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Quantum mechanics
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Quantum logic
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In mathematical physics and quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Quantum mechanics (QM or quantum theory) is a physical science dealing with the behavior of Matter and Energy on the scale of Atoms The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. An interpretation of quantum mechanics is a statement which attempts to explain how Quantum mechanics informs our Understanding of Nature. The Copenhagen interpretation is an interpretation of Quantum mechanics. The Ensemble Interpretation, or Statistical Interpretation of Quantum mechanics, is an interpretation that can be viewed as a minimalist interpretation it is a quantum Historically in Physics, hidden variable theories were espoused by a minority of Physicists who argued that the statistical nature of Quantum mechanics The transactional interpretation of Quantum mechanics ( TIQM) is an Interpretation of quantum mechanics that describes quantum interactions in terms of a The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, parallel universes, In Quantum mechanics, the consistent histories approach is intended to give a modern Interpretation of quantum mechanics, generalising the conventional Copenhagen Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Reasoning is the cognitive process of looking for Reasons for beliefs conclusions actions or feelings In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence This research area and its name originated in the 1936 paper by Garrett Birkhoff and John von Neumann, who were attempting to reconcile the apparent inconsistency of classical boolean logic with the facts concerning the measurement of complementary variables in quantum mechanics, such as position and momentum. Garrett Birkhoff ( January 19, 1911, Princeton, New Jersey, USA – November Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of In Physics, complementarity is a basic principle of quantum theory closely identified with the Copenhagen interpretation, and refers to effects such

Quantum logic can be formulated as a modified version of propositional logic. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" It has some properties which clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:

p and (q or r) = (p and q) or (p and r),

where the symbols p, q and r are propositional variables. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law To illustrate why the distributive law fails, consider a particle moving on a line and let

p = "the particle is moving to the right"
q = "the particle is to the left of the origin"
r = "the particle is to the right of the origin"

then the proposition "q or r" is true, so

p and (q or r) = true

On the other hand, the propositions "p and q" and "p and r" are both false, since they assert tighter restrictions on simultaneous values of position and momentum than is allowed by the uncertainty principle. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain So,

(p and q) or (p and r) = false

Thus the distributive law fails.

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. Hilary Whitehall Putnam (born July 31 1926 is an American Philosopher who has been a central figure in Western philosophy since the 1960s especially in Philosophy This thesis was an important ingredient in Putnam's paper Is Logic Empirical? in which he analysed the epistemological status of the rules of propositional logic. " Is logic empirical? " is the title of two articles that discuss the idea that the algebraic properties of logic may or should be empirically determined in particular they Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself to the physicist David Finkelstein. David Ritz Finkelstein (born July 19, 1929, New York City) is currently an emeritus professor of Physics at the Georgia Institute of Technology 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. George Whitelaw Mackey (February 1 1916 in St Louis, Missouri – March 15 2006 in Belmont, Massachusetts) was an American Mathematician

The more common view regarding quantum logic, however, is that it provides a formalism for relating observables, system preparation filters and states. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical 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. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. The similarities of the quantum logic formalism to a system of deductive logic may then be regarded more as a curiosity than as a fact of fundamental philosophical importance. Deductive reasoning is Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the

Contents

Introduction

In his classic treatise Mathematical Foundations of Quantum Mechanics, John von Neumann noted that projections on a Hilbert space can be viewed as propositions about physical observables. This article assumes some familiarity with Analytic geometry and the concept of a limit. 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) G. Mackey attempted to provide a set of axioms for this propositional system as an orthocomplemented lattice. George Whitelaw Mackey (February 1 1916 in St Louis, Missouri – March 15 2006 in Belmont, Massachusetts) was an American Mathematician In Lattice theory, a branch of the mathematical discipline called Order theory, an orthocomplemented lattice (or just ortholattice) is an Algebraic 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. In Lattice theory, a branch of the mathematical discipline called Order theory, an orthocomplemented lattice (or just ortholattice) is an Algebraic Closed may refer to Math Closure (mathematics Closed manifold Closed orbits Closed The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' 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. In Mathematics, spectral theory is an inclusive term for theories extending the Eigenvector and Eigenvalue theory of a single Square matrix. In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix However, the main ideas can be understood using the finite-dimensional spectral theorem.

Projections as propositions

The so-called Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical In physics the term dynamics customarily refers to the time evolution of physical processes 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. The propositions concerning a classical system are generated from basic statements 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. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. By logic in this context we mean the rules that relate set operations and ordering relations, such as de Morgan's laws. In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed 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 sets. In Mathematics, the Borel algebra (or Borel &sigma-algebra) on a Topological space X is a &sigma-algebra of Subsets of 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:

\operatorname{LUB}(\{E_i\}) = \bigcup_{i=1}^\infty E_i.

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 operator A on a Hilbert space H. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R. In Mathematics, particularly Functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint In particular, for any bounded Borel function f, the following equation holds:

f(A) = \int_{\mathbb{R}} f(\lambda) \, d \operatorname{E}(\lambda).

