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No-go results for non-topological solitons in some types of gauge field theories
Mikhail Smolyakov

Skobeltsyn Institute of Nuclear Physics, Moscow State University


The Derrick theorem
R.H. Hobart, Proc. Phys. Soc. 82 (1963) 201. G.H. Derrick, J. Math. Phys. 5 (1964) 1252.



The same result can be obtained by multiplying the corresponding equation of motion by

and integrating over the 3-volume

How to overcome this restriction? For example, one can consider


Gauge theories


Extra conditions · there are no sources which are external to the system described by the presented action · solutions to equations of motion are periodic in time with a period T up to a coordinate shift and a spatial rotation, i.e. for all fields on the solution the relation must hold for any t


One can always pass to a suitable coordinate system in which

One can use the effective action


(1) (2)

(3)

(4) (5) (6)


Non-topological solitons of form (1), (2), periodic in time up to a spatial rotation and a coordinate shift, with integrals (3)-(6) and integrals

finite, are absent in the theory if there exists for which the inequality

is fulfilled for any

.





Corollary

R.T. Glassey, W.A. Strauss, Commun. Math. Phys. 67 (1979) 51

G. Rosen, J. Math. Phys. 9 (1968) 999

S. Deser, Phys. Lett. B 64 (1976) 463; H. Pagels, Phys. Lett. B 68 (1977) 466; S.R. Coleman, Commun. Math. Phys. 55 (1977) 113; S.R. Coleman, L. Smarr, Commun. Math. Phys. 56 (1977) 1; R. Weder, Commun. Math. Phys. 57 (1977) 161; M. Magg, J. Math. Phys. 19 (1978) 991; R.T. Glassey, W.A. Strauss, Commun. Math. Phys. 65 (1979) 1


Charged massive vector field