Force

== Revisions of the force concept ==
At the beginning of the 20th century, new physical ideas emerged to explain experimental results in astronomical and submicroscopic realms. As discussed below, relativity alters the definition of momentum and quantum mechanics reuses the concept of "force" in microscopic contexts where Newton's laws do not apply directly.

=== Special theory of relativity ===
{{Main | Relativistic mechanics#Force}}
In the [[special theory of relativity]], mass and [[energy]] are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases, so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's Second Law,
<math display="block">\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t},</math>
remains valid because it is a mathematical definition.<ref name=Cutnell/>{{rp|855–876}} But for momentum to be conserved at  relativistic relative velocity, <math>v</math>, momentum must be redefined as:
<math display="block"> \vec{p} = \frac{m_0\vec{v}}{\sqrt{1 - v^2/c^2}},</math>
where <math>m_0</math> is the [[rest mass]] and <math>c</math> the [[speed of light]].

The expression relating force and acceleration for a particle with constant non-zero [[rest mass]] <math>m</math> moving in the <math>x</math> direction at velocity <math>v</math> is:<ref name=French1972>{{Cite book |last=French |first=A. P. |title=Special Relativity |date=1972 |publisher=Chapman & Hall |isbn=978-0-17-771075-9 |edition=reprint |series=The MIT introductory physics series |location=London}}</ref>{{rp|216}}
<math display="block">\vec{F} = \left(\gamma^3 m a_x, \gamma m a_y, \gamma m a_z\right),</math>
where
<math display="block" qid=Q599404> \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.</math>
is called the [[Lorentz factor]]. The Lorentz factor increases steeply as the relative velocity approaches the speed of light.  Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach <math>c</math>.<ref name=French1972/>{{rp|26}}<ref name=FeynmanVol1/>{{rp|at=§15–8}}
If <math>v</math> is very small compared to <math>c</math>, then <math>\gamma</math> is very close to 1 and
<math display="block">F = m a</math>
is a close approximation. Even for use in relativity, one can restore the form of
<math display="block">F^\mu = mA^\mu </math>
through the use of [[four-vectors]]. This relation is correct in relativity when <math>F^\mu</math> is the [[four-force]], <math>m</math> is the [[invariant mass]], and <math>A^\mu</math> is the [[four-acceleration]].<ref>{{cite web
|first=John B.
|last=Wilson
|title=Four-Vectors (4-Vectors) of Special Relativity: A Study of Elegant Physics
|work=The Science Realm: John's Virtual Sci-Tech Universe
|url=http://SciRealm.com/4Vectors.html
|archive-url=https://web.archive.org/web/20090626152836/http://www.austininc.com/SciRealm/4Vectors.html
|archive-date=26 June 2009
|url-status=dead
|access-date=2008-01-04
}}</ref>

The [[general relativity|''general'' theory of relativity]] incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully [[#Gravitational|below]].

=== Quantum mechanics ===
{{main|Quantum mechanics|Pauli exclusion principle}}
[[Quantum mechanics]] is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object ''has'', one considers what result might ''appear'' when a [[Measurement in quantum mechanics|measurement]] of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result.<ref>{{cite journal|last=Mermin|first=N. David|author-link=N. David Mermin|year=1993|title=Hidden variables and the two theorems of John Bell|journal=[[Reviews of Modern Physics]]|volume=65|issue=3|pages=803&ndash;815|arxiv=1802.10119|bibcode=1993RvMP...65..803M|doi=10.1103/RevModPhys.65.803|s2cid=119546199 |quote=It is a fundamental quantum doctrine that a measurement does not, in general, reveal a pre-existing value of the measured property.}}</ref><ref>{{Cite journal|last1=Schaffer|first1=Kathryn|last2=Barreto Lemos|first2=Gabriela|date=24 May 2019|title=Obliterating Thingness: An Introduction to the "What" and the "So What" of Quantum Physics|journal=[[Foundations of Science]] |volume=26 |pages=7–26 |language=en|arxiv=1908.07936|doi=10.1007/s10699-019-09608-5|issn=1233-1821|s2cid=182656563}}</ref> The [[Expectation value (quantum mechanics)|expectation value]] for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.<ref>{{Cite journal|last1=Marshman|first1=Emily|last2=Singh|first2=Chandralekha|author-link2=Chandralekha Singh|date=2017-03-01|title=Investigating and improving student understanding of the probability distributions for measuring physical observables in quantum mechanics|journal=[[European Journal of Physics]]|volume=38|issue=2|pages=025705|doi=10.1088/1361-6404/aa57d1|bibcode=2017EJPh...38b5705M |s2cid=126311599 |issn=0143-0807|doi-access=free}}</ref>

