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Formulation of singular theories in a partial Hamiltonian formalism using a new bracket and multi-time dynamics

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 نشر من قبل Steven Duplij
 تاريخ النشر 2013
  مجال البحث فيزياء
والبحث باللغة English
 تأليف Steven Duplij




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A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The noncanonical generalized velocities satisfy a system of linear algebraic equations, which allows an appropriate classification of singular theories (gauge and nongauge). A new antisymmetric bracket (similar to the Poisson bracket) is introduced, which describes the time evolution of physical quantities in a singular theory. The origin of constraints is shown to be a consequence of the (unneeded in our formulation) extension of the phase space. In this case the new bracket transforms into the Dirac bracket. Quantization is briefly discussed.



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286 - Steven Duplij 2013
We formulate singular classical theories without involving constraints. Applying the action principle for the action (27) we develop a partial (in the sense that not all velocities are transformed to momenta) Hamiltonian formalism in the initially re duced phase space (with the canonical coordinates $q_{i},p_{i}$, where the number $n_{p}$ of momenta $p_{i}$, $i=1,...,n_{p}$ (17) is arbitrary $n_{p}leq n$, where $n$ is the dimension of the configuration space), in terms of the partial Hamiltonian $H_{0}$ (18) and $(n-n_{p})$ additional Hamiltonians $H_{alpha}$, $alpha=n_{p}+1,...,n$ (20). We obtain $(n-n_{p}+1)$ Hamilton-Jacobi equations (25)-(26). The equations of motion are first order differential equations (33)-(34) with respect to $q_{i},p_{i}$ and second order differential equations (35) for $q_{alpha}$. If $H_{0}$, $H_{alpha}$ do not depend on $dot{q}_{alpha}$ (42), then the second order differential equations (35) become algebraic equations (43) with respect to $dot{q}_{alpha}$. We interpret $q_{alpha}$ as additional times by (45), and arrive at a multi-time dynamics. The above independence is satisfied in singular theories and $r_{W}leq n_{p}$ (58), where $r_{W}$ is the Hessian rank. If $n_{p}=r_{W}$, then there are no constraints. A classification of the singular theories is given by analyzing system (62) in terms of $F_{alphabeta}$ (63). If its rank is full, then we can solve the system (62); if not, some of $dot{q}_{alpha}$ remain arbitrary (sign of a gauge theory). We define new antisymmetric brackets (69) and (80) and present the equations of motion in the Hamilton-like form, (67)-(68) and (81)-(82) respectively. The origin of the Dirac constraints in our framework is shown: if we define extra momenta $p_{alpha}$ by (86), then we obtain the standard primary constraints (87), and the new brackets transform to the Dirac bracket. Quantization is discussed.
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