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The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups. [22] The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. [23]
In quantum mechanics a state space is a separable complex Hilbert space.The dimension of this Hilbert space depends on the system we choose to describe. [1] [2] The different states that could come out of any particular measurement form an orthonormal basis, so any state vector in the state space can be written as a linear combination of these basis vectors.
A state of the quantum system is a unit vector of , up to scalar multiples; or equivalently, a ray of the Hilbert space . The expectation value of an observable A for a system in a state ψ {\displaystyle \psi } is given by the inner product ψ , A ψ {\displaystyle \langle \psi ,A\psi \rangle } .
In quantum field theory, it is expected that the Hilbert space is also the space on the configuration space of the field, which is infinite dimensional, with respect to some Borel measure naturally defined. However, it is often hard to define a concrete Borel measure on the classical configuration space, since the integral theory on infinite ...
Von Neumman formalized quantum mechanics using the concept of Hilbert spaces and linear operators. [3] He acknowledged the previous work by Paul Dirac on the mathematical formalization of quantum mechanics, but was skeptical of Dirac's use of delta functions. He wrote the book in an attempt to be even more mathematically rigorous than Dirac. [4]
Let | be an arbitrary quantum state in some Hilbert space and let there be a physical process that transforms | | with = | |. If is independent of the input state | , then in the enlarged Hilbert space the mapping is of the form | | | | = | (| | ), where | is the initial state of the environment, | 's are the orthonormal basis of the environment Hilbert space and denotes the fact that one may ...
If a quantum field is free and Euclidean-invariant in the spatial dimensions, then that field's vacuum does not polarize. If two Poincaré-invariant quantum fields share the same vacuum, then their first four Wightman functions coincide. Moreover, if one such field is free, then the other must also be a free field of the same mass.
The relevant axioms are that as in quantum mechanics (i) the states of a system are represented by normalized vectors of a complex Hilbert space, and the observables are given by self-adjoint operators acting on that space, (ii) the expectation value of an observable is obtained in the manner as the expectation value in quantum mechanics, (iii ...