I was reading Schwartz's book on QFT. In chapter 14.5 at p.267, while speaking about path integral he says:
[...] the path integral (and field theories more generally) is only known to exist (i.e. have a precise mathematical definition) for free theories, and for $\phi^4$ theory in two or three dimensions. $\phi^4$ theory in five dimensions is known not to exist. In four dimensions, we do not know much, exactly. We do not know if QED exists, or if scalar $\phi^4$ exists, or even if asymptotically free or conformal field theories exist. In fact we do not know if any field theory exists, in a mathematically precise way, in four dimensions.
Is this because we have to deal with renormalisation? How can we say that we are not sure if a field theory exists in four dimensions, although we can get results from our calculation?