In quantum mechanics, interactions are typically described in terms of energy rather than force. The [[Ehrenfest theorem]] provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the [[Born rule]] is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.<ref name="Cohen-Tannoudji">{{cite book|last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics |author-link1=Claude Cohen-Tannoudji |publisher=John Wiley & Sons |year=2005 |isbn=0-471-16433-X |translator-first1=Susan Reid |translator-last1=Hemley |translator-first2=Nicole |translator-last2=Ostrowsky |translator-first3=Dan |translator-last3=Ostrowsky |page=242}}</ref><ref>{{cite book|last=Peres|first=Asher|author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods|publisher=[[Kluwer]]|year=1993|isbn=0-7923-2549-4|oclc=28854083 |page=302}}</ref>

Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the [[uncertainty principle]] limits the minimum extent of an electron probability distribution<ref name=Lieb-RMP>{{Cite journal |last=Lieb |first=Elliott H. |date=1976-10-01 |title=The stability of matter |url=https://link.aps.org/doi/10.1103/RevModPhys.48.553 |journal=Reviews of Modern Physics |language=en |volume=48 |issue=4 |pages=553–569 |doi=10.1103/RevModPhys.48.553 |issn=0034-6861 |quote=the fact that if one tries to compress a wave function ''anywhere'' then the kinetic energy will increase. This principle was provided by Sobolev (1938)...}}</ref> and the [[Pauli exclusion principle]] prevents electrons from sharing the same probability distribution.<ref name=Lieb-Bulletin>{{Cite journal |last=Lieb |first=Elliott H. |date=1990 |title=The stability of matter: from atoms to stars |url=https://www.ams.org/bull/1990-22-01/S0273-0979-1990-15831-8/ |journal=Bulletin of the American Mathematical Society |language=en |volume=22 |issue=1 |pages=1–49 |doi=10.1090/S0273-0979-1990-15831-8 |issn=0273-0979 |quote= bulk matter is stable, and has a volume proportional to the number of particles, because of the Pauli exclusion principle for fermions (Le., the electrons). Effectively the electrons behave like a fluid with energy density <math>\rho^{5/3}</math>, and this limits the compression caused by the attractive electrostatic forces. |doi-access=free }}</ref> This gives rise to an emergent pressure known as [[electron degeneracy pressure|degeneracy pressure]]. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids [[Stability of matter|stability]].<ref>{{cite book|last=Griffiths|title=Introduction to Quantum Mechanics, Second Edition|year=2005|publisher=[[Prentice Hall]]|location=London, UK|isbn=0131244051|pages=221–223}}</ref>

=== Quantum field theory ===
{{main|Quantum field theory}}
[[File:Beta Negative Decay.svg|thumb|Feynman diagram for the decay of a neutron into a proton. The [[W boson]] is between two vertices indicating a repulsion.]]
In modern [[particle physics]], forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying [[gauge boson]]s. With the development of [[quantum field theory]] and [[general relativity]], it was realized that force is a redundant concept arising from [[conservation of momentum]] ([[4-momentum]] in relativity and momentum of [[virtual particle]]s in [[quantum electrodynamics]]). The conservation of momentum can be directly derived from the homogeneity or [[Symmetry in physics|symmetry]] of [[space]] and so is usually considered more fundamental than the concept of a force. Thus the currently known [[fundamental forces]] are considered more accurately to be "[[fundamental interactions]]".<ref name="final theory">{{cite book |last=Weinberg |first=S. |year=1994 |title=Dreams of a Final Theory |publisher=Vintage Books |isbn=978-0-679-74408-5}}</ref>{{rp|199–128}}

While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of [[Feynman diagram]]s. In a Feynman diagram, each matter particle is represented as a straight line (see [[world line]]) traveling through time, which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction [[Vertex (graph theory)|vertices]], and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines and, in the case of virtual particle exchange, are absorbed at an adjacent vertex.<ref name=Shifman>{{cite book |first=Mikhail |last=Shifman |title=ITEP lectures on particle physics and field theory |publisher=World Scientific |year=1999 |isbn=978-981-02-2639-8}}</ref> The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of [[fundamental interaction]]s but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a [[neutron]] [[beta decay|decays]] into an [[electron]], [[proton]], and [[neutrino|antineutrino]], an interaction mediated by the same gauge boson that is responsible for the [[weak nuclear force]].<ref name="Shifman"/>