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Now, take the continuum limit by letting £ —► 0, N -+ oo, such that the length L = N£, mass per unit length fi — m/£, and string tension T = k£ are fixed. Then the displacement and energy of the string can be defined in terms of a continuous field 4>{z,t), where

The Lagrangian and Hamiltonian are ke

H = ke + pe where £ and H are the Lagrangian and Hamiltonian densities. In this example, the field function <p{z,t) is the displacement of an infinitesimal mass from its equilibrium position at z. In three dimensions, 4> would be a vector field.

where summation over repeated indices is implied. Assuming that 6 (V^) = Vj (64>) and integrating by parts (assuming boundary terms are zero because the

8 QUANTIZATION OF THE NONRELATIVISTIC STRING where periodicity requires

O.tth

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