Abstract The mathematical description of quantum theory rests on the idea of states and observables. However, the dynamical description of these two described in Section 6.4 is quite different.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
A physical theory often admits different mathematical descriptions. Classical mechanics has three familiar formulations: the traditional Newton’s formulation, the Lagrange’s and the Hamilton’s formulations. It is easy to see that these are physically equivalent. Quantum mechanics also admits many different formulations. This chapter discusses how different mathematical descriptions of a probability theory based on a Hilbert space can be physically equivalent. The prescription does not single out a unique Hilbert space nor a unique set of unit vectors and operators for the description of a given quantum system. This non-uniqueness does not matter since operators and unit vectors themselves are not physically measurable quantities. Two different mathematical descriptions of a quantum system are physically equivalent if the unit vectors and selfadjoint operators for the representation of states and observables in the two descriptions are related by a simultaneous unitary transformation.
Modern physical theories employ mathematical frameworks describing states, interactions, symmetries, and information with remarkable predictive success. Yet these frameworks often begin from conceptual assumptions that may already lie downstream from deeper structural conditions of physical reality. States assume persistence, interactions assume identifiable relata, and measurement assumes observers capable of registering outcomes. This paper proposes an alternative conceptual starting point based on resonant process. Rather than treating states or interactions as primitive, we suggest that persistent patterns of process arise through selective compatibility and reinforcement within ongoing dynamics. From this perspective, state descriptions and interaction descriptions emerge as complementary projections of stabilized resonant structure, while observation arises when resonant processes stabilize discriminable differences within subsystems capable of participating in further interaction. The purpose of this work is not to present a completed mathematical theory but to articulate a conceptual framework and the structural requirements that any future resonance-based formalism would need to satisfy. The approach connects naturally with existing ideas in quantum foundations, symmetry theory, and information-based interpretations of physics.
Abstract A statistical law for interaction processes in quantum theory is presented. It states that the final state of the scattered systems, evaluated on the basis of quantum theory, reduces alternatively to different substates provided correlation states are produced by the interaction. Such processes also underlie the measuring process (interaction system‐apparatus). To test the modified scattering theory experiments are proposed. The consequences for the description of the interaction between a system and a macroscopic body, in particular a measuring apparatus, are discussed. The reduction of the wave function needed here comes out quite naturally.