Abstract
The equation of motion for cohomogeneity-one Nambu–Goto strings in flat space Rn,1 has been investigated. We first classify possible forms of the Killing vector fields in Rn,1 after appropriate action of the Poincaré group. Then, all possible forms of the Hamiltonian for the cohomogeneity-one Nambu–Goto strings are determined. It has been shown that the system always has the maximum number of functionally independent, pair-wise commuting conserved quantities, i.e., it is completely integrable. We have also determined all the possible coordinate forms of the Killing vector basis for the two-dimensional noncommutative Lie algebra.
Highlights
According to the standard models of elementary particles, the quantum vacuum of certain scalar field depends on the environmental temperature
In order to find the canonical form of ξ, we study the transformation of fν under the subgroup GF of the Poincaré group IOn,1 that preserves Fμν
We have shown that the equation of motion for cohomogeneity-one Nambu–Goto strings in Rn,1 is completely integrable for n ≥ 1, which generalize the result of Koike et al for n = 3
Summary
According to the standard models of elementary particles, the quantum vacuum of certain scalar field depends on the environmental temperature. When the spacetime admits a Killing vector field, we can consider Nambu–Goto strings whose world sheets are invariant under the action of the isometry It is well-known that the general form of the Nambu–Goto strings in flat spacetime Rn, is given by the n + 1 harmonic functions This is because the equation of motion for the string is reduced to n + 1 linear wave equations when isothermal coordinates are taken as the world-sheet coordinates. We first enumerate the equivalence classes of pairs (Fμν, fν), which give the canonical classification of Killing vector fields in Rn,1 It has been shown in Ref. 4 that the equation of motion for cohomogeneity-one strings in R3,1 is completely integrable so that every solution to it is given by quadrature.
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