Static forces and virtual-particle exchange: Difference between revisions

Revision as of 14:45, 21 December 2023

Revision as of 19:11, 18 February 2024

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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

<math display="block"> J(x) = \left( J_1 +J_2,0,0,0 \right)</math>

<math display="block"> \begin{align}
J_1 &= a_1 \delta^3\left ( \vecmathbf x - \vecmathbf x_1 \right )</math> \\

<math display="block"> J_2 &= a_2 \delta^3\left ( \vecmathbf x - \vecmathbf x_2 \right )</math>

\end{align}</math>

where the delta functions are in space, the disturbances are located at <math> \vecmathbf x_1 </math> and <math> \vecmathbf x_2 </math>, and the coefficients <math> a_1 </math> and <math> a_2 </math> are the strengths of the disturbances.

If we neglect self-interactions of the disturbances then W becomes

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which can be written

<math display="block"> W\left ( J \right ) =

- T a_1 a_2\int \frac{d^3k}{(2 \pi )^3} \; \; D\left ( k \right )\mid_{k_0=0} \; \exp\left ( i \vecmathbf k \cdot \left ( \vecmathbf x_1 - \vecmathbf x_2 \right ) \right ).

</math>

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Finally, the change in energy due to the static disturbances of the vacuum is

<math display="block"> E = - \frac{W}{T} = a_1 a_2\int \frac{d^3k}{(2 \pi )^3} \; \; D\left ( k \right )\mid_{k_0=0} \; \exp\left ( i \vecmathbf k \cdot \left ( \vecmathbf x_1 - \vecmathbf x_2 \right ) \right ).</math>

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 [[Static forces and virtual-particle exchange#The Yukawa potential: The force between two nucleons in an atomic nucleus|Yukawa potential]], the [[Static forces and virtual-particle exchange#The Coulomb potential in a vacuum|Coulomb potential in a vacuum]], and the [[Static forces and virtual-particle exchange#Coulomb potential in a simple plasma or electron gas|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 [[Static forces and virtual-particle exchange#Darwin interaction in a vacuum|in a vacuum]] and [[Static forces and virtual-particle exchange#Darwin interaction in a plasma|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: [[Static forces and virtual-particle exchange#Two line charges embedded in a plasma or electron gas|two line charges embedded in a plasma or electron gas]], [[Static forces and virtual-particle exchange#Coulomb potential between two current loops embedded in a magnetic field|Coulomb potential between two current loops embedded in a magnetic field]], and the [[Static forces and virtual-particle exchange#Magnetic interaction between current loops in a simple plasma or electron gas|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 Hall effect|fractional quantum numbers]].

==Selected examples==

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If we add a disturbance the probability amplitude becomes

<math display="block"> Z =

\int D\varphi \; \exp \left \{ i \int d^4x4\mathbf{x}\; \left [ \frac{1}{2} \left ( \left ( \partial \varphi \right )^2 - m^2\varphi^2 \right ) + J\varphi \right ] \right \}.</math>

If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes

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From this it can be seen that

<math display="block">D\left ( k \right )\mid_{k_0=0} \; = \; -\frac{1}{\vec k^2 + m^2}.</math>

The energy due to the static disturbances becomes (see {{slink|Common integrals in quantum field theory#Yukawa Potential: The Coulomb potential with mass}})

<math display="block">E =-\frac{a_1 a_2}{4 \pi r} \exp \left ( -m r \right )</math>

with

<math display="block">r^2 = \left (\vecmathbf x_1 - \vecmathbf x_2 \right )^2</math>

which is attractive and has a range of <math display="block">\frac{1}{m}.</math>

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If we follow the same procedure as we did with the Yukawa potential we find that

<math display="block">\begin{align}

& -\frac{1}{4} \int d^4x F_{\mu \nu}F^{\mu \nu}

&= -\frac{1}{4}\int d^4x \left( \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} \right)\left( \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu} \right) \\

&= {} & \frac{1}{2}\int d^4x \; A_{\nu} \left( \partial^{2} A^{\nu} - \partial^{\nu} \partial_{\mu} A^{\mu} \right) \\

&= \frac{1}{2}\int d^4x \; A^{\mu} \left( \eta_{\mu \nu} \partial^{2} \right) A^{\nu},

\end{align}</math>

which implies

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This yields

<math display="block">D\left( k \right)\mid_{k_0=0}\; = \; \frac{1}{\vecmathbf k^2 + m^2}</math>

for the [[timelike]] propagator and

<math display="block">E = + \frac{a_1 a_2}{4 \pi r} \exp \left( -m r \right)</math>