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

The propositional lattice of a quantum mechanical system

This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics. This is essentially Mackey's Axiom VII:

Q is also sequentially complete: any pairwise disjoint sequence{Vi}i of elements of Q has a least upper bound. Here disjointness of W1 and W2 means W2 is a subspace of W1⊥. The least upper bound of {Vi}i is the closed internal direct sum.

Henceforth we identify elements of Q with self-adjoint projections on the Hilbert space H.

The structure of Q immediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a proposition p, the equations

I = p \vee q
0 = p\wedge q

have exactly one solution, namely the set-theoretic complement of p. In these equations I refers to the atomic proposition which is identically true and 0 the atomic proposition which is identically false. In the case of the lattice of projections there are infinitely many solutions to the above equations.

Having made these preliminary remarks, we turn everything around and attempt to define observables within the projection lattice framework and using this definition establish the correspondence between self-adjoint operators and observables : A Mackey observable is a countably additive homomorphism from the orthocomplemented lattice of the Borel subsets of R to Q. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with To say the mapping φ is a countably additive homomorphism means that for any sequence {Si}i of pairwise disjoint Borel subsets of R, {φ(Si)}i are pairwise orthogonal projections and

\varphi\left(\bigcup_{i=1}^\infty S_i\right) = \sum_{i=1}^\infty \varphi(S_i).

Theorem. There is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on H.

This is the content of the spectral theorem as stated in terms of spectral measures. In Mathematics, particularly Functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint

Statistical structure

Imagine a forensics lab which has some apparatus to measure the speed of a bullet fired from a gun. Under carefully controlled conditions of temperature, humidity, pressure and so on the same gun is fired repeatedly and speed measurements taken. This produces some distribution of speeds. Though we will not get exactly the same value for each individual measurement, for each cluster of measurements, we would expect the experiment to lead to the same distribution of speeds. In particular, we can expect to assign probability distributions to propositions such as {a ≤ speed ≤ b}. Probability is the likelihood or chance that something is the case or will happen This leads naturally to propose that under controlled conditions of preparation, the measurement of a classical system can be described by a probability measure on the state space. This same statistical structure is also present in quantum mechanics.

A quantum probability measure is a function P defined on Q with values in [0,1] such that P(0)=0, P(I)=1 and if {Ei}i is a sequence of pairwise orthogonal elements of Q then

\operatorname{P}\!\left(\sum_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \operatorname{P}(E_i).

The following highly non-trivial theorem is due to Andrew Gleason:

Theorem. Andrew Mattei Gleason (born November 4 1921 in Fresno, California, U Suppose H is a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure on Q there exists a unique trace class operator S such that

\operatorname{P}(E) = \operatorname{Tr}(S E)

for any self-adjoint projection E. In Mathematics, a trace class operator is a Compact operator for which a trace may be defined such that the trace is finite and independent of the choice

The operator S is necessarily non-negative (that is all eigenvalues are non-negative) and of trace 1. Such an operator is often called a density operator.

Physicists commonly regard a density operator as being represented by a (possibly infinite) density matrix relative to some orthonormal basis.

For more information on statistics of quantum systems, see quantum statistical mechanics. Quantum statistical mechanics is the study of Statistical ensembles of quantum mechanical systems.

Automorphisms

An automorphism of Q is a bijective mapping α:Q → Q which preserves the orthocomplemented structure of Q, that is

\alpha\!\left(\sum_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \alpha(E_i)

for any sequence {Ei}i of pairwise orthogonal self-adjoint projections. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself Note that this property implies monotonicity of α. If P is a quantum probability measure on Q, then E → α(E) is also a quantum probability measure on Q. By the Gleason theorem characterizing quantum probability measures quoted above, any automorphism α induces a mapping α* on the density operators by the following formula:

\operatorname{Tr}(\alpha^*(S) E) = \operatorname{Tr}(S \alpha(E)).

The mapping α* is bijective and preserves convex combinations of density operators. Gleason's theorem, named after Andrew Gleason, is a mathematical result of particular importance for Quantum logic. This means

\alpha^*(r_1 S_1 + r_2 S_2) = r_1\alpha^*(S_1) + r_2 \alpha^*(S_2) \quad

whenever 1 = r1 + r2 and r1, r2 are non-negative real numbers. Now we use a theorem of Richard Kadison:

Theorem. Suppose β is a bijective map from density operators to density operators which is convexity preserving. Then there is an operator U on the Hilbert space which is either linear or conjugate-linear, preserves the inner product and is such that

\beta(S) = U S U^* \,

for every density operator S. In the first case we say U is unitary, in the second case U is anti-unitary.

Remark. This note is included for technical accuracy only, and should not concern most readers. The result quoted above is not directly stated in Kadison's paper, but can be reduced to it by noting first that β extends to a positive trace preserving map on the trace class operators, then applying duality and finally applying a result of Kadison's paper.

The operator U is not quite unique; if r is a complex scalar of modulus 1, then r U will be unitary or anti-unitary if U is and will implement the same automorphism. In fact, this is the only ambiguity possible.

