The structure of physics

Bob Coecke

Ovidiu Cristinel Stoica

Abstract I point out an underdetermination of physics, in particular of Quantum Mechanics: that of the relation between the operators and their physical meaning. The set of operators for which we attribute physical meaning is, from relational, structural, and dynamical points of view, indistinguishable from any set of operators obtained from the former by a unitary transformation that commutes with the Hamiltonian. The same is true for the structures inhabiting the world, in particular there are structures isomorphic to the structure of an observer, but they would attribute the meaning of position and other physical meanings to different operators. In particular, space itself is underdetermined from the structure of the theory. Different structures that are isomorphic to an observer would call “position” different observables, and consequently would call “space” different structures. I show that they are highly non-unique. A solution would be that all possible assignments of physical meaning to the operators allowed by the symmetries of the formalism are equally valid. But this leads to a problem: there would be no correlation between the content of the mind of an observer about the external world, and the actual properties of the external world. This has unexpected consequences: it refutes structural realism, and it shows that, in addition to the structure, the substance underlying the structures matters for consciousness to exist.

Gianpiero Cattaneo, Claudio Garola, Giuseppe Nisticò

We introduce the mathematical structure PES (preparation‐effect structure), which is capable of a physical interpretation thus making the five axioms of PES physically plausible. In this way, PES can be considered a general mathematical structure that underlies many physical theories. We then introduce four more axioms to obtain a more specific structure [the pure states (PS) complete orthogonality (OG) PES satisfying the covering law]. This more specific structure can be considered as the basic mathematical structure for quantum physics (QP). We show that some definitions and results pertaining to different axiomatic approaches to QP can be interpreted into a PES. (This allows a better comparison of the approaches.) In particular, we consider the Jauch‐Piron (JP) approach to QP, reformulate it to make it completely axiomatical thus distinguishing between a basic underlying mathematical structure [i.e., the question‐preparation structure (QPS)] and its specific mathematical structure (i.e., the JP QPS), and show that it can be completely interpreted into a PS complete OG PES satisfying the covering law.

Bang-Hai Wang, Dongyang Long

The structural physical approximation (SPA) to a positive map is considered to be one of the most important methods to detect entanglement in the real physical world. We first show that an arbitrary entanglement witness (EW) $W$ can be constructed from a separable density matrix $\ensuremath{\sigma}$ in the form of $W=\ensuremath{\sigma}\ensuremath{-}{c}_{\ensuremath{\sigma}}I$, where ${c}_{\ensuremath{\sigma}}$ is a non-negative number and $I$ is the identity matrix. Following the general form of EWs from separable states, we show a sufficient condition and a sufficient and necessary condition in low dimensions of that SPAs to positive maps do not define entanglement-breaking channels. We show that either the SPA of an EW or the SPA of the partial transposition of the EW in low dimensions is an entanglement-breaking channel. We give sufficient conditions of violating the SPA conjecture [Phys. Rev. A 78, 062105 (2008)]. Our results indicate that the SPA conjecture is independent of whether or not positive maps are optimal.