Quantum structures and their future importance

Relativity theory, formulated in great part by one person, Albert Einstein [1], is founded on the concept of ‘event’, which is a concept that is physically well defined and understood [2]. Within relativity theory itself, the events are represented by the points of a four dimensional space-time continuum. In this way, relativity theory has a well defined physical and mathematical base. The development of quantum mechanics proceeded in a rather haphazard manner, with the introduction of many ill-defined and poorly understood new concepts. During its first years, from 1890 to 1925, quantum mechanics, commonly referred to as the ‘old quantum theory’, did not even possess a coherent mathematical basis. In 1926 Werner Heisenberg formulated matrix mechanics in a mainly technical effort to explain and describe the energy spectrum of the atoms [3]. Around the same time Erwin Schrödinger elaborated wave mechanics which seemed to have a more solid physical base: a general idea of wave-particle duality, in the spirit of Louis de Broglie or Niels Bohr [4]. But then Paul Adrien Maurice Dirac [5] and later John Von Neumann [6] proved that the matrix mechanics of Heisenberg and the wave mechanics of Schrödinger are equivalent: they can be constructed as two mathematical representations of one and the same vector space, the Hilbert space. This fundamental result indicated already that the ‘de Broglie wave’ and the ‘Bohr wave’ are not physical waves and that the state of a quantum entity is an abstract concept: a vector in an abstract vector space. The abstract Hilbert space formulation of quantum mechanics, as introduced by John von Neumann in 1932 [6], and now commonly referred to as ‘Standard Quantum Mechanics’ (SQM), uses an elaborate and sophisticated mathematical formalism, of which the basic concepts remain however vague and unclear from