Dirac bracket

Suppose we have a constrained system represented by a Lagrangian and the set of constraints $\phi_{m}$ $\approx$ 0. While computing the Hamiltonian and the Poisson brackets of the system, the constraints must be taken care of. This is done by the Dirac procedure, where instead of Poisson brackets, the brackets are the $Dirac$ $brackets$, defined by

$$\{f,g\}^* \equiv \{f,g\}-\{f,\phi_{m}\}C^{-1}_{mk}\{\phi_{k},g\}$$ (44)

where the constraint matrix $C_{mk} \equiv \{\phi_{m},\phi_{k}\}$ is assumed to be invertible.

A constrained Hamiltonian system is thus defined by a given Lagrangian, together with the Dirac brackets corresponding to the set of constraints. The physical phase space of the system is the constraint manifold, which is a submanifold of the ``naive'' phase space spanned by the $p$'s and $q$'s. The ``flat'' symplectic structure on the naive phase space induces a non-trivial symplectic structure on the physical phase space. This non-trivial symplectic structure is given by the Dirac brackets.

When the second class constraints have been taken care of by the Dirac brackets, only first class constraints remain. The Hamiltonian of this system is the extended Hamiltonian

$$H_{ext} = H'+\lambda_r \psi_r \approx H'$$ (45)

where $\lambda_r$ are Lagrange multipliers. The Hamiltonian together with the constraints obey the Poisson bracket algebras

${\hspace{60mm}}$ $\{\psi_n,\psi_m\} = U_{nm}^r\psi_r$ ${\hspace{7mm}}$ $\{\psi_n,H\} = V_n^r\psi_r$,

where $H'=H+c^m\psi_m$, $c^m \approx U^m+\lambda^rV_r^m$, and $U^m$ are particular solutions to the inhomogeneous equation $\{\psi_{j},H\}+ U^m\{\psi_{j},\psi_{m}\}$ $\approx$ 0. Likewise, the $V_r^m$ are the most general solution of $V^m\{\psi_{j},\psi_{m}\}$ $\approx$ 0. The phase space action is then

$$S = \int dt[ \dot{q}p - H - \lambda_r \psi_r],$$ (46)

and the corresponding equations of motion are

${\hspace{60mm}}$ $\dot{q} = \{q,H\} + \lambda_n\{q,\psi_n\}$ ${\hspace{3mm}}$
${\hspace{60mm}}$ $\dot{p} = \{p,H\} + \lambda_n\{p,\psi_n\}$ ${\hspace{3mm}}$
${\hspace{60mm}}$ $\psi_n(q,p) = 0$.

The Lagrange multipliers $\lambda_r$ have no equations of motion. The time evolution of the equations of motion for the $q$'s and $p$'s are therefore partly arbitrary, since the $\lambda_r$'s enter into their equations of motion.

In a gauge theory, given the initial conditions, the equations of motion do not determine all the dynamical variables for all times. This is because it is always possible to change frame and keep the initial conditions fixed, whereby a time evolution different from the first one will evolve. So in gauge theory the general solution of the equations of motion contains arbitrary functions of time.