Noncommutative quantum field theory
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 are noncommutative. One commonly studied version of such theories has the "canonical" commutation relation:
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 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-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.
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.
- ^ 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.
- Noncommutative quantum field theory, see statistics on arxiv.org
Quantum field theories StandardChern–Simons model · Chiral model · Kondo model · Conformal field theory · Noncommutative quantum field theory · Non-linear sigma model · Quartic interaction · Quantum chromodynamics · Quantum electrodynamics · Quantum Yang–Mills theory · Schwinger model · Sine–Gordon · Standard Model · String theory · Toda field theory · Topological quantum field theory · Wess–Zumino model · Wess–Zumino–Witten model · Yang–Mills theory · Yang–Mills–Higgs model · Yukawa model · Thirring–Wess model Four-fermion interactions
Wikimedia Foundation. 2010.
Look at other dictionaries:
List of quantum field theories — List of quantum field theories:*Chern Simons model *Chiral modelComplex Quantum Mechanicshttp://tech.groups.yahoo.com/group/complexquantummechanics*Gross Neveu *Kondo model *Lower dimensional quantum field theory *Minimal model *Nambu Jona… … Wikipedia
Noncommutative geometry — Not to be confused with Anabelian geometry. Noncommutative geometry (NCG) is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative… … Wikipedia
Quantum chromodynamics — Standard model of particle physics Standard Model … Wikipedia
Noncommutative standard model — In theoretical particle physics, the non commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise an extension of the Standard Model to include a modified form of general… … Wikipedia
Quantum gravity — is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature (electromagnetism, weak interaction, and strong interaction), with general relativity, the theory of the fourth… … Wikipedia
Quantum harmonic oscillator — The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential … Wikipedia
Quantum probability — was developed in the 1980s as a noncommutative analog of the Kolmogorovian stochastic processes theory. One of its aims is to clarify the probabilistic mathematical foundations of quantum theory and its statistical interpretation.Significant… … Wikipedia
Noncommutative algebraic geometry — is a branch of mathematics, and more specifically a direction in noncommutative geometry that studies the geometric properties of formal duals of non commutative algebraic objects such as rings as well as geometric objects derived from them (e.g … Wikipedia
Quantum group — In mathematics and theoretical physics, quantum groups are certain noncommutative algebras that first appeared in the theory of quantum integrable systems, and which were then formalized by Vladimir Drinfel d and Michio Jimbo. There is no single … Wikipedia
Introduction to gauge theory — This article is an accessible, non technical introduction to the subject. For the main encyclopedia article, see Gauge theory. Quantum field theory … Wikipedia