Types of quantum field thories
In the Bayesian interpretation of physics, the state of a system is not a property of reality but instead indicates an observer's knowledge about the system. A pure state gives maximal information about the system (which amounts to complete information in classical mechanics but not generally in quantum mechanics), while a mixed state is more general. A mixed state can be decomposed into a probability distribution on the space of pure states, although this decomposition is unique only for classical systems. In a frequentist interpretation of probability, a mixed state can describe only a statistical ensemble of systems; the real world is in one (generally unknown) pure state (possibly with additional hidden variables in the quantum case, depending on the interpretation of quantum physics).
States in the Schrödinger picture describe the state of the world at any given time and are subject to time evolution?, while in the Heisenberg picture a single state describes the entire history of the world.
The precise mathematical notion of state depends on what mathematical formalization of mechanics is used.
In classical Lagrangian mechanics, a pure state is a point in the state space? of the system, giving all of the (generalised) positions? and velocities?. In classical Hamiltonian mechanics, a pure state is a point in the phase space of the system, giving the positions and momenta. In either case, a mixed state is a probability distribution on the space of pure states.
In quantum mechanics given by a Hilbert space , a pure state is a ray in , which we often call the Hilbert space of states. Strictly speaking, the space of states is not but , or equivalently . A mixed state is then a density matrix on .
In AQFT, a quantum mechanical system is given by a -algebra , and a quantum state is usually defined as a linear function which satisfies positivity and normalization; see states in AQFT and operator algebra.
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.
In statistical physics, a pure state is a state of maximal information, while a mixed state is a state with less than maximal information. In the classical case, we may say that a pure state is a state of complete information, but this does not work in the quantum case; from the perspective of the information-theoretic or Bayesian interpretation of quantum physics, this inability to have complete information, even when having maximal information, is the key feature of quantum physics that distinguishes it from classical physics.
See pure state.
Here are some toy examples of spaces of states.
For a classical bit, a system with two distinct ways to be, the space of states is a line segment?; a state is given by a real number with . This is the probability that the system is in the first state, with the probability that it is in the second. The two pure states correspond to and .
For a quantum bit, a qubit, the space of states is shaped like an American football. A state is given by a matrix
with unit trace and nonnegative determinant; in other words, it's given by real numbers , , and satisfying the inequality
The pure states are those satisfying
forming the surface of the football (what one might call an American footsphere, although properly it is a lemon). If we graph where it is positive (from to ) and rotate this around the -axis, then we get this lemon.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|