Abstract
In this chapter the quantization of the bosonic string is discussed. This leads to the notion of a critical dimension (d= 26) in which the bosonic string can consistently propagate. Its discovery was of great importance for the further development of string theory. We will discuss both the quantization in so-called light-cone gauge and the covariant path integral quantization, which leads to the introduction of ghost fields.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Our notation does not distinguish between classical and quantum quantities and between operators and their eigenvalues. Only when confusion is possible we will denote operators by hatted symbols.
- 2.
These ghosts are not to be confused with the Faddeev-Popov ghosts of Sect. 3.4.
- 3.
In Chap. 4we will perform a Wick rotation to a Euclidean world-sheet and \(z \in \mathbb{C}\)with \(\overline{z}\)its complex conjugate.
- 4.
Note that the situation is very similar to the one in the quantization of electromagnetism. There we can only impose the positive frequency part of the gauge condition \(\partial \cdot A = 0\)on physical states. This suffices to get \(\langle \mathrm{phys}^\prime\vert \partial \cdot A\vert \mathrm{phys}\rangle = 0\). In this restricted Hilbert space longitudinal and scalar photons decouple.
- 5.
The proper way to go to light-cone gauge would be to use the local symmetries on the world-sheet to fix components of the world-sheet metric and X + . One then has to show that no propagating ghosts are introduced in this process of gauge fixing. Rather than going through these steps we take this a posteriori justifiable short-cut.
- 6.
Here and below a summation over \(i = 2,\ldots,d - 1\)is implied.
- 7.
The group Sp(2N) is defined as the group generated by 2N×2N-matrices with MJM T = Jwhere \(J = \left (\begin{array}{*{10}c} 0 & \nVdash \\ -\nVdash & 0 \end{array} \right )\)and \(\nVdash \)is the N×Nunit matrix. In our conventions Sp(2N) has rank N.
- 8.
The presence of a massless spin two particle is a priori not sufficient to have gravity. We will show in the last chapter that at low energies it couples to matter and to itself like the graviton of general relativity.
- 9.
Later we will compute scattering amplitudes as correlation functions with this partition function.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Blumenhagen, R., Lüst, D., Theisen, S. (2012). The Quantized Bosonic String. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-29497-6_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29496-9
Online ISBN: 978-3-642-29497-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)