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After this semester, I'll have a background up to a first course in QFT (first 5 or 6 chapters of Peskin and Schroeder).

The next step in QFT will be something specific to the Standard Model (Elementary particles, QCD, etc).

Is that needed before going to study String Theory or not?

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If one is a mathematician one can study any set of theories to his/her heart's content and end up with a QED at the end.

Physics is about studying understanding and modeling nature ( Physis in greek) preferably with mathematical models which are predictive of new behaviors.

The standard model (SM) is one such mathematical model . What does this mean? It means that it is a shorthand for a huge number of physical data painstakingly gathered from individual experiments. It fits the results and all the predictions it has given for the LHC are within the experimental errors.

Why are physicists interested in String Theories and not satisfied with the SM? Because they believe that all interactions in nature should be modeled by a single model and the SM describes and predicts data only for three out of the four. Gravity is not within SM. For many years now theoretical physicists have been studying various mathematical models that will include gravity in one unified theory with all four interactions.

String theories are the best candidate for modeling all interactions because their group structures can accommodate the groups of the SM, and thus all known data can be embedded in a string theory model.

If you do not know in depth what your are trying to model, you may be a good technician in mathematical modeling, but not a good physicist with a physicist's intuition. It all depends on your further goals, if you want to be a research physicist or are just studying some physics for other requirements. If the former, yes the SM course is necessary.

The work for theoretical physicists in string theory models lies in finding which specific branch and finally form of a string theory embeds best the SM and predicts new phenomena to be found and studied. If one has no data bank of the known phenomena and the way they are connected mathematically one will have developed no intuition on finding the needle in the haystack of string theory possibilities, or be able to suggest experiments that will lead to a validation for a specific ST model..

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To understand it mathematically, probably not -- the standard model is an empirical model. But it's useful to get a physical appreciation for the theory's usefulness.

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