Editing Quantum logic (section)

== Quantum logic as the logic of observables ==
The remainder of this article assumes the reader is familiar with the [[spectral theory]] of [[self-adjoint operator]]s on a Hilbert space. However, the main ideas can be under&shy;stood in the [[Dimension (vector space)|finite-dimensional]] case.

=== Logic of classical mechanics ===
The [[Hamiltonian mechanics|Hamiltonian]] formulations of [[classical mechanics]] have three ingredients: [[Classical mechanics|states]], [[observable]]s and [[Dynamics (mechanics)|dynamics]].  In the simplest case of a single particle moving in '''R'''<sup>3</sup>, the state space is the position–momentum space '''R'''<sup>6</sup>. An observable is some [[real-valued function]] ''f'' on the state space.  Examples of observables are position, momentum or energy of a particle.   For  classical systems, the value ''f''(''x''), that is the value of ''f'' for some particular system state ''x'', is obtained by a process of measurement of ''f''.

The [[proposition]]s concerning a classical system are generated from basic statements of the form
: "Measurement of ''f'' yields a value in the interval [''a'', ''b''] for some real numbers ''a'', ''b''."

through the conventional arithmetic operations and [[Limit (mathematics)|pointwise limits]].  It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to the [[Boolean algebra (structure)|Boolean algebra]] of [[Borel subset]]s of the state space.  They thus obey the laws of [[Classical logic|classical]] [[propositional logic]] (such as [[de&nbsp;Morgan's laws]]) with the set operations of union and intersection corresponding to the [[Boolean operator (Boolean algebra)|Boolean conjunctives]] and subset inclusion corresponding to [[Material implication (rule of inference)|material implication]].

In fact, a stronger claim is true: they must obey the [[infinitary logic]] {{Math|''L''<sub>&omega;<sub>1</sub>,&omega;</sub>}}.

We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguished ''orthocomplementation'' operation:  The lattice operations of ''meet'' and ''join'' are respectively set intersection and set union.  The orthocomplementation  operation is set complement.  Moreover, this lattice is ''sequentially complete'', in the sense that any sequence {''E''<sub>''i''</sub>}<sub>''i''∈'''N'''</sub> of elements of the lattice has a [[least upper bound]], specifically the set-theoretic union: <math display="block"> \operatorname{LUB}(\{E_i\}) = \bigcup_{i=1}^\infty E_i\text{.} </math>

=== Propositional lattice of a quantum mechanical system ===
In the [[Hilbert space]] formulation of quantum mechanics as presented by von&nbsp;Neumann, a physical observable is represented by some (possibly [[bounded operator|unbounded]]) densely defined [[self-adjoint operator]] ''A'' on a Hilbert space ''H''.  ''A'' has a [[Spectral theorem|spectral decomposition]], which is a [[projection-valued measure]] E defined on the Borel subsets of '''R'''.  In particular, for any bounded [[Borel function]] ''f'' on '''R''', the following extension of ''f'' to operators can be made: <math display=block> f(A) = \int_{\mathbb{R}} f(\lambda) \, d \operatorname{E}(\lambda).</math>

In case ''f'' is the indicator function of an interval [''a'', ''b''], the operator ''f''(''A'') is a self-adjoint projection onto the subspace of [[generalized eigenvector]]s of ''A'' with eigenvalue in {{closed-closed|''a'',''b''}}.  That subspace can be interpreted as the quantum analogue of the classical proposition
* Measurement of ''A'' yields a value in the interval [''a'', ''b''].

This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey's ''Axiom VII'':
* The propositions of a quantum mechanical system correspond to the lattice of closed subspaces of ''H''; the negation of a proposition ''V'' is the orthogonal complement ''V''<sup>⊥</sup>.

The space ''Q'' of quantum propositions is also sequentially complete: any pairwise-disjoint sequence {''V''<sub>''i''</sub>}<sub>''i''</sub> of elements of ''Q'' has a least upper bound.  Here disjointness of ''W''<sub>1</sub> and ''W''<sub>2</sub> means ''W''<sub>2</sub> is a subspace of ''W''<sub>1</sub><sup>⊥</sup>. The least upper bound of {''V''<sub>''i''</sub>}<sub>''i''</sub> is the closed internal [[direct sum]].

=== Standard semantics ===
The standard semantics of quantum logic is that quantum logic is the logic of [[projection operator]]s in a [[separable space|separable]] [[Hilbert space|Hilbert]] or [[pre-Hilbert space]], where an observable ''p'' is associated with the [[eigenspace|set of quantum states]] for which ''p'' (when measured) has [[eigenvalue]] 1.  From there,
* ''¬p'' is the [[orthogonal complement]] of ''p'' (since for those states, the probability of observing ''p'', P(''p'') = 0),
* ''p''∧''q'' is the intersection of ''p'' and ''q'', and
* ''p''∨''q'' = ¬(¬''p''∧¬''q'') refers to states that [[quantum superposition|superpose]] ''p'' and ''q''.

This semantics has the nice property that the pre-Hilbert space is complete (i.e., Hilbert) if and only if the propositions satisfy the orthomodular law, a result known as the [[Solèr theorem]].<ref>{{harvnb|Dalla Chiara|Giuntini|2002}} and {{harvnb|de&nbsp;Ronde|Domenech|Freytes}}.  Despite suggestions otherwise in Josef Jauch, ''Foundations of Quantum Mechanics'', Addison-Wesley Series in Advanced Physics; Addison-Wesley, 1968, this property cannot be used to deduce a vector space structure, because it is not peculiar to (pre-)Hilbert spaces.  An analogous claim holds in most [[Category (math)|categories]]; see John Harding, "[https://www.ams.org/journals/tran/1996-348-05/S0002-9947-96-01548-6/S0002-9947-96-01548-6.pdf Decompositions in Quantum Logic]," ''Transactions of the AMS'', vol.&nbsp;348, no.&nbsp;5, 1996.  pp.&nbsp;1839-1862.</ref> Although much of the development of quantum logic has been motivated by the standard semantics, it is not characterized by the latter; there are additional properties satisfied by that lattice that need not hold in quantum logic.{{sfn|Dalla&nbsp;Chiara|Giuntini|2002}}