Beginning with an entangled state of a time-independent (TI) quantum system coupled to its TI quantum environment, we show that a time-dependent Schrödinger equation (TDSE) for the quantum system alone can be derived in the limit that one of the environment variables becomes a classical variable. In the same limit the TI amplitude of the environment wavefunction becomes the TD amplitude of an eigenfunction expansion of the system TD wavefunction. Similarly, the phase of the TI environment wavefunction goes over into the TD phase of the system amplitude. By considering that more and more environment variables become successively classical, each provides a classical clock to give a multiple-time TDSE for the quantum system. Two examples of two-time TDSE given in the literature are derived. When all clocks are synchronized the multiple-time TDSE reduces to the usual one-time TDSE.
Canonical quantization applied to closed systems leads to static equations, the Wheeler-deWitt equation in Quantum Gravity and the time independent Schrödinger equation in Quantum Mechanics. How to restore time is the Problem of Time(s). Integrating developments are: a) entanglement of a microscopic system with its classical environment accords it a time evolution description, the time dependent Schrödinger equation, where t is the laboratory time measured by clocks; b) canonical quantization of Special Relativity yields both the Dirac Hamiltonian and a self adjoint "time" operator, restoring to position and time the equivalent footing accorded to energy and momentum in Relativistic Quantum Mechanics. It introduces an intrinsic time property τ associated with the mass of the system, and a basis additional to the usual configuration, momentum and energy basis. As a generator of momentum displacements and consequently of energy, it invalidates Pauli's objection to the existence of a time operator. It furthermore complies with the requirements to condition the other observables in the conditional interpretation of QG. As Pauli's objection figures explicit or implicitly in most current developments of QM and QG, its invalidation opens to research the effect of this new two times perspective on such developments.
Jacobi's action principle is known to lead to a problem of time. For example, the timelessness of the Wheeler-DeWitt equation can be seen as resulting from using Jacobi's principle to define the dynamics of 3-geometries through superspace. In addition, using Jacobi's principle for nonrelativistic particles is equivalent classically to Newton's theory but leads to a time-independent Schr\"odinger equation upon Dirac quantization. In this paper, we study the mechanism for the disappearance of time as a result of using Jacobi's principle in these simple particle models. We find that the path integral quantization very clearly elucidates the physical mechanism for the timeless of the quantum theory as well as the emergence of duration at the classical level. Physically, this is the result of a superposition of clocks, which occurs in the quantum theory due to a sum over all histories. Mathematically, the timelessness is related to how the gauge fixing functions impose the boundary conditions in the path integral.
Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density
ρ=R2 coupled to Hamilton's principal function
S. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair
(R,S) into a single complex field. Starting from a completely general complex ansatz
ψ=f(R,S)eig(R,S), and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose
∣κ∣→0 limit reproduces the HJ formulation exactly. Remarkably, when
Re(κ)=0, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.
It is shown that the optics-mechanics analogy which originally led Schrödinger to his famous equation stops at the time-independent level. For potentials involving time explicitly, the time-dependent Schrödinger equation cannot be deduced in a similar manner. Instead, a variational principle for the time-dependent Schrödinger equation is established. It minimizes the total quantum fluctuations of a newly defined Hamilton-Jacobi function about its limiting classical value over space-time, thereby demonstrating the existence of a deeper relationship between classical and quantum mechanics beyond the simple optics-mechanics analogy. This principle is similar in spirit to Feynman's space-time approach to quantum mechanics.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.