A local and operational framework for the foundations of physics

Abstract

We discuss a novel framework for physical theories that is based on the principles of locality and operationalism. It generalizes and unifies previous frameworks, including the standard formulation of quantum theory, the convex operational framework and Segal's approach to quantum field theory. It is capable of encoding both classical and quantum (field) theories, implements spacetime locality in a manifest way and contains the complete modern notion of measurement in the quantum case. Its mathematical content can be condensed into a set of axioms that are similar to those of Atiyah and Segal. This is supplemented by two basic rules for extracting probabilities or expectation values for measurement processes. The framework, called the positive formalism, is derived in three completely different ways. One derivation is from first principles, one starts with classical field theory and one with quantum field theory. The latter derivation arose previously in the programme of the general boundary formulation of quantum theory. As in the convex operational framework, the difference between classical and quantum theories essentially arises from certain partially ordered vector spaces being either lattices or anti-lattices. If we add the ad hoc ingredient of imposing anti-lattice structures, the derivation from first principles may be seen as a reconstruction of quantum theory. Among other things, the positive formalism suggests a statistical approach to classical field theories with dynamical metric, provides a common ground for quantum information theory and quantum field theory, introduces a notion of local measurement into quantum field theory, and suggests a new perspective on quantum gravity by removing the incompatibility with general relativistic principles. The positive formalism as a framework for quantum theory is in conflict with various interpretations or modifications of quantum theory, including physical collapse theories, manyworlds interpretations, and non-local hidden variable theories.

Toolbox for reconstructing quantum theory from rules on information acquisition

“…We remind the reader that the time evolution groups T N for density matrices in rebit and qubit quantum theory are projective because they correspond to ρ → U ρ U † , where for (1) rebits ρ is a 2 N × 2 N real symmetric matrix, U ∈ SO(2 N ) and † denotes matrix transpose; and (2) for qubits ρ is a 2 N × 2 N hermitian matrix, U ∈ SU(2 N ) and † denotes hermitian conjugation. Furthermore, using one additional rule on Q N which allows O to ask S any question that "makes (probabilistic) sense", we show in [1,2] that 1. in the rebit case [2], Q N is (isomorphic to) the set of projective measurements onto the +1 eigenspaces of Pauli operators 25 over R 2 N . This is the set of rank-(2 N −1 )projectors.…”

“…Our ambition is to develop a novel informational framework for deriving the formalism and structure of quantum theory for systems of arbitrarily many qubits from elemen-tary operational postulates -a task which is completed in the companion paper [1]. While neither this question nor the fact that one can reconstruct quantum theory from elementary axioms is new and has been extensively explored before in various contexts [14][15][16][17][18][19][20][21][22][23][24][25], we shall approach both from a novel constructive perspective and with a stronger emphasis on the conceptual content of the theory. The ultimate goal of this work is therefore very rudimentary: to redo a well established theory -albeit in a novel way which is especially engineered for exposing its informational and logical structure, physical content and distinctive phenomena more clearly.…”

“…It is fair to say that the hope alluded to under point 4 has not been realized thus far because the existing successful reconstructions [14][15][16][17][18][19][20][21][22][23][24][25] are fairly neutral as far as an interpretation is concerned. While most of them emphasize the operational character of the theory, a particular interpretation of quantum theory is not strongly suggested.…”

Gravitational $SL(2,\mathbb{R})$ Algebra on the Light Cone

“…Even if such a construction succeeds, the task remains how to recover the ordinary S matrix in this framework. In fact, the S matrix is a highly entangled boundary state (linear functional) [48,49] on the tensor product Hilbert space H + ⊗ H † − , where H ± is the Fock space for future (past) null infinity. Defining this tensor product amounts to join the two asymptotic boundaries into a single Hilbert space.…”

“…However, this non-uniqueness is mild. In fact, if the dyad (m, m) on N is given, the pull-back of Γ to the null generators is unique, see(49) below.…”