Quantum states: an analysis via the orthogonality relation

From the Hilbert space formalism we note that five simple conditions are satisfied by the orthogonality relation between the (pure) states of a quantum system. We argue, by proving a mathematical theorem, that they capture the essentials of this relation. Based on this, we investigate the rationale behind these conditions in the form of six physical hypotheses. Along the way, we reveal an implicit theoretical assumption in theories of physics and prove a theorem which formalizes the idea that the Superposition Principle makes quantum physics different from classical physics. The work follows the paradigm of mathematical foundations of quantum theory, which I will argue by methodological reflection that it exemplifies a formal approach to analysing concepts in theories.

Automated summary

By utilizing the Hilbert space formalism, we examine the orthogonality relation between pure states in a quantum system and identify the essential conditions governing this relation. We then explore the physical hypotheses underlying these conditions and demonstrate how the Superposition Principle distinguishes quantum physics from classical physics.