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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?

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There are a couple of points to be precise:

  • in four spacetime dimensions there is no scalar relativistic interacting field theory that can be rigorously defined (i.e. in which the unitary dynamics can be constructed), at least for the moment. This does not mean it is not possible, but we have not the mathematical tools to do it. The physical calculations are perturbative, and even if they strongly suggest that a mathematical theory is definable, there is no rigorous result as your book says.

  • The triviality of the scalar field theories in higher dimensions is due to a result of Aizenmann and Frohlich, however it is only a result that holds when you take the limit of lattice theories (and so in principle it does not exclude to obtain field theories by other methods)

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