Quantum Mechanics in symmetry language

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.