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The [[dispersion relation]] for [[plasma wave]]s is<ref name="Chen">{{cite book | first = Francis F. | last = Chen | title=Introduction to Plasma Physics| publisher= Plenum Press| year=1974 | isbn=0-306-30755-3}}</ref>{{rp|pp=75–82}}

<math display="block">\omega^2 = \omega_p^2 + \gamma\left( \omega \right) \frac{T_e}{m} \vecmathbf k^2.</math>

where <math>\omega </math> is the angular frequency of the wave,

<math display="block">\omega_p^2 = \frac{4\pi n e^2}{m}</math>

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For low frequencies, the dispersion relation becomes

<math display="block"> \vecmathbf k^2 + \vecmathbf k_D^2 = 0</math>

where

<math display="block"> k_D^2= \frac{4\pi n e^2}{T_e}</math>

is the Debye number, which is the inverse of the [[Debye length]]. This suggests that the propagator is

<math display="block">D\left ( k \right )\mid_{k_0=0} \; = \; \frac{1}{\vec k^2 + k_D^2}.</math>

In fact, if the retardation effects are not neglected, then the dispersion relation is

<math display="block"> -k_0^2 +\vec k^2 + k_D^2 -\frac{m}{T_e} k_0^2 = 0,</math>

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

<math display="block">E = \frac{a_1 a_2}{4 \pi r} \exp \left ( -k_D r \right ).</math>

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<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''-plane from the line of charge, <math>L_B</math> 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

<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; \frac{1}{\vecmathbf 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.

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For <math> k_{Ds} r_{12} \ll 1</math>, we have

<math display="block">K_0 \left( k_{Ds} r_{12} \right) \rightarrowto -\ln \left(\frac{k_{Ds} r_{12}}{2}\right) + 0.5772.</math>

====Coulomb potential between two current loops embedded in a magnetic field====

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======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>l\ell = l\ell'</math>. Lengths are in units are in <math>r_lr_\ell</math>, and the energy is in units of <math display="inline"> \frac{e^2}{L_B}</math>. Here <math>r = r_{12}</math>. Note that there are local minima for large values of <math> k_{B}</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{l\ell}{l\ell'} = 1 </math>. The highest energy plotted is for <math display="inline">\theta = 0.90\frac{\pi}{4}</math>. Lengths are in units of <math>r_{l\ell l\ell'}</math>.]]

[[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> l\ell = l\ell' </math> or <math display="inline"> \frac{l\ell}{l\ell^*} = \frac{1}{2} </math>. When the total angular momentum is odd, there are no integer values of angular momenta that will lie in the energy minimum. Therefore, there are two states that lie on either side of the minimum. Because <math> l\ell \ne l\ell' </math>, the total energy is higher than the case when <math> l\ell = l\ell' </math> for a given value of <math> l\ell^* </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">r_{l\ell} = \sqrt{l\ell}\;r_B\; \; \; l\ell=0,1,2, \ldots</math>

where <math>l\ell</math> is the angular momentum [[quantum number]]. When <math> l\ell = 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> l\ell > 0 </math> and <math>l\ell' \ge l\ell </math> interact, and we assume the charge densities are delta functions at radius <math>r_{l\ell}</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_{l\ell}^2}

\;\mathcal J_0 \left ( k \right) \;\mathcal J_0 \left ( \sqrt{\frac{l\ell'}{l\ell}} \;k \right) \;\mathcal J_0 \left ( k \frac{r_{12}}{r_{l\ell}} \right).</math>

The interaction energy for <math> l\ell = l\ell'</math> is given in Figure 1 for various values of <math>k_B r_{l}\ell</math>. 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

<math display="block">r_{12} = r_{l\ell l\ell'} = \sqrt{l\ell + l\ell'} \; r_B.</math>

This suggests that the pair of particles that are bound and separated by a distance <math>r_{l\ell l\ell'} </math> act as a single [[quasiparticle]] with angular momentum <math> l\ell + l\ell'</math>.

