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\begin{document}

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\section*{state}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{AlgebraicDefinition}{Algebraic definition}\dotfill \pageref*{AlgebraicDefinition} \linebreak
\noindent\hyperlink{in_classical_mechanics}{In classical mechanics}\dotfill \pageref*{in_classical_mechanics} \linebreak
\noindent\hyperlink{in_geometric_quantization}{In geometric quantization}\dotfill \pageref*{in_geometric_quantization} \linebreak
\noindent\hyperlink{in_hilbertspace_quantum_mechanics}{In Hilbert-space quantum mechanics}\dotfill \pageref*{in_hilbertspace_quantum_mechanics} \linebreak
\noindent\hyperlink{in_aqft}{In AQFT}\dotfill \pageref*{in_aqft} \linebreak
\noindent\hyperlink{in_fqft}{In FQFT}\dotfill \pageref*{in_fqft} \linebreak
\noindent\hyperlink{pure_and_mixed_states}{Pure and mixed states}\dotfill \pageref*{pure_and_mixed_states} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{state} of a system in [[physics]] is\ldots{}

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 \emph{[[pure state]]} gives maximal information about the system (which amounts to complete information in [[classical mechanics]] but not generally in [[quantum mechanics]]), while a \emph{[[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.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

The precise mathematical notion of \emph{state} depends on what mathematical formalization of mechanics is used.

\hypertarget{AlgebraicDefinition}{}\subsubsection*{{Algebraic definition}}\label{AlgebraicDefinition}

Quite generally, both in [[classical physics]] as well as in [[quantum physics]], one may define states as assignments of [[expectation values]] to [[observables]] in an [[algebra of observables]]. This is the definition used in \emph{[[quantum probability theory]]} (which subsumes ordinary [[probability theory]]). See at

\begin{itemize}%
\item \emph{[[state in AQFT and operator algebra]]}

\end{itemize}
for details.

\hypertarget{in_classical_mechanics}{}\subsubsection*{{In classical mechanics}}\label{in_classical_mechanics}

In classical [[Lagrangian mechanics]], a pure state is a [[global element|point]] in the [[state space]] of the system, giving all of the (generalised) [[generalized position|positions]] and [[velocity|velocities]]. In classical [[Hamiltonian mechanics]], a pure state is a point in the [[phase space]] of the system, giving the positions and [[momentum|momenta]]. In either case, a mixed state is a [[probability distribution]] on the space of pure states.

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

\hypertarget{in_geometric_quantization}{}\subsubsection*{{In geometric quantization}}\label{in_geometric_quantization}

\begin{itemize}%
\item [[space of states (in geometric quantization)]]

\end{itemize}
\hypertarget{in_hilbertspace_quantum_mechanics}{}\subsubsection*{{In Hilbert-space quantum mechanics}}\label{in_hilbertspace_quantum_mechanics}

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

\hypertarget{in_aqft}{}\subsubsection*{{In AQFT}}\label{in_aqft}

In [[AQFT]], a [[quantum mechanical system]] is given by a $C^*$-[[C-star-algebra|algebra]] $A$, and a [[quantum state]] is usually defined as a linear function $\rho\colon A \to \mathbb{C}$ which satisfies \emph{positivity} and \emph{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 \emph{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.

\hypertarget{in_fqft}{}\subsubsection*{{In FQFT}}\label{in_fqft}

In the [[FQFT]] formulation of [[quantum field theory]], a physical system is given by a [[cobordism]] [[representation]]

\begin{displaymath}
Bord_n^S \to \mathcal{C}
  \,.
\end{displaymath}
In this formulation the [[k-morphism|(n-1)-morphism]] in $\mathcal{C}$ assigned to an $(n-1)$-dimensional [[manifold]] $\Sigma_{n-1}$ is the \emph{space of states} over that manifold. A state is accordingly a [[generalized element]] of this object.

\hypertarget{pure_and_mixed_states}{}\subsection*{{Pure and mixed states}}\label{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 \emph{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]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Here are some toy examples of spaces of states.

For an impossible system, the space of states is [[empty space|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 $t$ with $0 \leq t \leq 1$. This $t$ is the probability that the system is in the first state, with $1 - t$ the probability that it is in the second. The two pure states correspond to $t = 0$ and $t = 1$.

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

\begin{displaymath}
\begin{pmatrix} a & b + \mathrm{i} c \\ b - \mathrm{i} c & d \end{pmatrix}
\end{displaymath}
with unit trace and nonnegative determinant; in other words, it's given by real numbers $a$, $b$, and $c$ satisfying the inequality

\begin{displaymath}
a^2 + b^2 + c^2 \leq a .
\end{displaymath}
The pure states are those satisfying

\begin{displaymath}
a^2 + b^2 + c^2 = a ,
\end{displaymath}
forming the surface of the football (what one might call a gridiron footsphere, although properly it is a \href{http://mathworld.wolfram.com/Lemon.html}{lemon}). If we graph $a - a^2$ where it is positive (from $0$ to $1$) and rotate this around the $a$-axis, then we get this lemon.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{state}

\begin{itemize}%
\item [[classical state]],


\item [[quantum state]]

\begin{itemize}%
\item [[wave function]]


\item [[superposition]]


\item [[space of states (in geometric quantization)]]


\item [[state in AQFT and operator algebra]]



\end{itemize}

\item [[ground state]]



\end{itemize}

\item [[observable]]

\begin{itemize}%
\item [[algebra of observables]]


\item [[GNS construction]]


\item [[quantum operator (in geometric quantization)]]



\end{itemize}


\end{itemize}
[[!include Isbell duality - table]]

[[!redirects state]] [[!redirects states]] [[!redirects physical state]] [[!redirects physical states]]

[[!redirects space of states]] [[!redirects spaces of states]]

[[!redirects space of physical states]] [[!redirects spaces of physicsal states]]



\end{document}