Los campos de fuerza estática son campos, como campos eléctricos , magnéticos o gravitacionales simples , que existen sin excitaciones. El método de aproximación más común que utilizan los físicos para los cálculos de dispersión se puede interpretar como fuerzas estáticas que surgen de las interacciones entre dos cuerpos mediados por partículas virtuales , partículas que existen solo por un corto tiempo determinado por el principio de incertidumbre . [1] Las partículas virtuales, también conocidas como portadoras de fuerza , son bosones , con diferentes bosones asociados con cada fuerza. [2]
La descripción de partículas virtuales de las fuerzas estáticas es capaz de identificar la forma espacial de las fuerzas, como el comportamiento del cuadrado inverso en la ley de Newton de la gravitación universal y en la ley de Coulomb . También puede predecir si las fuerzas son atractivas o repulsivas para cuerpos similares.
La formulación de la integral de trayectoria es el lenguaje natural para describir los portadores de fuerza. Este artículo utiliza la fórmula integral de trayectoria para describir los portadores de fuerza para los campos de espín 0, 1 y 2. Los piones , fotones y gravitones se incluyen en estas categorías respectivas.
Existen límites para la validez de la imagen de partículas virtuales. La formulación de partículas virtuales se deriva de un método conocido como teoría de la perturbación, que es una aproximación asumiendo que las interacciones no son demasiado fuertes, y fue pensada para problemas de dispersión, no estados ligados como átomos. Para la fuerza fuerte que une a los quarks en nucleones a bajas energías, nunca se ha demostrado que la teoría de la perturbación produzca resultados de acuerdo con los experimentos, [3] por lo tanto, la validez de la imagen de la "partícula mediadora de fuerza" es cuestionable. De manera similar, para los estados ligados, el método falla. [4] En estos casos, la interpretación física debe ser reexaminada. Por ejemplo, los cálculos de la estructura atómica en física atómica o de la estructura molecular en química cuántica no podrían repetirse fácilmente, si es que se repiten, utilizando la imagen de la "partícula mediadora de fuerzas". [ cita requerida ]
El uso de una imagen de "partículas mediadoras de fuerza" (FMPP) es innecesario en la mecánica cuántica no relativista , y la ley de Coulomb se usa como se indica en la física atómica y la química cuántica para calcular tanto los estados ligados como los de dispersión. Una teoría cuántica relativista no perturbativa , en la que se conserva la invariancia de Lorentz, se puede lograr evaluando la ley de Coulomb como una interacción de 4 espacios usando el vector de posición de 3 espacios de un electrón de referencia que obedece a la ecuación de Dirac y la trayectoria cuántica de un segundo electrón que depende solo del tiempo escalado. La trayectoria cuántica de cada electrón en un conjunto se infiere a partir de la corriente de Dirac para cada electrón ajustándolo a un campo de velocidad multiplicado por una densidad cuántica, calculando un campo de posición a partir de la integral de tiempo del campo de velocidad y, finalmente, calculando una trayectoria cuántica. del valor esperado del campo de posición. Las trayectorias cuánticas son, por supuesto, dependientes del espín, y la teoría puede validarse comprobando que se obedece el principio de exclusión de Pauli para una colección de fermiones .
Fuerzas clásicas
La fuerza ejercida por una masa sobre otra y la fuerza ejercida por una carga sobre otra son sorprendentemente similares. Ambos caen como el cuadrado de la distancia entre los cuerpos. Ambos son proporcionales al producto de las propiedades de los cuerpos, masa en el caso de la gravitación y carga en el caso de la electrostática.
También tienen una diferencia sorprendente. Dos masas se atraen, mientras que dos cargas iguales se repelen.
En ambos casos, los cuerpos parecen actuar entre sí a distancia. El concepto de campo se inventó para mediar en la interacción entre los cuerpos, eliminando así la necesidad de actuar a distancia . La fuerza gravitacional está mediada por el campo gravitacional y la fuerza de Coulomb está mediada por el campo electromagnético .
Fuerza gravitacional
La fuerza gravitacional sobre una masa. ejercido por otra masa es
donde G es la constante gravitacional , r es la distancia entre las masas yes el vector unitario de la masa a misa .
La fuerza también se puede escribir
dónde es el campo gravitacional descrito por la ecuación de campo
dónde es la densidad de masa en cada punto del espacio.
Fuerza de coulomb
La fuerza electrostática de Coulomb sobre una carga ejercido por una carga es ( unidades SI )
where is the vacuum permittivity, is the separation of the two charges, and is a unit vector in the direction from charge to charge .
The Coulomb force can also be written in terms of an electrostatic field:
where
being the charge density at each point in space.