It follows that automorphisms of Q are in bijective correspondence to unitary or anti-unitary operators modulo multiplication by scalars of modulus 1. Moreover, we can regard automorphisms in two equivalent ways: as operating on states (represented as density operators) or as operating on Q.

Non-relativistic dynamics

In non-relativistic physical systems, there is no ambiguity in referring to time evolution since there is a global time parameter. Moreover an isolated quantum system evolves in a deterministic way: if the system is in a state S at time t then at time s > t, the system is in a state Fs,t(S). Determinism is the philosophical Proposition that every event including human cognition and behaviour decision and action is causally determined Moreover, we assume

By Kadison's theorem, there is a 1-parameter family of unitary or anti-unitary operators {Ut}t such that

\operatorname{F}_{s,t}(S) = U_{s-t} S U_{s-t}^*

In fact,

Theorem. Under the above assumptions, there is a strongly continuous 1-parameter group of unitary operators {Ut}t such that the above equation holds.

Note that it easily from uniqueness from Kadison's theorem that

Ut + s = σ(t,s)UtUs

where σ(t,s) has modulus 1. Now the square of an anti-unitary is a unitary, so that all the Ut are unitary. The remainder of the argument shows that σ(t,s) can be chosen to be 1 (by modifying each Ut by a scalar of modulus 1. )

Pure states

A convex combinations of statistical states S1 and S2 is a state of the form S = p1 S1 +p2 S2 where p1, p2 are non-negative and p1 + p2 =1. Considering the statistical state of system as specified by lab conditions used for its preparation, the convex combination S can be regarded as the state formed in the following way: toss a biased coin with outcome probabilities p1, p2 and depending on outcome choose system prepared to S1 or S2

Density operators form a convex set. The convex set of density operators has extreme points; these are the density operators given by a projection onto a one-dimensional space. To see that any extreme point is such a projection, note that by the spectral theorem S can be represented by a diagonal matrix; since S is non-negative all the entries are non-negative and since S has trace 1, the diagonal entries must add up to 1. Now if it happens that the diagonal matrix has more than one non-zero entry it is clear that we can express it as a convex combination of other density operators.

The extreme points of the set of density operators are called pure states. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. If S is the projection on the 1-dimensional space generated by a vector ψ of norm 1 then

\operatorname{Tr}(S E) = \langle E \psi | \psi \rangle

for any E in Q. In physics jargon, if

S = | \psi \rangle \langle \psi | ,

where ψ has norm 1, then

\operatorname{Tr}(S E) = \langle \psi | E | \psi \rangle .

Thus pure states can be identified with rays in the Hilbert space H.

The measurement process

Consider a quantum mechanical system with lattice Q which is in some statistical state given by a density operator S. This essentially means an ensemble of systems specified by a repeatable lab preparation process. The result of a cluster of measurements intended to determine the truth value of proposition E, is just as in the classical case, a probability distribution of truth values T and F. Say the probabilities are p for T and q = 1 - p for F. By the previous section p = Tr(S E) and q = Tr(S (I-E)).

Perhaps the most fundamental difference between classical and quantum systems is the following: regardless of what process is used to determine E immediately after the measurement the system will be in one of two statistical states:

\frac{1}{\operatorname{Tr}(E S)} E S E.
\frac{1}{\operatorname{Tr}((I-E) S)}(I- E) S (I- E).

(We leave to the reader the handling of the degenerate cases in which the denominators may be 0. ) We now form the convex combination of these two ensembles using the relative frequencies p and q. We thus obtain the result that the measurement process applied to a statistical ensemble in state S yields another ensemble in statistical state:

\operatorname{M}_E(S) = E S E + (I - E) S (I - E)

We see that a pure ensemble becomes a mixed ensemble after measurement. Measurement, as described above, is a special case of quantum operations. In Quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo

Limitations of quantum logic

Quantum logic provides a satisfactory foundation for a theory of reversible quantum processes. Examples of such processes are the covariance transformations relating two frames of reference, such as change of time parameter or the transformations of special relativity. Quantum logic also provides a satisfactory understanding of density matrices. Quantum logic can be stretched to account for some kinds of measurement processes corresponding to answering yes-no questions about the state of a quantum system. However, for more general kinds of measurement operations (that is quantum operations), a more complete theory of filtering processes is necessary. Such an approach is provided by the consistent histories formalism. In Quantum mechanics, the consistent histories approach is intended to give a modern Interpretation of quantum mechanics, generalising the conventional Copenhagen

In any case, these quantum logic formalisms must be generalized in order to deal with super-geometry (which is needed to handle Fermi-fields) and non-commutative geometry (which is needed in string theory and quantum gravity theory). Both of these theories use a partial algebra with an "integral" or "trace". The elements of the partial algebra are not observables; instead the "trace" yields "greens functions" which generate scattering amplitudes. One thus obtains a local S-matrix theory (see D. Edwards).

Since around 1978 the Flato school ( see F. Bayen ) has been developing an alternative to the quantum logics approach called deformation quantization (see Weyl quantization ). In Mathematics and Physics, in the area of Quantum mechanics, Weyl quantization is a method for associating a "quantum mechanical" Hermitian

See also

References

External links

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