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  Quantization of strings, string Fock space and transition to QFT

+ 2 like - 0 dislike

I am not an expert of string theory and am quiet uncertain about these basic ideas of string theory I am going to ask and would appreciate some hints of more experienced physicists.

What I am trying to understand is how string theory can describe particles as quantum fields as it is done by quantum field thoery. After all it is said that QFT can be seen in low energy string theory.

What I would like to do is to compare second quantization in QFT to string theory. In QFT, if the wave function of a quantum mechanical particle can be described as a superposition of a complete set of wave functions $\{\phi_i(x)\}$, then we can define creation (annihilation) $a_i^+$($a_i$) operators which generate symmetrized/antisymmetrized products of corresponding one-particles states. So, by this construction we have for example $|100\dots\rangle = a_1^+|0\rangle = |\phi_1\rangle$ where again $\langle x|\phi_1\rangle= \phi_1(x)$. The commutation relations for $a_i^+$ and $a_i$ follow by construction, this is then called canonical quantization. Finally, changing the basis from $\{|\phi_i\rangle\}$ to $\{|x\rangle\}$ we obtain the field operators $\psi(x) = \sum_i\phi_i(x)a_i$ and $\psi(x)^+ = \sum_i\phi_i^*(x)a_i^+$, such that $\psi(x)^+|0\rangle = |x\rangle = \sum_i\phi_i^*(x)|\phi_i\rangle$. So, to sum up, field operators $\psi(x)^+$ create superposition states with a probability distribution which is equal to a delta function: $$ \langle x'|x\rangle = \sum_i\psi_i(x')\psi_i^*(x) = \delta(x-x')$$ And again, commutation relations for field operators follow by definition.

Now in string theory, the coordinates $(\sigma,\tau)$ on the world sheet which parametrize the embedding $X^\mu(\sigma,\tau)$ of the string into the space-time play the role of space-time coordinates $(x,t)$ in QFT, and the embedded string $X^\mu(\sigma,\tau)$ plays the same role as field operators $\psi(x)$, and with these identifications the quantization is done along the same lines. But with this construction it is not clear to me, what these operators $X^\mu$ really represent. If in QFT $\psi(x)|0\rangle$ was a localized wave function of a one-particle state, what is $X^\mu(\sigma,\tau)|0\rangle$? A wave-function localized on the world-sheet? How can we then identify something that lives on some fictitions two-dimensional parameter space $(\sigma,\tau)$ with particles in 4-dimensional space-time? And what is the Fock space and the states therein in string theory? If someone tells me that something like $\alpha^\mu|0\rangle$ ($\alpha^\mu$ being the modes of $X^\mu$) can for example be seen as a photon state, the only thing that I see in common with the photon is the vector index $\mu$. Photons that I know are bosons described by a probability distribution over the space-time. In this sense, how can I make the identification with $\alpha^\mu|0\rangle$?

I would very much appreciate any help to disentangle these ideas!

This post imported from StackExchange Physics at 2014-06-03 16:40 (UCT), posted by SE-user Stan
asked May 28, 2014 in Theoretical Physics by Stan (60 points) [ no revision ]

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