Abstract
Relativistic field theories with a power law decay in r−k at spatial infinity generically possess an infinite number of conserved quantities because of Lorentz invariance. Most of these are not related in any obvious way to symmetry transformations of which they would be the Noether charges. We discuss the issue in the case of a massless scalar field. By going to the dual formulation in terms of a 2-form (as was done recently in a null infinity analysis), we relate some of the scalar charges to symmetry transformations acting on the 2-form and on surface degrees of freedom that must be added at spatial infinity. These new degrees of freedom are necessary to get a consistent relativistic description in the dual picture, since boosts would otherwise fail to be canonical transformations. We provide explicit boundary conditions on the 2-form and its conjugate momentum, which involves parity conditions with a twist, as in the case of electromagnetism and gravity. The symmetry group at spatial infinity is composed of “improper gauge transformations”. It is abelian and infinite-dimensional. We also briefly discuss the realization of the asymptotic symmetries, characterized by a non trivial central extension and point out vacuum degeneracy.
Highlights
Which is the model considered in a similar context in [1, 2]
By going to the dual formulation in terms of a 2-form, we relate some of the scalar charges to symmetry transformations acting on the 2-form and on surface degrees of freedom that must be added at spatial infinity
We show that for the interpretation of the scalar charges to be symmetry generators, does one need to go to the dual formulation where there are constraints, but one must introduce surface degrees of freedom at infinity and modify the symplectic form by surface terms
Summary
2.1 Action in Hamiltonian form — boundary conditions The Hamiltonian form of the action for the scalar field reads. The reflection is written r → r, xA → −xA ( if the angles are the standard polar angles, one has θ → π − θ and φ → φ + π) These parity conditions make the logarithmic divergence in the kinetic term of the action d3x π. Alternative parity conditions where φ would be odd and π would be even would fulfill the same purpose of making the symplectic form finite They are incompatible with spherical symmetry for φ, these boundary conditions are mathematically consistent. This computation uses the parity conditions, since otherwise an unwanted surface term at infinity remains in the variation of the symplectic form (independently of the value of ∂tφ). Parity conditions are needed for finiteness of the Poincare charges
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