Noncommutative quantum field theory

Noncommutative quantum field theory (or quantum field theory on noncommutative space-time) is a branch of quantum field theory in which the spatial coordinates¹ do not commute. One (commonly studied) version of such theories has the "canonical" commutation relation:

<math>

[x^{\mu}, x^{\nu}]=i \theta^{\mu \nu} \,\! <math>

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 mixing² phenomenon in which the physics at high energies effects 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 theory³. Causality condition is possibly modified from that of the commutative theories.

There is a good review (http://arxiv.org/abs/hep-th/0109162) of noncommutative quantum field theories freely available on the web.

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 paper (http://arxiv.org/abs/hep-th/9908142) 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 (http://arxiv.org/abs/hep-th/0303037) 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.

[1] 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

[2] See for example this paper (http://arxiv.org/abs/hep-th/9912072) and this paper (http://arxiv.org/abs/hep-th/0002075).

[3] See this paper (http://arxiv.org/abs/hep-th/0409096).

[4] See the reviews[1] (http://prola.aps.org/abstract/RMP/v73/i4/p977_1?qid=a81527af6e5a2fa2&qseq=1&show=10) and [2] (http://www.sciencedirect.com/science?_ob=ArticleURL&_aset=V-WA-A-W-B-MsSAYZW-UUA-U-AAWAUEWUDA-AAAYZDWYDA-AZCYZBDYY-B-U&_rdoc=1&_fmt=full&_udi=B6TVP-484126R-1&_coverDate=05%2F31%2F2003&_cdi=5540&_orig=search&_st=13&_sort=d&view=c&_acct=C000053663&_version=1&_urlVersion=0&_userid=1553430&md5=d4a4b58df37c651b83736df5bbec3964).