Martin Gardner (an admirer of Russell) may have been thinking, in part, of Russell's example when he wrote this about the length of the Loch Ness monster: books.google.com/…

To be precise, in quantum field theory, "regularization" is the replacement of divergent integrals over unconstrained intermediate virtual states with finite expressions that depend on an adjustable cutoff parameter. "Renormalization" is the tuning of the cutoff parameters and the other underlying parameters of the theory to match a set of physical observables for the theory. The variation of the underlying parameters according to the momentum scale at which those physical observables are chosen is the Gellman-Low version of the "renormalization group."

@StevenStadnicki Why would rep-tile tilings be aperiodic? I mean, I suppose they could be, but (non-fractal) rep-tile tilings are typically periodic.

+1 for the correct spelling of "Gordan"

It seems like this should be derivable as a special case from the usual analysis of Hill's equation.

I think there is a "not" missing from this definition somewhere.

You can see the issue with the Fock state by looking at the Lee Model, which a toy model that has an exactly solvable spectrum, yet which still requires renormalization. If you try to set up the Fock space at the beginning, you will find that it does not really correspond to the state space of the theory, because the renormalization changes this fundamentally.

There is really nothing fundamental about making a connection between quantum mechanics and free quantum field theory. The relationship between the two is not subtle, and it can be expressed rigorously in many different fashions. It is only when there are nontrivial interactions that quantum field theory becomes a fundamentally different "kind" of thing from quantum mechanics, so you are never going to get the key insights into what makes quantum field theory interesting in its own right by looking at free theories. And how to treat the interacting cases with rigor is, generally, unknown.

Just a word of warning: Different authors can have very different views on what the fundamental underlying mathematical structure is in quantum mechanics, and it can be hard to find a balanced treatment. Part of the issue comes down to the infamous measurement problem, and the associated question of whether mixed states really "exist."

@YCor I fully admit that this is not a particularly interesting answer. However, it is absurd to claim that it is not actually an answer to the question (as appearing in the title or in the body text: "Is it always the case that there are more conjugacy classes in the kernel of $\phi$ than conjugacy classes not in the kernel of $\phi$?").