Noncommutative quantum field theory: Difference between revisions

Browse history interactively ← Previous edit Next edit →Content deleted Content addedVisual WikitextInline

Revision as of 04:45, 25 July 2009

In physics, noncommutative quantum field theory (or quantum field theory on noncommutative space-time) is a branch of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the spatial coordinates do not commute. One commonly studied version of such theories has the "canonical" commutation relation:

=i\theta ^{\mu \nu }\,\!

which 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 theory or grounds for ruling them out.

One of the novel features of noncommutative field theories is the UV/IR mixing 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. The causality condition is modified from that of the commutative theories.

History and motivation

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 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 algebraic description of noncommutative space-times, and 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. 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-Schwartz 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.

A paper by Sergio Doplicher, Klaus Fredenhagen and John Roberts set out another motivation for the possible noncommutativity of space-time. Their arguments goes 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 commutation relation for the coordinates.

References

It is possible to have a noncommuting time coordinate, 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. However, string theory derivations of noncommutative coordinates excludes time-space noncommutativity.
See, for example, Shiraz Minwalla, Mark Van Raamsdonk, Nathan Seiberg (2000) "Noncommutative Perturbative Dynamics," Journal of High Energy Physics, and Alec Matusis, Leonard Susskind, Nicolaos Toumbas (2000) "The IR/UV Connection in the Non-Commutative Gauge Theories," Journal of High Energy Physics.
M. Chaichian, P. Prešnajder, A. Tureanu (2005) "New concept of relativistic invariance in NC space-time: twisted Poincaré symmetry and its implications," Phys. Rev. Letters 94: .
Seiberg, N. and E. Witten (1999) "String Theory and Noncommutative Geometry," Journal of High Energy Physics .
Sergio Doplicher, Klaus Fredenhagen, John E. Roberts (1995) "The quantum structure of spacetime at the Planck scale and quantum fields," Commun. Math. Phys. 172: 187-220.

M.R. Douglas and N. A. Nekrasov (2001) "Noncommutative field theory," Rev. Mod. Phys. 73: 977 - 1029.
Szabo, R. J. (2003) "Quantum Field Theory on Noncommutative Spaces," Physics Reports 378: 207-99. An expository article on noncommutative quantum field theories.

Also see statistics for Noncommutative quantum field theory on arxiv.org

Quantum field theories

Theories

Models

Regular	Born–Infeld Euler–Heisenberg Ginzburg–Landau Non-linear sigma Proca Quantum electrodynamics Quantum chromodynamics Quartic interaction Scalar electrodynamics Scalar chromodynamics Soler Yang–Mills Yang–Mills–Higgs Yukawa
Low dimensional	2D Yang–Mills Bullough–Dodd Gross–Neveu Schwinger Sine-Gordon Thirring Thirring–Wess Toda
Conformal	2D free massless scalar Liouville Minimal Polyakov Wess–Zumino–Witten
Supersymmetric	4D N = 1 N = 1 super Yang–Mills Seiberg–Witten Super QCD Wess–Zumino
Superconformal	6D (2,0) ABJM N = 4 super Yang–Mills
Supergravity	Pure 4D N = 1 4D N = 1 4D N = 8 Higher dimensional Type I Type IIA Type IIB 11D
Topological	BF Chern–Simons
Particle theory	Chiral Fermi MSSM Nambu–Jona-Lasinio NMSSM Standard Model Stueckelberg

See also:

Template:Quantum mechanics topics

Categories:

