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Using the deWitt condensed notation, we formally invert components of the 2-form $$\Omega~=~\frac{1}{2} \mathrm{d}x^I \Omega_{IJ} \wedge \mathrm{d}x^J \tag{2.6.1}$$ into the Poisson bracket $$\frac{[\hat{A},\hat{B}]}{i\hbar}~\longleftrightarrow~\{A,B\}_{PB}~=~(\partial_IA) (\Omega^{-1})^{IJ} (\partial_JB).\tag{2.6.2}$$ [Ref. 1 puts Planck's constant \$\hbar=...

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Assuming from the notation $$\dot{x}^i~=~f^i(x,p,t), \qquad \dot{p}_i~=~g_j(x,p,t), \tag{1}$$ that the symplectic structure is the standard canonical symplectic structure $$\omega = \sum_{i=1}^n\mathrm{d}p_i\wedge \mathrm{d}x^i,\tag{2}$$ we get that \begin{align}\mathrm{d}H(x,p,t)- \frac{\partial H(x,p,t)}{\partial t}\mathrm{d}t ~=~&\sum_{i=1}^n\... 2 Hints: The point is to construct the corresponding Hamiltonian formulation of the Lagrangian (2.86). The Dirac-Bergmann analysis reveals a primary constraintC\approx 0.\tag{2.87}$$[When we try to isolate the two velocities \dot{q}^i in the equation for momenta p_i=f_i(q,\dot{q},t), we discover that the two momenta p_i are not independent.] OP has ... 1 Well, given$$ \dot{p} = - \frac{\partial H}{\partial x} \quad \text{and} \quad \dot{q} = \frac{\partial H}{\partial p} $$we have$$ H = - \int dx_{i} \, g_{i}(x,p,t) $$and$$ H = \int dp_i \, f_{i} (x, p, t). $$You deal with the constants of integration using the obvious constraint:$$ - \int dx_{i} \, g_{i} (x, p, t) = \int d p_{i} \, f_{i} (x,...

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The way I inverted the symplectic form was to write it out in coordinate notation. Not keeping track of overall constants (homework exercise!) we can write $$\Omega \sim \int d^2 z du \partial_u \delta A_z \wedge \delta A_{\bar z} \sim \int du du' d^2 z d^2 z' \partial_u \delta(u-u') \delta^2(z-z') \delta A_z(u) \wedge \delta A_{{\bar z}'} (u')$$ In matrix ...

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