Static forces and virtual-particle exchange: Difference between revisions

Revision as of 11:05, 29 October 2014

Revision as of 15:46, 11 February 2015

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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.

As with any physical theory, 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>[http://www.hep.phy.cam.ac.uk/theory/research/hadronic.html]</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. 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.{{fact|date=October 2014}}

<!-- Text below hidden for the time being (vs. being deleted) because it seems valuable, but needs rewriting in an encyclopedic form. Admonishment to look critically and avoid fallacy is unwikipedic. The CERN experiments are cited, but then criticized in a way appearing to be original research. Please reinstate with improvements. -->

<!-- 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. One should be careful not to make the logical error known as [[Reification (fallacy)|reification]], which confuses concept and reality. -->

Line 75:

The probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in the [[path integral formulation]] by

:<math> 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{S} [\varphi] \right )\;

= \exp\left ( i W \right )

</math>

Line 99:

The path integral often can be converted to the form

:<math> Z=

\int \exp\left[ i \int d^4x \left ( \frac 1 2 \varphi \hat O \varphi + J \varphi \right) \right ] D\varphi

Line 109:

:<math> Z \propto

\exp\left( i W\left ( J \right )\right)

</math>

Line 128:

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> J\left ( x \right )

= \left( J_1 +J_2,0,0,0 \right)

Line 159:

Here <math> D\left ( k \right ) </math> is the Fourier transform of

:<math> {1\over 2} \left [ D\left ( x-y \right ) + D\left ( y-x \right )\right ]

</math>.

Line 186:

Consider the [[Spin (physics)|spin]]-0 Lagrangian density<ref>Zee, pp. 21-29</ref>

:<math>

\mathcal{L} [\varphi (x)]

= {1\over 2} \left [ \left ( \partial \varphi \right )^2 -m^2 \varphi^2 \right ]

</math>.

The equation of motion for this Lagrangian is the [[Klein–Gordon equation]]

:<math>

\partial^2 \varphi + m^2 \varphi =0

</math>.

If we add a disturbance the probability amplitude becomes

:<math>

Z =

\int D\varphi \; \exp \left \{ i \int d^4x\; \left [ {1\over 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

:<math>

Z =

\int D\varphi \; \exp \left \{ i \int d^4x\; \left [ -{1\over 2}\varphi \left ( \partial^2 + m^2\right )\varphi + J\varphi \right ] \right \}

</math>.

With the amplitude in this form it can be seen that the propagator is the solution of

:<math>

-\left ( \partial^2 + m^2\right ) D\left ( x-y \right ) = \delta^4\left ( x-y \right )

</math>.

Line 290:

:<math>

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

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

,</math>

which implies

:<math>

\eta_{\mu \alpha} \left ( \partial^2 + m^2\right ) D^{\alpha \nu}\left ( x-y \right ) = \delta_{\mu }^{ \nu} \delta^4\left ( x-y \right )

</math>

Line 328:

<math>

E =

{a_1 a_2 \over 4 \pi r }

.</math>

|}

Line 448:

:<math>

E=

\left( { a_1\, a_2 \over 2 \pi L_B}\right) \int_0^{\infty} {{k\;dk \;} \over

k^2 + k_{Ds}^2 }

\mathcal J_0 \left ( kr_{12} \right)

= \left( { a_1\, a_2 \over 2 \pi L_B}\right) K_0 \left( k_{Ds} r_{12} \right)

</math>

|}

Line 469:

are [[Bessel function]]s and <math> r_{12}</math> 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#Integration of the cylindrical propagator with mass|Common integrals in quantum field theory]])

:<math>

\int_0^{2 \pi} {d\varphi \over 2 \pi} \exp\left( i p \cos\left( \varphi \right) \right)

=

Line 478:

:<math>

\int_0^{\infty} {{k\;dk \;} \over

k^2 + m^2 }

\mathcal J_0 \left ( kr \right)

= K_0 \left( m r \right)

.</math>

Line 505:

:<math>

r_{B1}

=

{\sqrt{4 \pi}m_1v_1\over a_1 B}

Line 522:

:<math>

v_{1}

=

\sqrt {2 \hbar \omega_c \over m_1}

Line 548:

is a [[Bessel function]] of the first kind. In obtaining the interaction energy we made use of the integral

:<math>

\int_0^{2 \pi} {d\varphi \over 2 \pi} \exp\left( i p \cos\left( \varphi \right) \right)

=

Line 566:

where <math>

-e\varphi

</math> is the [[potential energy]] of an electron in an [[electric potential]] and <math>

N_0

</math> and <math>

N

</math> are the number of particles in the [[electron gas]] in the absence of and in the presence of an electrostatic potential, respectively.

Line 581:

where <math>

A_M

</math> is the area of the material in the plane perpendicular to the magnetic field.

