The amplitude of production of n on-mass-shell scalar bosons by a highly virtual field φ is considered in a λφ4 theory with weak coupling λ and spontaneously broken symmetry. The amplitude of this process is known to have an n! growth when the produced bosons are exactly at rest. Here it is shown that for n1/λ the process goes through "quantum bubbles," i.e., quantized droplets of a different vacuum phase, which are nonperturbative resonant states of the field φ. The bubbles provide a form factor for the production amplitude, which rapidly decreases above the threshold. As a result the probability of the process may be heavily suppressed and may decrease with energy E as exp(-const×Ea), where the power a depends on the number of space dimensions. Also discussed are the quantized states of bubbles and the amplitudes of their formation and decay.