Language Research Foundation, Boston. Mathematical Foundations of Quantum Mechanics. O'Connor and E. See 5. See 8. The New York Times. December 27, It is widely used in particle physics and condensed matter physics. Most theories in modern particle physics, including the Standard Model of elementary particles and their interactions, are formulated as relativistic quantum field theories. In condensed matter physics, quantum field theories are used in many circumstances, especially those where the number of particles is allowed to fluctuate—for example, in the BCS theory of superconductivity.
In quantum field theory QFT the forces between particles are mediated by other particles. The electromagnetic force between two electrons is caused by an exchange of photons. Intermediate vector bosons mediate the weak force and gluons mediate the strong force. There is currently no complete quantum theory of the remaining fundamental force, gravity, but many of the proposed theories postulate the existence of a graviton particle which mediates it. These force-carrying particles are virtual particles and, by definition, cannot be detected while carrying the force, because such detection will imply that the force is not being carried.
In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta - necessarily chunked ripples in a field that 'look like' particles. Fermions, like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field field quanta.
In , Max Born, Pascual Jordan, and Werner Heisenberg constructed such a theory by expressing the field's internal degrees of freedom as an infinite set of harmonic oscillators and by employing the usual procedure for quantizing those oscillators canonical quantization. This theory assumed that no electric charges or currents were present and today would be called a free field theory.
This quantum field theory could be used to model important processes such as the emission of a photon by an electron dropping into a quantum state of lower energy, a process in which the number of particles changes — one atom in the initial state becomes an atom plus a photon in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.
It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had after all grown out of the study of classical electromagnetism. This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory.
Pascual Jordan and Wolfgang Pauli showed in that quantum fields could be made to behave in the way predicted by special relativity during coordinate transformations specifically, they showed that the field commutators were Lorentz invariant , and in Niels Bohr and Leon Rosenfeld showed that this result could be interpreted as a limitation on the ability to measure fields at space-like separations, exactly as required by relativity. A further boost for quantum field theory came with the discovery of the Dirac equation, a single-particle equation obeying both relativity and quantum mechanics, when it was shown that several of its undesirable properties such as negative-energy states could be eliminated by reformulating the Dirac equation as a quantum field theory.
The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In , Jordan tried to extend the canonical quantization of fields to the many-body wavefunctions of identical particles, a procedure that is sometimes called second quantization. In , Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle.
This thread of development was incorporated into many-body theory, and strongly influenced condensed matter physics and nuclear physics. Despite its early successes, quantum field theory was plagued by several serious theoretical difficulties. Many seemingly-innocuous physical quantities, such as the energy shift of electron states due to the presence of the electromagnetic field, gave infinity — a nonsensical result — when computed using quantum field theory. This "divergence problem" was solved during the s by Bethe, Tomonaga, Schwinger, Feynman, and Dyson, through the procedure known as renormalization.
This phase of development culminated with the construction of the modern theory of quantum electrodynamics QED. Beginning in the s with the work of Yang and Mills, QED was generalized to a class of quantum field theories known as gauge theories. The s and s saw the formulation of a gauge theory now known as the Standard Model of particle physics, which describes all known elementary particles and the interactions between them.
The weak interaction part of the standard model was formulated by Sheldon Glashow, with the Higgs mechanism added by Steven Weinberg and Abdus Salam. The theory was shown to be renormalizable and hence consistent by Gerardus 't Hooft and Martinus Veltman. Also during the s, parallel developments in the study of phase transitions in condensed matter physics led Leo Kadanoff, Michael Fisher and Kenneth Wilson extending work of Ernst Stueckelberg, Andre Peterman, Murray Gell-Mann and Francis Low to a set of ideas and methods known as the renormalization group.
The study of quantum field theory is alive and flourishing, as are applications of this method to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to many branches of physics. Furthermore, each observable corresponds, in a technical sense, to the classical idea of a degree of freedom. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators and.
