In string theory, we quantize the two dimensional field theory on the string's worldsheet. I have a question about this kind of quantization of string theory: did we have similar theory for point-like particle theory, i.e., we quantize the classical theory on worldline, then we get the usual quantum mechanics?

If the string theory is the two dimensional quantum field theory on its worldsheet, then what does the momentum operator measure? Does it measure the expected momentum from initial time to final time represented by the worldsheet (because usually the moment operator measure the moment of particle at some specified time)?


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Yes, one may discuss ordinary quantum mechanics based on point-like particles using the 0+1-dimensional quantum field theory (i.e. quantum mechanics) defined on the world line. This is actually how several textbooks of string theory, including Polchinski's textbook, begin.

The Green's functions $G(x,y)$ may be calculated as a traditional "Feynman sum over all world lines" and they get mapped to the transition amplitude for the particle to get from one point to another. However, to make the point-like particles interact, like in a QFT, one needs singular world lines with junctions – that look just like the vertices in Feynman diagrams or the whole diagrams. It's no coincidence: the Feynman diagrams do describe the topology of the world lines in the relevant histories in which the point-like particles merge and split!

An advantage of string theory, and a reason why it's ultimately UV finite, is that the world sheets are smooth and non-singular even if the particles' (strings') interactions are allowed i.e. if the strings join and split.

The world sheet momentum along the $\sigma$ direction of the world sheet is known as $L_0-\tilde L_0$, and this operator vanishes. Only states whose eigenvalue under this operator is zero are physical states of a string (there are additional conditions).

There is a simple reason why the momentum has to be zero. The theory on the world sheet is a 2-dimensional theory of gravity – because the choice of the coordinates $(\sigma,\tau)$ on the world sheet is and has to be unphysical (diffeomorphism symmetry). And like in other theories of gravity, one may derive something like Einstein's equations. In $d=2$ dimensions, the Einstein tensor $R_{ab}-Rg_{ab}/2$ is identically equal to zero, so Einstein's equations reduce to $$ T_{ab} = 0$$ which also implies that the momentum density $T_{++}, T_{--}$ and the total momentum as well, among other things, has to vanish.

On the basis of "discrete excitations of the string", the condition $L_0-\tilde L_0=0$ for the vanishing of the total momentum gets translated to the condition that the "total excitation of the left-moving quanta" is the same as it is for the "right-moving quanta" for a closed string (additive shifts may arise in this equation due to the sum of all integers etc.). For open strings, the corresponding condition doesn't exist because the translational symmetry along the $\sigma$ direction of the world sheet is explicitly broken by the open string's endpoints.

A general theory of Einstein-like gravity would always be ill-defined at the quantum level due to various divergences. The world sheet theory describing perturbative string theory is a special 1+1-dimensional theory of quantum gravity that avoids this problem because it doesn't contain any physical degrees of freedom of the metric tensor field at all. It's because the three components of the metric tensor $h_{\tau\tau},h_{\tau\sigma},h_{\sigma\sigma}$ may be locally set to predetermined non-singular values by 3 parameters that are functions of $\sigma,\tau$ as well, namely by two parameters for a diffeomorphism and one parameter for the Weyl scaling (different for each point). That's why the Weyl symmetry is necessary for string theory's consistency in the spacetime-Lorentz-covariant formalism. The conformal symmetry is the residual symmetry that's left from the diffeomorphism and Weyl symmetries even after we gauge-fixed a privileged form of the world sheet metric. Conformal transformations are those that preserve the angles i.e. preserve the metric up to a Weyl scaling (which may also be done because it's a symmetry).


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