nLab force

Idea

Physical bodies interact. The effects of binary interactions between point particles are known to be summarizable in vector quantities called forces applied to any body in an interaction; namely the forces on one body from each of several other bodies add up as vectors and such vector sum of forces on the body is proportional to the acceleration of the body (hence Newton's second law $F = m a$ is not a definition of force but a real law).

Thus force is a vector-like quantity which is a manifestation of the interaction between bodies. If body $A$ acts with force $F$ on body $B$ then $B$ acts on $A$ with the force $-F$ of the opposite vector value. These two forces do not cancel as they act on different bodies and at different points in space, though they are along the same line.

Sometimes one abstracts the forces on a system of bodies from the background by “potential energy” of the particles. Then the background acts on each particle with a force equal to the negative gradient of the potential energy.

In quantum field theory the forces appear mediated by particles which get exchanged between the bodies in interaction. For example, the strong nuclear force is mediated by gluons. There are 4 known fundamental forces in nature and all others are derived from them: the electromagnetic, weak, strong and gravitational force; and the first three are unified in the standard model of particle physics.

In nuclei there are also effective forces which are not of vector but of tensorial nature, and effective forces involving more than two bodies. But such quantum systems are far from classical mechanical systems.

Related concepts

• gauge field

• Lorentz force

• gravity,

• electroweak field, weak nuclear force, strong nuclear force

• matter

• entropic force

References

General

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Classification of long-range forces

Classification of possible long-range forces, hence of scattering processes of massless fields, by classification of suitably factorizing and decaying Poincaré-invariant S-matrices depending on particle spin, leading to uniqueness statements about Maxwell/photon-, Yang-Mills/gluon-, gravity/graviton- and supergravity/gravitino-interactions:

• Steven Weinberg, Feynman Rules for Any Spin. 2. Massless Particles, Phys. Rev. 134 (1964) B882 (doi:10.1103/PhysRev.134.B882)

• Steven Weinberg, Photons and Gravitons in $S$-Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass, Phys. Rev. 135 (1964) B1049 (doi:10.1103/PhysRev.135.B1049)

• Steven Weinberg, Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations,” Phys. Rev. 138 (1965) B988 (doi:10.1103/PhysRev.138.B988)

• Paolo Benincasa, Freddy Cachazo, Consistency Conditions on the S-Matrix of Massless Particles (arXiv:0705.4305)

• David A. McGady, Laurentiu Rodina, Higher-spin massless S-matrices in four-dimensions, Phys. Rev. D 90, 084048 (2014) (arXiv:1311.2938, doi:10.1103/PhysRevD.90.084048)

Review:

• Claus Kiefer, section 2.1.3 of: Quantum Gravity, Oxford University Press 2007 (doi:10.1093/acprof:oso/9780199585205.001.0001, cds:1509512)

• Daniel Baumann, What long-range forces are allowed?, 2019 (pdf)