A previous blog post, Patterns in Physics, discussed alternative “representations” in physics as akin to languages; an underlying quantum reality described in either a position or a momentum representation. Both are equally capable of a complete description, the underlying reality itself residing in a complex space with the very concepts of position/momentum or wave/particle only relevant in a “classical limit”. The history of physics has progressively separated such incidentals of our description from what is essential to the physics itself. We will consider this for time itself here.
Thus, consider the simple instance of the motion of a ball from being struck by a bat (A) to being caught later at a catcher’s hand (B). The specific values given for the locations of A and B or the associated time instants are immediately seen as dependent on each person in the stadium being free to choose the origin of his or her coordinate system. Even the direction of motion, whether from left to right or vice versa, is of no significance to the physics, merely dependent on which side of the stadium one is sitting.
All spectators sitting in the stands and using their own “frame of reference” will, however, agree on the distance of separation in space and time of A and B. But, after Einstein, we have come to recognize that these are themselves frame dependent. Already in Galilean and Newtonian relativity for mechanical motion, it was recognized that all frames travelling with uniform velocity, called “inertial frames”, are equivalent for physics so that besides the seated spectators, a rider in a blimp moving overhead with uniform velocity in a straight line, say along the horizontal direction of the ball, is an equally valid observer of the physics.
Einstein’s Special Theory of Relativity, in extending the equivalence of all inertial frames also to electromagnetic phenomena, recognized that the spatial separation between A and B or, even more surprisingly to classical intuition, the time interval between them are different in different inertial frames. All will agree on the basics of the motion, that ball and bat were coincident at A and ball and catcher’s hand at B. But one seated in the stands and one on the blimp will differ on the time of travel or the distance travelled.
Even on something simpler, and already in Galilean relativity, observers will differ on the shape of the trajectory of the ball between A and B, all seeing parabolas but of varying “tightness”. In particular, for an observer on the blimp travelling with the same horizontal velocity as that of the ball as seen by the seated, the parabola degenerates into a straight up and down motion, the ball moving purely vertically as the stadium itself and bat and catcher slide by underneath so that one or the other is coincident with the ball when at ground level.
There is no “trajectory of the ball’s motion” without specifying as seen by which observer/inertial frame. There is a motion, but to say that the ball simultaneously executes many parabolic trajectories would be considered as foolishly profligate when that is simply because there are many observers. Every observer does see a trajectory, but asking for “the real trajectory”, “What did the ball really do?”, is seen as an invalid, or incomplete, question without asking “as seen by whom”. Yet what seems so obvious here is the mistake behind posing as quantum mysteries and then proposing as solutions whole worlds and multiple universes(!). What is lost sight of is the distinction between the essential physics of the underlying world and our description of it.
The same simple problem illustrates another feature, that physics works equally well in a local time-dependent or a global, time-independent description. This is already true in classical physics in what is called the Lagrangian formulation. Focusing on the essential aspects of the motion, namely the end points A and B, a single quantity called the action in which time is integrated over (later, in quantum field theory, a Lagrangian density with both space and time integrated over) is considered over all possible paths between A and B. Among all these, the classical motion is the one for which the action takes an extreme (technically, stationary) value. This stationary principle, a global statement over all space and time and paths, turns out to be exactly equivalent to the local Newtonian description from one instant to another at all times in between A and B.
There are many sophisticated aspects and advantages of the Lagrangian picture, including its natural accommodation of basic conservation laws of energy, momentum and angular momentum. But, for our purpose here, it is enough to note that such stationary formulations are possible elsewhere and throughout physics. Quantum scattering phenomena, where it seems natural to think in terms of elapsed time during the collisional process, can be described instead in a “stationary state” picture (fixed energy and standing waves), with phase shifts (of the wave function) that depend on energy, all experimental observables such as scattering cross-sections expressed in terms of them.
“The concept of time has vexed humans for centuries, whether layman, physicist or philosopher”
No explicit invocation of time is necessary although if desired so-called time delays can be calculated as derivatives of the phase shifts with respect to energy. This is because energy and time are quantum-mechanical conjugates, their product having dimensions of action, and Planck’s quantum constant with these same dimensions exists as a fundamental constant of our Universe. Indeed, had physicists encountered quantum physics first, time and energy need never have been invoked as distinct entities, one regarded as just Planck’s constant times the derivative (“gradient” in physics and mathematics parlance) of the other. Equally, position and momentum would have been regarded as Planck’s constant times the gradient in the other.
The concept of time has vexed humans for centuries, whether layman, physicist or philosopher. But, making a distinction between representations and an underlying essence suggests that space and time are not necessary for physics. Together with all the other concepts and words we perforce have to use, including particle, wave, and position, they are all from a classical limit with which we try to describe and understand what is actually a quantum world. As long as that is kept clearly in mind, many mysteries and paradoxes are dispelled, seen as artifacts of our pushing our models and language too far and “identifying” them with the underlying reality that is in principle out of reach.
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