What is Quantum Mechanics? A Minimal Formulation

R. McWeeny

Ralf Otte

Quantum mechanics is formulated on complex Hilbert spaces, where the complex numbers constitute the underlying scalar algebra. In this work, the structural question is investigated whether this scalar structure is algebraically minimal or whether a consistent extension exists that preserves the analytical infrastructure of the theory. We formulate this problem as a minimality question for finite-dimensional real, commutative and associative algebras that contain the complex numbers as a subalgebra and at the same time possess a nontrivial ideal that is isomorphic to C as a real algebra and carries a complex-like exponential structure. It is shown that such structures necessarily require at least four real dimensions. The bicomplex algebra B=C ⊗R C realizes these requirements in minimal dimension and is unique up to isomorphism under the stated conditions. For the extended state space M=H ⊗C B one obtains a canonical decomposition M≅H⊕H, so that the bicomplex scalar extension structurally corresponds to two parallel complex quantum sectors. The Born rule remains unchanged. The present work provides a structural analysis of possible extensions of the scalar algebra underlying quantum mechanics.

Martin Pieper

Michael T. Strickland

This chapter introduces the path integral formulation of quantum mechanics as a rigorous and systematic alternative to canonical quantization. It begins by defining the transition probability amplitude (propagator) and deriving the path integral by discretizing time and summing over all possible intermediate paths between initial and final spacetime points. The chapter shows the final form of the propagator as a functional integral over all paths, weighted by the exponential of the action. This formulation provides an intuitive understanding that paths near the classical trajectory (which minimizes the action) interfere constructively, dominating the overall amplitude.

Edoardo Piparo

This paper is the first of several parts introducing a new powerful algebra: the algebra of the pseudo-observables. This is a C*-algebra whose set is formed by formal expressions involving observables. The algebra is constructed by applying the Occam's razor principle, in order to obtain the minimal description of physical reality. Proceeding in such a manner, every aspect of quantum mechanics acquires a clear physical interpretation or a logical explanation, providing, for instance, in a natural way the reason for the structure of complex algebra and the matrix structure of the formulation of Werner Heisenberg of quantum mechanics. Last but not least, the very general hypotheses assumed, allow one to state that quantum mechanics is the unique minimal description of physical reality.