Editing Noncommutative quantum field theory

{{technical|date=May 2023}}

{{Short description|Quantum field theory using noncommutative mathematics}}
{{redirect|Noncommutative field|mathematical objects known as "non-commutative fields"|Division ring}}
In [[mathematical physics]], '''noncommutative quantum field theory''' (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the [[spacetime]] of [[quantum field theory]] that is an outgrowth of [[noncommutative geometry]] and [[index theory]] in which the coordinate functions<ref>It is possible to have a noncommuting time coordinate as in the paper by Doplicher, Fredenhagen and Roberts mentioned below, but this causes many problems such as the violation of [[unitarity]] of the [[S-matrix]]. Hence most research is restricted to so-called "space-space" noncommutativity. There have been attempts to avoid these problems by redefining the [[perturbation theory (quantum mechanics)|perturbation theory]]. However, [[string theory]] derivations of noncommutative coordinates excludes time-space noncommutativity.</ref> are [[noncommutative]]. One commonly studied version of such theories has the "canonical" commutation relation:
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:<math>
[x^{\mu}, x^{\nu}]=i \theta^{\mu \nu} \,\!
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where <math>x^{\mu}</math> and <math>x^{\nu}</math> are the hermitian generators of a noncommutative <math>C^*</math>-algebra of "functions on spacetime". That means that (with any given set of axes), it is impossible to accurately measure the position of a particle with respect to more than one axis. In fact, this leads to an uncertainty relation for the coordinates analogous to the [[Heisenberg uncertainty principle]].

Various lower limits have been claimed for the noncommutative scale, (i.e. how accurately positions can be measured) but there is currently no experimental evidence in favour of such a theory or grounds for ruling them out.

One of the novel features of noncommutative field theories is the [[UV/IR mixing]]<ref>See, for example, 
Shiraz Minwalla, Mark Van Raamsdonk, Nathan Seiberg (2000) "[https://arxiv.org/abs/hep-th/9912072 Noncommutative Perturbative Dynamics]," ''Journal of High Energy Physics'', and Alec Matusis, [[Leonard Susskind]], Nicolaos Toumbas (2000) "[https://arxiv.org/abs/hep-th/0002075 The IR/UV Connection in the Non-Commutative Gauge Theories]," ''Journal of High Energy Physics''.</ref> phenomenon in which the physics at high energies affects the physics at low energies which does not occur in quantum field theories in which the coordinates commute.

Other features include violation of [[Lorentz invariance]] due to the preferred direction of noncommutativity. [[Relativistic invariance]] can however be retained in the sense of twisted [[Poincaré invariance]] of the theory.<ref>M. Chaichian, P. Prešnajder, A. Tureanu (2005)
"[https://arxiv.org/abs/hep-th/0409096 New concept of relativistic invariance in NC space-time: twisted Poincaré symmetry and its implications]," ''Physical Review Letters'' 94: .</ref> The [[causality conditions|causality condition]] is modified from that of the commutative theories.

==History and motivation==

[[Werner Heisenberg|Heisenberg]] was the first to suggest extending noncommutativity to the coordinates as a possible way of removing the infinite quantities appearing in field theories before the [[renormalization]] procedure was developed and had gained acceptance. The first paper on the subject was published in 1947 by [[Hartland Snyder]]. The success of the renormalization method resulted in little attention being paid to the subject for some time. In the 1980s, mathematicians, most notably [[Alain Connes]], developed [[noncommutative geometry]]. Among other things, this work generalized the notion of [[differential structure]] to a noncommutative setting. This led to an [[operator algebra]]ic description of noncommutative [[space-time]]s, with the problem that it classically corresponds to a manifold with positively defined [[metric tensor]], so that there is no description of (noncommutative) causality in this approach. However it also led to the development of a [[Yang–Mills theory]] on a noncommutative [[torus]].

