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Quantum physics

Quantum mechanics

Introduction to... Mathematical formulation of... Image File history File links Question_book-3. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Quantum mechanics (QM, or quantum theory) is a physical science dealing with the behaviour of matter and energy on the scale of atoms and subatomic particles / waves. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ...

Fundamental concepts

Decoherence · Interference Uncertainty · Exclusion Transformation theory Ehrenfest theorem · Measurement Superposition · Entanglement In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ... For other uses, see Interference (disambiguation). ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... The term transformation theory refers to a procedure used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927. ... The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ... It has been suggested that Quantum coherence be merged into this article or section. ...

Experiments

Double-slit experiment Davisson-Germer experiment Stern–Gerlach experiment Bell's inequality experiment Popper's experiment Schrödinger's cat Slit experiment redirects here. ... In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow moving electrons at a crystalline Nickel target. ... In quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is a celebrated experiment in 1920 on deflection of particles, often used to illustrate basic principles of quantum mechanics. ... In quantum mechanics, Bells Theorem states that a Bell inequality must be obeyed under any local hidden variable theory but can in certain circumstances be violated under quantum mechanics (QM). ... Poppers experiment is an experiment proposed by the 20th century philosopher of science Karl Popper, to test the standard interpretation (the Copenhagen interpretation) of Quantum mechanics. ... Schrödingers Cat: When the nucleus (bottom left) decays, the Geiger counter (bottom centre) may sense it and trigger the release of the gas. ...

Equations

Schrödinger equation Pauli equation Klein-Gordon equation Dirac equation For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The Pauli equation is a Schrödinger equation which handles spin. ... The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation. ...

Advanced theories

Quantum field theory Wightman axioms Quantum electrodynamics Quantum chromodynamics Quantum gravity Feynman diagram Quantum field theory (QFT) is the quantum theory of fields. ... In physics the Wightman axioms are an attempt of mathematically stringent, axiomatic formulation of quantum field theory. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ...

Interpretations

Copenhagen · Ensemble Hidden variables · Transactional Many-worlds · Consistent histories Quantum logic Consciousness causes collapse It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ... The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ... The Ensemble Interpretation, or Statistical Interpretation of Quantum Mechanics, is an interpretation that can be viewed as a minimalist interpretation. ... In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event. ... The transactional interpretation of quantum mechanics (TIQM) by Professor John Cramer is an unusual interpretation of quantum mechanics that describes quantum interactions in terms of a standing wave formed by retarded (forward in time) and advanced (backward in time) waves. ... The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome to every event to... In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. ... In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ... Consciousness causes collapse is the name given to the claim that observation by a conscious observer is responsible for the wavefunction collapse in quantum mechanics. ...

Scientists

Planck · Schrödinger Heisenberg · Bohr · Pauli Dirac · Bohm · Born de Broglie · von Neumann Einstein · Feynman Everett · Penrose · Others “Planck” redirects here. ... Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Niels Henrik David Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... This article is about the Austrian-Swiss physicist. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... David Bohm. ... Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ... For other persons named John Neumann, see John Neumann (disambiguation). ... “Einstein” redirects here. ... This article is about the physicist. ... Hugh Everett III (November 11, 1930 – July 19, 1982) was an American physicist who first proposed the many-worlds interpretation(MWI) of quantum physics, which he called his relative state formulation. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Below is a list of famous physicists. ...

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In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. The equation demands the existence of antiparticles and actually predated their experimental discovery, making the discovery of the positron, the antiparticle of the electron, one of the greatest triumphs of modern theoretical physics. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Two-dimensional analogy of space-time curvature described in General Relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Year 1928 (MCMXXVIII) was a leap year starting on Sunday (link will display full calendar) of the Gregorian calendar. ... For the novel, see The Elementary Particles. ... In quantum mechanics, spin is an intrinsic property of all elementary particles. ... For other uses, see Electron (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Corresponding to most kinds of particle, there is an associated antiparticle with the same mass and opposite charges. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ...