If we scale the lengths as <math> r_{l\ell l\ell'} </math>, then the interaction energy becomes

<math display="block"> E = \frac{2 e^2}{L_B} \int_0^{\infty} \frac{k\,dk}{k^2 + k_B^2 r_{l\ell l\ell'}^2}

\;\mathcal J_0 \left ( \cos \theta \, k \right) \;\mathcal J_0 ( \sin \theta \,k ) \;\mathcal J_0{\left( k \frac{r_{12}}{r_{l\ell l\ell'}} \right)}</math>

where

<math display="block">\tan \theta = \sqrt{\frac{l\ell}{l\ell'}}.</math>

The value of the <math> r_{12} </math> at which the energy is minimum, <math>r_{12} = r_{l\ell l\ell'} </math>, is independent of the ratio <math display="inline"> \tan \theta = \sqrt{{l\ell}/{l\ell'}}</math>. However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when

<math display="block"> \frac{l\ell}{l\ell'} = 1.</math>

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"> l\ell = l\ell' = 1</math>

or

<math display="block"> \frac{l\ell}{l\ell^*} = \frac{1}{2} </math>

where the total angular momentum is written as

<math display="block"> l\ell^* = l\ell + l\ell'. </math>

When the total angular momentum is odd, the minima cannot occur for <math> l\ell = l\ell' . </math> The lowest energy states for odd total angular momentum occur when

<math display="block"> \frac{l\ell}{l\ell^*} = \; \frac{l\ell^*\pm 1}{2l2\ell^*}</math>

or

<math display="block">\frac{l\ell}{l\ell^*} = \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \text{etc.,} </math>

and

<math display="block">\frac{l\ell}{l\ell^*} = \frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \text{etc.,} </math>

which also appear as series for the filling factor in the [[fractional quantum Hall effect]].

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<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 ( l\ell + 1, 1, -\frac{k^2}{4} \right)} \;M {\left ( l\ell' + 1, 1, -\frac{k^2}{4} \right)} \;\mathcal J_0 {\left ( k \frac{r_{12}}{r_{B}} \right)}

</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( e^{-r^2\right)} J_0(kr)

= M\left( n+1, 1, -\frac{k^2}{4}\right). </math>

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{l\ell}{l\ell^*} = \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \text{etc.,} </math>

and

<math display="block">\frac{l\ell}{l\ell^*} = \frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \text{etc.,} </math>

appear as well in the case of charges spread by the wave function.

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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">\vecmathbf J_1{\left( \vecmathbf x \right)} = a_1 \vecmathbf v_1 \delta^3 {\left( \vecmathbf x - \vecmathbf x_1 \right)}</math>

with a comparable expression for <math> \vecmathbf J_2 </math>.

The Fourier transform of this current is

<math display="block">\vecmathbf J_1{\left( \vecmathbf k \right)} = a_1 \vecmathbf v_1 \exp\left( i \vecmathbf k \cdot \vecmathbf x_1 \right).</math>

The current can be decomposed into a transverse and a longitudinal part (see [[Helmholtz decomposition]]).

<math display="block">\vecmathbf J_1{\left( \vecmathbf k \right)} = a_1 \left[ 1 - \hat\mathbf k \hat\mathbf k \right ] \cdot \vecmathbf v_1 \exp\left( i \vecmathbf k \cdot \vecmathbf x_1 \right) + a_1 \left[ \hat\mathbf k \hat\mathbf k \right ] \cdot \vecmathbf v_1 \exp\left( i \vecmathbf k \cdot \vecmathbf x_1 \right).</math>

The hat indicates a [[unit vector]]. The last term disappears because

<math display="block">\vecmathbf k \cdot \vecmathbf J = -k_0 J^0 \rightarrowto 0,</math>

which results from charge conservation. Here <math>k_0 </math> vanishes because we are considering static forces.

With the current in this form the energy of interaction can be written

<math display="block"> E = a_1 a_2\int \frac{d^3\mathbf{k}}{(2 \pi )^3} \; \; D\left ( k \right )\mid_{k_0=0} \;

\vecmathbf v_1 \cdot \left[ 1 - \hat\mathbf k \hat\mathbf k \right ] \cdot \vecmathbf v_2 \; \exp\left ( i \vecmathbf k \cdot \left ( \mathbf x_1 - \mathbf x_2 \right ) \right ) .</math>

a_1 a_2\int \frac{d^3k}{(2 \pi )^3} \; \; D\left ( k \right )\mid_{k_0=0} \;

\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 ) .</math>

The propagator equation for the Proca Lagrangian is

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The [[spacelike]] solution is

<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; -\frac{1}{\vec k^2 + m^2},</math>

which yields

<math display="block"> E =

- a_1 a_2 \int \frac{d^3k3\mathbf{k}}{(2 \pi )^3} \; \;

