Content deleted Content added
No edit summary |
\mathit interferes with prime and offers no other visual advantage |
||
Line 7:
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 (physics)|spin]] 0, 1, and 2 fields. [[Pion]]s, [[photon]]s, and [[graviton]]s 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]] 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]]s into [[nucleon]]s at low energies, perturbation theory has never been shown to yield results in accord with experiments,<ref>{{cite web |url=http://www.hep.phy.cam.ac.uk/theory/research/hadronic.html |title=Archived copy |accessdate=2010-08-31 |url-status=dead |archiveurl=https://web.archive.org/web/20110717002648/http://www.hep.phy.cam.ac.uk/theory/research/hadronic.html |archivedate=2011-07-17 }}</ref> thus, the validity of the "force-mediating particle" picture is questionable. Similarly, for [[bound state]]s the method fails.<ref>{{cite web| url=http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Time_Ind_PT.htm|title=Time-Independent Perturbation Theory| work=virginia.edu}}</ref> In these cases, the physical interpretation must be re-examined.
<!-- Text below hidden for the time being (vs. being deleted) because it seems valuable, but needs rewriting in an encyclopedic form.
<!-- Additionally, one should look critically{{fact|date=October 2014}} at the recent CERN experiments{{fact|date=October 2014}} in which evidence is shown supporting the physical reality of the Higgs boson, which is a force-mediating particle.
The use of "force-mediating particle" picture (FMPP) is unnecessary in [[Quantum mechanics|nonrelativistic quantum mechanics]], and Coulomb's law is used as given in atomic physics and quantum chemistry to calculate both bound and scattering states.
==Classical forces==
Line 171:
\end{align}</math>
which implies
<math display="block"> \eta_{\mu \alpha} \left ( \partial^2 + m^2\right ) D^{\alpha \nu}\left ( x-y \right ) = \delta_{\mu}^{
and
<math display="block">D_{\mu \nu}\left ( k \right )\mid_{k_0=0} \; = \; \eta_{\mu \nu}\frac{1}{- k^2 + m^2}.</math>
Line 215:
where the inverse of the Thomas–Fermi screening length is
<math display="block"> k_s^2 = \frac{6\pi n e^2}{\varepsilon_F}</math>
and <math>\varepsilon_F</math> is the [[Fermi energy]] <math display="inline">\varepsilon_F = \frac{\hbar^2}{2m} \left( {3 \pi^2 n} \right)^{2/3} .</math>
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
Line 226 ⟶ 225:
We consider a line of charge with axis in the ''z'' direction embedded in an electron gas
<math display="block"> J_1\left( x\right) = \frac{a_1}{L_B} \frac{1}{2 \pi r} \delta^2\left( r \right)</math>
where <math>r</math> is the distance in the ''xy
<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; \frac{1}{\vec k^2 + k_{Ds}^2}</math>
where <math>k_{Ds} </math> is either the inverse [[Debye–Hückel equation|Debye-Hückel screening length]] or the inverse [[Thomas–Fermi screening]] length.
Line 236 ⟶ 235:
<math display="block"> \int_0^{2 \pi} \frac{d\varphi}{2 \pi} \exp\left( i p \cos\left( \varphi \right) \right) = \mathcal J_0 ( p ) </math>
and
<math display="block"> \int_0^{\infty} \frac{k\
For <math> k_{Ds} r_{12} \ll 1</math>, we have
Line 246 ⟶ 245:
We consider a charge density in tube with axis along a magnetic field embedded in an electron gas
<math display="block"> J_1\left( x\right) = \frac{a_1}{L_b} \frac{1}{2 \pi r} \delta^2{\left( r - r_{B1}\right)}</math>
where <math>r</math> is the distance from the [[guiding center]], <math>L_B</math> is the width of the material in the direction of the magnetic field
<math display="block"> r_{B1} = \frac{\sqrt{4 \pi} m_1 v_1}{a_1 B} = \sqrt{\frac{2 \hbar}{m_1 \omega_c}} </math>
Line 261 ⟶ 260:
where <math>r_{12}</math> is the distance between the centers of the current loops and <math> \mathcal J_n ( x ) </math> is a [[Bessel function]] of the first kind. In obtaining the interaction energy we made use of the integral
<math display="block">
\int_0^{2 \pi} \frac{d\varphi}{2 \pi} \exp\left( i p \cos
=====Electric field due to a density perturbation=====
Line 297 ⟶ 295:
======Delta function currents======
[[Image:100927c Angular momentum 11.jpg|thumb|250px|right|Figure 1. Interaction energy vs. r for angular momentum states of value one. The curves are identical to these for any values of <math>
[[Image:100927 Angular momentum 15.jpg|thumb|250px|right|Figure 2. Interaction energy vs. r for angular momentum states of value one and five.]]
[[Image:101005 energy by theta.jpg|thumb|250px|right|Figure 3. Interaction energy vs. r for various values of theta. The lowest energy is for <math display="inline">\theta = \frac{\pi}{4}</math> or <math> \frac{
[[Image:101011 energy picture.jpg|thumb|250px|right|Figure 4. Ground state energies for even and odd values of angular momenta. Energy is plotted on the vertical axis and r is plotted on the horizontal. When the total angular momentum is even, the energy minimum occurs when <math>
Unlike classical currents, quantum current loops can have various values of the [[Larmor radius]] for a given energy.<ref name="Ezewa">{{cite book | first = Zyun F. | last = Ezewa | title=Quantum Hall Effects: Field Theoretical Approach And Related Topics | edition = Second | publisher= World Scientific| year=2008 | isbn=978-981-270-032-2}}</ref>{{rp|pp=187–190}} [[Landau level]]s, the energy states of a charged particle in the presence of a magnetic field, are multiply [[Degenerate energy level|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
<math display="block">
where <math>
▲<math display="block"> r_{\mathit l} = \sqrt{\mathit l}\;r_B\; \; \; \mathit l=0,1,2, \ldots</math>
▲where <math>\mathit l</math> is the angular momentum [[quantum number]]. When <math>\mathit l=1</math> we recover the classical situation in which the electron orbits the magnetic field at the [[Larmor radius]]. If currents of two angular momentum <math>\mathit l >0 </math> and <math>\mathit l' \ge \mathit l </math> interact, and we assume the charge densities are delta functions at radius <math>r_{\mathit l}</math>, then the interaction energy is
<math display="block"> E =
\left( \frac{2 e^2}{L_B}\right) \int_0^{\infty} \frac{k\;dk \;}{k^2 + k_B^2 r_{
\;\mathcal J_0 \left ( k \right) \;\mathcal J_0 \left ( \sqrt{\frac{
The interaction energy for <math>
======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
<math display="block">r_{12} = r_{
This suggests that the pair of particles that are bound and separated by a distance <math>r_{
If we scale the lengths as <math> r_{
<math display="block"> E = \frac{2 e^2}{L_B} \int_0^{\infty} \frac{k\,dk}{k^2 + k_B^2 r_{l l'}^2}
\;\mathcal J_0 \left ( \cos \theta \
▲\;\mathcal J_0 \left ( \cos \theta \; k \right) \;\mathcal J_0 \left ( \sin \theta \;k \right) \;\mathcal J_0 \left ( k \frac{r_{12}}{r_{\mathit l \mathit l'}} \right)</math>
where
<math display="block">\tan \theta = \sqrt{\frac{
The value of the <math> r_{12} </math> at which the energy is minimum, <math>r_{12} = r_{
<math display="block"> \frac{
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)
<math display="block">
or
<math display="block"> \frac{
where the total angular momentum is written as
<math display="block">
When the total angular momentum is odd, the minima cannot occur for <math>
<math display="block"> \frac{
or
<math display="block">\frac{
and
<math display="block">\frac{
which also appear as series for the filling factor in the [[fractional quantum Hall effect]].
