(o) Classical foreplay

One may adopt a somewhat old fashioned point of view and start with a classical Lagrangian description for a particular model and attempt to quantize it. In general one encounters the usual problems of QFT, namely renormalization and is therefore thrown back to the well-known perturbative approach. With regard to the above mentioned motivation such an approach is not very progressive as it ignores the special features which take place in IQFT. Nonetheless, some accidental magic happens for special integrable models in which one luckily bypasses the renormalization procedure. For instance for affine Toda field theories (ATFTs) [ 103 ] related to simply laced Lie algebras the classical mass ratios are preserved after renormalization [3 ,76 ]. This is not the case for theories related to non-simply laced algebras, which is just one illustration of the limitation of this approach. Similar things can be said about the classical fusing coefficient [ 76 ] and higher couplings [60 ]. Concerning the solutions of the classical equations of motion one can exploit some old result of Eisenbud and Wigner [ 45 ] which relates the time-delays in multisoliton solutions to a semi-classical quantum scattering matrix. Since for the sine-Gordon model these results could be extrapolated [47 ] successfully, the computation for generic ATFT [ 65 ] are useful with regard to more generic theories. Nonetheless, this approach requires a lot of guesswork when aiming at a full QFT. A similar characterization also holds for other quantities which can be computed classically. For instance, in the context of integrable models it is well known that classically one may encode the equations of motion into a zero curvature condition (Lax-pair), which allows directly for a computation of conserved quantities. Needless to say that the quantization of these charges also does not bypass the procedure of renormalization, even in 1+1 dimensions. Extrapolations to higher dimensions are of course doomed to fail as they will be in conflict with restrictions imposed by the aforementioned Coleman-Mandula theorem. Despite this somewhat pessimistic view on classical Lagrangians, they do provide some form of intuition and are useful as a starting point for seeking the corresponding quantum field theory, although one should always keep in mind that a consistent formulation of QFT does not require any classical theory (examples for this are several CFTs and also the models proposed in [ 70 ,22 ,29 ]) and many results on the classical level, even though mathematically often appealing, remain useless on the quantum level.