This work reflects on mechanics as an epistemological framework on the state of a physical system to regard dynamics as the distribution of mechanical properties over spacetime coordinates. The resulting distribution is taken to be the partition function of the relevant physical quantities over a space-time parametrized by coordinates. The partition yields a probabilistic interpretation that, based on Feynman’s path integral formulation, leads to a dynamical law that generalizes the Schrödinger equation. A variety of systems can be put into the form proposed here, including particles in potentials, as well as matter and interaction fields. The main advantage of the proposed framework is that it presents the space of properties separately from that of the space of coordinates, whereas the dynamical law can be interpreted as the equation of two differential structures, one from each of these spaces. The resulting framework shows possibilities to further study physical quantities that relate directly to the space-time coordinates, whose dynamics is best described in thermodynamical, rather than Hamiltonian, terms. A notable case being the theory of general relativity.
This work reflects on mechanics as an epistemological framework on the state of a physical system to regard dynamics as the distribution of mechanical properties over spacetime coordinates. The resulting distribution is taken to be the partition function of the relevant physical quantities over a space-time parametrized by coordinates. The partition yields a probabilistic interpretation that, based on Feynman’s path integral formulation, leads to a dynamical law that generalizes the Schrödinger equation. A variety of systems can be put into the form proposed here, including particles in potentials, as well as matter and interaction fields. The main advantage of the proposed framework is that it presents the space of properties separately from that of the space of coordinates, whereas the dynamical law can be interpreted as the equation of two differential structures, one from each of these spaces. The resulting framework shows possibilities to further study physical quantities that relate directly to the space-time coordinates, whose dynamics is best described in thermodynamical, rather than Hamiltonian, terms. A notable case being the theory of general relativity.
This work reflects on mechanics as an epistemological framework on the state of a physical system to regard dynamics as the distribution of mechanical properties over spacetime coordinates. The resulting distribution is taken to be the partition function of the relevant physical quantities over a space-time parametrized by coordinates. The partition yields a probabilistic interpretation that, based on Feynman’s path integral formulation, leads to a dynamical law that generalizes the Schrödinger equation. A variety of systems can be put into the form proposed here, including particles in potentials, as well as matter and interaction fields. The main advantage of the proposed framework is that it presents the space of properties separately from that of the space of coordinates, whereas the dynamical law can be interpreted as the equation of two differential structures, one from each of these spaces. The resulting framework shows possibilities to further study physical quantities that relate directly to the space-time coordinates, whose dynamics is best described in thermodynamical, rather than Hamiltonian, terms. A notable case being the theory of general relativity.
This work reflects on mechanics as an epistemological framework on the state of a physical system to regard dynamics as the distribution of mechanical properties over spacetime coordinates. The resulting distribution is taken to be the partition function of the relevant physical quantities over a space-time parametrized by coordinates. The partition yields a probabilistic interpretation that, based on Feynman’s path integral formulation, leads to a dynamical law that generalizes the Schrödinger equation. A variety of systems can be put into the form proposed here, including particles in potentials, as well as matter and interaction fields. The main advantage of the proposed framework is that it presents the space of properties separately from that of the space of coordinates, whereas the dynamical law can be interpreted as the equation of two differential structures, one from each of these spaces. The resulting framework shows possibilities to further study physical quantities that relate directly to the space-time coordinates, whose dynamics is best described in thermodynamical, rather than Hamiltonian, terms. A notable case being the theory of general relativity.
This work reflects on mechanics as an epistemological framework on the state of a physical system to regard dynamics as the distribution of mechanical properties over spacetime coordinates. The resulting distribution is taken to be the partition function of the relevant physical quantities over a spacetime parametrized by coordinates. The partition yields a probabilistic interpretation that, based on Feynman's path integral formulation, leads to a dynamical law that generalizes the Schrödinger equation. A variety of systems can be put into the form proposed here, including particles in potentials, as well as matter and interaction fields. The main advantage of the proposed framework is that it presents the space of properties separately from that of the space of coordinates, whereas the dynamical law can be interpreted as the equation of two differential structures, one from each of these spaces. The resulting framework shows possibilities to further study physical quantities that relate directly to the spacetime coordinates, whose dynamics is best described in thermodynamical, rather than Hamiltonian, terms. A notable example is the theory of general relativity, in which the case of a scalar field in a Robertson-Walker metric is explored.
This paper elucidates the dual structure of the Schrödinger dynamics in two correlated stages: (1) We first derive the real-valued Schrödinger equation from scratch without referring to classical mechanics, wave mechanics, nor optics, and thereby attain a concrete and clear interpretation of the Schrödinger (wave) function. Beginning with a factorization of the density distribution function of the particles to two component vectors in configuration space, we impose very simple conditions on them such as translational invariance of space-time and the conservation of flux under a given potential function. A real-valued path-integral is formulated as a Green function for the real-valued Schrödinger equation. (2) We then study a quantum stochastic path dynamics in a manner compatible with the Schrödinger equation. The relation between them is like the Langevin dynamics with the diffusion equation. Each quantum path describes a \textquotedblleft trajectory\textquotedblright\ in configuration space representing, for instance, a singly launched electron in the double-slit experiment that leaves a spot one by one at the measurement board, while accumulated spots give rise to the fringe pattern as predicted by the absolute square of the Schrödinger function. We start from the relationship between the Ito stochastic differential equation, the Feynman-Kac formula, and the associated parabolic partial differential equations, to one of which\ the Schrödinger equation is transformed. The physical significance of the quantum intrinsic stochasticity and the indirect correlation among the quantum paths and so on are discussed. The self-referential nonlinear interrelationship between the Schrödinger functions (regarded as a whole) and the quantum paths (as its parts) is identified as the ultimate mystery in quantum dynamics.
The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.