Line 601:

D\left( k \right) \mid_{k_0=k_B=0}

=

{1 \over

k^2 + k_B^2 }

Line 612:

:<math>

E=

\left( { a_1\, a_2 \over 2 \pi L_B}\right) \int_0^{\infty} {{k\;dk \;} \over

k^2 + k_B^2 }

\mathcal J_0 \left ( kr_{B1} \right) \mathcal J_0 \left ( kr_{B2} \right) \mathcal J_0 \left ( kr_{12} \right)

=

\left( { 2 e^2 \over L_B}\right) \int_0^{\infty} {{k\;dk \;} \over

k^2 + k_B^2 r_B^2 }

\mathcal J_0^2 \left ( k \right) \mathcal J_0 \left ( k{r_{12}\over r_B} \right)

Line 646:

:<math>

E=

\left( { 2 e^2 \over L_B}\right) \int_0^{\infty} {{k\;dk \;} \over

k^2 + k_B^2 r_{\mathit l}^2 }

\;\mathcal J_0 \left ( k \right) \;\mathcal J_0 \left ( \sqrt{{\mathit l^{\prime}}\over {\mathit l}} \;k \right) \;\mathcal J_0 \left ( k{r_{12}\over r_{\mathit l}} \right)

Line 661:

:<math>

r_{12}

=r_{\mathit l \mathit l^{\prime}}

= \sqrt{\mathit l + \mathit l^{\prime}}\;r_B

.</math>

Line 673:

:<math>

E=

\left( { 2 e^2 \over L_B}\right) \int_0^{\infty} {{k\;dk \;} \over

k^2 + k_B^2 r_{\mathit l \mathit l^{\prime}}^2 }

\;\mathcal J_0 \left ( \cos \theta \; k \right) \;\mathcal J_0 \left ( \sin \theta \;k \right) \;\mathcal J_0 \left ( k{r_{12}\over r_{\mathit l \mathit l^{\prime}}} \right)

Line 721:

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<ref>Ezewa, p. 189</ref>

: <math>

{1 \over \pi r_B^2 L_B}

{1 \over n!}

\left( {r \over r_B} \right)^{2 \mathit l}

Line 734:

:<math>

E=

\left( { 2 e^2 \over L_B}\right) \int_0^{\infty} {{k\;dk \;} \over

k^2 + k_B^2 r_{B}^2 }

\; M \left ( \mathit l + 1, 1, -{k^2 \over 4} \right) \;M \left ( \mathit l^{\prime} + 1, 1, -{k^2 \over 4} \right) \;\mathcal J_0 \left ( k{r_{12}\over r_{B}} \right)

Line 743:

where <math>M</math> 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#Integration over a magnetic wave function|Common integrals in quantum field theory]])

:<math>

{2 \over n!}

\int_0^{\infty} { dr }\;r^{2n+1}\exp\left( -r^2\right) J_{0} \left( kr \right)

Line 804:

The propagator equation for the Proca Lagrangian is

:<math>

\eta_{\mu \alpha} \left ( \partial^2 + m^2\right ) D^{\alpha \nu}\left ( x-y \right ) = \delta_{\mu }^{ \nu} \delta^4\left ( x-y \right )

.</math>

Line 825:

:<math> E =

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

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

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

Line 835:

|

<math> E =

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

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

</math>

Line 862:

|

<math> E =

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

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

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

{2 \over \left( \omega_p r \right)^2 } \left( e^{\omega_p r} -1 \right) - {2\over \omega_p r} \right \}

.</math>

Line 883:

:<math>

r_{B1}

=

{\sqrt{4 \pi}m_1v_1\over a_1 B}

Line 889:

and <math>

\hat 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]].

Line 908:

is a [[Bessel function]] of the first kind. In obtaining the interaction energy we made use of the integrals

:<math>

\int_0^{2 \pi} {d\varphi \over 2 \pi} \exp\left( i p \cos\left( \varphi \right) \right)

=

Line 916:

and

:<math>

\int_0^{2 \pi} {d\varphi \over 2 \pi} \cos\left( \varphi \right) \exp\left( i p \cos\left( \varphi \right) \right)

=

Line 935:

D\left( k \right) \mid_{k_0=k_B=0}\;

= \;

-\left( {1\over \vec k^2 + k_X^2}\right)

</math>

Line 941:

:<math>

k_X \equiv {\omega_p^2 \over \omega_H}

</math>

Line 970:

<math>

E=

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

\mathcal J_1^2 \left ( kr_{B} \right) \mathcal J_0 \left ( kr_{12} \right)

</math>

Line 983:

<math>

E=

- E_0 \;

I_1 \left( \mu \right)K_1 \left( \mu \right)

</math>

Line 1,007:

We have made use of the integral (see [[Common integrals in quantum field theory#Integration of the cylindrical propagator with mass|Common integrals in quantum field theory]])

:<math>

\int_o^{\infty} {k\; dk \over k^2 +m^2} \mathcal J_1^2 \left( kr \right)

=

Line 1,015:

For small mr the integral becomes

:<math>

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

\rightarrow

Line 1,023:

For large mr the integral becomes

:<math>

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

\rightarrow

Line 1,035:

:<math>

\mu =

{\omega_p^2 r_B\over \omega_H c}

= \left( {2e^2r_B\over L_B \hbar c }\right) {\nu \over \sqrt{1+{\omega_p^2\over \omega_c^2}}}

= 2 \alpha \left( { r_B\over L_B }\right) \left({1 \over \sqrt{1+{\omega_p^2\over \omega_c^2}}}\right) \nu

Line 1,063:

:<math>

E_0=

{4\pi}{ e^2 \over L_B}{v^2\over c^2}

= {8\pi}{ e^2 \over L_B}\left( {\hbar \omega_c\over m c^2}\right)

</math>

Line 1,070:

:<math>

{1\over 2} m v^2

= \hbar \omega_c

.</math>