Maximal estimates for orthonormal systems of wave equations with sharp regularity

We study maximal estimates for the wave equation with orthonormal initial data. In dimension $d=3$, we establish optimal results with the sharp regularity exponent up to the endpoint. In higher dimensions $d\ge 4$ and also in $d=2$, we obtain sharp bounds for the Schatten exponent (summability index) $\beta \in [2,\mathrm{\infty }]$ when $d\ge 4$, and $\beta \in [1,2]$ when $d=2$, improving upon the previous estimates due to Kinoshita–Ko–Shiraki. Our approach is based on a novel analysis of a key integral arising in the case $\beta =2$, which allows us to refine existing techniques and achieve the optimal estimates.