Mathematical formulation

The Dirac Equation in the form originally proposed by Dirac is:

where

m is the rest mass of the electron,

c is the speed of light,

p is the momentum operator,

is the reduced Planck's constant,

x and t are the space and time coordinates.

The new elements in this equation are the 4x4 matrices α_k and β, and the four-component wavefunction ψ. The matrices are all Hermitian and have squares equal to the identity matrix, and they all mutually anticommute: For other uses, see Mass (disambiguation). ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... This article is about momentum in physics. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... This article is about the idea of space. ... Look up time in Wiktionary, the free dictionary. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

where i and j are distinct and range from 1 to 3. These matrices, and the form of the wavefunction, have a deep mathematical significance. The algebraic structure represented by the Dirac matrices had been created some 50 years earlier by the English mathematician W. K. Clifford, which in turn had been based on the mid-19th century work of the German mathematician Hermann Grassmann in his "Lineare Ausdehnungslehre" (Theory of Linear Extensions). The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, in such a direct physical manner, amounts to one of the most remarkable chapters in the history of physics. William Kingdon Clifford. ... Hermann Günther Grassmann (April 15, 1809, Stettin – September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ...

Comparison with the Schrödinger equation

The Dirac equation is superficially similar to the Schrödinger equation for a free mass: For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

The left side represents the square of the momentum operator divided by twice the mass, which, classically speaking, is the kinetic energy. If one wants to get a relativistic generalization of this equation, then the space and time derivatives must enter symmetrically, as they do in the Maxwell theory of the electromagnetic field, which is known to be relativistically invariant - that is, the derivatives must be of the same order in space and time. Now, in relativity, the momentum and the energy are each part of an invariant object, the 4-momentum, and they are connected by the relativistically invariant relation

with m now representing the rest mass. If we replace E and p by their operator equivalents in the Schrödinger theory, we get a differential equation that is a valid relativistic generalization of the Schrödinger equation:

where it is assumed that the wave function is now a relativistic scalar. In fact Schrödinger, who was well acquainted with relativity, tried this equation before the one that bears his name, but found it unsuitable. Because the time derivative is second order, one must specify both the initial value of as well as φ itself when solving the equation. This is typical in the solution of problems of wave propagation, as in electrodynamics. However, in quantum theory, one is interested not in the actual motion as such, rather, the energy spectrum - mathematically, what is needed is a well-defined eigenvalue problem. As in electrodynamics, there will be advanced waves that appear to be propagating backward in time toward the source - these can be safely discarded as unphysical in electrodynamics, but not here, because one needs all the solutions in order to be able to express any solution as an expansion in terms of energy eigenfunctions and the corresponding eigenvalues. Erwin Schrödinger, as depicted on the former Austrian 1000 Schilling bank note. ...

There was an even more serious objection to be raised - in the Schrödinger theory, the probability density is given by the positive definite expression

ρ = φ ^* φ

and its current by

with the conservation of probability density expressed as

In a relativistic theory, the form of the probability density must match that of the current when we replace by , and in order that the conservation of probability current be a relativistically invariant expression, must form the 0-component of a 4-vector - thus we must have

Everything is now perfectly relativistic, but the probability density is not positive definite, because one may freely choose the initial values of both φ and . Such a theory would not have a simple, immediate physical interpretation, and so Schrodinger abandoned it. (Though it was short-lived as a single-particle equation, it is resurrected in quantum field theory, where it is known as the Klein-Gordon equation, and describes particles of spin-0.)

Dirac's coup

What is needed, then, is an equation that is first-order in both space and time. One could formally take the relativistic expression for the energy , replace p by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.

As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator thus:

On multiplying out the right side, we see that in order to get all the cross-terms such as to vanish, we must assume

AB + BA = 0,...

with

A² = B² = ... = 1

Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if A, B... are matrices, with the implication that the wave function has multiple components. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least 4x4 matrices to set up a system with the properties desired - so the wave function had four components, not two, as in the Pauli theory. Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...

