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**1 - 2**of**2**### Decoherence: Basic Concepts and their Interpretation

, 2001

"... The superposition principle forms the most fundamental kinematical element of quantum theory. Its universality seems to have first been postulated by Dirac as part of the definition of his “ket-vectors”, which he proposed as a complete 1 and general concept to characterize quantum states regardless ..."

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The superposition principle forms the most fundamental kinematical element of quantum theory. Its universality seems to have first been postulated by Dirac as part of the definition of his “ket-vectors”, which he proposed as a complete 1 and general concept to characterize quantum states regardless of any “basis of representation”. Hilbert space. The inner product (also needed to define a Hilbert space, and formally indicated by the distinction between “bra ” and “ket ” vectors) is not required as part of the kinematics of quantum states, but only for their probability interpretation (applicable in measurements). The third Hilbert space axiom (closure with respect to Cauchy series) is merely mathematically convenient, since one cannot decide in principle whether the number of linearly independent physical states is infinite in reality, or just very large. According to this kinematical superposition principle, any two physical states, |1 〉 and |2〉, whatever their meaning, can be superposed in the form c 1|1 〉 + c 2|2〉, with complex numbers c 1 and c 2, to form a new physical state. By induction, the principle can be applied to more than two and even an infinite number of states, and appropriately extended to apply to a continuum of states. After postulating the linear Schrödinger equation in a general form, one may furthermore conclude that the superposition of two (or more) of its solutions forms again a solution. This is the dynamical version of the superposition principle. A dynamical superposition principle (though in general with respect to real numbers only) is also known from classical waves which obey a linear wave equation. Its validity is then restricted to cases where these equations apply, while the quantum superposition principle is meant to be universal and exact.

### 2.1 The Phenomenon of Decoherence

, 2002

"... The superposition principle forms the most fundamental kinematical concept of quantum theory. Its universality seems to have first been postulated by Dirac as part of the definition of his “ket-vectors”, which he proposed as a complete 1 and general concept to characterize quantum states regardless ..."

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The superposition principle forms the most fundamental kinematical concept of quantum theory. Its universality seems to have first been postulated by Dirac as part of the definition of his “ket-vectors”, which he proposed as a complete 1 and general concept to characterize quantum states regardless of any basis of representation. They were later recognized by von Neumann as forming an abstract Hilbert space. The inner product (also needed to define a Hilbert space, and formally indicated by the distinction between “bra” and “ket ” vectors) is not part of the kinematics proper, but required for the probability interpretation, which may be regarded as dynamics (as will be discussed). The third Hilbert space axiom (closure with respect to Cauchy series) is merely mathematically convenient, since one can never decide empirically whether the number of linearly independent physical states is infinite in reality, or just very large. According to this kinematical superposition principle, any two physical states, |1 〉 and |2〉, whatever their meaning, can be superposed in the form c1|1〉+c2|2〉, with complex numbers c1 and c2, to form a new physical state (to be be distinguished from a state of information). By induction, the principle can be applied to more than two, and even an infinite number, of states, and appropriately generalized to apply to a continuum of states. After postulating the linear Schrödinger equation in a general form, one may furthermore