The Quantized Bosonic String
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 stri
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The Quantized Bosonic String
Abstract In this chapter the quantization of the bosonic string is discussed. This leads to the notion of a critical dimension (d 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.
3.1 Canonical Quantization of the Bosonic String In this section we will discuss first the quantization of the bosonic string in terms of operators, i.e. we will consider the functions X .; / as quantum mechanical operators. This is equivalent to the transition from classical mechanics to quantum mechanics via canonical commutation relations for the coordinates and their canonically conjugate momenta. We replace Poisson brackets by commutators according to f ; gP:B: !
1 Π; : i
(3.1)
In this way we obtain for the equal time commutators1 ŒX .; /; XP . 0 ; / D 2 i ˛ 0 ı. 0 / ; ŒX .; /; X . 0 ; / D ŒXP .; /; XP . 0 ; / D 0 :
(3.2)
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.
R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 3, © Springer-Verlag Berlin Heidelberg 2013
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3 The Quantized Bosonic String
The Fourier expansion coefficients in Eqs. (2.72a), (2.72b), etc. are now operators for which the following commutation relations hold: Œx ; p D i ; Œ˛m ; ˛n D Œ˛ m ; ˛ n D m ımCn;0 ;
Œ˛ m ; ˛n D 0 :
(3.3)
For the open string the ˛ m are, of course, absent. The reality conditions (cf. (2.73)) become hermiticity conditions / D ˛m ; .˛m
.˛ m / D ˛ m
(3.4)
which follow from requiring hermiticity of the operators X .; /. If we rescale the ˛m ’s and define, for m > 0, am D p1m ˛m ; .am / D p1m ˛m , then the am sat
isfy the familiar harmonic oscillator commutation relations Œam ; .an / D ım;n . One defines the oscillator ground state as the state which is annihilated by all positive modes ˛m ; m > 0. They are annihilation operators while the negative modes ˛m m > 0 are creation operators. This does not yet completely specify the state; we can choose it to be an eigenstate of the center of mass momentum operator with eigenvalue p . If we denote this state by j0I p i, we have j0I p i D 0 ˛m
for m > 0 ;
pO j0I p i D p j0I p i :
(3.5)
The number operator for the m’th mode (m > 0) is NO m DW ˛m ˛m WD ˛m ˛m , where the normal ordering symbol which, like the number operator, is defined w.r.t. the vacuum j0i, instructs us to put annihilation operators to the right of creation operators. The number operator satisfies NO n ˛˙m D ˛˙m .Nn ın;m /. From now on we drop the hat on NO and denote by N both
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