Intercambio de partículas virtuales
In perturbation theory, forces are generated by the exchange of virtual particles. 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 field.
The probability amplitude
Using natural units, , the probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in the path integral formulation by
where is the Hamiltonian operator, is elapsed time, is the energy change due to the disturbance, is the change in action due to the disturbance, is the field of the virtual particle, the integral is over all paths, and the classical action is given by
where is the Lagrangian density.
Here, the spacetime metric is given by
The path integral often can be converted to the form
where is a differential operator with and functions of spacetime. 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 Common integrals in quantum field theory)
where
is the change in the action due to the disturbances and the propagator is the solution of
- .
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
where the delta functions are in space, the disturbances are located at and , and the coefficients and are the strengths of the disturbances.
If we neglect self-interactions of the disturbances then W becomes
- ,
which can be written
- .
Here is the Fourier transform of
- .
Finally, the change in energy due to the static disturbances of the vacuum is
.
If this quantity is negative, the force is attractive. If it is positive, the force is repulsive.
Examples of static, motionless, interacting currents are Yukawa Potential, The Coulomb potential in a vacuum, and 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 Darwin interaction in a vacuum and Darwin interaction 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 are Two line charges embedded in a plasma or electron gas, Coulomb potential between two current loops embedded in a magnetic field, and 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.
Ejemplos seleccionados
The Yukawa potential: The force between two nucleons in an atomic nucleus
Consider the spin-0 Lagrangian density[5]
- .
The equation of motion for this Lagrangian is the Klein–Gordon equation
- .
If we add a disturbance the probability amplitude becomes
- .
If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes
- .
With the amplitude in this form it can be seen that the propagator is the solution of
- .
From this it can be seen that
- .
The energy due to the static disturbances becomes (see Common integrals in quantum field theory)
with
which is attractive and has a range of
- .
Yukawa 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-1 Proca Lagrangian with a disturbance[6]
where
- ,
charge is conserved
- ,
and we choose the Lorenz gauge
- .
Moreover, we assume that there is only a time-like component 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
which implies
and
This yields
for the timelike propagator and
which has the opposite sign to the Yukawa case.
In the limit of zero photon mass, the Lagrangian reduces to the Lagrangian for electromagnetism
Therefore the energy reduces to the potential energy for the Coulomb force and the coefficients and 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 waves is[7]
where is the angular frequency of the wave,
is the plasma frequency, is the magnitude of the electron charge, is the electron mass, is the electron temperature (Boltzmann's constant equal to one), and 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 and is equal to three. At low frequencies, the compression is an isothermal process and is equal to one. Retardation effects have been neglected in obtaining the plasma-wave dispersion relation.
For low frequencies, the dispersion relation becomes
where
is the Debye number, which is the inverse of the Debye length. This suggests that the propagator is
- .
In fact, if the retardation effects are not neglected, then the dispersion relation is
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
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 to yield[8]
where the inverse of the Thomas–Fermi screening length is
and is the Fermi energy
This expression can be derived from the chemical potential for an electron gas and from Poisson's equation. The chemical potential for an electron gas near equilibrium is constant and given by
where is the electric potential. 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.
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
where is the distance in the xy plane from the line of charge, is the width of the material in the z direction. The superscript 2 indicates that the Dirac delta function is in two dimensions. The propagator is
where is either the inverse Debye-Hückel screening length or the inverse Thomas–Fermi screening length.
The interaction energy is
where
and
are Bessel functions and is the distance between the two line charges. In obtaining the interaction energy we made use of the integrals (see Common integrals in quantum field theory)
and
For , we have
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
where is the distance from the guiding center, is the width of the material in the direction of the magnetic field
where the cyclotron frequency is (Gaussian units)
and
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 the quantum treatment of a charged particle in a magnetic field.
In this geometry, the interaction energy can be written
where is the distance between the centers of the current loops and
is a Bessel function of the first kind. In obtaining the interaction energy we made use of the integral
Electric field due to a density perturbation
The chemical potential near equilibrium, is given by
where is the potential energy of an electron in an electric potential and and 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
where is the area of the material in the plane perpendicular to the magnetic field.
Poisson's equation yields
where
The propagator is then
and the interaction energy becomes
where in the second equality (Gaussian units) we assume that the vortices had the same energy and the electron charge.
In analogy with plasmons, the force carrier is the quantum version of the upper hybrid oscillation which is a longitudinal plasma wave 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 for a given energy.[9] Landau levels, the energy states of a charged particle in the presence of a magnetic field, are multiply degenerate. The current loops correspond to angular momentum states of the charged particle that may have the same energy. Specifically, the charge density is peaked around radii of
where is the angular momentum quantum number. When we recover the classical situation in which the electron orbits the magnetic field at the Larmor radius. If currents of two angular momentum and interact, and we assume the charge densities are delta functions at radius , then the interaction energy is
The interaction energy for is given in Figure 1 for various values of . 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
This suggests that the pair of particles that are bound and separated by a distance act as a single quasiparticle with angular momentum .
