Abstract
We consider two non-relativistic strings and their Galilean symmetries. These strings are obtained as the two possible non-relativistic (NR) limits of a relativistic string. One of them is non-vibrating and represents a continuum of non-relativistic massless particles, and the other one is a non-relativistic vibrating string. For both cases we write the generator of the most general point transformation and impose the condition of Noether symmetry. As a result we obtain two sets of non-relativistic Killing equations for the vector fields that generate the symmetry transformations. Solving these equations shows that NR strings exhibit two extended, infinite dimensional space-time symmetries which contain, as a subset, the Galilean symmetries. For each case, we compute the associated conserved charges and discuss the existence of non-central extensions.
Highlights
The first check that holographic ideas are working is to verify that the symmetries of the bulk and the screen coincide
As a first step towards using non-relativistic strings in non-relativistic holography, we study in this paper the general Noether Galilean symmetries of non-relativistic strings
For a massive non-relativistic particle, the maximal set of symmetries is larger than the Galilei group, and it is the Schrodinger group [20, 21], which is the group corresponding to the z = 2 case of an infinite set of z-Galilean conformal algebras [22,23,24]
Summary
We will study the possible non-relativistic limits of an extended object, a relativistic p-brane. We assume that the Lagrangian density L is pseudo-invariant under a given set of relativistic symmetries δR, δRL = ∂ · F,. For the first orders in the expansion parameter we have δ0L2 = ∂ · F2, δ0L0 + δ−2L2 = ∂ · F0, δ0L2 + δ−2L0 + δ−4L2 = ∂ · F−2 From these one sees that the highest order Lagrangian density L2, which appears with a divergent factor ω2, is always (pseudo)invariant under δ0, while the finite one, L0 is only invariant if δ−2L2 is a divergence. In the limit ω → ∞ all the terms with negative powers of ω vanish Let us apply these ideas to a relativistic string in flat space-time described by the. One can consider two non-relativistic limits, which depend on whether one or two embedding coordinates are scaled by ω. According to the general discussion, the Lagrangian density −T (tx − t x ) will be, at least, pseudo-invariant under Galilean transformations. The term in the ω expansion will be NR invariant, and it results in the standard NR particle action
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