  The relationship between expectation and price is commonly established with
two principles: no-arbitrage, which asserts that both maps are positive; and
equivalence, which asserts that the maps share the same null events.
Constructed from the Arrow-Debreu securities, classical and quantum models of
economics are then distinguished by their respective use of classical and
quantum logic, following the program of von Neumann.
  In this essay, the operations and axioms of quantum groups are discovered in
the minimal preconditions of stochastic and functional calculus, making this
the natural domain for the axiomatic development of mathematical finance.
Quantum economics emerges from the twin pillars of the Gelfand-Naimark-Segal
construction, implementing the principle of no-arbitrage, and the Radon-Nikodym
theorem, implementing the principle of equivalence.
  Exploiting quantum group duality, a holographic principle that exchanges the
roles of state and observable creates two distinct economic models from the
same set of elementary valuations. Advocating on the grounds that this contains
and extends classical economics, noncommutativity is presented as a modelling
resource, with novel applications in the pricing of options and other
derivative securities.
