A covariant formulation for the Newton-Hooke particle is presented by following an algorithm developed by us cite{BMM1, BMM2, BMM3}. It naturally leads to a coupling with the Newton-Cartan geometry. From this result we provide an understanding of gra
vitation in a Newtonian geometric background. Using Diracs constrained analysis a canonical formulation for the Newton-Hooke covariant action is done in both gauge independent and gauge fixed approaches. While the former helps in identifying the various symmetries of the model, the latter is able to define the physical variables. From this analysis a path to canonical quantisation is traced and the Schroedinger equation is derived which is shown to satisfy various consistency checks. Some consequences of this equation are briefly mentioned.
Recent discussions of higher rank symmetric (fractonic) gauge theories have revealed the important role of Gauss constraints. This has prompted the present study where a detailed hamiltonian analysis of such theories is presented. Besides a general t
reatment, the traceless scalar charge theory is considered in details. A new form for the action is given which, in 2+1 dimensions, yields area preserving diffeomorphisms. Investigation of global symmetries reveals that this diffeomorphism invariance induces a noncommuting charge algebra that gets exactly mapped to the algebra of coordinates in the lowest Landau level problem. Connections of this charge algebra to noncommutative fluid dynamics and magnetohydrodynamics are shown.
We discuss a new formalism for constructing a non-relativistic (NR) theory in curved background. Named as galilean gauge theory, it is based on gauging the global galilean symmetry. It provides a systematic algorithm for obtaining the covariant curve
d space time generalisation of any NR theory defined in flat space time. The resulting background is just the Newton- Cartan manifold. The example of NR free particle is explicitly demonstrated.
A new approach to the study of nonrelativistic bosonic string in flat space time is introduced, basing on a holistic hamiltonian analysis of the minimal action for the string. This leads to a structurally new form of the action which is, however, equ
ivalent to the known results since, under appropriate limits, it interpolates between the minimal action (Nambu Goto type) where the string metric is taken to be that induced by the embedding and the Polyakov type of action where the world sheet metric components are independent fields. The equivalence among different actions is established by a detailed study of symmetries using constraint analysis. Various vexing issues in the existing literature are clarified. The interpolating action mooted here is shown to reveal the geometry of the string and may be useful in analyzing nonrelativistic string coupled with curved background.
We systematically derive an action for a nonrelativistic spinning partile in flat background and discuss its canonical formulation in both Lagrangian and Hamiltonian approaches. This action is taken as the starting point for deriving the correspondin
g action in a curved background. It is achieved by following our recently developed technique of localising the flat space galilean symmetry cite{BMM1, BMM3, BMM2}. The coupling of the spinning particle to a Newton-Cartan background is obtained naturally. The equation of motion is found to differ from the geodesic equation, in agreement with earlier findings. Results for both the flat space limit and the spinless theory (in curved background) are reproduced. Specifically, the geodesic equation is also obtained in the latter case.
A detailed canonical treatment of a new action for a nonrelativistic particle coupled to background gravity, recently given by us, is performed both in the Lagrangian and Hamiltonian formulations. The equation of motion is shown to satisfy the geodes
ic equation in the Newton-Cartan background, thereby clearing certain confusions. The Hamiltonian analysis is done in the gauge independent as well as gauge fixed approaches, following Diracs analysis of constraints. The physical (canonical) variables are identified and the path to canonical quantisation is outlined by explicitly deriving the Schroedinger equation.
We show a direct connection between Kubos fluctuation-dissipation relation and Hawking effect that is valid in any dimensions for any stationary or static black hole. The relevant correlators corresponding to the fluctuating part of the force, comput
ed from the known expressions for the anomalous stress tensor related to gravitational anomalies, are shown to satisfy the Kubo relation, from which the temperature of a black hole as seen by an observer at an arbitrary distance is abstracted. This reproduces the Tolman temperature and hence the Hawking temperature as that measured by an observer at infinity.
A specific mapping is introduced to reduce the Dirac action to the non-relativistic (Pauli - Schrodinger) action for spinors. Using this mapping, the structures of the vector and axial vector currents in the non-relativistic theory are obtained. The
implications of the relativistic Ward identities in the non-relativistic limit are discussed. A new non-abelian type of current in the Pauli - Schrodinger theory is obtained. As we show, this is essential for the closure of the algebra among the usual currents. The role of parity in the non-relativistic theory is also discussed.
We obtain a new form for the action of a nonrelativistic particle coupled to Newtonian gravity. The result is different from that existing in the literature which, as shown here, is riddled with problems and inconsistencies. The present derivation is
based on the formalism of galilean gauge theory, introduced by us as an alternative method of analysing nonrelativistic symmetries in gravitational background.
We provide an exact mapping between the Galilian gauge theory, recently advocated by us cite{BMM1, BMM2, BM}, and the Poincare gauge theory. Applying this correspondence we provide a vielbein approach to the geometric formulation of Newtons gravity w
here no ansatze or additional conditions are required.
