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21 ; . 16) Toeholds. We observe at this stage that, if the system is invariant under Boundary reparametrizations, it is insensitive to the place on the Boundary where the Area variation takes place and the Area Functional Derivative can be traded with an ordinary (partial) derivative. 2). We observe that under the term “variation”, we understand not only a variation of the fields defined on the fixed domain Σ, but also a variation of Σ itself. As we have already pointed out, this is related to the dynamical role which is assigned to the Boundary S1 ≈ Γ = ∂Σ in this formulation: we do not push it to infinity, or consider it as a non–dynamical, purely geometrical quantity, on the contrary, the physical, evolving object is the Boundary itself.

Some new features of string Dynamics can be better understood in this way. In particular it turns out that the Quantum Dynamics of the object is, consistently described as a Shadow Dynamics of areas parametrized by an areal time. Eguchi, who was the first in suggesting such a non–standard string quantization method, and give sound motivations for the choice of the Lagrangian density. This is the subject of the next two sections; after, we will apply all that to the Functional Schr¨ odinger Quantization of the bosonic string; the possible relations between this approach and the more traditional one will be pointed out in chapter 9.

5) for the parametrization of the boundary. We will also call Y µ = Y µ (s) a Parametrization of the Classsical Closed Bosonic String. Moreover we set C = C˜ ◦ ξ . Of course, at least in the computations related to the Classical Dynamics of the objects, we always assume that all the defined maps are sufficiently regular in such a way that all the derivations and integrations can be unambiguously carried out. The problem of defining the most adequate functional class for the desciption of the Quantum Dynamics of the objects will be tackled in a more deep way in chapter 10 and in two appendices, B and C.

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Boundary versus Bulk Dynamics of Extended Objects and the Fractal Structure of Quantum Spacetime by Ansoldi, S

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