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 and the other by D. V. Ahluwalia, 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 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 coordinates 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 Further reading M.R. Douglas and N. A. Nekrasov (2001) "Noncommutative field theory," Rev. Mod. Phys. 73: 977–1029. Szabo, R. (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 V. Moretti (2003), "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 Noncommutative geometry Quantum field theory Mathematical quantization