# Quantum theory of a single worldline

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In introductions to string theory it is common to start with the quantum theory of a single worldline by quantization of the Polyakov action for some $0$-brane.

What does this theory correspond to, exactly? I know it's not the theory of a single point particle, since those are not well defined for a variety of reasons, and from some arguments of ACuriousMind$^{[1]}$ it may not have a position operator defined. I also know that the full theory of worldlines can be considered as equivalent to QFT, with every worldline diagram corresponding to a Feynman diagram.

Would the theory of a single worldline correspond to the theory of a single QFT particle state because of this?

$[1]$ private correspondance

This post imported from StackExchange Physics at 2017-10-11 16:29 (UTC), posted by SE-user Slereah
@ACuriousMind please elaborate. What does it mean to have a position operator? Doesn't passing to the Gelfand triple given by the Schwartz space on $R^4$ (together with its algebraic dual and $L_2[R^4]$) make the position operator well-defined? I can also promote the constraint to an operator acting on this algebraic dual space to give the reparametrization invariant dynamics. Or are we talking about something completely different?
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