In the theory of a classical bosonic string, we have expressions like:

$$ \{\alpha^\mu_m,\alpha^\nu_n \} = - i m \delta_{m,-n} \eta^{\mu \nu} $$

were $\alpha^\mu_n$ are the Fourier modes of the string. How is this Poisson bracket defined? The definition from analytical mechanics involves partial derivatives with respect to the generalized coordinates and momenta. Since the modes $ \alpha^\mu_n $ are constants, their partial derivatives are zero, so there must be something I'm missing here.


1 Answer 1


Concerning OP's last sentence (v1), the Fourier modes $\alpha^{\mu}_{m}$ are (some of) the fundamental variables of the string. Phrased equivalently, the Poisson bracket reads

$$ \{F(\alpha),G(\alpha)\}~=~ \sum_{m\in\mathbb{Z}} \frac{\partial F(\alpha)}{\partial \alpha^{\mu}_{m}} (-im \eta^{\mu\nu}) \frac{\partial G(\alpha)}{\partial \alpha^{\nu}_{-m}}.$$

The above Poisson bracket can be derived via the usual Legendre transformation from Lagrangian to Hamiltonian formulation, and by imposing CCR. This is done in many textbooks on string theory.


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