# Noether and Belinfante stress-energy tensors for theories with arbitrary Lagrangians of tensor fields

## TL;DR

This paper explores the relationship between Noether and Belinfante stress-energy tensors in Poincaré-invariant tensor field theories with complex Lagrangians, providing explicit formulas and surface integral expressions for their differences.

## Contribution

It derives an exact relation between Noether and Belinfante stress-energy tensors for theories with arbitrary tensor fields and derivatives, extending previous results to a broad class of actions.

## Key findings

- Derived the explicit difference between the two stress-energy tensors.
- Expressed the difference in integrals of motion as a surface integral at infinity.
- Established the connection in theories with arbitrary tensor fields and derivatives.

## Abstract

We investigate the connection between stress-energy tensor (SET) arising from Noether's theorem and Belinfante SET which can be obtained as a right-hand side of the Einstein's equation in the flat metric limit. This question is studied in the wide class of Poincar\`e-invariant field theories with actions which depend on the tensor fields of arbitrary rank and their derivatives of arbitrary order. For this class we derive the relation between these SET and present the exact expression for the difference between them. We also show that the difference between corresponding integrals of motion can be expressed as a surface integral over 2-dimensinal infinitely remote surface.