Galilean conformal mechanics from nonlinear realizations

Abstract

We apply the nonlinear realizations method for constructing new Galilean conformal mechanics models. Our starting point is the Galilean conformal algebra which is a nonrelativistic contraction of its relativistic counterpart. We calculate Maurer-Cartan oneforms, examine various choices of the relevant coset spaces and consider the geometric inverse Higgs-type constraints which reduce the number of the independent coset parameters and, in some cases, provide dynamical equations. New Galilean conformally invariant actions are derived in arbitrary space-time dimension D = d+1 (no central charges), as well as in the special dimension D = 2+1 with one "exotic" central charge. We obtain new classical mechanics models which extend the standard (D = 0+1) conformal mechanics in the presence of d non-vanishing space dimensions.

Effective field theory for spacetime symmetry breaking

“…For simplicity, let us consider the case α 3 (z) = 0 in this subsection. 8 Then, the brane profile is characterized by the free function α 1 (z) in the effective action. Generically, α 1 (z) is related to the order parameterφ (z) as…”

“…Such a coset construction was also extended to spacetime symmetry breaking [4,5] accompanied by the inverse Higgs constraints [6] and has been applied to various systems (see, e.g., [7][8][9][10][11][12][13][14][15][16][17][18][19] for recent discussions). Although the coset construction captures certain aspects of spacetime symmetry breaking, its understanding seems incomplete compared to the internal symmetry case and as a result generated a lot of recent research activities .…”

“…Just as the linear order discussions, the α 3 term plays an important role: 8) where note that the factor, √ −h, does not contain ∂ z π. As we have discussed, the mass term for the gapped modes, π λ i , arises from this interaction at the second order action level.…”

Nonlinear realizations, the orbit method and Kohnʼs theorem

“…Using the orbit method it is not difficult to generalize their method to other dynamical systems [12,13]. The relevant equations reads…”

Quantizations of $$D=3$$ D = 3 Lorentz symmetry

“…The o(2, 1) Lie algebra has been used as D = 1 conformal algebra describing basic symmetries in conformal classical and quantum mechanics [15]; in such a case the o(2, 1) algebra is realized as a nonlinear realization on the one-dimensional time axis [16,17] and can be extended to osp(1|2) describing D = 1 N = 2 supersymmetric conformal algebra [18]. In field-theoretic framework the o(2, 1) Lie algebra describes Lorentz symmetries of three-dimensional relativistic systems with planar d = 2 space sector, which are often discussed as simplified version of the four-dimensional relativistic case.…”