To what extent will your choice of PhD topic restrict your career? Obviously, if you choose to either do a PhD in English literature, or in mathematics, this choice will heavily determine the possible career paths you can later take.

But to what extent does the choice between two different topics within the same discipline influence your career possibilities?


Physics: Quantum computing vs high energy particle physics 
Math: differential topology vs abstract algebra
Economics: business cycle theory vs industrial organization.
Psychology: Evolutionary psychology vs Neuro-psychology


Would such a choice significantly affect your future career, or is there generally such an overlap in the skills you learn doing different topics that you will be able to easily switch between these topics within the same discipline?
Very few people are static in their careers, in and out of academia. My PhD was in the area of molecular simulations in chemical engineering. My post doc was with a physicist working in computational materials, then I took a staff position in environmental engineering doing fluid flow and geochemistry studies, I have since moved into renewable energy applications and biomedical, with side work in algorithm development and kinetics.

All of this is to say, any limits on what direction your career can take are placed by the researcher, not the field of study. If you’re not willing to branch out, your choice of jobs will be limited. If you’re willing to explore, the options are as infinite as your curiosity and salesmanship will allow.