nLab local observable

Idea

In physics (field theory) a local observable is an observable which is an average of a function of the values of the fields and their derivatives at each fixed spacetime point.

If $\Phi$ is a field configuration over some spacetime $\Sigma$ then a local observable is a function of $\Phi$ which, assigns values of the form

$$\Phi \;\mapsto\; \int_\Sigma f(\Phi(x), \nabla \Phi(x), \nabla^2 \Phi(x), \cdots) b(x) dvol_\Sigma(x)$$

where $f$ is some smooth function of its arguments and $b$ is some bump function on spacetime.

This is in contrast to an observable which combines the values of fields at different spacetime points other than by forming spacetime averages. For instance if $x \neq y \in \Sigma$ are two distinct spacetime points, then

$$\Phi \mapsto \Phi(x) \Phi(y)$$

is a smooth function on the space of fields, hence an observable, but it is not a local observables.

More in detail, if

$$\array{ E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }$$

is a fiber bundle regarded as the field bundle that defines the given field theory, then a local observables is a function on the the space of field configurations, hence the space of sections $\Gamma_\Sigma(E)$ (or else on the subspace of those which solve the equations of motion, the shell) which arises as the transgression of a horizontal differential form of degree $dim(\Sigma)$ on the jet bundle $J^\infty_\Sigma(E)$ of $E$.

Products of local observables are called multilocal observables.

types of observables in perturbative quantum field theory:

$$\array{ && \text{local} \\ && & \searrow \\ \text{field} &\longrightarrow& \text{linear} &\longrightarrow& \text{microcausal} &\longrightarrow& \text{polynomial} &\longrightarrow& \text{general} \\ && & \nearrow \\ && \text{regular} }$$

Definition

For the moment see at geometry of physics – perturbative quantum field theory this def.

Properties

Example

(polynomial local observables are polynomial observables)

A local observable which comes from a horizontal differential form which is a polynomial in the fields and their jets times the volume form on spacetime is a polynomial observable.

These happen to be also microcausal observables (this example).

The resultiing inclusion

$$LocPolyObs(E) \hookrightarrow PolyObs(E)_{cm}$$

of the local polynomial observables into the microcausal polynomial observables is a dense subspace-inclusion. (Fredenhagen-Rejzner 12, p. 23)

Related concepts

• Wick algebra

• spacetime support

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

• Romeo Brunetti, Klaus Fredenhagen, section 5.4.1 of Quantum Field Theory on Curved Backgrounds, chapter 5 in Christian Bär, Klaus Fredenhagen (eds.), Quantum Field Theory on Curved Spacetime, Springer 2009

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, section 3.2 of_Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

• Klaus Fredenhagen, Katarzyna Rejzner, Perturbative algebraic quantum field theory, In Mathematical Aspects of Quantum Field Theories, Springer 2016 (arXiv:1208.1428)

• Katarzyna Rejzner, def. 3.14 in Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)

• Michael Dütsch, def. 1.2 in From classical field theory to perturbative quantum field theory, 2018