Partial Hamiltonian formalism, multi-time dynamics and singular theories

TL;DR

This paper introduces a partial Hamiltonian formalism for singular classical theories that avoids constraints, develops multi-time dynamics, and connects to Dirac constraints and quantization.

Contribution

It formulates a novel partial Hamiltonian approach for singular theories, enabling multi-time dynamics and a new perspective on constraints and quantization.

Findings

Developed a partial Hamiltonian formalism without involving constraints.

Derived multi-time Hamilton-Jacobi equations and equations of motion.

Connected the formalism to Dirac constraints and discussed quantization.

Abstract

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 reduced 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 $F_{\alpha\beta}$ (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.