Ordinary quantum mechanics deals with systems such as this, which possess a small set of degrees of freedom. It is important to note, at this point, that this article does not use the word "particle" in the context of wave—particle duality. In quantum field theory, "particle" is a generic term for any discrete quantum mechanical entity, such as an electron, which can behave like classical particles or classical waves under different experimental conditions. A quantum field is a quantum mechanical system containing a large, and possibly infinite, number of degrees of freedom. This is not as exotic a situation as one might think.
A classical field contains a set of degrees of freedom at each point of space; for instance, the classical electromagnetic field defines two vectors — the electric field and the magnetic field — that can in principle take on distinct values for each position. When the field as a whole is considered as a quantum mechanical system, its observables form an infinite in fact uncountable set, because is continuous. Furthermore, the degrees of freedom in a quantum field are arranged in "repeated" sets.
For example, the degrees of freedom in an electromagnetic field can be grouped according to the position , with exactly two vectors for each. Note that is an ordinary number that "indexes" the observables; it is not to be confused with the position operator encountered in ordinary quantum mechanics, which is an observable. Thus, ordinary quantum mechanics is sometimes referred to as "zero-dimensional quantum field theory", because it contains only a single set of observables. It is also important to note that there is nothing special about because, as it turns out, there is generally more than one way of indexing the degrees of freedom in the field.
In the following sections, we will show how these ideas can be used to construct a quantum mechanical theory with the desired properties. We will begin by discussing single-particle quantum mechanics and the associated theory of many-particle quantum mechanics. Then, by finding a way to index the degrees of freedom in the many-particle problem, we will construct a quantum field and study its implications.
Quantum field theory We wish to consider how this problem generalizes to particles. There are two motivations for studying the many-particle problem.
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The first is a straightforward need in condensed matter physics, where typically the number of particles is on the order of Avogadro's number 6. The second motivation for the many-particle problem arises from particle physics and the desire to incorporate the effects of special relativity.
If one attempts to include the relativistic rest energy into the above equation, the result is either the Klein-Gordon equation or the Dirac equation. It turns out that such inconsistencies arise from neglecting the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles.
For example, an electron and a positron can annihilate each other to create photons. Thus, a consistent relativistic quantum theory must be formulated as a many-particle theory. Furthermore, we will assume that the particles are indistinguishable. As described in the article on identical particles, this implies that the state of the entire system must be either symmetric bosons or antisymmetric fermions when the coordinates of its constituent particles are exchanged. These multi-particle states are rather complicated to write.
For example, the general quantum state of a system of bosons is written as. In general, this is a sum of factorial distinct terms, which quickly becomes unmanageable as increases. The way to simplify this problem is to turn it into a quantum field theory. Second quantization In this section, we will describe a method for constructing a quantum field theory called second quantization. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states.
It is based on the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as the Feynman path integral , which uses a Lagrangian formulation. For an overview, see the article on quantization. Second quantization of bosons For simplicity, we will first discuss second quantization for bosons, which form perfectly symmetric quantum states.
Let us denote the mutually orthogonal single-particle states by and so on. For example, the 3-particle state with one particle in state and two in state is. The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the single-particle states etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as. The next step is to expand the -particle state space to include the state spaces for all possible values of. This extended state space, known as a Fock space, is composed of the state space of a system with no particles the so-called vacuum state , plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth.
It is easy to see that there is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space. At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number , indicating which of the single-particle states it refers to. The properties of this quantum field can be explored by defining creation and annihilation operators, which add and subtract particles.
They are analogous to "ladder operators" in the quantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator and creation operator have the following effects:.
It can be shown that these are operators in the usual quantum mechanical sense, i. Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the commutation relation. These are precisely the relations obeyed by the ladder operators for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.
For instance, the Hamiltonian of a field of free non-interacting bosons is. Second quantization of fermions It turns out that a different definition of creation and annihilation must be used for describing fermions. According to the Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers can only take on the value 0 or 1.
The fermionic annihilation operators and creation operators are defined by. One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle. Field operators We have previously mentioned that there can be more than one way of indexing the degrees of freedom in a quantum field. Second quantization indexes the field by enumerating the single-particle quantum states.
