nLab composite system

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).

Related concepts

• subsystem, independent subsystem

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

linear homotopy type theory	generalized cohomology theory	quantum theory
linear type	(module-)spectrum
multiplicative conjunction	smash product of spectra	composite system
dependent linear type	module spectrum bundle
Frobenius reciprocity	six operation yoga in Wirthmüller context
dual type (linear negation)	Spanier-Whitehead duality
invertible type	twist	prequantum line bundle
dependent sum	generalized homology spectrum	space of quantum states (“bra”)
dual of dependent sum	generalized cohomology spectrum	space of quantum states (“ket”)
linear implication	bivariant cohomology	quantum operators
exponential modality		Fock space
dependent sum over finite homotopy type (of twist)	suspension spectrum (Thom spectrum)
dualizable dependent sum over finite homotopy type	Atiyah duality between Thom spectrum and suspension spectrum
(twisted) self-dual type	Poincaré duality	inner product
dependent sum coinciding with dependent product	ambidexterity, semiadditivity
dependent sum coinciding with dependent product up to invertible type	Wirthmüller isomorphism
$(\sum_f \dashv f^\ast)$-counit	pushforward in generalized homology
(twisted-)self-duality-induced dagger of this counit	(twisted-)Umkehr map/fiber integration
linear polynomial functor	correspondence	space of trajectories
linear polynomial functor with linear implication	integral kernel (pure motive)	prequantized Lagrangian correspondence/action functional
composite of this linear implication with daggered-counit followed by unit	integral transform	motivic/cohomological path integral
trace	Euler characteristic	partition function

Referenced

Early discussion of composite quantum systems and their quantum entanglement:

• Erwin Schrödinger, Discussion of Probability Relations between Separated Systems, Mathematical Proceedings of the Cambridge Philosophical Society, 31 4 (1935) 555-563 [doi:10.1017/S0305004100013554]