Contents

Contents

Idea

In physics given two physical system their composite is meant to be both systems regarded as a single system, but trivially so, without the two interacting.

In classical mechanics forming a composite in this sense amounts to taking the cartesian product of their phase spaces.

In quantum mechanics forming the composite amounts to taking the (non-cartesian) tensor product of the spaces of quantum states, hence their “multiplicative conjunction” in the language of linear logic/quantum logic. The non-cartesian nature of this tensor product is the source of the phenomenon of quantum entanglement and all that goes with this (cf. the EPR paradox and Bell's inequalities but also phenomena like quantum materials and topological order).

twisted generalized cohomology theory is conjecturally ∞-categorical semantics of linear homotopy type theory:

linear homotopy type theorygeneralized cohomology theoryquantum theory
linear type(module-)spectrum
multiplicative conjunctionsmash product of spectracomposite system
dependent linear typemodule spectrum bundle
Frobenius reciprocitysix operation yoga in Wirthmüller context
dual type (linear negation)Spanier-Whitehead duality
invertible typetwistprequantum line bundle
dependent sumgeneralized homology spectrumspace of quantum states (“bra”)
dual of dependent sumgeneralized cohomology spectrumspace of quantum states (“ket”)
linear implicationbivariant cohomologyquantum operators
exponential modalityFock space
dependent sum over finite homotopy type (of twist)suspension spectrum (Thom spectrum)
dualizable dependent sum over finite homotopy typeAtiyah duality between Thom spectrum and suspension spectrum
(twisted) self-dual typePoincaré dualityinner product
dependent sum coinciding with dependent productambidexterity, semiadditivity
dependent sum coinciding with dependent product up to invertible typeWirthmüller isomorphism
$(\sum_f \dashv f^\ast)$-counitpushforward in generalized homology
(twisted-)self-duality-induced dagger of this counit(twisted-)Umkehr map/fiber integration
linear polynomial functorcorrespondencespace of trajectories
linear polynomial functor with linear implicationintegral kernel (pure motive)prequantized Lagrangian correspondence/action functional
composite of this linear implication with daggered-counit followed by unitintegral transformmotivic/cohomological path integral
traceEuler characteristicpartition function

Referenced

Early discussion of composite quantum systems and their quantum entanglement:

Last revised on November 12, 2022 at 15:20:50. See the history of this page for a list of all contributions to it.