Given the factorization in terms of these matrices, one can now write down immediately an equation

with κ to be determined. Applying again the matrix operator on either side yields

On taking we find that all the components of the wave function individually satisfy the relativistic energy-momentum relation. Thus the sought-for equation that is first-order in both space and time is

With (A,B,C) = iβα_k and D = β, we get the Dirac equation.

Comparison with the Pauli theory

The necessity of introducing half-integral spin goes back experimentally to the results of the Stern-Gerlach experiment. A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two - the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into 3 parts, corresponding to atoms with Lz = -1, 0, and +1. The conclusion is that silver atoms have net intrinsic angular momentum of 1/2. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so: In quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is a celebrated experiment in 1920 on deflection of particles, often used to illustrate basic principles of quantum mechanics. ... This article is about Austrian-Swiss physicist Wolfgang Pauli. ... In physics, Hamiltons principle is an alternative formulation of the differential equations of motion for a physical system as an equivalent integral equation, using the calculus of variations. ...

Here A^μ is the applied electromagnetic field, and the three sigmas are Pauli matrices. e is the charge of the particle, e.g. e = − e₀ for the electron. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual Hamiltonian of a charged particle interacting with an applied field: Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. ...

This Hamiltonian is now a 2x2 matrix, so the Schrödinger equation based on it,

must use a two-component wave function. Pauli had introduced the sigma matrices

as pure phenomenology - Dirac now had a theoretical argument that implied that spin was somehow the consequence of the marriage of quantum theory to relativity. The term phenomenology in modern science, especially in physics, is used to describe a body of knowledge which relates several different empirical observations of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. ...

The Pauli matrices share the same properties as the Dirac matrices - they are all Hermitian, square to 1, and anticommute. This allows one to immediately find a representation of the Dirac matrices in terms of the Pauli matrices:

The Dirac equation now may be written as an equation coupling two-component spinors:

Notice that on the diagonal we find the rest energy of the particle. If we set the momentum to zero - that is, bring the particle to rest - then we have

The equations for the individual two-spinors are now decoupled, and we see that the "top" and "bottom" two-spinors are individually eigenfunctions of the energy with eigenvalues equal to plus and minus the rest energy, respectively. The appearance of this negative energy eigenvalue is completely consistent with relativity.

It should be strongly emphasized that this separation in the rest frame is not an invariant statement - the "bottom" two-spinor does not represent antimatter as such in general. The entire four-component spinor represents an irreducible whole - in general, states will have an admixture of positive and negative energy components. If we couple the Dirac equation to an electromagnetic field, as in the Pauli theory, then the positive and negative energy parts will be mixed together, even if they are originally decoupled. Dirac's main problem was to find a consistent interpretation of this mixing. As we shall see below, it brings a new phenomenon into physics - matter/antimatter creation and annihilation.

Covariant form and relativistic invariance

The covariant form of the Dirac equation is (employing the Einstein summation convention)

In the above, γ⁰ is Hermitian, and the γ^k are anti-Hermitian, with the definition

γ⁰ = β

γ^k = γ⁰α_k

This may be summarized using the Minkowski metric on spacetime in the form Hermann Minkowski. ...

{γ^μ,γ^ν} = 2g^μν

where the bracket expression {a, b} means ab + ba, the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-d space with metric signature (+---). Note that one may also employ the metric form (-+++) by multiplying all the gammas by a factor of i. At an elementary level, the choice may be regarded as conventional, but there are specific reasons for preferring the former, both mathematically and for convenience in calculation and physical interpretation. In the literature, one almost always finds the convention (+---) in use. The specific Clifford algebra employed in the Dirac equation is known as the Dirac algebra. In mathematics, Clifford algebras are a type of associative algebra. ... In mathematical physics, the Dirac algebra is the Clifford algebra Cℓ1,3(C) which is generated by matrix multiplication and real and complex linear combination over the Dirac gamma matrices, introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles. ...

The Dirac equation may be interpreted as an eigenvalue expression, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportion being the speed of light in vacuo: In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

P_opψ = mcψ

In practice, physicists often use units of measure such that and c are equal to 1, known as "natural" units. The equation is then multiplied through by − i and takes the simple form

A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:

If in addition the matrices are all unitary, as are the Dirac set, then S itself is unitary; A unitary transformation is an isomorphism between two Hilbert spaces. ...

The transformation U is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the derivative operators, which form a covariant vector. In order that the operator remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form

If we now define the transformed spinor

then we have the transformed Dirac equation

Thus, once we settle on a unitary representation of the gammas, it is final providing we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation.

These considerations reveal the origin of the gammas in geometry, hearkening back to Grassmann's original motivation - they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as γ_μγ_ν represent oriented surface elements, and so on. With this in mind, we can find the form the unit volume element on spacetime in terms of the gammas as follows. By definition, it is

In order that this be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of , where g is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus

V = iγ⁰γ¹γ²γ³

This matrix is given the special symbol γ₅, owing to its importance when one is considering improper transformations of spacetime, that is, those that change the orientation of the basis vectors. In the representation we are using for the gammas, it is

Also note that could as easily have taken the negative square root of the determinant of g - the choice amounts to an initial handedness convention.

Adjoint equation and Dirac Current

By defining the adjoint spinor In mathematics, the term adjoint applies in several situations. ...

and noticing that

,

we obtain, by taking the Hermitian conjugate of the Dirac equation and multiplying from the right by γ⁰, the adjoint equation:

where is understood to act to the left. Multiplying the Dirac equation by from the left, and the adjoint equation by ψ from the right, and adding, produces the law of conservation of the Dirac current in covariant form:

Now we see the great advantage of the first-order equation over the one Schrödinger had tried - this is the conserved probability current density required by relativistic invariance, only now its 0-component is positive definite:

The Dirac equation and its adjoint are the Euler-Lagrange equations of the 4-d invariant action integral In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ...

where the scalar L is the Dirac Lagrangian

and for the purposes of variation, ψ and are regarded as independent fields. The relativistic invariance also follows immediately from the variational principle.

Coupling to an electromagnetic field

To consider problems in which an applied electromagnetic field interacts with the particles described by the Dirac equation, one uses the correspondence principle, and takes over into the theory the corresponding expression from classical mechanics, whereby the total momentum of a charged particle in an external field is modified as so: In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ...

In natural units, the Dirac equation then takes the form

This validity of this prescription is confirmed experimentally with great precision. It is known as minimal coupling, and is found throughout particle physics. Indeed, while the introduction of the electromagnetic field in this way is essentially phenomenological in this context, it rises to a fundamental principle in quantum field theory. Quantum field theory (QFT) is the quantum theory of fields. ...

Now as stated above, the transformation U is defined only up to a phase factor e^iθ. Also, the fundamental observable of the Dirac theory, the current, is unchanged if we multiply the wave function by an arbitrary phase. We may exploit this to get the form of the mutual interaction of a Dirac particle and the electromagnetic field, as opposed to simply considering a Dirac particle in an applied field, by assuming this arbitrary phase factor to depend continuously on position:

Notice now that

In order to preserve minimal coupling, we must add to the potential a term proportional to the gradient of the phase. But we know from electrodynamics that this leaves the electromagnetic field itself invariant. The value of the phase is arbitrary, but not how it changes from place to place. This is the starting point of gauge theory, which is the main principle on which quantum field theory is based. The simplest such theory, and the one most thoroughly understood, is known as quantum electrodynamics. The equations of field theory thus have invariance under both Lorentz transformations and gauge transformations. In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

Curved Spacetime Dirac Equation

Dirac equation can be written in curved spacetime using vierbein fields. Vierbeins describe a local frame that enables to define Dirac matrices at every point. Contracting these matrices with vierbeins give the right transformation properties. This way Dirac equation takes the following form in curved spacetime ^[1]: This article is being considered for deletion in accordance with Wikipedias deletion policy. ... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... In general relativity, a frame field (also called a tetrad or vierbein) is an orthonormal set of four vector fields, one timelike and three spacelike, defined on a Lorentzian manifold which is physically interpreted as a model of spacetime. ... The Dirac equation is a relativistic quantum mechanical wave equation invented by Paul Dirac in 1928. ... In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ...

