Editing Quantum field theory (section)

===Theoretical background===
[[File:Magnet0873.png|thumb|200px|[[Magnetic field lines]] visualized using [[iron filings]]. When a piece of paper is sprinkled with iron filings and placed above a bar magnet, the filings align according to the direction of the magnetic field, forming arcs allowing viewers to clearly see the poles of the magnet and to see the magnetic field generated.]]

Quantum field theory results from the combination of [[classical field theory]], [[quantum mechanics]], and [[special relativity]].<ref name="peskin"/>{{rp|xi}} A brief overview of these theoretical precursors follows.

The earliest successful classical field theory is one that emerged from [[Newton's law of universal gravitation]], despite the complete absence of the concept of fields from his 1687 treatise ''[[Philosophiæ Naturalis Principia Mathematica]]''. The force of gravity as described by Isaac Newton is an "[[action at a distance]]"—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with [[Richard Bentley]], however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact".<ref name=Hobson/>{{rp|4}} It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a [[vector (mathematics and physics)|vector]] in the case of [[gravitational field]]) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a mathematical trick.<ref name="weinberg">{{cite journal |last=Weinberg |first=Steven |author-link=Steven Weinberg |date=1977 |title=The Search for Unity: Notes for a History of Quantum Field Theory |journal=Daedalus |volume=106 |issue=4 |pages=17–35 |jstor=20024506 }}</ref>{{rp|18}}

Fields began to take on an existence of their own with the development of [[electromagnetism]] in the 19th century. [[Michael Faraday]] coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day.<ref name=Hobson>{{cite journal
 | last =Hobson
 | first =Art
 | title =There are no particles, there are only fields 
 | journal =[[American Journal of Physics]]
 | volume =81
 | issue =211
 | pages =211–223
 | year =2013
 | doi =10.1119/1.4789885
 | arxiv =1204.4616
 | bibcode =2013AmJPh..81..211H
 | s2cid =18254182
 }}</ref><ref name="Heilbron2003">{{Cite book |url=https://archive.org/details/oxfordcompaniont0000unse_s7n3 |title=The Oxford companion to the history of modern science |date=2003 |publisher=[[Oxford University Press]] |isbn=978-0-19-511229-0 |editor-last=Heilbron |editor-first=J. L. |editor-link=John L. Heilbron |location=Oxford; New York}}</ref>{{rp|301}}<ref name="Thomson1893">{{Cite book |last1=Thomson |first1=Joseph John |author-link1=Joseph John Thomson |url=https://archive.org/details/notesonrecentres00thom |title=Notes on recent researches in electricity and magnetism, intended as a sequel to Professor Clerk-Maxwell's 'Treatise on Electricity and Magnetism' |last2=Maxwell |first2=James Clerk |publisher=[[Clarendon Press]] |year=1893}}</ref>{{rp|2}}

The theory of [[classical electromagnetism]] was completed in 1864 with [[Maxwell's equation]]s, which described the relationship between the [[electric field]], the [[magnetic field]], [[electric current]], and [[electric charge]]. Maxwell's equations implied the existence of [[electromagnetic waves]], a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the [[speed of light]]. Action-at-a-distance was thus conclusively refuted.<ref name=Hobson/>{{rp|19}}

Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in [[emission spectrum|atomic spectra]], nor for the distribution of [[blackbody radiation]] in different wavelengths.<ref name="weisskopf">{{cite journal |last=Weisskopf |first=Victor |author-link=Victor Weisskopf |date=November 1981 |title=The development of field theory in the last 50 years |journal=[[Physics Today]] |volume=34 |issue=11 |pages=69–85 |doi=10.1063/1.2914365 |bibcode=1981PhT....34k..69W }}</ref> [[Max Planck]]'s study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit [[electromagnetic radiation]], as tiny [[oscillator]]s with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as [[quantum harmonic oscillator]]s. This process of restricting energies to discrete values is called quantization.<ref name="Heisenberg1999">{{Cite book |last=Heisenberg |first=Werner |author-link=Werner Heisenberg |url=https://archive.org/details/physics-and-philosophy-the-revolution-in-modern-scirnce-werner-heisenberg-f.-s.-c.-northrop |title=Physics and philosophy: the revolution in modern science |publisher=[[Prometheus Books]] |year=1999 |isbn=978-1-57392-694-2 |series=Great minds series |location=Amherst, N.Y}}</ref>{{rp|Ch.2}} Building on this idea, [[Albert Einstein]] proposed in 1905 an explanation for the [[photoelectric effect]], that light is composed of individual packets of energy called [[photon]]s (the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles.<ref name="weisskopf" />

In 1913, [[Niels Bohr]] introduced the [[Bohr model]] of atomic structure, wherein [[electrons]] within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In 1924, [[Louis de Broglie]] proposed the hypothesis of [[wave–particle duality]], that microscopic particles exhibit both wave-like and particle-like properties under different circumstances.<ref name="weisskopf" /> Uniting these scattered ideas, a coherent discipline, [[quantum mechanics]], was formulated between 1925 and 1926, with important contributions from [[Max Planck]], [[Louis de Broglie]], [[Werner Heisenberg]], [[Max Born]], [[Erwin Schrödinger]], [[Paul Dirac]], and [[Wolfgang Pauli]].{{r|weinberg|page1=22–23}}

In the same year as his paper on the photoelectric effect, Einstein published his theory of [[special relativity]], built on Maxwell's electromagnetism. New rules, called [[Lorentz transformations]], were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred.{{r|weinberg|page1=19}} It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.

Two difficulties remained. Observationally, the [[Schrödinger equation]] underlying quantum mechanics could explain the [[stimulated emission]] of radiation from atoms, where an electron emits a new photon under the action of an external electromagnetic field, but it was unable to explain [[spontaneous emission]], where an electron spontaneously decreases in energy and emits a photon even without the action of an external electromagnetic field. Theoretically, the Schrödinger equation could not describe photons and was inconsistent with the principles of special relativity—it treats time as an ordinary number while promoting spatial coordinates to [[linear operator]]s.<ref name="weisskopf" />