Wigner's theoremWigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert space of states. The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor, i.e. by the complex line or ray the vector spans.
Rigged Hilbert spaceIn mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place. A function such as is an eigenfunction of the differential operator on the real line R, but isn't square-integrable for the usual Borel measure on R.
Projective Hilbert spaceIn mathematics and the foundations of quantum mechanics, the projective Hilbert space of a complex Hilbert space is the set of equivalence classes of non-zero vectors in , for the relation on given by if and only if for some non-zero complex number . The equivalence classes of for the relation are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space. The physical significance of the projective Hilbert space is that in quantum theory, the wave functions and represent the same physical state, for any .
Haag's theoremWhile working on the mathematical physics of an interacting, relativistic, quantum field theory, Rudolf Haag developed an argument against the existence of the interaction picture, a result now commonly known as Haag’s theorem. Haag’s original proof relied on the specific form of then-common field theories, but subsequently generalized by a number of authors, notably Hall & Wightman, who concluded that no single, universal Hilbert space representation can describe both free and interacting fields.
Phase factorFor any complex number written in polar form (such as r eiθ), the phase factor is the complex exponential factor (eiθ). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on the unit circle in the complex plane). The phase factor is a unit complex number, i.e. a complex number of absolute value 1. It is commonly used in quantum mechanics. The variable θ appearing in such an expression is generally referred to as the phase.
SuperselectionIn quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originally introduced by Wick, Wightman, and Wigner to impose additional restrictions to quantum theory beyond those of selection rules. Mathematically speaking, two quantum states and are separated by a selection rule if for the given Hamiltonian , while they are separated by a superselection rule if for all physical observables .