Here is the vierbein and D_μ is the covariant derivative for fermion fields, defined as follows This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ...

where η_ac is the Lorentzian metric, σ^ab is the commutator of Dirac matrices: In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

and is the spin connection: For the theory, see Cartan connection applications For the chromosomal formation, see meiosis. ...

where is the Christoffel symbol. Note that here, latin letters denote the "Lorentzian" indeces and greek ones denote "Riemannian" indices. In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

Physical Interpretation

The Dirac theory, while providing a wealth of information that is accurately confirmed by experiments, nevertheless introduces a new physical paradigm that appears at first difficult to interpret and even paradoxical. Some of these issues of interpretation must be regarded as open questions. Here we will see how the Dirac theory brilliantly answered some of the outstanding issues in physics at the time it was put forward, while posing others that are still the subject of debate.

Identification of Observables

The critical physical question in a quantum theory is - what are the physically observable quantities defined by the theory? According to general principles, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. If we wish to maintain this interpretation on passing to the Dirac theory, we must take the Hamiltonian to be

This looks promising, because we see by inspection the rest energy of the particle and, in case A = 0, the energy of a charge placed in an electric potential eA⁰. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is

Thus the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and we must take great care to correctly identify what is an observable in this theory. Much of the apparent paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables. Let us now describe one such effect. (cont'd)

History

Since the Dirac equation was originally invented to describe the electron, we will generally speak of "electrons" in this article. The equation also applies to quarks, which are also elementary spin-½ particles. A modified Dirac equation can be used to approximately describe protons and neutrons, which are not elementary particles (they are made up of quarks), but have a net spin of ½. Another modification of the Dirac equation, called the Majorana equation, is thought to describe neutrinos — also spin-½ particles. For other uses, see Quark (disambiguation). ... For other uses, see Proton (disambiguation). ... This article or section does not adequately cite its references or sources. ... The Majorana equation is a relativistic wave equation similar to the Dirac equation but includes the charge conjugate ψc of spinor ψ. It is named after the Italian Ettore Majorana, and in natural units it is written in Feynman notation, where the charge conjugate is defined as Equation (1) can alternatively... For other uses, see Neutrino (disambiguation). ...

The Dirac equation describes the probability amplitudes for a single electron. This is a single-particle theory; in other words, it does not account for the creation and destruction of the particles. It gives a good prediction of the magnetic moment of the electron and explains much of the fine structure observed in atomic spectral lines. It also explains the spin of the electron. Two of the four solutions of the equation correspond to the two spin states of the electron. The other two solutions make the peculiar prediction that there exist an infinite set of quantum states in which the electron possesses negative energy. This strange result led Dirac to predict, via a remarkable hypothesis known as "hole theory," the existence of particles behaving like positively-charged electrons. Dirac thought at first these particles might be protons. He was chagrined when the strict prediction of his equation (which actually specifies particles of the same mass as the electron) was verified by the discovery of the positron in 1932. When asked later why he hadn't actually boldly predicted the yet unfound positron with its correct mass, Dirac answered "Pure cowardice!" He shared the Nobel Prize anyway, in 1933. In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... In atomic physics, the fine structure describes the splitting of the spectral lines of atoms. ... Properties For other meanings of Atom, see Atom (disambiguation). ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... Year 1932 (MCMXXXII) was a leap year starting on Friday (the link will display full 1932 calendar) of the Gregorian calendar. ...