Line 352 ⟶ 345:
\frac{1}{\pi r_B^2 L_B}
\frac{1}{n!}
\left( \frac{r}{r_B} \right)^{
\exp \left( -\frac{r^2}{r_B^2} \right).</math>
Line 358 ⟶ 351:
<math display="block"> E =
\left( \frac{2 e^2}{L_B}\right) \int_0^{\infty} \frac{k\;dk \;}{k^2 + k_B^2 r_{B}^2}
\; M {\left (
</math>
where <math>M</math> is a [[confluent hypergeometric function]] or [[Kummer function]]. In obtaining the interaction energy we have used the integral (see {{slink|Common integrals in quantum field theory#Integration over a magnetic wave function}})
<math display="block">
\frac{2}{n!} \int_0^{\infty} dr \;r^{2n+1}\exp\left( -r^2\right) J_0(kr)
As with delta function charges, the value of <math>r_{12}</math> 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
<math display="block">\frac{
and
<math display="block">\frac{
appear as well in the case of charges spread by the wave function.
Line 382 ⟶ 373:
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 Lagrangian|Darwin interaction]]. To calculate this, consider the electrical currents in space generated by a moving charge
<math display="block">\vec J_1{\left( \vec x \right)} = a_1 \vec v_1 \delta^3 {\left( \vec x - \vec x_1 \right)}</math>
with a comparable expression for <math> \vec J_2 </math>.
The Fourier transform of this current is
<math display="block">\vec J_1{\left( \vec k \right)} = a_1 \vec v_1 \exp\left( i \vec k \cdot \vec x_1 \right).</math>
The current can be decomposed into a transverse and a longitudinal part (see [[Helmholtz decomposition]]).
<math display="block">\vec J_1{\left( \vec k \right)} = 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).</math>
The hat indicates a [[unit vector]]. The last term disappears because
Line 401 ⟶ 392:
The propagator equation for the Proca Lagrangian is
<math display="block"> \eta_{\mu \alpha} \left ( \partial^2 + m^2\right ) D^{\alpha \nu}\left ( x-y \right ) = \delta_{\mu}^{
The [[spacelike]] solution is
Line 439 ⟶ 430:
Consider a tube of current rotating in a magnetic field embedded in a simple [[Plasma (physics)|plasma]] or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as
<math display="block">\vec J_1( \vec x) = a_1 v_1 \frac{1}{2 \pi r L_B} \; \delta^ 2 {\left( r - r_{B1} \right)
\left( \hat b \times \hat r \right)</math>
where
Line 446 ⟶ 437:
The energy of interaction is
<math display="block"> E = \left( \frac{a_1\, a_2}{2 \pi L_B}\right) v_1\, v_2\, \int_0^{\infty} {k\;dk \;} D\left( k \right) \mid_{k_0=k_B=0} \mathcal J_1 {\left ( kr_{B1} \right)} \mathcal J_1 {\left ( kr_{B2} \right)} \mathcal J_0 {\left ( kr_{12} \right)}</math>
where <math>r_{12}</math> is the distance between the centers of the current loops and <math> \mathcal J_n ( x )</math> is a [[Bessel function]] of the first kind. In obtaining the interaction energy we made use of the integrals
<math display="block">
Line 458 ⟶ 449:
A current in a plasma confined to the plane perpendicular to the magnetic field generates an [[Electromagnetic electron wave#X wave|extraordinary wave]].<ref name="Chen"/>{{rp|pp=110–112}} This wave generates [[Hall current]]s that interact and modify the electromagnetic field. The [[dispersion relation]] for extraordinary waves is<ref name="Chen"/>{{rp|112}}
<math display="block"> -k_0^2 +\vec k^2 + \omega_p^2 \frac{
which gives for the propagator
<math display="block"> D\left( k \right) \mid_{k_0=k_B=0}\;= \;-\left( \frac{1}{\vec k^2 + k_X^2}\right)</math>
| |||