The Classical Bosonic String
Even though we will eventually be interested in a quantum theory of interacting strings, it will turn out to be useful to start two steps back and treat the free classical string. We will set up the Lagrangian formalism which is essential for the path int
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The Classical Bosonic String
Abstract Even though we will eventually be interested in a quantum theory of interacting strings, it will turn out to be useful to start two steps back and treat the free classical string. We will set up the Lagrangian formalism which is essential for the path integral quantization which we will treat in Chap. 3. We will then solve the classical equations of motion for single free closed and open strings. These solutions will be used for the canonical quantization which we will discuss in detail in the next chapter.
2.1 The Relativistic Particle Before treating the relativistic string we will, as a warm up exercise, first study the free relativistic particle of mass m moving in a d -dimensional Minkowski spacetime. Its action is simply the length of its world-line1 Z S D m
s1 s0
Z dx D m
1
0
1=2 dx dx d ; d d
(2.1)
where is an arbitrary parametrization along the world-line, whose embedding in d -dimensional Minkowski space is described by d real functions x ./; D 0; : : : ; d 1. We use the metric D diag.1; C1; : : : ; C1/. The action (2.1) is invariant under -reparametrizations ! ./. Q Under infinitesimal reparametrizations ! C ./, x transforms like ıx ./ D ./ @ x ./ :
(2.2)
1 It is easy to generalize the action to the case of a particle moving in a curved background by simply replacing the Minkowski metric by a general metric G .x/.
R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 2, © Springer-Verlag Berlin Heidelberg 2013
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2 The Classical Bosonic String
The action is invariant as long as .0 / D .1 / D 0. The momentum conjugate to x ./ is p D
@L xP D mp ; @xP xP 2
(2.3)
where xP D @ x and xP 2 D x x . Equation (2.3) immediately leads to the following constraint equation p 2 C m2 D 0 :
(2.4)
Constraints which, as the one above, follow from the definition of the conjugate momenta without the use of the equations of motion are called primary constraints. @p Their number equals the number of zero eigenvalues of the Hessian matrix @xP D 2 @ L which, in the case of the free relativistic particle, is one, the corresponding @xP @xP eigenvector being xP . The absence of zero eigenvalues is necessary (via the inverse function theorem) to express the ‘velocities’ xP uniquely in terms of the ‘momenta’ 2 and ‘coordinates’, p and x . Systems where the rank of @xP@ @LxP is not maximal, thus implying the existence of primary constraints, are called singular. P For singular systems the -evolution is governed by the Hamiltonian H D Hcan C ck k , where Hcan is the canonical Hamiltonian, the k an irreducible set of primary constraints and the ck are constants in the coordinates and momenta. This is so since the Hamiltonian is well defined only on the submanifold of phase space defined by the primary constraints and can be arbitrarily extended off that submanifold. For the free relativistic particle w
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