Quantum Bosonic String Energy-Momentum Tensor in Minkowski Space-Time

TL;DR

This paper calculates the quantum energy-momentum tensor for bosonic strings in Minkowski space, showing it is finite and analyzing its behavior for different string states, revealing quantum smearing effects and propagation characteristics.

Contribution

It provides the first explicit computation of the quantum energy-momentum tensor expectation value for open and closed bosonic strings with wave-packet states, demonstrating finiteness and quantum smearing effects.

Findings

The quantum energy-momentum tensor is finite, indicating removal of classical divergences.

Massive string states have a vanishing expectation value at leading order.

Massless string states produce a non-zero, propagating energy-momentum distribution.

Abstract

The quantum energy-momentum tensor ${\hat T}^{\mu\nu}(x)$ is computed for strings in Minkowski space-time. We compute its expectation value for different physical string states both for open and closed bosonic strings. The states considered are described by normalizable wave-packets in the center of mass coordinates. We find in particular that ${\hat T}^{\mu\nu}(x)$ is {\bf finite} which could imply that the classical divergence that occurs in string theory as we approach the string position is removed at the quantum level as the string position is smeared out by quantum fluctuations. For massive string states the expectation value of ${\hat T}^{\mu\nu}(x)$ vanishes at leading order (genus zero). For massless string states it has a non-vanishing value which we explicitly compute and analyze for both spherically and cylindrically symmetric wave packets. The energy-momentum tensor components propagate outwards as a massless lump peaked at $ r = t $.