Despite these successes, Dirac's theory is flawed by its neglect of the possibility of creating and destroying particles, one of the basic consequences of relativity. This difficulty is resolved by reformulating it as a quantum field theory. Adding a quantized electromagnetic field to this theory leads to the theory of quantum electrodynamics (QED). Moreover the equation cannot fully account for particles of negative energy but is restricted to positive energy particles. Quantum field theory (QFT) is the quantum theory of fields. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

A similar equation for spin 3/2 particles is called the Rarita-Schwinger equation. In theoretical physics, the Rarita-Schwinger equation is the field equation of spin-3/2 fermions. ...

Hole theory

The negative E solutions found in the preceding section are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, we cannot simply ignore them, for once we include the interaction between the electron and the electromagnetic field, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy by emitting excess energy in the form of photons. Real electrons obviously do not behave in this way. In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates. Look up Vacuum in Wiktionary, the free dictionary. ... The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles possessing negative energy. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ...

Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a positive energy, since energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but Hermann Weyl pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932. Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... Carl Anderson at LBNL 1937 Carl David Anderson (3 September 1905 – 11 January 1991) was a U.S. experimental physicist. ... Year 1932 (MCMXXXII) was a leap year starting on Friday (the link will display full 1932 calendar) of the Gregorian calendar. ...

It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons has to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the net electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it. Jellium is the theory of interacting electrons in which a uniform background of positive charge exists. ... Quantum field theory (QFT) is the quantum theory of fields. ... In theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an isomorphism of the CCR/CAR algebra. ...

In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively-charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively-charged ionic lattice of the material. Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Electrical conduction is the current (movement of charged particles) through a material in response to an electric field. ... In science and engineering, conductors, such as copper or aluminum, are materials with atoms having loosely held valence electrons. ... In physics and Fermi-Dirac statistics, the Fermi energy (EF) of a system of non-interacting fermions is the smallest possible increase in the ground state energy when exactly one particle is added to the system. ... In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a...

Dirac bilinears

There are five different (neutral) Dirac bilinear terms not involving any derivatives:

• (S)calar: (scalar, P-even)
• (P)seudoscalar: (scalar, P-odd)
• (V)ector: (vector, P-even)
• (A)xial: (vector, P-odd)
• (T)ensor: (antisymmetric tensor, P-even),

where and .

A Dirac mass term is an S coupling. A Yukawa coupling may be S or P. The electromagnetic coupling is V. The weak interactions are V-A.

See also

• Breit equation
• Klein-Gordon equation
• Quantum electrodynamics
• Rarita-Schwinger equation
• Feynman checkerboard
• Theoretical and experimental justification for the Schrödinger equation

The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the first order in perturbation theory. ... The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... In theoretical physics, the Rarita-Schwinger equation is the field equation of spin-3/2 fermions. ... The Feynman Checkerboard is a discrete representation of the Dirac equation in (1+1)-dimensional spacetime. ... The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. ...

References

1. ^ Lawrie, Ian D.. A Unified Grand Tour of Theoretical Physics. 

Selected papers

• P.A.M. Dirac "The Quantum Theory of the Electron", Proc. R. Soc. A117) link to the volume of the Proceedings of the Royal Society of London containing the article at page 610
• P.A.M. Dirac "A Theory of Electrons and Protons", Proc. R. Soc. A126) link to the volume of the Proceedings of the Royal Society of London containing the article at page 360
• C.D. Anderson, Phys. Rev. 43, 491 (1933)
• R. Frisch and O. Stern, Z. Phys. 85, 4 (1933)

Textbooks

• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN. 
• Dirac, P.A.M., Principles of Quantum Mechanics, 4th edition (Clarendon, 1982)
• Shankar, R., Principles of Quantum Mechanics, 2nd edition (Plenum, 1994)
• Bjorken, J D & Drell, S, Relativistic Quantum mechanics
• Thaller, B., The Dirac Equation, Texts and Monographs in Physics (Springer, 1992)
• Schiff, L.I., Quantum Mechanics, 3rd edition (McGraw-Hill, 1955)