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

TL;DR

This paper characterizes the form of local conservation laws in certain 1+1 dimensional relativistic field theories, showing they can be transformed into specific polynomial forms under given assumptions.

Contribution

It provides a classification of local conservation laws for a class of relativistic field equations, revealing their structure under specified conditions.

Findings

Conservation laws can be expressed as polynomial forms in derivatives of fields.

Such laws can be transformed into a standard polynomial form or its parity-transformed version.

The results depend on assumptions about the functions defining the equations and the field space topology.

Abstract

We investigate the possible form of local translation invariant conservation laws associated with the relativistic field equations $\partial\bar\partial\phi_i=-v_i(\bphi)$ for a multicomponent field $\bphi$. Under the assumptions that (i)~the $v_i$'s can be expressed as linear combinations of partial derivatives $\partial w_j/\partial\phi_k$ 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 $\partial{\bar{\cal P}}=\bar\partial\sum_j w_j {\cal Q}_j$ (where $\bar{\cal P}$ and ${\cal Q}_j$ are homogeneous polynomials in the variables $\bar\partial\phi_i$, $\bar\partial^2\phi_i$,\ldots), or to the parity transformed version of this expression $\partial\equiv(\partial_t+\partial_x)/ \sqrt{2}\rightleftharpoons\bar\partial \equiv (\partial_t-\partial_x)/\sqrt{2}$.