Noncommutative quantum field theory: Difference between revisions

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

Revision as of 06:41, 29 March 2009 editHenry Delforn (old) (talk \| contribs)2,675 edits noncommutative geometry/Index theory← Previous edit		Revision as of 06:55, 29 March 2009 edit undoHenry Delforn (old) (talk \| contribs)2,675 edits Quantum Tori/referencesNext 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		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:
	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:
	<!--		<!--
	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 16:		Line 15:
	].</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.		].</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. Relativistic invariance can however be retained in the sense of twisted ] invariance of the theory<ref>See ].</ref>. ] condition is possibly 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>See ].</ref>. ] condition is possibly modified from that of the commutative theories.

	==History and motivation==		==History and motivation==
Line 22:		Line 21:
	The idea of extension of noncommutativity to the coordinates was first suggested by ] as a possible solution for removing the infinite quantities of 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 renormalization theory however drained interest from the subject for some time. In 1980s ] was studied and developed by mathematicians, most notably ]. The notion of ] was generalized to a noncommutative setting. This led to an ] description of noncommutative space-times and a ] on a noncommutative ] was developed.		The idea of extension of noncommutativity to the coordinates was first suggested by ] as a possible solution for removing the infinite quantities of 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 renormalization theory however drained interest from the subject for some time. In 1980s ] was studied and developed by mathematicians, most notably ]. The notion of ] was generalized to a noncommutative setting. This led to an ] description of noncommutative space-times and a ] on a noncommutative ] was developed.

		⚫	The recent interest by the particle physics community was driven by a 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 presented 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 recent interest by the particle physics community was driven by a
⚫	by ] and ]. 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 presented 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.

	Another possible motivation for the noncommutativity of space-time was presented by ], ] and ]. Their arguments is as follows: According to ], when the energy density grows sufficiently large, a ] 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 separation. Thus energy of scale corresponding 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 black hole is formed, preventing any information to escape the region. Thus a lower limit is introduced for the measurement of length. A sufficient condition for preventing the gravitational collapse can be expressed as a form of uncertainty relation for the coordinates. This relation in turn can be derived from a commutation relation for the coordinates.		Another paper <ref>http://arxiv.org/abs/hep-th/0303037</ref> possible motivation for the noncommutativity of space-time was presented by Sergio Doplicher, Klaus Fredenhagen and John Roberts. Their arguments is as follows: According to ], when the energy density grows sufficiently large, a ] 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 separation. Thus energy of scale corresponding 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 black hole is formed, preventing any information to escape the region. Thus a lower limit is introduced for the measurement of length. A sufficient condition for preventing the gravitational collapse can be expressed as a form of uncertainty relation for the coordinates. This relation in turn can be derived from a ] relation for the coordinates.

	==See also==		==See also==
	*]		*]
	*]		*]
	*]		*]
			*]

	== References ==		== References ==

Revision as of 06:55, 29 March 2009

In physics, noncommutative quantum field theory (or quantum field theory on noncommutative space-time) is a branch of quantum field theory born from 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. Causality condition is possibly modified from that of the commutative theories.

History and motivation

The idea of extension of noncommutativity to the coordinates was first suggested by Heisenberg as a possible solution for removing the infinite quantities of 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 renormalization theory however drained interest from the subject for some time. In 1980s noncommutative geometry was studied and developed by mathematicians, most notably Alain Connes. The notion of differential structure was generalized to a noncommutative setting. This led to an operator algebraic description of noncommutative space-times and a Yang-Mills theory on a noncommutative torus was developed.

The recent interest by the particle physics community was driven by a 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 presented 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.

Another paper possible motivation for the noncommutativity of space-time was presented by Sergio Doplicher, Klaus Fredenhagen and John Roberts. Their arguments is 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 separation. Thus energy of scale corresponding 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, preventing any information to escape the region. Thus a lower limit is introduced for the measurement of length. A sufficient condition for preventing the gravitational collapse can be expressed as a form of uncertainty relation for the coordinates. This relation in turn can 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, and most research is restricted to so-called "space-space" noncommutativity. There have been attempts to avoid these problems by redefining the perturbation theory. String theoretical derivation of noncommutative coordinates however excludes time-space noncommutativity.
See, for example, arXiv:hep-th/9912072 and arXiv:hep-th/0002075.
See arXiv:hep-th/0409096.
http://arxiv.org/abs/hep-th/9908142 paper]
http://arxiv.org/abs/hep-th/0303037

R.J. Szabo, Quantum Field Theory on Noncommutative Spaces, arXiv:hep-th/0109162 (A good review of noncommutative quantum field theories)
M.R. Douglas, N.A. Nekrasov, Noncommutative field theory Rev. Mod. Phys. 73, 977 - 1029 (2001)

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: