Bogolyubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, Algebra of Observables and State Space [doi:10.1007/978-94-009-0491-0_6], Chapter in: General principles of quantum field theory, Mathematical Physics and Applied Mathematics 10, Kluwer (1990) [doi:10.1007/978-94-009-0491-0]

Lecture notes in perturbative quantum field theory:

• Urs Schreiber, geometry of physics – perturbative quantum field theory – Observables

In geometric quantization

See also the references at geometric quantization.

Standard facts are recalled for instance around p. 35 of

• Matthias Blau, Symplectic Geometry and Geometric Quantization (pdf)

Computation of quantum observables by index maps in equivariant K-theory is in (see specifically around p. 8 and 9)

• Michèle Vergne, Geometric quantization and equivariant cohomology (pdf)

Quantum observable algebras as Groupoid algebras

The original derivation of the Heisenberg picture of quantum mechanics, introducing $N \times N$ matrix algebra for transitions between $N$ measurable states of atomic spectra, by

• Werner Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik 33 (1925) 879–893 [doi:10.1007/BF01328377, Engl. pdf]

was argued by

• Alain Connes, pp 43-44 of: Noncommutative geometry, Academic Press (1994) [ISBN:9780121858605, pdf]

to be fruitfully understood as the groupoid convolution algebra of the pair groupoid of transitions between $N$ elements.

Later but more generally, the “algebra of (selective) measurement” originally envision by

• Julian Schwinger: Quantum Kinematics and Dynamics, CRC Press (1969, 1991) [ISBN:9780738203034, pdf]

• Julian Schwinger (ed.: Berthold-Georg Englert): Quantum Mechanics – Symbolism of Atomic Measurements, Springer (2001) [doi:10.1007/978-3-662-04589-3]

is argued to be, in modern language, the groupoid convolution algebras of groupoids whose morphisms reflect transitions between possible quantum measurement-outcomes – by:

• Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo: Schwinger’s Picture of Quantum Mechanics I: Groupoids, Int. J. Geometric Methods in Modern Physics 16 08 (2019) 1950119 [doi:10.1142/S0219887819501196, arXiv:1905.12274]

• Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo: Schwinger’s Picture of Quantum Mechanics II: Algebras and Observables, Int. J. Geometric Methods in Modern Physics 16 09 (2019) 1950136 [doi:10.1142/S0219887819501366, arXiv:1907.03883]

• Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo: Schwinger’s Picture of Quantum Mechanics II: Algebras and Observables, Int. J. Geometric Methods in Modern Physics 16 11 (2019) 1950165 [doi:10.1142/S0219887819501652, arXiv:1909.07265]

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo: Schwinger’s Picture of Quantum Mechanics IV: Composition and independence, Int. J. Geometric Methods in Modern Physics 17 04 (2020) 2050058 [doi:10.1142/S0219887820500589, arXiv:2004.02472]

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo: Schwinger’s picture of Quantum Mechanics, Int. J. Geometric Methods in Modern Physics 17 04 (2020) 2050054 [doi:10.1142/S0219887820500541, arXiv:2002.09326]

and as such further developed in:

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo: Schrödinger’s problem with cats: measurements and states in the Groupoid Picture of Quantum Mechanics, Entropy 22 11 (2020) [doi:10.3390/e22111292, arXiv:2012.10284]

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Luca Schiavone: Schwinger’s picture of quantum mechanics: 2-groupoids and symmetries, Journal of Geometric Mechanics 13 3 (2021) [doi:10.3934/jgm.2021008, arXiv:2104.13880]

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo: Quantum Tomography and Schwinger’s Picture of Quantum Mechanics, Journal of Physics A: Mathematics and Theoretical 55 27 (2022) [doi:10.1088/1751-8121/ac7591, arXiv:2205.00170]

• Florio M. Ciaglia, Fabio Di Cosmo, Paolo Facchi, Alberto Ibort, Arturo Konderak, Giuseppe Marmo: Groupoid and algebra of the infinite quantum spin chain, Journal of Geometry and Physics 191 (2023) [doi:10.1016/j.geomphys.2023.104901, arXiv:2302.01050],

Proof that non-perturbative quantum observables on Yang-Mill fluxes through a surface form a convolution algebra:

• Hisham Sati, Urs Schreiber, Thm 1 in: Quantum Observables of Quantized Fluxes, Annales Henri Poincaré (2024) [doi:10.1007/s00023-024-01517-z, arXiv:2312.13037]