Static forces and virtual-particle exchange: Difference between revisions

Revision as of 19:12, 13 June 2023

Revision as of 00:09, 15 June 2023

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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]]. 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 (particle 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=High Energy Physics Group - Hadronic Physics |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. 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.{{citation needed|date=October 2014}}

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

Consider the [[Spin (particle physics)|spin]]-0 Lagrangian density<ref name="Zee"/>{{rp|pp=21–29}}

<math display="block">

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

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====The Coulomb potential in a vacuum====

Consider the [[Spin (particle physics)|spin]]-1 [[Proca action|Proca Lagrangian]] with a disturbance<ref name="Zee"/>{{rp|pp=30–31}}

<math display="block">\mathcal{L} [\varphi (x)] = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2} m^2 A_{\mu} A^{\mu} + A_{\mu} J^{\mu}</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 (particle 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