However, as we have discussed, it is more natural to think about a "field", such as the electromagnetic field, as a set of degrees of freedom indexed by position. To this end, we can define field operators that create or destroy a particle at a particular point in space. In particle physics, these operators turn out to be more convenient to work with, because they make it easier to formulate theories that satisfy the demands of relativity.
Single-particle states are usually enumerated in terms of their momenta as in the particle in a box problem. We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator is. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.
It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. This looks remarkably like an expression for the expectation value of the energy, with playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.
Unification of fields and particles The "second quantization" procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the electromagnetic field must be a quantum field theory, because it is impossible for various reasons to define a wavefunction for a single photon. In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the corresponding quantum theory.
The quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as well-defined creation and annihilation operators obeying commutation or anticommutation relations. Quantum field theory thus provides a unified framework for describing "field-like" objects such as the electromagnetic field, whose excitations are photons and "particle-like" objects such as electrons, which are treated as excitations of an underlying electron field.
Physical meaning of particle indistinguishability The second quantization procedure relies crucially on the particles being identical. We would not have been able to construct a quantum field theory from a distinguishable many-particle system, because there would have been no way of separating and indexing the degrees of freedom. Many physicists prefer to take the converse interpretation, which is that quantum field theory explains what identical particles are. In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric bosonic or antisymmetric fermionic states, and the need for such states is simply regarded as an empirical fact.
From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical? Particle conservation and non- conservation During second quantization, we started with a Hamiltonian and state space describing a fixed number of particles , and ended with a Hamiltonian and state space for an arbitrary number of particles.
Of course, in many common situations is an important and perfectly well-defined quantity, e. From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the number operator , which measures the total number of particles present. In that case, the quantum state is trapped in the -particle subspace of the total Fock space, and the situation could equally well be described by ordinary -particle quantum mechanics.
For example, we can see that the free-boson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator is immediately put back by the creation operator. On the other hand, it is possible, and indeed common, to encounter quantum states that are not eigenstates of , which do not have well-defined particle numbers.
Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as quantum superpositions of states having different values of. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as. This "interaction term" describes processes in which a fermion in state either absorbs or emits a boson, thereby being kicked into a different eigenstate.
In fact, this type of Hamiltonian is used to describe interaction between conduction electrons and phonons in metals. The interaction between electrons and photons is treated in a similar way, but is a little more complicated because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will typically end up with a superposition of states with different numbers of bosons at later times.
The number of fermions, however, is conserved in this case. In condensed matter physics, states with ill-defined particle numbers are particularly important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers. Axiomatic approaches The preceding description of quantum field theory follows the spirit in which most physicists approach the subject. However, it is not mathematically rigorous.
Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes. The first class of axioms, first proposed during the s, include the Wightman, Osterwalder-Schrader, and Haag-Kastler systems. They attempted to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis, and enjoyed limited success.
It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the CPT theorem. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the Standard Model, satisfied these axioms. The construction of theories satisfying one of these sets of axioms falls in the field of constructive quantum field theory. Important work was done in this area in the s by Segal, Glimm, Jaffe and others.
During the s, a second set of axioms based on geometric ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as topological quantum field theories, is associated most closely with Michael Atiyah and Graeme Segal, and was notably expanded upon by Edward Witten, Richard Borcherds, and Maxim Kontsevich. However, most physically-relevant quantum field theories, such as the Standard Model, are not topological quantum field theories; the quantum field theory of the fractional quantum Hall effect is a notable exception.
The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in representation theory, algebraic topology, and differential geometry. Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the Millennium Prize Problems—proving the existence of a mass gap in Yang-Mills theory—is linked to this issue. Phenomena associated with quantum field theory In the previous part of the article, we described the most general properties of quantum field theories.
Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections. Renormalization Early in the history of quantum field theory, it was found that many seemingly innocuous calculations, such as the perturbative shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. The reason is that the perturbation theory for the shift in an energy involves a sum over all other energy levels, and there are infinitely many levels at short distances which each give a finite contribution.
Many of these problems are related to failures in classical electrodynamics that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field which it carries around with it. The energy carried by a single electron—its self energy—is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons.
