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In particular, quantization, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself. Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical ...
In functional analysis, a discipline within mathematics, given a -algebra, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic -representations of and certain linear functionals on (called states).
Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms. [2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot.
In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, † and . These operators change the eigenvalues of the number operator , N = ∑ i n i = ∑ i a i † a i , {\displaystyle N=\sum _{i}n_{i}=\sum _{i}a_{i}^{\dagger }a_{i}^{\,},} by one, in analogy to the harmonic oscillator.
The category of locally compact quantum groups allows for a dual construction with which one can prove that the bi-dual of a locally compact quantum group is isomorphic to the original one. This result gives a far-reaching generalization of Pontryagin duality for locally compact Hausdorff abelian groups.
In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise mathematical structures. This demonstration requires new mathematics, in a sense analogous to classical real analysis, putting calculus on a mathematically rigorous foundation.
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states.
One of the explicit hypotheses in the construction of RQM is that quantum mechanics is a complete theory, that is it provides a full account of the world. Moreover, the Bohmian view seems to imply an underlying, "absolute" set of states of all systems, which is also ruled out as a consequence of RQM.