| • Science | • People | • Locations | • Timeline |
| Contents | ||
Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopher Hilary Putnam, at least at one point in his career. Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself, to the physicist David Finkelstein. It should be noted, however, that this idea had been around for some time and had been revived several years earlier by George Mackey's work on group representations and symmetry.
This logic has some unusual properties; for instance, the distributive law of propositional logic,
fails in this logic.
The more common view regarding quantum logic, however, is that it provides a formalism for relating observables, system preparation filters and states. In this view, the quantum logic approach resembles more closely the C*-algebraic approach to quantum mechanics; in fact with some minor technical assumptions it can be subsumed by it. The similarities of the quantum logic formalism to a system of deductive logic is regarded more as a curiosity than as a fact of fundamental philosophical importance.
In his classic treatise Mathematical Foundations of Quantum Mechanics, von Neumann noted that projections on a Hilbert space can be viewed as propositions about physical observables. The set of principles for manipulating these quantum propositions was called quantum logic by von Neumann and Birkhoff. In his book (also called Mathematical Foundations of Quantum Mechanics) Mackey attempted to provide a set of axioms for this propositional system as an orthocomplemented partially ordered set. Mackey viewed elements of this set as potential yes or no questions an observer might ask about the state of a physical system, questions that would be settled by some measurement. Moreover Mackey defined a physical observable in terms of these basic questions. Mackey's axiom system is somewhat unsatisfactory though, since it assumes that the partially ordered set is actually given as the orthocomplemented closed subspace lattice of a separable Hilbert space. Piron , Ludwig and others have attempted to give axiomatizations which do not require such explicit relations to the lattice of subspaces.
The remainder of the following article assumes the reader is familiar with the spectral theory of self-adjoint operators on a Hilbert space. However, the main ideas can be understood using the finite-dimensional spectral theorem.
The so-called Hamiltonian formulations of classical mechanicsClassical mechanics is a model of the physics of forces acting upon bodies. It is often referred to as Newtonian mechanics after Newton and his laws of motion. Classical mechanics is subdivided into statics (which models objects at rest), kinematics (whic have three ingredients: stateThis article discusses states as sovereign political entities. For other meanings, see state (disambiguation). In international law and international relations, a state is a political entity possessing sovereignty, i. not being subject to any higher polits, observables and dynamicsThe word dynamics can refer to: a branch of mechanics; see dynamics (mechanics) the volume of music; see dynamics (music) When used referring to mechanics, it is referring to the study of the motion of both rigid bodies and particles. The field of dynamic. In the simplest case of a single particle moving in R3, the state space is the position-momentum space R6. We will merely note here that 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. A propositionIn modern logic, a proposition or ansatz is what is asserted as the result of uttering a sentence. In other words, it is the meaning of the sentence, rather than the sentence itself. Different sentences can express the same proposition, if they have the s concerning a classical system is a statement of the form
It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to that of some boolean algebra of subsets of the state space. By logic in this context we mean the rules that relate set operations and ordering relations, such as de Morgan's lawsIn logic, De Morgan's laws (or De Morgan's theorem), named for nineteenth century logician and mathematician Augustus De Morgan, are the two rules of propositional logic, boolean algebra and set theory :not (P and Q) (not P) or (not Q) :not (P or Q) (not. These are analogous to the rules relating boolean conjunctives and material implication in classical propositional logic. For technical reasons, we will also assume that the algebra of subsets of the state space is that of all Borel setTopology In mathematics, the Borel algebra (or Borel σ-algebra on a topological space is either of two σ-algebras on a topological space X The minimal σ-algebra containing the open sets. The minimal σ-algebra containing the compacts. The set of propositions is ordered by the natural ordering of sets and has a complementation operation. In terms of observables, the complement of the proposition {f ≥ a} is {f < a}.
We summarize these remarks as follows:
In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely-defined self-adjoint operatorOn a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. By the finite-dimensional A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measureIn mathematics, projection-valued measures are used to express results in spectral theory. A projection-valued measure on a measurable space X M is a mapping π from M to the set of self-adjoint projections on a Hilbert space H such that : and for every E defined on the Borel subsets of R. In particular, for any bounded Borel function f, the following equation holds:
In case f is the indicator function of an interval [a, b], the operator f(A) is a self-adjoint projection, and can be interpreted as the quantum analogue of the classical proposition