The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by Weisskopf, in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere. The solution to the problem, presciently suggested by Stueckelberg, independently by Bethe after the crucial experiment by Lamb, implemented at one loop by Schwinger, and systematically extended to all loops by Feynman and Dyson, with converging work by Tomonaga in isolated postwar Japan, is called renormalization.
The technique of renormalization recognizes that the problem is essentially purely mathematical, that extremely short distances are at fault. In order to define a theory on a continuum, first place a cutoff on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice. The rule is that one computes physical quantities in terms of the observable parameters such as the physical mass, not the bare parameters such as the bare mass.
The main point is not that of getting finite quantities any regularization procedure does that , but to eliminate the regularization parameters by a suitable addition of counterterms to the original Lagrangian. The main requirements on the counterterms are a Locality polynomials in the fields and their derivatives and b Finiteness number of monomials in the Lagrangian that remain finite after the introduction of all the necessary counterterms.
The reason for b is that each new counterterm leaves behind a free parameter of the theory like physical mass. There is no way such a parameter can be fixed other than by its experimental value, so one gets not a single theory but a family of theories parameterized by as many free parameters as the counterterms added to the Lagrangian. Since a theory with an infinite number of free parameters has virtually no predictive power the finiteness of the number of counterterms is required. On a lattice, every quantity is finite but depends on the spacing.
When taking the limit of zero spacing, we make sure that the physically-observable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. Hopefully, by allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a continuum limit. The renormalization procedure only works for a certain class of quantum field theories, called renormalizable quantum field theories.
A theory is perturbatively renormalizable when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined non-perturbatively, the problems only show up at distance scales which are exponentially small in the inverse coupling for weak couplings. Of the three components, quantum electrodynamics is believed to not have a continuum limit, while the asymptotically free SU 2 and SU 3 weak hypercharge and strong color interactions are nonperturbatively well defined.
The renormalization group describes how renormalizable theories emerge as the long distance low-energy effective field theory for any given high-energy theory. Because of this, renormalizable theories are insensitive to the precise nature of the underlying high-energy short-distance phenomena. This is a blessing because it allows physicists to formulate low energy theories without knowing the details of high energy phenomenon. It is also a curse, because once a renormalizable theory like the standard model is found to work, it gives very few clues to higher energy processes.
The only way high energy processes can be seen in the standard model is when they allow otherwise forbidden events, or if they predict quantitative relations between the coupling constants. Gauge freedom A gauge theory is a theory that admits a symmetry with a local parameter.
Consequently, the theory is invariant under a global change of phases adding a constant to the phase of all wave functions, everywhere ; this is a global symmetry. In quantum electrodynamics, the theory is also invariant under a local change of phase, that is - one may shift the phase of all wave functions so that the shift may be different at every point in space-time. This is a local symmetry. However, in order for a well-defined derivative operator to exist, one must introduce a new field, the gauge field, which also transforms in order for the local change of variables the phase in our example not to affect the derivative.
In quantum electrodynamics this gauge field is the electromagnetic field. The change of local gauge of variables is termed gauge transformation. In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics. The degrees of freedom in quantum field theory are local fluctuations of the fields.
The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or gauge artifacts; usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version i. In other words, a gauge symmetry cannot have a quantum anomaly.
If a gauge symmetry is anomalous i. In general, the gauge transformations of a theory consist several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators.
Therefore the number of gauge bosons is the group dimension i. All the fundamental interactions in nature are described by gauge theories. The gauge boson is the photon. The gauge bosons are eight gluons. Supersymmetry Supersymmetry assumes that every fundamental fermion has a superpartner that is a boson and vice versa.
It was introduced in order to solve the so-called Hierarchy Problem, that is, to explain why particles not protected by any symmetry like the Higgs boson do not receive radiative corrections to its mass driving it to the larger scales GUT, Planck It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity Supergravity , and it is a key ingredient for the consistency of string theory. The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.
