Entropic uncertainty is a well-known concept to formulate uncertainty
relations for continuous variable quantum systems with finitely many
degrees of freedom. Typically, the bounds of such relations scale with
the number of oscillator modes, preventing a straight-forward
generalization to quantum field theories. In this work, we overcome this
difficulty by introducing the notion of a functional relative entropy
and show that it has a meaningful field theory limit. We present the
first entropic uncertainty relation for a scalar quantum field theory
and exemplify its behavior by considering few particle excitations and
the thermal state.