Wordline formalism and QCD The worldline formalism of QFT (as I understand it) is a first quantisation approach to particle physics. We consider '0+1 dimensional QFT happening on the worldline of the particle' in the same way that in string theory we say that there is a 1+1 CFT living on the string worldsheet.
It has been shown that the Feynman diagrams corresponding to scattering particles in perturbative treatment of QFT (a.k.a the second quantization approach) can be reproduced in this worldline formalism, and so according to https://ncatlab.org/nlab/show/worldline+formalism these formalism are completely equivalent, and neither is more fundamental. I have also seen answers on this site saying that neither fields nor particles are more fundamental, which one you use is a matter of whichever is easiest for a given problem.
This all makes sense perturbative where you normally assign a single particle to a single field but we know in strongly interacting QFT's this simple picture breaks down. Now is there a nonperturbative generalization of the worldline formalism for QCD or other strongly interacting field theories? How does non-perturbative effects emerge in the worldline formalism? If the field theoretic approach can explain/demonstrate such behavior but the worldline formalism can not then surely the field theoretic approach should be considered more fundamental?
This line of thought came about because I was wondering how perturbative string theory scattering gave rise to feynman diagrams at low energies. It makes sense from a weakily interacting perspective, in the infinite tension limit the worldsheet diagrams look like worldline diagrams (when we are not sensitive to the stringy nature of the particles), but I didn't see how strongly interacting effects emerge (which seem manifestly as a consequence of a more general field theoretic formalism).
 A: Alright so I think that yes one may be able to consider the Feynman formalism and worldline QFT as equivalent just due to the fact that any Feynman diagram can be constructed from a specific worldline (As far as I can remember there is a chapter in David Skinners Advanced Quantum field theory notes discussing some of the subtleties of this and I can't remember what conclusion he comes to at the end) However both of these theories (as the source you link states) are perturvative theories by design. As such neither of them captures non-perturbative effects. So I believe you should be able to generalize the formalism to QCD however only in the high energy/low coupling limit. But exactly the same is true for Feynman diagrams anyhow as they only capture these limits correctly. So e.g. for the strong coupling limit of QCD both formalisms break down fully (as far as I'm aware it is expected that in this limit the Quantum field theory does not have a Lagrangian formulation though do not trust this information as I can't remember the reasoning)
So field theories are more powerful as they can campture non perturbavative effects (e.g. Instanton contributions) while worldline QFT can't (at least not in the same obvious way).
A: The worldline formalism in QFT has been nicely clarified in Strassler's 1992 paper Field Theory Without Feynman Diagrams:
One-Loop Effective Actions . The lessons that one can take from this paper is

*

*For free theories the worldline formalism is fully equivalent to the field-Lagrangian formalism.

*For interacting theories, the perturbative expansion of the field-Lagrangian path integral can be used to derive the perturbative worldline path integral and vice versa.

That is, if you have all the terms of the perturbative worldline formulation, you can usually derive the field Lagrangian, at least if the Lagrangian is a local function of the fields with a convergent expansion in the interaction strength parameter.
Note that this is a different statement from statements about the convergence of the perturbative expansion of the QFT dynamics. Consider the following example. If you perturbatively compute the partition function of a particle in a potential $V(x) = x^2  - 2 x^4 + x^6$ around the harmonic potential ground state $\langle x\rangle = 0$, you will miss the contribution of the instantons around the minima $ x = \pm 1$ (tunelling to the other equilibria at equal energy level) in the naive approach. However, the potential function itself expanded about $x= 0$ has a globally convergent analytical expansion/Taylor series.
Similarly, the QCD Lagrangian has a convergent analytical expansion in the interaction strength, even though the dynamics themselves exhibit well-known non-perturbative effects such as quark confinement. As such, it can be obtained from the perturbative expansion of the partition function in the worldline formulation, at least if all terms of the perturbative expansion are known. In the next step, one can discretize the Lagrangian to obtain the Lattice QCD formulas such as the Wilson plaquette action. So, in principle, all the perturbative and non-perturbative results of QCD can be derived from the worldline formalism for QCD and neither of the formulations can be considered as more fundamental.
