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Measure and probability theory



A state is a configuration of a system in physics, together with enough information about its time evolution? (typically, the first derivative is sufficient) to allow one to propagate it arbitrarily.

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 mechanics

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.

More generally, a classical state is a linear function ρ:A\rho\colon A \to \mathbb{R} on the Poisson algebra AA underlying the classical mechanical system which satisfies positivity and normalization.

In geometric quantization

In Hilbert-space quantum mechanics

In quantum mechanics given by a Hilbert space HH, a pure state is a ray in HH, which we often call the Hilbert space of states. Strictly speaking, the space of states is not HH but (H{0})/(H \setminus \{0\})/\mathbb{C}, or equivalently S(H)/U(1)S(H)/\mathrm{U}(1). A mixed state is then a density matrix on HH.


In AQFT, a quantum mechanical system is given by a C *C^*-algebra AA, and a quantum state is usually defined as a linear function ρ:A\rho\colon A \to \mathbb{C} 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 the FQFT formulation of quantum field theory, a physical system is given by a cobordism representation

Bord n S𝒞. Bord_n^S \to \mathcal{C} \,.

In this formulation the (n-1)-morphism in 𝒞\mathcal{C} assigned to an (n1)(n-1)-dimensional manifold Σ n1\Sigma_{n-1} is the space of states over that manifold. A state is accordingly a generalized element of this object.

Pure and mixed states

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 an impossible system, the space of states is empty; for a trivial system (with a unique way to be), then space of states is the point. This unique state is pure.

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 tt with 0t10 \leq t \leq 1. This tt is the probability that the system is in the first state, with 1t1 - t the probability that it is in the second. The two pure states correspond to t=0t = 0 and t=1t = 1.

For a quantum bit, a qubit, the space of states is shaped like a gridiron (American or Canadian) football. A state is given by a matrix

(a b+ic bic d) \begin{pmatrix} a & b + \mathrm{i} c \\ b - \mathrm{i} c & d \end{pmatrix}

with unit trace and nonnegative determinant; in other words, it's given by real numbers aa, bb, and cc satisfying the inequality

a 2+b 2+c 2a. a^2 + b^2 + c^2 \leq a .

The pure states are those satisfying

a 2+b 2+c 2=a, a^2 + b^2 + c^2 = a ,

forming the surface of the football (what one might call a gridiron footsphere, although properly it is a lemon). If we graph aa 2a - a^2 where it is positive (from 00 to 11) and rotate this around the aa-axis, then we get this lemon.

duality between algebra and geometry in physics:

Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

Revised on February 15, 2017 15:54:25 by Toby Bartels (