If we scale the lengths as , then the interaction energy becomes
where
The value of the at which the energy is minimum, , is independent of the ratio . However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when
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)
or
where the total angular momentum is written as
When the total angular momentum is odd, the minima cannot occur for The lowest energy states for odd total angular momentum occur when
or
and
which also appear as series for the filling factor in the fractional quantum Hall effect.
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[10]
The interaction energy becomes
where is a confluent hypergeometric function or Kummer function. In obtaining the interaction energy we have used the integral (see Common integrals in quantum field theory)
As with delta function charges, the value of 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
and
appear as well in the case of charges spread by the wave function.
The Laughlin wavefunction is an ansatz for the quasiparticle wavefunction. If the expectation value of the interaction energy is taken over a Laughlin wavefunction, 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
with a comparable expression for .
The Fourier transform of this current is
The current can be decomposed into a transverse and a longitudinal part (see Helmholtz decomposition).
The hat indicates a unit vector. The last term disappears because
which results from charge conservation. Here vanishes because we are considering static forces.
With the current in this form the energy of interaction can be written
- .
The propagator equation for the Proca Lagrangian is
The spacelike solution is
which yields
which evaluates to (see Common integrals in quantum field theory)
which reduces to
in the limit of small m. 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 is[11] ()
which implies
Here is the plasma frequency. The interaction energy is therefore
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 or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as
where
and is the unit vector in the direction of the magnetic field. Here 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.
The energy of interaction is
where is the distance between the centers of the current loops and
is a Bessel function of the first kind. In obtaining the interaction energy we made use of the integrals
and
See Common integrals in quantum field theory.
A current in a plasma confined to the plane perpendicular to the magnetic field generates an extraordinary wave.[12] This wave generates Hall currents that interact and modify the electromagnetic field. The dispersion relation for extraordinary waves is[13]
which gives for the propagator
where
in analogy with the Darwin propagator. Here, the upper hybrid frequency is given by
the cyclotron frequency is given by (Gaussian units)
and the plasma frequency (Gaussian units)
Here n is the electron density, e is the magnitude of the electron charge, and m is the electron mass.
The interaction energy becomes, for like currents,
Limit of small distance between current loops
In the limit that the distance between current loops is small,
where
and
and I and K 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 Common integrals in quantum field theory)
For small mr the integral becomes
For large mr the integral becomes
Relation to the quantum Hall effect
The screening wavenumber can be written (Gaussian units)
where is the fine-structure constant and the filling factor is
and N is the number of electrons in the material and A 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 filling factor is the fraction of occupied Landau states at the ground state energy.
For cases of interest in the quantum Hall effect, is small. In that case the interaction energy is
where (Gaussian units)
is the interaction energy for zero filling factor. We have set the classical kinetic energy to the quantum energy
Gravitation
A gravitational disturbance is generated by the stress–energy tensor ; consequently, the Lagrangian for the gravitational field is spin-2. If the disturbances are at rest, then the only component of the stress–energy tensor that persists is the 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
and
,
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.[14]
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.[15]
Referencias
- ^ Jaeger, Gregg (2019). "Are virtual particles less real?". Entropy. 21 (2): 141. Bibcode:2019Entrp..21..141J. doi:10.3390/e21020141.
- ^ A. Zee (2003). Quantum Field Theory in a Nutshell. Princeton University. ISBN 0-691-01019-6. pp. 16-37
- ^ "Archived copy". Archived from the original on 2011-07-17. Retrieved 2010-08-31.CS1 maint: archived copy as title (link)
- ^ "Time-Independent Perturbation Theory". virginia.edu.
- ^ Zee, pp. 21-29
- ^ Zee, pp. 30-31
- ^ F. F. Chen (1974). Introduction to Plasma Physics. Plenum Press. ISBN 0-306-30755-3. pp. 75-82
- ^ C. Kittel (1976). Introduction to Solid State Physics (Fifth ed.). John Wiley and Sons. ISBN 0-471-49024-5. pp. 296-299.
- ^ Z. F. Ezewa (2008). Quantum Hall Effects, Second Edition. World Scientific. ISBN 978-981-270-032-2. pp. 187-190
- ^ Ezewa, p. 189
- ^ Chen, pp. 100-103
- ^ Chen, pp. 110-112
- ^ Chen, p. 112
- ^ Zee, pp. 32-37
- ^ Zee, p. 35