Since no superpartners have yet been observed, if supersymmetry exists it must be broken through a so-called soft term, which breaks supersymmetry without ruining its helpful features. The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the Large Hadron Collider. Notes  Weinberg, S. Quantum Field Theory, Vols. Pais recounts how his astonishment at the rapidity with which Feynman could calculate using his method. Feynman's method is now part of the standard methods for physicists.
Review article written by a master of Q. Introduction to relativistic Q. A monumental treatise on Q. Gauge Theory of Weak Interactions. Kane Modern Elementary Particle Physics. Perseus Books. Quantum Field Theory. Quantum Fields. Benjamin-Cummings Pub. Bogoliubov, A.
Logunov, A. Oksak, I. Dordrecht; Boston, Kluwer Academic Publishers. ISBN X. Click on the link "Introduction" for a simplified introduction to QFT suitable for someone familiar with quantum mechanics. Algebraic quantum field theory The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those. Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms. Minkowski space has a causal structure.
If an open set V lies in the causal complement of an open set U, then the image of the maps. If is the causal completion of an open set U, then is an isomorphism primitive causality. If we have a state over , we can take the "partial trace" to get states associated with for each open set via the net monomorphism. It's easy to show the states over the open sets form a presheaf structure. According to the GNS construction, for each state, we can associate a Hilbert space representation of Pure states correspond to irreducible representations and mixed states correspond to reducible representations.
Each irreducible up to equivalence is called a superselection sector. This is the vacuum sector. Local quantum field theory Algebraic logic In mathematical logic, algebraic logic formalizes logic using the methods of abstract algebra.
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Algebras as models of logics Algebraic logic treats algebraic structures, often bounded lattices, as models interpretations of certain logics, making logic a branch of order theory. Modus ponens remains valid, but is seldom employed. In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structure which are its models are shown on the right in the same row. Some of these structures are either Boolean algebras or proper extensions thereof. Modal and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators.
Classical sentential logic Lindenbaum-Tarski algebra Two-element Boolean algebra. First-order logic Cylindric algebra Polyadic algebra Predicate functor logic. On the postwar history, see Maddux and Quine Perhaps surprisingly, algebraic logic is the oldest approach to formal logic, arguably beginning with a number of memoranda Leibniz wrote in the s, some of which were published in the 19th century and translated into English by Clarence Lewis in But nearly all of Leibniz's known work on algebraic logic was published only in , after Louis Couturat discovered it in Leibniz's Nachlass.
Parkinson and Loemker translated selections from Couturat's volume into English. Brady discusses the rich historical connections between algebraic logic and model theory. The founders of model theory, Ernst Schroder and Leopold Loewenheim, were logicians in the algebraic tradition. Modern mathematical logic began in , with two pamphlets whose respective authors were Augustus DeMorgan and George Boole.
They, and later C. Whitehead all shared Leibniz's dream of combining symbolic logic, mathematics, and philosophy. Relation algebra is arguably the culmination of Leibniz's approach to logic. With the exception of some writings by Leopold Loewenheim and Thoralf Skolem, algebraic logic went into eclipse soon after the publication of Principia Mathematica, not to revive until Tarski's reexposition of relation algebra.
Leibniz had no influence on the rise of algebraic logic because his logical writings were little studied before the Parkinson and Loemker translations. Our present understanding of Leibniz the logician stems mainly from the work of Wolfgang Lenzen, summarized in Lenzen From Peirce to Skolem: A neglected chapter in the history of logic. The Search for Mathematical Roots. Princeton Univ. North-Holland: Leibniz: Logical Papers. Oxford Uni. Harvard Univ.
Press: This research area and its name originated in the 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. Quantum logic can be formulated either as a modified version of propositional logic or as a non-commutative and non-associative many-valued MV logic    . 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.
This thesis was an important ingredient in Putnam's paper Is Logic Empirical? Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself to the physicist David Finkelstein.
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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. The more common view regarding quantum logic, however, is that it provides a formalism for relating observables, system preparation filters and states. 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.
