Examples of topological solitons include the screw dislocation in a crystalline lattice, the Dirac string and the magnetic monopole in electromagnetism, the Skyrmion and the Wess – Zumino – Witten model in quantum field theory, and cosmic strings and domain walls in cosmology.

* Dirac string, a fictitious one-dimensional curve in space, stretching between two magnetic monopoles

A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as a Dirac string of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole " charge " g. The Dirac string starts from, and terminates on, a magnetic monopole.

Like the electromagnetic potential A the Dirac string is not gauge invariant ( it moves around with fixed endpoints under a gauge transformation ) and so is also not directly measurable.

In physics, a Dirac string is a fictitious one-dimensional curve in space, conceived of by the physicist Paul Dirac, stretching between two Dirac magnetic monopoles with opposite magnetic charges, or from one magnetic monopole out to infinity.

The gauge potential cannot be defined on the Dirac string, but it is defined everywhere else.

The quantization forced by the Dirac string can be understood in terms of the cohomology of the fibre bundle representing the gauge fields over the base manifold of space-time.

Informally, one might say that the Dirac string carries away the " excess curvature " that would otherwise prevent F from being a closed form, as one has that everywhere except at the location of the monopole.

The vector potential A is used when studying the Lagrangian in classical mechanics and in quantum mechanics ( see Schrödinger equation for charged particles, Dirac equation, Aharonov-Bohm effect ).

Time-dependent perturbation theory, developed by Paul Dirac, studies the effect of a time-dependent perturbation V ( t ) applied to a time-independent Hamiltonian.

The Kapitsa – Dirac effect effectively demonstrated during 2001 uses standing waves of light to affect a beam of particles.

The micromechanical cleavage technique led directly to the first observation of the anomalous quantum Hall effect in graphene, which provided direct evidence of the theoretically predicted pi Berry's phase of massless Dirac fermions in graphene.

These simple experiments started after the researchers watched colleagues who were looking for quantum Hall effect and Dirac fermions in bulk graphite.

Graphitic layers on the carbon face of silicon carbide show a clear Dirac spectrum in angle-resolved photoemission experiments, and the anomalous quantum Hall effect is observed in cyclotron resonance and tunneling experiments.

This effect can be approximated by a Dirac delta measure ( flash ) and a constant finite rectangular window, in combination.

However, one can measure the position ( alone ) of a moving free particle, creating an eigenstate of position with a wavefunction that is very large ( a Dirac delta ) at a particular position x, and zero everywhere else.

In quantum mechanics their importance lies in the Dirac – von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space.

The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation ( at the speed of light ) of the position of an electron around the median, with a circular frequency of, or approximately 1. 6 Hz.

is the Dirac hamiltonian ( see Dirac equation ) for particle i at position r < sub > i </ sub > and φ ( r < sub > i </ sub >) is the scalar potential at that position ; q < sub > i </ sub > is the charge of the particle, thus for electrons q < sub > i </ sub >

Paul Dirac observed in 1931 that, because electricity and magnetism show a certain symmetry, just as quantum theory predicts that individual positive or negative electric charges can be observed without the opposing charge, isolated South or North magnetic poles should be observable.

It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation.

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 Pauli and Dirac matrices should then depend on the metric as:

According to Dirac, the and orbitals should have the same energies.

It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation.

Since differences in 4-space and conventional Dirac solutions would not show up in the spectrum, which provided the original motivation and experimental confirmation of Dirac theory, reconciliation of the two forms of Dirac theory should motivate new experiments to confirm or falsify Zitterbewegung behavior in the 4-space wave functions.

Dirac ’ s two preceding remarks suggest that we should start searching for a realistic regularization in the case of quantum electrodynamics ( QED ) in the four-dimensional Minkowski spacetime, starting with the original QED Lagrangian density.

Sometimes, however, one of these negative-energy particles could be lifted out of this Dirac sea to become a positive-energy particle.

It can also be found in Fermi – Dirac statistics ( for particles of half-integer spin ) and Bose – Einstein statistics ( for particles of integer spin ).

Augustin-Louis Cauchy exploited such a density function in 1827 with an infinitesimal scale parameter, defining what would now be called a Dirac delta function.

Again the Dirac equation may be solved analytically in the special case of a two-body system, such as the hydrogen atom.

For these developments, it was essential that the solution of the Dirac equation for the hydrogen atom could be worked out exactly, such that any experimentally observed deviation had to be taken seriously as a signal of failure of the theory.

Compactification does not produce group actions on chiral fermions except in very specific cases: the dimension of the total space must be 2 mod 8 and the G-index of the Dirac operator of the compact space must be nonzero.

Attempted replacements for the Schrödinger equation, such as the Klein-Gordon equation or the Dirac equation, have many unsatisfactory qualities ; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a ground state.

Associated with the fact that the electron can be polarized is another small necessary detail which is connected with the fact that an electron is a Fermion and obeys Fermi – Dirac statistics.

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra, into which the spin group Spin ( p, q ) may be embedded.

On a 2k-or 2k + 1-dimensional space a Dirac spinor may be represented as a vector of 2 < sup > k </ sup > complex numbers.

The Dirac, Lorentz, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.

Recently, the interference of molecules as heavy as 6910 u could be demonstrated in a Kapitza – Dirac – Talbot – Lau interferometer.

While from this perspective the Dirac delta can usually be manipulated as though it were a function, formally it must be defined as a distribution that is also a measure.

The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

The Dirac delta function can be rigorously defined either as a distribution or as a measure.

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, C and D are matrices, with the implication that the wave function has multiple components.

The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:

This matrix will also be found to anticommute with the other four Dirac matrices.

The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner.

Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities.