Static forces and virtual-particle exchange
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Static force fields are fields, such as a simple electric,
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
or
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
s, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by
virtual particle A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturbat ...
s, particles that exist for only a short time determined by the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. The virtual particles, also known as
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical field ...
s, are bosons, with different bosons associated with each force. The virtual-particle description of static forces is capable of identifying the spatial form of the forces, such as the inverse-square behavior in Newton's law of universal gravitation and in
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
. It is also able to predict whether the forces are attractive or repulsive for like bodies. The path integral formulation is the natural language for describing force carriers. This article uses the path integral formulation to describe the force carriers for
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
0, 1, and 2 fields. Pions,
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
s, and gravitons fall into these respective categories. There are limits to the validity of the virtual particle picture. The virtual-particle formulation is derived from a method known as
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
which is an approximation assuming interactions are not too strong, and was intended for scattering problems, not bound states such as atoms. For the strong force binding
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s into nucleons at low energies, perturbation theory has never been shown to yield results in accord with experiments, thus, the validity of the "force-mediating particle" picture is questionable. Similarly, for bound states the method fails. In these cases, the physical interpretation must be re-examined. As an example, the calculations of atomic structure in atomic physics or of molecular structure in quantum chemistry could not easily be repeated, if at all, using the "force-mediating particle" picture. Use of the "force-mediating particle" picture (FMPP) is unnecessary in nonrelativistic quantum mechanics, and Coulomb's law is used as given in atomic physics and quantum chemistry to calculate both bound and scattering states. A non-perturbative
relativistic quantum theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, in which Lorentz invariance is preserved, is achievable by evaluating Coulomb's law as a 4-space interaction using the 3-space position vector of a reference electron obeying Dirac's equation and the quantum trajectory of a second electron which depends only on the scaled time. The quantum trajectory of each electron in an ensemble is inferred from the Dirac current for each electron by setting it equal to a velocity field times a quantum density, calculating a position field from the time integral of the velocity field, and finally calculating a quantum trajectory from the expectation value of the position field. The quantum trajectories are of course spin dependent, and the theory can be validated by checking that
Pauli's exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulate ...
is obeyed for a collection of
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s.


Classical forces

The force exerted by one mass on another and the force exerted by one charge on another are strikingly similar. Both fall off as the square of the distance between the bodies. Both are proportional to the product of properties of the bodies, mass in the case of gravitation and charge in the case of electrostatics. They also have a striking difference. Two masses attract each other, while two like charges repel each other. In both cases, the bodies appear to act on each other over a distance. The concept of
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
was invented to mediate the interaction among bodies thus eliminating the need for action at a distance. The gravitational force is mediated by the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
and the Coulomb force is mediated by the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
.


Gravitational force

The gravitational force on a mass m exerted by another mass M is \mathbf = - G \frac \, \hat\mathbf = m \mathbf \left ( \mathbf \right ), where is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, is the distance between the masses, and \hat\mathbf is the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
from mass M to mass m . The force can also be written \mathbf = m \mathbf \left ( \mathbf \right ), where \mathbf \left ( \mathbf \right ) is the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
described by the field equation \nabla\cdot \mathbf = -4\pi G\rho_m, where \rho_m is the
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
at each point in space.


Coulomb force

The electrostatic
Coulomb force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
on a charge q exerted by a charge Q is (
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
) \mathbf = \frac\frac\mathbf, where \varepsilon_0 is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
, r is the separation of the two charges, and \mathbf is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
in the direction from charge Q to charge q . The Coulomb force can also be written in terms of an
electrostatic field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
: \mathbf = q \mathbf \left ( \mathbf \right ), where \nabla \cdot \mathbf = \frac ; \rho_q being the
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
at each point in space.


Virtual-particle exchange

In perturbation theory, forces are generated by the exchange of
virtual particle A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturbat ...
s. The mechanics of virtual-particle exchange is best described with the path integral formulation of quantum mechanics. There are insights that can be obtained, however, without going into the machinery of path integrals, such as why classical gravitational and electrostatic forces fall off as the inverse square of the distance between bodies.


Path-integral formulation of virtual-particle exchange

A virtual particle is created by a disturbance to the vacuum state, and the virtual particle is destroyed when it is absorbed back into the vacuum state by another disturbance. The disturbances are imagined to be due to bodies that interact with the virtual particle’s field.


The probability amplitude

Using natural units, \hbar = c = 1 , the probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in the path integral formulation by Z \equiv \langle 0 , \exp\left ( -i \hat H T \right ) , 0 \rangle = \exp\left ( -i E T \right ) = \int D\varphi \; \exp\left ( i \mathcal
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
\right )\; = \exp\left ( i W \right ) where \hat H is the Hamiltonian operator, T is elapsed time, E is the energy change due to the disturbance, W = - E T is the change in action due to the disturbance, \varphi is the field of the virtual particle, the integral is over all paths, and the classical action is given by \mathcal
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
= \int \mathrm^4x\; where \mathcal varphi (x) is the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
density. Here, the
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
metric is given by \eta_ = \begin 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end. The path integral often can be converted to the form Z = \int \exp\left i \int d^4x \left ( \frac 1 2 \varphi \hat O \varphi + J \varphi \right) \right D\varphi where \hat O is a differential operator with \varphi and J functions of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. The first term in the argument represents the free particle and the second term represents the disturbance to the field from an external source such as a charge or a mass. The integral can be written (see ) Z \propto \exp\left( i W\left ( J \right )\right) where W\left ( J \right ) = -\frac \iint d^4x \; d^4y \; J\left ( x \right ) D\left ( x-y \right ) J\left ( y \right ) is the change in the action due to the disturbances and the
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In ...
D\left ( x-y \right ) is the solution of \hat O D\left ( x - y \right ) = \delta^4 \left ( x - y \right ).


Energy of interaction

We assume that there are two point disturbances representing two bodies and that the disturbances are motionless and constant in time. The disturbances can be written J(x) = \left( J_1 +J_2,0,0,0 \right) J_1 = a_1 \delta^3\left ( \vec x - \vec x_1 \right ) J_2 = a_2 \delta^3\left ( \vec x - \vec x_2 \right ) where the delta functions are in space, the disturbances are located at \vec x_1 and \vec x_2 , and the coefficients a_1 and a_2 are the strengths of the disturbances. If we neglect self-interactions of the disturbances then W becomes W\left ( J \right ) = - \iint d^4x \; d^4y \; J_1\left ( x \right ) \frac \left D\left ( x-y \right ) + D\left ( y-x \right )\right J_2\left ( y \right ), which can be written W\left ( J \right ) = - T a_1 a_2\int \frac \; \; D\left ( k \right )\mid_ \; \exp\left ( i \vec k \cdot \left ( \vec x_1 - \vec x_2 \right ) \right ). Here D\left ( k \right ) is the Fourier transform of \frac \left D\left ( x-y \right ) + D\left ( y-x \right )\right Finally, the change in energy due to the static disturbances of the vacuum is E = - \frac = a_1 a_2\int \frac \; \; D\left ( k \right )\mid_ \; \exp\left ( i \vec k \cdot \left ( \vec x_1 - \vec x_2 \right ) \right ). If this quantity is negative, the force is attractive. If it is positive, the force is repulsive. Examples of static, motionless, interacting currents are the Yukawa potential, the Coulomb potential in a vacuum, and the Coulomb potential in a simple plasma or electron gas. The expression for the interaction energy can be generalized to the situation in which the point particles are moving, but the motion is slow compared with the speed of light. Examples are the Darwin interaction in a vacuum and in a plasma. Finally, the expression for the interaction energy can be generalized to situations in which the disturbances are not point particles, but are possibly line charges, tubes of charges, or current vortices. Examples include: two line charges embedded in a plasma or electron gas, Coulomb potential between two current loops embedded in a magnetic field, and the magnetic interaction between current loops in a simple plasma or electron gas. As seen from the Coulomb interaction between tubes of charge example, shown below, these more complicated geometries can lead to such exotic phenomena as fractional quantum numbers.


Selected examples


The Yukawa potential: The force between two nucleons in an atomic nucleus

Consider the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
-0 Lagrangian density \mathcal varphi (x)= \frac \left \left ( \partial \varphi \right )^2 -m^2 \varphi^2 \right The equation of motion for this Lagrangian is the Klein–Gordon equation \partial^2 \varphi + m^2 \varphi =0. If we add a disturbance the probability amplitude becomes Z = \int D\varphi \; \exp \left \. If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes Z = \int D\varphi \; \exp \left \. With the amplitude in this form it can be seen that the propagator is the solution of -\left ( \partial^2 + m^2\right ) D\left ( x-y \right ) = \delta^4\left ( x-y \right ). From this it can be seen that D\left ( k \right )\mid_ \; = \; -\frac. The energy due to the static disturbances becomes (see ) E =-\frac \exp \left ( -m r \right ) with r^2 = \left (\vec x_1 - \vec x_2 \right )^2 which is attractive and has a range of \frac.
Yukawa Yukawa (written: 湯川) is a Japanese surname, but is also applied to proper nouns. People * Diana Yukawa (born 1985), Anglo-Japanese solo violinist. She has had two solo albums with BMG Japan, one of which opened to #1 * Hideki Yukawa (1907–1 ...
proposed that this field describes the force between two nucleons in an atomic nucleus. It allowed him to predict both the range and the mass of the particle, now known as the pion, associated with this field.


Electrostatics


The Coulomb potential in a vacuum

Consider the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
-1
Proca Lagrangian In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass ''m'' in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca a ...
with a disturbance \mathcal varphi (x)= -\frac F_ F^ + \frac m^2 A_ A^ + A_ J^ where F_ = \partial_ A_ - \partial_ A_, charge is conserved \partial_ J^ = 0, and we choose the
Lorenz gauge In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...
\partial_ A^ = 0. Moreover, we assume that there is only a time-like component J^0 to the disturbance. In ordinary language, this means that there is a charge at the points of disturbance, but there are no electric currents. If we follow the same procedure as we did with the Yukawa potential we find that \begin & -\frac \int d^4x F_F^ = -\frac\int d^4x \left( \partial_ A_ - \partial_ A_ \right)\left( \partial^ A^ - \partial^ A^ \right) \\ = & \frac\int d^4x \; A_ \left( \partial^ A^ - \partial^ \partial_ A^ \right) = \frac\int d^4x \; A^ \left( \eta_ \partial^ \right) A^, \end which implies \eta_ \left ( \partial^2 + m^2\right ) D^\left ( x-y \right ) = \delta_^ \delta^4\left ( x-y \right ) and D_\left ( k \right )\mid_ \; = \; \eta_\frac. This yields D\left( k \right)\mid_\; = \; \frac for the timelike propagator and E = + \frac \exp \left( -m r \right) which has the opposite sign to the Yukawa case. In the limit of zero
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
mass, the Lagrangian reduces to the Lagrangian for
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
E = \frac. Therefore the energy reduces to the potential energy for the Coulomb force and the coefficients a_1 and a_2 are proportional to the electric charge. Unlike the Yukawa case, like bodies, in this electrostatic case, repel each other.


Coulomb potential in a simple plasma or electron gas


=Plasma waves

= The dispersion relation for
plasma wave In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electron ...
s is \omega^2 = \omega_p^2 + \gamma\left( \omega \right) \frac \vec k^2. where \omega is the angular frequency of the wave, \omega_p^2 = \frac is the plasma frequency, e is the magnitude of the electron charge, m is the
electron mass The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of a ...
, T_e is the electron
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
( Boltzmann's constant equal to one), and \gamma\left( \omega \right) is a factor that varies with frequency from one to three. At high frequencies, on the order of the plasma frequency, the compression of the electron fluid is an
adiabatic process In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
and \gamma\left( \omega \right) is equal to three. At low frequencies, the compression is an
isothermal process In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...
and \gamma\left( \omega \right) is equal to one. Retardation effects have been neglected in obtaining the plasma-wave dispersion relation. For low frequencies, the dispersion relation becomes \vec k^2 + \vec k_D^2 = 0 where k_D^2= \frac is the Debye number, which is the inverse of the Debye length. This suggests that the propagator is D\left ( k \right )\mid_ \; = \; \frac. In fact, if the retardation effects are not neglected, then the dispersion relation is -k_0^2 +\vec k^2 + k_D^2 -\frac k_0^2 = 0, which does indeed yield the guessed propagator. This propagator is the same as the massive Coulomb propagator with the mass equal to the inverse Debye length. The interaction energy is therefore E = \frac \exp \left ( -k_D r \right ). The Coulomb potential is screened on length scales of a Debye length.


=Plasmons

= In a quantum electron gas, plasma waves are known as plasmons. Debye screening is replaced with
Thomas–Fermi screening Thomas–Fermi screening is a theoretical approach to calculate the effects of electric field screening by electrons in a solid.N. W. Ashcroft and N. D. Mermin, ''Solid State Physics'' (Thomson Learning, Toronto, 1976) It is a special case of the mo ...
to yield pp. 296-299. E = \frac \exp \left ( -k_s r \right ) where the inverse of the Thomas–Fermi screening length is k_s^2 = \frac and \varepsilon_F is the Fermi energy \varepsilon_F = \frac \left( \right)^ . This expression can be derived from the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
for an electron gas and from
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
. The chemical potential for an electron gas near equilibrium is constant and given by \mu = -e\varphi + \varepsilon_F where \varphi is the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
. Linearizing the Fermi energy to first order in the density fluctuation and combining with Poisson's equation yields the screening length. The force carrier is the quantum version of the
plasma wave In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electron ...
.


=Two line charges embedded in a plasma or electron gas

= We consider a line of charge with axis in the ''z'' direction embedded in an electron gas J_1\left( x\right) = \frac \frac \delta^2\left( r \right) where r is the distance in the ''xy''-plane from the line of charge, L_B is the width of the material in the z direction. The superscript 2 indicates that the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
is in two dimensions. The propagator is D\left ( k \right )\mid_\; = \; \frac where k_ is either the inverse Debye–Hückel screening length or the inverse
Thomas–Fermi screening Thomas–Fermi screening is a theoretical approach to calculate the effects of electric field screening by electrons in a solid.N. W. Ashcroft and N. D. Mermin, ''Solid State Physics'' (Thomson Learning, Toronto, 1976) It is a special case of the mo ...
length. The interaction energy is E = \left( \frac\right) \int_0^ \frac \mathcal J_0 ( kr_ ) = \left( \frac\right) K_0 \left( k_ r_ \right) where \mathcal J_n ( x ) and K_0 ( x ) are
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s and r_ is the distance between the two line charges. In obtaining the interaction energy we made use of the integrals (see ) \int_0^ \frac \exp\left( i p \cos\left( \varphi \right) \right) = \mathcal J_0 ( p ) and \int_0^ \frac \mathcal J_0 (kr) = K_0 (mr). For k_ r_ \ll 1, we have K_0 \left( k_ r_ \right) \rightarrow -\ln \left(\frac\right) + 0.5772.


Coulomb potential between two current loops embedded in a magnetic field


=Interaction energy for vortices

= We consider a charge density in tube with axis along a magnetic field embedded in an electron gas J_1\left( x\right) = \frac \frac \delta^2 where r is the distance from the guiding center, L_B is the width of the material in the direction of the magnetic field r_ = \frac = \sqrt where the
cyclotron frequency Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
is (
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
) \omega_c = \frac and v_1 = \sqrt is the speed of the particle about the magnetic field, and B is the magnitude of the magnetic field. The speed formula comes from setting the classical kinetic energy equal to the spacing between
Landau levels In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, call ...
in the quantum treatment of a charged particle in a magnetic field. In this geometry, the interaction energy can be written E = \left( \frac\right) \int_0^ D\left( k \right) \mid_ \mathcal J_0 \left ( kr_ \right) \mathcal J_0 \left ( kr_ \right) \mathcal J_0 \left ( kr_ \right) where r_ is the distance between the centers of the current loops and \mathcal J_n ( x ) is a
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
of the first kind. In obtaining the interaction energy we made use of the integral \int_0^ \frac \exp\left( i p \cos(\varphi) \right) = \mathcal J_0 ( p ) .


=Electric field due to a density perturbation

= The
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
near equilibrium, is given by \mu = -e\varphi + N\hbar \omega_c = N_0\hbar \omega_c where -e\varphi is the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
of an electron in an
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and N_0 and N are the number of particles in the electron gas in the absence of and in the presence of an electrostatic potential, respectively. The density fluctuation is then \delta n = \frac where A_M is the area of the material in the plane perpendicular to the magnetic field.
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
yields \left( k^2 + k_B^2 \right) \varphi = 0 where k_B^2 = \frac. The propagator is then D\left( k \right) \mid_ = \frac and the interaction energy becomes E = \left( \frac\right) \int_0^ \frac \mathcal J_0 \left ( kr_ \right) \mathcal J_0 \left ( kr_ \right) \mathcal J_0 \left ( kr_ \right) = \left( \frac\right) \int_0^ \frac \mathcal J_0^2 \left ( k \right) \mathcal J_0 \left ( k\frac \right) where in the second equality (
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
) we assume that the vortices had the same energy and the electron charge. In analogy with plasmons, the
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical field ...
is the quantum version of the
upper hybrid oscillation In plasma physics, an upper hybrid oscillation is a mode of oscillation of a magnetized plasma. It consists of a longitudinal motion of the electrons perpendicular to the magnetic field with the dispersion relation : \omega^2 = \omega_^2 + \ome ...
which is a longitudinal
plasma wave In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electron ...
that propagates perpendicular to the magnetic field.


=Currents with angular momentum

=


Delta function currents

Unlike classical currents, quantum current loops can have various values of the
Larmor radius The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. In SI units, the non-relativistic gyroradius is given by :r_ ...
for a given energy.
Landau level In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, call ...
s, the energy states of a charged particle in the presence of a magnetic field, are multiply degenerate. The current loops correspond to
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
states of the charged particle that may have the same energy. Specifically, the charge density is peaked around radii of r_ = \sqrt\;r_B\; \; \; l=0,1,2, \ldots where l is the angular momentum
quantum number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
. When l = 1 we recover the classical situation in which the electron orbits the magnetic field at the
Larmor radius The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. In SI units, the non-relativistic gyroradius is given by :r_ ...
. If currents of two angular momentum l > 0 and l' \ge l interact, and we assume the charge densities are delta functions at radius r_, then the interaction energy is E = \left( \frac\right) \int_0^ \frac \;\mathcal J_0 \left ( k \right) \;\mathcal J_0 \left ( \sqrt \;k \right) \;\mathcal J_0 \left ( k \frac \right). The interaction energy for l= l' is given in Figure 1 for various values of k_B r_. The energy for two different values is given in Figure 2.


Quasiparticles

For large values of angular momentum, the energy can have local minima at distances other than zero and infinity. It can be numerically verified that the minima occur at r_ = r_ = \sqrt \; r_B. This suggests that the pair of particles that are bound and separated by a distance r_ act as a single quasiparticle with angular momentum l + l'. If we scale the lengths as r_ , then the interaction energy becomes E = \frac \int_0^ \frac \;\mathcal J_0 \left ( \cos \theta \, k \right) \;\mathcal J_0 ( \sin \theta \,k ) \;\mathcal J_0 where \tan \theta = \sqrt. The value of the r_ at which the energy is minimum, r_ = r_ , is independent of the ratio \tan \theta = \sqrt. However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when \frac = 1. When the ratio differs from 1, then the energy minimum is higher (Figure 3). Therefore, for even values of total momentum, the lowest energy occurs when (Figure 4) l = l' = 1 or \frac = \frac where the total angular momentum is written as l^* = l + l'. When the total angular momentum is odd, the minima cannot occur for l = l' . The lowest energy states for odd total angular momentum occur when \frac = \; \frac or \frac = \frac, \frac, \frac, \text and \frac = \frac, \frac, \frac, \text which also appear as series for the filling factor in the
fractional quantum Hall effect The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...
.


Charge density spread over a wave function

The charge density is not actually concentrated in a delta function. The charge is spread over a wave function. In that case the electron density is \frac \frac \left( \frac \right)^ \exp \left( -\frac \right). The interaction energy becomes E = \left( \frac\right) \int_0^ \frac \; M \;M \;\mathcal J_0 where M is a
confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
or Kummer function. In obtaining the interaction energy we have used the integral (see ) \frac \int_0^ dr \;r^\exp\left( -r^2\right) J_0(kr) = M\left( n+1, 1, -\frac\right). As with delta function charges, the value of r_ in which the energy is a local minimum only depends on the total angular momentum, not on the angular momenta of the individual currents. Also, as with the delta function charges, the energy at the minimum increases as the ratio of angular momenta varies from one. Therefore, the series \frac = \frac, \frac, \frac, \text and \frac = \frac, \frac, \frac, \text appear as well in the case of charges spread by the wave function. The
Laughlin wavefunction In condensed matter physics, the Laughlin wavefunction pp. 210-213 is an ansatz, proposed by Robert Laughlin for the ground state of a two-dimensional electron gas placed in a uniform background magnetic field in the presence of a uniform jellium ...
is an ansatz for the quasiparticle wavefunction. If the expectation value of the interaction energy is taken over a
Laughlin wavefunction In condensed matter physics, the Laughlin wavefunction pp. 210-213 is an ansatz, proposed by Robert Laughlin for the ground state of a two-dimensional electron gas placed in a uniform background magnetic field in the presence of a uniform jellium ...
, these series are also preserved.


Magnetostatics


Darwin interaction in a vacuum

A charged moving particle can generate a magnetic field that affects the motion of another charged particle. The static version of this effect is called the Darwin interaction. To calculate this, consider the electrical currents in space generated by a moving charge \vec J_1 = a_1 \vec v_1 \delta^3 with a comparable expression for \vec J_2 . The Fourier transform of this current is \vec J_1 = a_1 \vec v_1 \exp\left( i \vec k \cdot \vec x_1 \right). The current can be decomposed into a transverse and a longitudinal part (see Helmholtz decomposition). \vec J_1 = a_1 \left 1 - \hat k \hat k \right \cdot \vec v_1 \exp\left( i \vec k \cdot \vec x_1 \right) + a_1 \left \hat k \hat k \right \cdot \vec v_1 \exp\left( i \vec k \cdot \vec x_1 \right). The hat indicates a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
. The last term disappears because \vec k \cdot \vec J = -k_0 J^0 \rightarrow 0, which results from charge conservation. Here k_0 vanishes because we are considering static forces. With the current in this form the energy of interaction can be written E = a_1 a_2\int \frac \; \; D\left ( k \right )\mid_ \; \vec v_1 \cdot \left 1 - \hat k \hat k \right \cdot \vec v_2 \; \exp\left ( i \vec k \cdot \left ( x_1 - x_2 \right ) \right ) . The propagator equation for the Proca Lagrangian is \eta_ \left ( \partial^2 + m^2\right ) D^\left ( x-y \right ) = \delta_^ \delta^4\left ( x-y \right ). The spacelike solution is D\left ( k \right )\mid_\; = \; -\frac, which yields E = - a_1 a_2 \int \frac \; \; \frac \; \exp\left ( i \vec k \cdot \left ( x_1 - x_2 \right ) \right ) which evaluates to (see ) E = - \frac \frac e^ \left\ \vec v_1 \cdot \left 1 + \rightcdot \vec v_2 which reduces to E = - \frac \frac \vec v_1 \cdot \left 1 + \right\cdot \vec v_2 in the limit of small . The interaction energy is the negative of the interaction Lagrangian. For two like particles traveling in the same direction, the interaction is attractive, which is the opposite of the Coulomb interaction.


Darwin interaction in a plasma

In a plasma, the dispersion relation for an
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
is (c = 1) k_0^2 = \omega_p^2 +\vec k^2, which implies D\left ( k \right )\mid_\; = \; -\frac. Here \omega_p is the plasma frequency. The interaction energy is therefore E = - \frac \frac \vec v_1 \cdot \left 1 + \rightcdot \vec v_2 \; e^ \left\.


Magnetic interaction between current loops in a simple plasma or electron gas


=The interaction energy

= Consider a tube of current rotating in a magnetic field embedded in a simple
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...
or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as \vec J_1( \vec x) = a_1 v_1 \frac \; \delta^ 2 \left( \hat b \times \hat r \right) where r_ = \frac and \hat b is the unit vector in the direction of the magnetic field. Here L_B indicates the dimension of the material in the direction of the magnetic field. The transverse current, perpendicular to the wave vector, drives the
transverse wave In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
. The energy of interaction is E = \left( \frac\right) v_1\, v_2\, \int_0^ D\left( k \right) \mid_ \mathcal J_1 \mathcal J_1 \mathcal J_0 where r_ is the distance between the centers of the current loops and \mathcal J_n ( x ) is a
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
of the first kind. In obtaining the interaction energy we made use of the integrals \int_0^ \frac \exp\left( i p \cos\left( \varphi \right) \right) = \mathcal J_0 \left( p \right) and \int_0^ \frac \cos\left( \varphi \right) \exp\left( i p \cos\left( \varphi \right) \right) = i\mathcal J_1 \left( p \right) . See . A current in a plasma confined to the plane perpendicular to the magnetic field generates an extraordinary wave. This wave generates
Hall current The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was disco ...
s that interact and modify the electromagnetic field. The dispersion relation for extraordinary waves is -k_0^2 +\vec k^2 + \omega_p^2 \frac =0, which gives for the propagator D\left( k \right) \mid_\;= \;-\left( \frac\right) where k_X \equiv \frac in analogy with the Darwin propagator. Here, the upper hybrid frequency is given by \omega_H^2 = \omega_p^2 + \omega_c^2, the
cyclotron frequency Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
is given by (
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
) \omega_c = \frac, and the plasma frequency (
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
) \omega_p^2 = \frac. Here is the electron density, is the magnitude of the electron charge, and is the electron mass. The interaction energy becomes, for like currents, E = - \left( \frac\right) v^2\, \int_0^ \frac \mathcal J_1^2 \left ( kr_ \right) \mathcal J_0 \left ( kr_ \right)


=Limit of small distance between current loops

= In the limit that the distance between current loops is small, E = - E_0 \; I_1 \left( \mu \right) K_1 \left( \mu \right) where E_0 = \left( \frac\right) v^2 and \mu =\frac= k_X \;r_B and and are modified Bessel functions. we have assumed that the two currents have the same charge and speed. We have made use of the integral (see ) \int_o^ \frac \mathcal J_1^2 \left( kr \right) = I_1 \left( mr \right)K_1 \left( mr \right) . For small the integral becomes I_1 \left( mr \right)K_1 \left( mr \right) \rightarrow \frac\left 1- \frac\left( mr \right)^2 \right. For large the integral becomes I_1 \left( mr \right)K_1 \left( mr \right) \rightarrow \frac\;\left( \frac\right) .


=Relation to the quantum Hall effect

= The screening
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
can be written (
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
) \mu = \frac = \left( \frac\right) \frac = 2 \alpha \left( \frac\right) \left(\frac\right) \nu where \alpha is the fine-structure constant and the filling factor is \nu = \frac and is the number of electrons in the material and is the area of the material perpendicular to the magnetic field. This parameter is important in the quantum Hall effect and the
fractional quantum Hall effect The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...
. The filling factor is the fraction of occupied Landau states at the ground state energy. For cases of interest in the quantum Hall effect, \mu is small. In that case the interaction energy is E = - \frac \left 1- \frac\mu^2\right/math> where (
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
) E_0 = \frac\frac = \frac\left( \frac\right) is the interaction energy for zero filling factor. We have set the classical kinetic energy to the quantum energy \frac m v^2 = \hbar \omega_c.


Gravitation

A gravitational disturbance is generated by the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
T^ ; consequently, the Lagrangian for the gravitational field is
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
-2. If the disturbances are at rest, then the only component of the stress–energy tensor that persists is the 00 component. If we use the same trick of giving the graviton some mass and then taking the mass to zero at the end of the calculation the propagator becomes D\left ( k \right )\mid_\; = \; - \frac \frac and E = -\frac\frac \exp \left ( -m r \right ), which is once again attractive rather than repulsive. The coefficients are proportional to the masses of the disturbances. In the limit of small graviton mass, we recover the inverse-square behavior of Newton's Law. Unlike the electrostatic case, however, taking the small-mass limit of the boson does not yield the correct result. A more rigorous treatment yields a factor of one in the energy rather than 4/3.


References

{{Reflist Quantum field theory