  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. Also, we show that the relation implies the
multidimensional Heisenberg uncertainty relation.
