On the form of local conservation laws for some relativistic field theories in 1+1 dimensions

Abstract

We investigate the possible form of local translation invariant conservation laws associated with the relativistic field equations $\partial\bar\partialϕ_i=-v_i(\bphi)$ for a multicomponent field $\bphi$. Under the assumptions that (i)~the $v_i$'s can be expressed as linear combinations of partial derivatives $\partial w_j/\partial {\varphi }_k$ of a set of functions $w_j(\bphi)$, (ii)~the space of functions spanned by the $w_j$'s is closed under partial derivations, and (iii)~the fields $\bphi$ take values in a simply connected space, the local conservation laws can either be transformed to the form $\partial \bar{\mathcal{P}}=\bar{\partial }{\sum }_j{w}_j{\mathcal{Q}}_j$ (where $\bar{\mathcal{P}}$ and ${\mathcal{Q}}_j$ are homogeneous polynomials in the variables $\bar{\partial }{\varphi }_i$, ${\bar{\partial }}^2{\varphi }_i$,\ldots), or to the parity transformed version of this expression $\partial \equiv ({\partial }_t+{\partial }_x)/\sqrt{2}\rightleftharpoons \bar{\partial }\equiv ({\partial }_t-{\partial }_x)/\sqrt{2}$.

12 pages, Latex

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