Time as a dynamical variable

G. Bhamathi, E. C. G. Sudarshan

Hou Y. Yau

Jan Hilgevoord

The meaning and scope of a recent type of uncertainty relation of a very general character are elucidated using the notions of time-indicating dynamical variables (clock variables) and place-indicating dynamical variables (position variables). It is shown that if the total energy (momentum) of a system is certain, all time-indicating (place-indicating) dynamical variables are completely uncertain. The quantum clock is discussed as an illustration of the energy–time uncertainty relation. The relations can be successfully applied to the thought experiments that Einstein introduced into his debate with Bohr about the uncertainty principle(s) and, in particular, to the famous photon-box experiment. It is shown that due to this general relation the photon box can never serve its purpose, independent of the details of the experiment.

Pedro Pereyra Padilla

Gerard ’t Hooft

Most classical mechanical systems are based on dynamical variables whose values are real numbers. Energy conservation is then guaranteed if the dynamical equations are phrased in terms of a Hamiltonian function, which then leads to differential equations in the time variable. If these real dynamical variables are instead replaced by integers, and also the time variable is restricted to integers, it appears to be hard to enforce energy conservation unless one can also derive a Hamiltonian formalism for that case. We here show how the Hamiltonian formalism works here, and how it may yield the usual Hamilton equations in the continuum limit. The question was motivated by the author's investigations of special quantum systems that allow for a deterministic interpretation. The 'discrete Hamiltonian formalism' appears to shed new light on these approaches.

Julian Schwinger

A measurement is a physical operation in space and in time. The properties of a system are described in relation to measurements at a given time, and no value of the time is intrinsically distinguished from another by the results of measurements on an isolated physical system. The operators symbolizing analogous properties at different times must be related by a unitary transformation. The propagation in time of the disturbance produced by a measurement implies that physical quantities referring to different times are incompatible, in general. The development of the fundamental dynamical variables through an infinitesimal time interval is described by an infinitesimal unitary transformation, which implies first order differential equations of motion for these variables. The necessity of an even number of variables and the possibility of dividing them into two complementary sets also appears for variables of the second kind.