Quantum logic 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. The set of principles for manipulating these quantum propositions was called quantum logic by von Neumann and Birkhoff. Mackey attempted to provide a set of axioms for this propositional system as an orthocomplemented lattice. 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 this 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. Projections as propositions The so-called Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. 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. 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 laws. 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. The set of propositions is ordered by the natural ordering of sets and has a complementation operation. The orthocomplementation operation is set complement. 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.
A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R. In particular, for any bounded Borel function f, the following equation holds:. The propositional lattice of a quantum mechanical system This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics.
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. 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. 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. 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. 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. The following highly non-trivial theorem is due to Andrew Gleason: Theorem. 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. 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. This means. 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. In the first case we say U is unitary, in the second case U is anti-unitary.
This note is included for technical accuracy only, and should not concern most readers. 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. In fact, Theorem. Note that it easily from uniqueness from Kadison's theorem that. Now the square of an anti-unitary is a unitary, so that all the Ut are unitary. 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. 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.
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 see that a pure ensemble becomes a mixed ensemble after measurement. Measurement, as described above, is a special case of quantum operations. Limitations Quantum logic derived from propositional 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 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. Since around the Flato school see F. Bayen has been developing an alternative to the quantum logics approach called deformation quantization see Weyl quantization. In , Prakash Panangaden described how to capture the kinematics of quantum causal evolution using System BV, a deep inference logic originally developed for use in structural proof theory.
Quantum Logic. Quantum LMn Algebraic Logic. Algebra, Lichnerowicz and D. Birkhoff and J. This is a thorough but elementary and well-illustrated introduction, suitable for advanced undergraduates. Benjamin, paperback reprint by Dover Reprinted in paperback form. An extraordinarily lucid discussion of some logical and philosophical issues of quantum mechanics, with careful attention to the history of the subject. Also discusses consistent histories. Piron, Foundations of Quantum Physics, W. Benjamin, Putnam, Is Logic Empirical? Quantum computer A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data.
The basic principle behind quantum computation is that quantum properties can be used to  represent data and perform operations on these data. Although quantum computing is still in its infancy, experiments have been carried out in which quantum computational operations were executed on a very small number of qubits quantum binary digits. Both practical and theoretical research continues with interest, and many national government and military The Bloch sphere is a representation of funding agencies support quantum computing research a qubit, the fundamental building to develop quantum computers for both civilian and block of quantum computers.
If large-scale quantum computers can be built, they will be able to solve certain problems much faster than any of our current classical computers for example Shor's algorithm. Quantum computers are different from other computers such as DNA computers and traditional computers based on transistors. Some computing architectures such as optical computers may use classical superposition of electromagnetic waves.
Without some specifically quantum mechanical resources such as entanglement, it is conjectured that an exponential advantage over classical computers is not possible. Basis A classical computer has a memory made up of bits, where each bit holds either a one or a zero.
A quantum computer maintains a sequence of qubits. A single qubit can hold a one, a zero, or, crucially, any quantum superposition of these; moreover, a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8. In general a quantum computer with qubits can be in an arbitrary superposition of up to different states simultaneously this compares to a normal computer that can only be in one of these states at any one time. A quantum computer operates by manipulating those qubits with a fixed sequence of quantum logic gates.
The sequence of gates to be applied is called a quantum algorithm. An example of an implementation of qubits for a quantum computer could start with the use of particles with two spin states: "down" and "up" typically written and , or and. But in fact any system possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit.
Bits vs. The state of the computer at any time is a probability distribution over the different three-bit strings , , If it is a deterministic computer, then it is in exactly one of these states with probability 1. However, if it is a probabilistic computer, then it may have a chance in being in a number of different states. There is a restriction that these probabilities sum to 1. Qubits are made up of controlled particles and the means of control e. The state of a three-qubit quantum computer is similarly described by an eight-dimensional vector a,b,c,d,e,f,g,h , called a wavefunction.
However, instead of adding to one, the sum of the squares of the coefficient magnitudes, , must equal one. Moreover, the coefficients are complex numbers. Since states are represented by complex wavefunctions, two states being added together will undergo interference. This is a key difference between quantum computing and probabilistic classical computing. Thesa organizes topics and quotations like a thesaurus organizes words and their meanings.
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