Editing Mass (section)

=== Tachyonic particles and imaginary (complex) mass ===
{{main|Tachyonic field|Tachyon#Mass}}

A [[tachyonic field]], or simply [[tachyon]], is a [[quantum field]] with an [[imaginary number|imaginary]] mass.<ref name=Randall/> Although [[tachyon]]s ([[particle]]s that move [[faster-than-light|faster than light]]) are a purely hypothetical concept not generally believed to exist,<ref name=Randall>Lisa Randall, ''Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions'', p.286: "People initially thought of tachyons as particles travelling faster than the speed of light...But we now know that a tachyon indicates an instability in a theory that contains it.  Regrettably for science fiction fans, tachyons are not real physical particles that appear in nature."</ref><ref name=Tipler>{{cite book| last1 = Tipler| first1 = Paul A.|last2 = Llewellyn| first2 = Ralph A.| title = Modern Physics| url = https://archive.org/details/modernphysicsfif00paul| url-access = limited| publisher = W.H. Freeman & Co.| date = 2008| edition = 5th| location = New York | isbn = 978-0-7167-7550-8| page = [https://archive.org/details/modernphysicsfif00paul/page/n71 54]|quote = ... so existence of particles v > c ... Called tachyons ... would present relativity with serious ... problems of infinite creation energies and causality paradoxes.}}</ref> [[field (physics)|fields]] with imaginary mass have come to play an important [[Higgs boson|role]] in modern physics<ref name=Kutasov>{{cite journal | author = Kutasov, David | author2 = Marino, Marcos | author3 = Moore, Gregory W. | name-list-style = amp | title= Some exact results on tachyon condensation in string field theory | journal=   Journal of High Energy Physics| volume= 2000 | issue = 10 | page= 045 | date= 2000 | doi=10.1088/1126-6708/2000/10/045|arxiv = hep-th/0009148 |bibcode = 2000JHEP...10..045K | s2cid = 15664546 }}</ref><ref name=Sen>{{Cite journal | doi=10.1088/1126-6708/2002/04/048| arxiv=hep-th/0203211| bibcode=2002JHEP...04..048S| title=Rolling Tachyon| year=2002| last1=Sen| first1=Ashoke| journal=Journal of High Energy Physics| volume=2002| issue=4| pages=048| s2cid=12023565}}</ref><ref name=Gibbons>{{cite journal | last1 = Gibbons | first1 = G.W. | year = 2002 | title = Cosmological evolution of the rolling tachyon | journal = Phys. Lett. B | volume = 537 | issue = 1–2| pages = 1–4 | doi=10.1016/s0370-2693(02)01881-6|arxiv = hep-th/0204008 |bibcode = 2002PhLB..537....1G | s2cid = 119487619 }}</ref> and are discussed in popular books on physics.<ref name=Randall/><ref name=Greene>{{cite book |title=The Elegant Universe |title-link=The Elegant Universe |last=Greene |first=Brian |author-link=Brian Greene |publisher=Vintage Books|date=2000}}</ref> Under no circumstances do any excitations ever propagate faster than light in such theories—the presence or absence of a tachyonic mass has no effect whatsoever on the maximum velocity of signals (there is no violation of [[causality]]).<ref name="susskind">{{cite journal |volume = 182 |journal = Phys. Rev. |last1 = Aharonov |first1 = Y. |last2 = Komar |first2 = A. |last3= Susskind |first3 = L. |title = Superluminal Behavior, Causality, and Instability |date = 1969 |doi = 10.1103/PhysRev.182.1400|issue = 5|pages = 1400–1403|bibcode = 1969PhRv..182.1400A }}</ref> While the ''field'' may have imaginary mass, any physical particles do not; the "imaginary mass" shows that the system becomes unstable, and sheds the instability by undergoing a type of [[phase transition]] called [[tachyon condensation]] (closely related to second order phase transitions) that results in [[symmetry breaking]] in [[Standard Model|current models]] of [[particle physics]].

The term "[[tachyon]]" was coined by [[Gerald Feinberg]] in a 1967 paper,<ref name="feinberg67"/> but it was soon realized that Feinberg's model in fact did not allow for [[superluminal]] speeds.<ref name="susskind"/> Instead, the imaginary mass creates an instability in the configuration:- any configuration in which one or more field excitations are tachyonic will spontaneously decay, and the resulting configuration contains no physical tachyons.  This process is known as tachyon condensation.  Well known examples include the [[Higgs mechanism|condensation]] of the [[Higgs boson]] in [[particle physics]], and [[ferromagnetism]] in [[condensed matter physics]].

Although the notion of a tachyonic [[imaginary number|imaginary]] mass might seem troubling because there is no classical interpretation of an imaginary mass, the mass is not quantized.  Rather, the [[scalar field]] is; even for tachyonic [[quantum field theory|quantum fields]], the [[field operator]]s at [[Minkowski space|spacelike]] separated points still [[Canonical commutation relation|commute (or anticommute)]], thus preserving causality. Therefore, information still does not propagate faster than light,<ref name="feinberg67">{{cite journal | first = Gerald | last = Feinberg | author-link = Gerald Feinberg | title = Possibility of Faster-Than-Light Particles | journal = Physical Review | volume = 159 | year = 1967 | pages = 1089–1105 | doi = 10.1103/PhysRev.159.1089|bibcode = 1967PhRv..159.1089F | issue = 5 }}</ref> and solutions grow exponentially, but not superluminally (there is no violation of [[causality]]). [[Tachyon condensation]] drives a physical system that has reached a local limit and might naively be expected to produce physical tachyons, to an alternate stable state where no physical tachyons exist. Once the tachyonic field reaches the minimum of the potential, its quanta are not tachyons any more but rather are ordinary particles with a positive mass-squared.<ref name = "Peskin" />

This is a special case of the general rule, where unstable massive particles are formally described as having a [[complex number|complex]] mass, with the real part being their mass in the usual sense, and the imaginary part being the [[Particle decay#Decay rate|decay rate]] in [[natural units]].<ref name="Peskin">
{{cite book
 |last1=Peskin |first1=M.E.
 |last2=Schroeder |first2=D.V.
 |date=1995
 |title=An Introduction to Quantum Field Theory
 |publisher=Perseus Books
}}</ref> However, in [[quantum field theory]], a particle (a "one-particle state") is roughly defined as a state which is constant over time; i.e., an [[eigenvalue]] of the [[Hamiltonian (quantum mechanics)|Hamiltonian]]. An [[Particle decay|unstable particle]] is a state which is only approximately constant over time; If it exists long enough to be measured, it can be formally described as having a complex mass, with the real part of the mass greater than its imaginary part. If both parts are of the same magnitude, this is interpreted as a [[resonance]] appearing in a scattering process rather than a particle, as it is considered not to exist long enough to be measured independently of the scattering process. In the case of a tachyon, the real part of the mass is zero, and hence no concept of a particle can be attributed to it.

In a [[Lorentz invariant]] theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "[[bradyon]]s" in discussions of tachyons) must also apply to tachyons. In particular the [[energy–momentum relation]]:
: <math>E^2 = p^2c^2 + m^2c^4 \;</math>
(where '''p''' is the relativistic [[momentum]] of the bradyon and '''m''' is its [[rest mass]]) should still apply, along with the formula for the total energy of a particle:
: <math>E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}.</math>
This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy.
When ''v'' is larger than ''c'', the denominator in the equation for the energy is [[imaginary number|"imaginary"]], as the value under the [[square root|radical]] is negative. Because the total [[energy]] must be [[real number|real]], the numerator must ''also'' be imaginary:  i.e. the [[rest mass]] '''m''' must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.