\frac{\vecmathbf v_1 \cdot \left[ 1 - \hat\mathbf k \hat\mathbf k \right ] \cdot \vecmathbf v_2}{\vec k^2 + m^2} \; \exp\left ( i \vecmathbf k \cdot \left (\mathbf x_1 - \mathbf x_2 \right ) \right ),</math>

whichwhere <math display="inline">k = |\mathbf k|</math>. The integral evaluates to (see {{slink|Common integrals in quantum field theory#Transverse potential with mass}})

<math display="block"> E =

- \frac{1}{2} \frac{a_1 a_2}{4 \pi r} e^{ - m r} \left\{

\frac{2}{\left( mr \right)^2} \left( e^{mr} -1 \right) - \frac{2}{mr} \right \}

\vecmathbf v_1 \cdot \left[ 1 + {\hat\mathbf r} {\hat\mathbf r}\right]\cdot \vecmathbf v_2

</math>

which reduces to

<math display="block"> E = - \frac{1}{2} \frac{a_1 a_2}{4 \pi r} \vecmathbf v_1 \cdot \left[ 1 + {\hat\mathbf r} {\hat\mathbf r}\right] \cdot \vecmathbf v_2 </math>

in the limit of small {{mvar|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.

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In a plasma, the [[dispersion relation]] for an [[electromagnetic wave]] is<ref name="Chen"/>{{rp|pp=100–103}} (<math>c = 1</math>)

<math display="block">k_0^2 = \omega_p^2 +\vec k^2,</math>

which implies

<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; -\frac{1}{\vec k^2 + \omega_p^2}.</math>

Here <math>\omega_p</math> is the [[plasma frequency]]. The interaction energy is therefore

<math display="block"> E =

- \frac{1}{2} \frac{a_1 a_2}{4 \pi r}

\vecmathbf v_1 \cdot \left[ 1 + {\hat\mathbf r} {\hat\mathbf r}\right]\cdot \vecmathbf v_2

\; e^{ - \omega_p r } \left\{

\frac{2}{\left( \omega_p r \right)^2} \left( e^{\omega_p r} -1 \right) - \frac{2}{\omega_p r} \right \}.</math>

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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">\vecmathbf J_1( \vecmathbf x) = a_1 v_1 \frac{1}{2 \pi r L_B} \; \delta^ 2 {\left( r - r_{B1} \right)}

\left( \hat\mathbf b \times \hat\mathbf r \right)</math>

where

<math display="block"> r_{B1} = \frac{\sqrt{4 \pi}m_1 v_1}{a_1 B}</math>

and <math>\hat\mathbf b</math> is the unit vector in the direction of the magnetic field. Here <math>L_B</math> 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

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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{k_0^2 - \omega_p^2}{k_0^2- \omega_H^2} =0,</math>

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>

where

<math display="block">k_X \equiv \frac{\omega_p^2}{\omega_H}</math>

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The interaction energy becomes, for like currents,

<math display="block"> E =

- \left( \frac{a^2}{2 \pi L_B}\right) v^2\, \int_0^{\infty} \frac{k\;dk}{\vec k^2 + k_X^2}

\mathcal J_1^2 \left ( kr_{B} \right) \mathcal J_0 \left ( kr_{12} \right)</math>

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In the limit that the distance between current loops is small,

<math display="block"> E = - E_0 \; I_1 {\left( \mu \right)} K_1 {\left( \mu \right)}</math>

where

<math display="block"> E_0 = \left( \frac{a^2}{2 \pi L_B}\right) v^2</math>

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For small {{math|''mr''}} the integral becomes

<math display="block">

I_1 {\left( mr \right)} K_1 {\left( mr \right)}

\to

\rightarrow

\frac{1}{2}\left[ 1- \frac{1}{8}\left( mr \right)^2 \right] . </math>

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===Gravitation===

A gravitational disturbance is generated by the [[stress–energy tensor]] <math> T^{\mu \nu} </math>; consequently, the Lagrangian for the gravitational field is [[Spin (physics)|spin]]-2. If the disturbances are at rest, then the only component of the stress–energy tensor that persists is the <math> 00 </math> 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

<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; - \frac{4}{3} \frac{1}{\vec k^2 + m^2}</math>

and

<math display="block">E = -\frac{4}{3}\frac{a_1 a_2}{4 \pi r} \exp \left ( -m r \right ),</math>