Revision as of 04:17, 25 July 2009 edit123.255.28.37 (talk) →References: reformat← Previous edit		Revision as of 04:45, 25 July 2009 edit undo123.255.28.37 (talk) flesh out citations and arXiv linksNext edit →
Line 1:		Line 1:
	In ], '''] quantum field theory''' (or quantum field theory on noncommutative ]) is a branch of ] ~~born~~ ~~from~~ ] and ] in which the spatial coordinates<ref>It is possible to have a noncommuting time coordinate but this causes many problems, such as the violation of ] of the ], ~~and~~ most research is restricted to so-called "space-space" noncommutativity. There have been attempts to avoid these problems by redefining the ]. ~~String~~ ~~theoretical~~ ~~derivation~~ of noncommutative coordinates ~~however~~ excludes time-space noncommutativity.</ref> do not commute. One (commonly studied) version of such theories has the "canonical" commutation relation:		In ], '''] quantum field theory''' (or quantum field theory on noncommutative ]) is a branch of ] that is an outgrowth of ] and ] in which the spatial coordinates<ref>It is possible to have a noncommuting time coordinate, but this causes many problems such as the violation of ] of the ]. Hence most research is restricted to so-called "space-space" noncommutativity. There have been attempts to avoid these problems by redefining the ]. However, ] derivations of noncommutative coordinates excludes time-space noncommutativity.</ref> do not commute. One commonly studied version of such theories has the "canonical" commutation relation:
	<!--		<!--
	The \,\! is to keep the formula rendered as PNG instead of HTML to		The \,\! is to keep the formula rendered as PNG instead of HTML to
Line 12:		Line 12:
	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 theory or grounds for ruling them out.		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 theory or grounds for ruling them out.

	One of the novel features of noncommutative field theories is the ]<ref>See, for example, ~~] and~~		One of the novel features of noncommutative field theories is the ]<ref>See, for example,
	].</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.		Shiraz Minwalla, Mark Van Raamsdonk, Nathan Seiberg (2000) "" ''Journal of High Energy Physics'', and Alec Matusis, ], Nicolaos Toumbas (2000) "" ''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 ] due to the preferred direction of noncommutativity. ] can however be retained in the sense of twisted ] invariance of the theory<ref>~~See ].</ref>~~. ~~The~~ ] ~~condition~~ is ~~modified~~ ~~from that of the commutative theories.~~		Other features include violation of ] due to the preferred direction of noncommutativity. ] can however be retained in the sense of twisted ] invariance of the theory<ref>M. Chaichian, P. Prešnajder, A. Tureanu (2005)
			"" ''Phys. Rev. Letters'' 94: .</ref>. The ] condition is modified from that of the commutative theories.

	==History and motivation==		==History and motivation==

	] 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 ] procedure was developed and had gained acceptance. The first paper on the subject was published in 1947 by ]. The success of the renormalization method ~~drained~~ ~~the~~ ~~subject~~ of ~~any~~ ~~interest~~ for some time. In the 1980s, mathematicians, most notably ], developed ]. Among other things, this work generalized the notion of ] to a noncommutative setting. This led to an ] description of noncommutative ]s, and the development of a ] on a noncommutative ].		] 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 ] procedure was developed and had gained acceptance. The first paper on the subject was published in 1947 by ]. The success of the renormalization method resulted in little attention being paid to to the subject for some time. In the 1980s, mathematicians, most notably ], developed ]. Among other things, this work generalized the notion of ] to a noncommutative setting. This led to an ] description of noncommutative ]s, and the development of a ] on a noncommutative ].

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

	A paper by Sergio Doplicher, Klaus Fredenhagen and John Roberts<ref>http://arxiv.org/abs/hep-th/0303037</ref> set out another motivation for the possible noncommutativity of space-time. Their arguments goes as follows: According to ], when the energy density grows sufficiently large, a ] is formed. On the other hand according to the Heisenberg ], 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 ] of the system is reached and a ] 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 ] relation for the coordinates.		A paper by Sergio Doplicher, Klaus Fredenhagen and John Roberts<ref>Sergio Doplicher, Klaus Fredenhagen, John E. Roberts (1995) "" ''Commun. Math. Phys''. 172: 187-220.</ref> set out another motivation for the possible noncommutativity of space-time. Their arguments goes as follows: According to ], when the energy density grows sufficiently large, a ] is formed. On the other hand according to the Heisenberg ], 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 ] of the system is reached and a ] 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 ] relation for the coordinates.

	==See also==		==See also==