The particle physics community became interested in the noncommutative approach because of a paper by [[Nathan Seiberg]] and [[Edward Witten]].<ref>Seiberg, N. and E. Witten (1999) "[https://arxiv.org/abs/hep-th/9908142 String Theory and Noncommutative Geometry]," ''Journal of High Energy Physics'' .</ref> They argued in the context of [[string theory]] that the coordinate functions of the endpoints of open strings constrained to a [[D-brane]] in the presence of a constant Neveu–Schwarz B-field—equivalent to a constant [[magnetic field]] on the brane—would satisfy the noncommutative algebra set out above. The implication is that a quantum field theory on noncommutative spacetime can be interpreted as a low energy limit of the theory of open strings.

Two papers, one by [[Sergio Doplicher]], [[Klaus Fredenhagen]] and John Roberts<ref>Sergio Doplicher, Klaus Fredenhagen, John E. Roberts (1995) "[https://arxiv.org/abs/hep-th/0303037 The quantum structure of spacetime at the Planck scale and quantum fields]," ''Commun. Math. Phys''. 172: 187-220.</ref> 
and the other by D. V. Ahluwalia,<ref>D. V. Ahluwalia (1993) "[https://arxiv.org/abs/gr-qc/9308007 Quantum Measurement, Gravitation, and Locality]," ``Phys. Lett. B339:301-303,1994. A look at preprint dates shows that this work takes priority over Doplicher et al. publication by eight months</ref>
set out another motivation for the possible noncommutativity of space-time. 
The arguments go as follows: According to [[general relativity]], when the energy density grows sufficiently large, a [[black hole]] is formed. On the other hand, according to the Heisenberg [[uncertainty principle]], a measurement of a space-time separation causes an uncertainty in momentum inversely proportional to the extent of the separation. Thus energy whose scale corresponds to the uncertainty in momentum is localized in the system within a region corresponding to the uncertainty in position. When the separation is small enough, the [[Schwarzschild radius]] of the system is reached and a [[black hole]] is formed, which prevents any information from escaping the system. Thus there is a lower bound for the measurement of length. A sufficient condition for preventing gravitational collapse can be expressed as an uncertainty relation for the coordinates. This relation can in turn be derived from a [[Commutativity|commutation]] relation for the coordinates.

It is worth stressing that, differently from other approaches, in particular those relying upon Connes' ideas, here the noncommutative spacetime is a proper spacetime, i.e. it extends the idea of a four-dimensional [[pseudo-Riemannian manifold]].   On the other hand, differently from Connes' noncommutative geometry, the proposed model turns out to be coordinate-dependent from scratch.
In Doplicher Fredenhagen Roberts' paper noncommutativity of coordinates concerns all four spacetime coordinates and not only spatial ones.

==See also==
*[[Moyal product]]
*[[Noncommutative geometry]]
*[[Noncommutative standard model]]
*[[Wigner–Weyl transform]]

== Footnotes ==
{{reflist}}

==Further reading==
*{{cite book | last=Grensing | first=Gerhard | year=2013 | title=Structural Aspects of Quantum Field Theory and Noncommutative Geometry | publisher=World Scientific | doi=10.1142/8771 | isbn=978-981-4472-69-2 }}
* M. R. Douglas and N. A. Nekrasov,  (2001). ''[https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.73.977 Noncommutative field theory]''. Rev. Mod. Phys., 73(4), 977.
* Richard J. Szabo (2003) "[https://arxiv.org/abs/hep-th/0109162 Quantum Field Theory on Noncommutative Spaces]," ''Physics Reports'' 378: 207-99. An expository article on noncommutative quantum field theories.
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=2&index1=-173391 Noncommutative quantum field theory, see statistics] on arxiv.org
* Valter Moretti (2003), "[https://arxiv.org/abs/gr-qc/0203095 Aspects of noncommutative  Lorentzian  geometry for globally hyperbolic spacetimes]," Rev. Math. Phys. 15: 1171-1218. An expository paper (also) on the difficulties to extend non-commutative geometry to the Lorentzian case describing causality
{{Quantum field theories}}

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[[Category:Noncommutative geometry]]
[[Category:Quantum field theory]]
[[Category:Mathematical quantization]]