Types of quantum field thories
The precise mathematical notion of state depends on what mathematical formalization of quantum mechanics is used.
In the simple formulation over a Hilbert space , a pure state is a ray in . Thus, the pure states form the space , where we mod out by the action of on by scalar multiplication; equivalently, we can use , the unit sphere? in modulo the action of the unitary group . Often by abuse of language, one calls the ‘space of states’.
The mixed states are density matrices on . Every pure state may be interpreted as a mixed state; taking a representative normalised vector from a ray in Hilbert space, the operator is a density matrix.
In principle, any quantum mechanical system can be treated using Hilbert spaces, by imposing superselection rules that identify only some self-adjoint operators on as observables. Alternatively, one may take a more abstract approach, as follows.
it is positive: for every we have ;
it is normalized: .
See also state in AQFT and operator algebra.
If is a Hilbert space, then the bounded operators on form a -algebra , and states on the Hilbert space correspond directly to states on . Classical mechanics can also be formulated in AQFT; the classical space of states gives rise to a commutative von Neumann algebra as the algebra of observables.
Arguably, the correct notion of state to use is that of quasi-state; every state gives rise to a unique quasi-state, but not conversely. However, when either classical mechanics or Hilbert-space quantum mechanics is formulated in AQFT, every quasi-state is a state (at least if the Hilbert space is not of very low dimension, by Gleason's theorem). See also the Idea-section at Bohr topos for a discussion of this point.
|classical mechanics||semiclassical approximation||…||formal deformation quantization||quantum mechanics|
|order of Planck's constant|
|states||classical state||semiclassical state||quantum state|
|observables||classical observable||quantum observable|