theta vacuum (changes) in nLab

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Contents

Idea

In Yang-Mills theory on 4d Minkowski spacetime the theta vacuum is a vacuum state which is a superposition of each Yang-Mills instanton-sector weighted by a complex phase proportional to the theta angle times the instanton number.

• instanton sea

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

Survey is in

• Richard Healey, Gauge Theory and the Theta Vacuum, 2007 (web, pdf)

Detailed argument for the theta vacuum from chiral symmetry breaking is offered in

• Curtis Callan, R.F. Dashen, David Gross, The Structure of the Gauge Theory Vacuum, Phys.Lett. 63B (1976) 334-340 (spire)

• Giovanni Morchio, Franco Strocchi, Chiral symmetry breaking and theta vacuum structure in QCD, Annals Phys.324:2236-2254, 2009 (arXiv:0907.2522)

Discussion in a context of algebraic non-perturbative quantum field theory:

• Franco Strocchi, §8 of: An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

See also:

• Wei-Yang Liu: Generic framework for non-perturbative QCD in light hadrons [[arXiv:2501.07776](https://arxiv.org/abs/2501.07776)]