nLab beable

Idea

The term beable was proposed in (Bell 75) as a replacement in quantum physics of the traditional term observable. While the latter is typically given a precise mathematical meaning which does express the way in which the physical system may be, the word “observable” alludes to the complicated and subtle issue of something (a quantum measurement device) or even somebody (a concious experimentor) “observing” these ways of the system to be. The point of the term “beable” is to help conceptually cleanly separate the being of quantum systems from whatever it means to observe them.

In (Bell 75), to further justify this, recourse is made to Niels Bohr‘s expressed view that whatever quantum physics really is, it must be possible to communicate statements about it in terms of classical logic. Hence beables mostly refer to sets of commuting operators, classical contexts. This same argument was much later used to motivate Bohr toposes. In this context, the Kochen-Specker theorem proves the absence of certain global sections (Proposition 1). Halvorson and Clifton define a beable subalgebra (p9) with respect to a given state $\rho$. We first consider the important special case of the definite algebra $D_\rho$ for a state $\rho$. This is the subalgebra on which $\rho$ is dispersion free. This gives, for each commutative subalgebra of $C\subset D_\rho$, a choice of an element of its spectrum. These choices are compatible. Hence, we obtain not a global section, but only local section. The general notion of beable subalgebra only requires $\rho$ to be a mixture of dispersion-free states.

In practice of theoretical physics most everytime one writes “observable” it should, by this logic, rather be “beable”. While this might be reasonable, as a convention of conversation it has not been picked up.

Related concepts

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

The original discussion of beables in the context of causal locality in AQFT is in

• John Bell, The theory of local beables, Epistemological Letters (1975) (pdf)

and further in

• John Bell, Beables for quantum field theory (1984) (pdf)

A survey is in

• Scholarpedia, Bell’s theorem - beables

A more technical development can be found here:

• Hans Halvorson, Rob Clifton, Maximal Beable Subalgebras of Quantum-Mechanical Observables (arXiv:quant-ph/9905042)

Discussion in the context of interpretation of quantum mechanics includes

• Yuichiro Kitajima, Interpretations of Quantum Mechanics in Terms of Beable Algebras, (pdf)