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Theoretical and mathematical physics

Mathematical Description of Infinitely Extended Quantum Systems

Franco Strocchi

Published Jan 1, 2021• 0 citations
Publisher

Abstract

From the discussion of the previous Chapter, it appears that the description of infinite systems looks much more difficult than in the finite-dimensional case, above all because of the existence of (too) many possible representations of the algebra of canonical variables.
DOI
10.1007/978-3-662-62166-0_14

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The received view in the philosophy of physics is that a quantum theory is given its mathematical formulation in a so-called Hilbert space, with the states of a quantum system given by the elements of this space and observables—measurable quantities of the system—given by so-called operators on this space. But the status of this fundamental postulate becomes threatened when one moves to complex systems, like the quantum fields underlying fundamental particle physics or statistical systems composed of infinite collections of quantum particles. In these more complex cases, one finds many inequivalent Hilbert space representations that appear to be competing quantum theories. In looking for a mathematical structure of the theory to interpret as representing the physical world, one is faced with a choice: either arbitrarily pick one Hilbert space representation (out of the many competing representations) as a Hilbert Space Conservative, or else look for the abstract algebraic structure that all of these competing representations have in common as an Algebraic Imperialist. This dissertation provides an extended argument for the second route.