We present a non‐technical introduction to the general boundary formulation (GBF) of quantum theory. We emphasize the main motivation for the GBF: Formulating the quantum theory in way compatible with the principles of general relativity. We give an overview of the basic structures of the GBF, defining the notions of state space, amplitude, probability interpretation and expectation value of observables in the GBF. Finally, we briefly mention some of the results obtained so far and indicate future lines of research.
We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure---which we call subsystem space---that we believe merits further study. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text.
Abstract Inspired by classical (‘actual’) quantum theory over C and modal quantum theory (MQT), which is a model of quantum theory over certain finite fields, we introduce general quantum theory as a quantum theory—in the København interpretation—over general division rings with involution, in which the inner product ‘is’ a ( σ , 1)-Hermitian form φ . This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical quantum theory over C and in some MQTs, hold for any general quantum theory. Our approach is totally different (and more general) than Hardy’s axiomatic approach [15], and generalized probability theories. On the other hand, in many general quantum theories, a geometrical object which we call ‘quantum kernel’ arises, which is invariant under the unitary group U ( V , φ ), and which carries the geometry of a so-called polar space. This object cannot be seen in classical quantum theory over C , but it is present, for instance, in all known MQTs. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic 0, every general quantum theory over an algebraically closed field behaves like classical quantum theory over C at many levels, and that all such theories share one model, which we pin down as the ‘minimal model,’ which is countable and defined over Q ¯ . Moreover, to make the analogy with classical quantum theory even more striking, we show that Born’s rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for general quantum theories in characteristic 0 which allows one to extend any such theory over algebraically closed fields (such as classical complex quantum theory) to larger theories in which a quantum kernel is present. In this sense, these singular objects are always virtually around in abundance.
Inspired by classical ("actual") Quantum Theory over
C and Modal Quantum Theory (MQT), which is a model of Quantum Theory over certain finite fields, we introduce General Quantum Theory as a Quantum Theory -- in the København interpretation -- over general division rings with involution, in which the inner product "is" a
(σ,1)-Hermitian form
φ. This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical Quantum Theory over
C and in some MQTs, hold for any General Quantum Theory. On the other hand, in many General Quantum Theories, a geometrical object which we call "quantum kernel" arises, which is invariant under the unitary group
U(V,φ), and which carries the geometry of a so-called polar space. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic
0, every General Quantum Theory over an algebraically closed field behaves like classical Quantum Theory over
C at many levels, and that all such theories share one model, which we pin down as the "minimal model," which is countable and defined over
Q. Moreover, to make the analogy with classical Quantum Theory even more striking, we show that Born's rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for General Quantum Theories in characteristic
0 which allows one to extend any such theory over algebraically closed fields (such as classical complex Quantum Theory) to larger theories in which a quantum kernel is present.
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.