We analyse the Noether charges for scalar and Maxwell fields on light cones on a de Sitter, Minkowski, and anti-de Sitter backgrounds. Somewhat surprisingly, under natural asymptotic conditions all charges for the Maxwell fields on both the de Sitter and anti-de Sitter backgrounds are finite. On the other hand, one needs to renormalise the charges for the conformally-covariant scalar field when the cosmological constant does not vanish. In both cases well-defined renormalised charges, with well-defined fluxes, are obtained. Again surprisingly, a Hamiltonian analysis of a suitably rescaled scalar field leads to finite charges, without the need to renormalise. Last but not least, we indicate natural phase spaces where the Poisson algebra of charges is well defined.

1.
Bialynicki-Birula
,
I.
and
Bialynicka-Birula
,
Z.
,
Quantum Electrodynamics
(
Pergamon Press
,
1975
).
2.
Blanchet
,
L.
and
Damour
,
T.
, “
Radiative gravitational fields in general relativity. I. General structure of the field outside the source
,”
Philos. Trans. R. Soc. London, Ser. A
320
,
379
430
(
1986
).
3.
Brown
,
J. D.
and
Henneaux
,
M.
, “
On the Poisson brackets of differentiable generators in classical field theory
,”
J. Math. Phys.
27
,
489
491
(
1986
).
4.
Chruściel
,
P. T.
,
Hoque
,
S. J.
,
Maliborski
,
M.
, and
Smołka
,
T.
, “
On the canonical energy of weak gravitational fields with a cosmological constant ΛR
,”
Eur. Phys. J. C
81
,
696
(
2021
); arXiv:2103.05982v2 [gr-qc].
5.
Chruściel
,
P. T.
and
Ifsits
,
L.
, “
The cosmological constant and the energy of gravitational radiation
,”
Phys. Rev. D
93
,
124075
(
2016
); arXiv:1603.07018 [gr-qc].
6.
Chruściel
,
P. T.
,
Jezierski
,
J.
, and
Kijowski
,
J.
,
Hamiltonian Field Theory in the Radiating Regime
,
Lecture Notes in Physics Vol. m70
(
Springer
,
Berlin, Heidelberg, New York
,
2002
).
7.
Compère
,
G.
,
Fiorucci
,
A.
, and
Ruzziconi
,
R.
, “
The Λ-BMS4 charge algebra
,”
J. High Energy Phys.
2020
,
205
; arXiv:2004.10769 [hep-th].
8.
Fischer
,
K.
, “
Interpretation of Einstein’s theory of gravitation including the cosmological term as a de Sitter-invariant field theory on the de Sitter space
,”
Z. Phys. A
229
,
33
43
(
1969
).
9.
Freidel
,
L.
, “
A canonical bracket for open gravitational system
,” arXiv:2111.14747 [hep-th] (
2021
).
10.
Jezierski
,
J.
,
Kijowski
,
J.
, and
Waluk
,
P.
, “
Gauge-invariant quadratic approximation of quasi-local mass and its relation with Hamiltonian for gravitational field
,”
Classical Quantum Gravity
38
,
095006
(
2021
).
11.
Kijowski
,
J.
, “
A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity
,”
Gen. Relativ. Gravitation
29
,
307
343
(
1997
).
12.
Kijowski
,
J.
and
Tulczyjew
,
W. M.
,
A Symplectic Framework for Field Theories
,
Lecture Notes in Physics Vol. 107
(
Springer
,
New York, Heidelberg, Berlin
,
1979
).
13.
Poole
,
A.
,
Skenderis
,
K.
, and
Taylor
,
M.
, “
Charges, conserved quantities, and fluxes in de Sitter spacetime
,”
Phys. Rev. D
106
(
6
),
L061901
(
2022
); arXiv:2112.14210 [hep-th].
14.
Soloviev
,
V. O.
, “
Boundary values as Hamiltonian variables. I. New Poisson brackets
,”
J. Math. Phys.
34
,
5747
5769
(
1993
).
15.

References to numbering in 4 are to the arXiv version.

16.

Note that the singularity at r = 0 in (1.12) is integrable for fields ϕ which are smooth at the origin. While the presence of an unbounded integrand might be aesthetically unpleasant, it does not present difficulties as far as calculus of variations is concerned.

17.

We take this opportunity to correct a misprint in [4, Eq. (2.68)], where the terms involving ϕ(2) are missing.

You do not currently have access to this content.