Quantum Mechanics in symmetry language. (arXiv:1305.4349v6 [quant-ph] UPDATED)

We consider symmetry as a foundational concept in quantum mechanics and
rewrite quantum mechanics and measurement axioms in this description. We argue
that issues related to measurements and physical reality of states can be
better understood in this view. In particular, the abstract concept of symmetry
provides a basis-independent definition for observables. Moreover, we show that
the apparent projection/collapse of the state as the final step of measurement
or decoherence is the result of breaking of symmetries. This phenomenon is
comparable with a phase transition by spontaneous symmetry breaking, and makes
the process of decoherence and classicality a natural fate of complex systems
consisting of many interacting subsystems. Additionally, we demonstrate that
the property of state space as a vector space representing symmetries is more
fundamental than being an abstract Hilbert space, and its $L2$ integrability
can be obtained from the imposed condition of being a representation of a
symmetry group and general properties of probability distributions.

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