Quantum Reference Frame - Reference Frame in Classical Mechanics and Inertial Frame

Reference Frame in Classical Mechanics and Inertial Frame

Consider a simple physics problem: a car is moving such that it covers a distance of 1 mile in every 2 minutes, what is its velocity in metres per second? With some conversion and calculation, one can come up with the answer "13.41m/s"; on the other hand, one can instead answer "0, relative to itself". The first answer is correct because it recognises a reference frame is implied implicitly in the problem. The second one, albeit pedantic, is also correct because it exploits the fact that there is not a particular reference frame specified by the problem. This simple problem illustrates the importance of a reference frame: a reference frame is quintessential in a clear description of a system, whether it is included implicitly or explicitly.

A reference frame is a physical system in which physical quantities are defined, such as position, momentum, spin, time, etc. Some obvious examples of reference frame are metre stick for distance and clock for time. While some standards are widely accepted and used like the metric and imperial system, there is no constraint on what physical system a reference frame has to be, so it is perfectly valid, though peculiar, to use Tom Cruise (who is 1.70m high) as a reference frame and to describe Katie Holmes as 1.029 Tom Cruise high. Regardless of what reference frame is used, it is always relational, not absolute, just as in this TomKat example.

When speaking of a car moving towards east, one is referring to a particular point on the surface of the Earth; moreover, as the Earth is rotating, the car is actually moving towards a changing direction, with respect to the Sun. In fact, this is the best one can do: describing a system in relation to some reference frame. Describing a system with respect to an absolute space does not make much sense because an absolute space, if it exists, is unobservable. Hence, it is impossible to describe the path of the car in the above example with respect to some absolute space. This notion of absolute space troubled a lot of physicists over the centuries, including Newton. Indeed, Newton was fully aware of this stated that all inertial frames are observationally equivalent to each other. Simply put, relative motions of a system of bodies do not depend on the inertial motion of the whole system.

An inertial reference frame (or inertial frame in short) is a frame in which all the physical laws hold. For instance, in a rotating reference frame, Newton's laws have to be modified because there is an extra Coriolis force (such frame is an example of non-inertial frame). Here, "rotating" means "rotating with respect to some inertial frame". Therefore, although it is true that a reference frame can always be chosen to be any physical system for convenience, any system has to be eventually described by an inertial frame, directly or indirectly. Finally, one may ask how an inertial frame can be found, and the answer lies in the Newton's laws, at least in Newtonian mechanics: the first law guarantees the existence of an inertial frame while the second and third law are used to examine whether a given reference frame is an inertial one or not.

It may appear an inertial frame can now be easily found given the Newton's laws as empirical tests are accessible. Quite the contrary; an absolutely inertial frame is not and will most likely never be known. Instead, inertial frame is approximated. As long as the error of the approximation is undetectable by measurements, the approximately inertial frame (or simply "effective frame") is reasonably close to an absolutely inertial frame. With the effective frame and assuming the physical laws are valid in such frame, descriptions of systems will ends up as good as if the absolutely inertial frame was used. As a digression, the effective frame Astronomers use is a system called "International Celestial Reference Frame" (ICRF), defined by 212 radio sources and with an accuracy of about radians. However, it is likely that a better one will be needed when a more accurate approximation is required.

Reconsidering the problem at the very beginning, one can certainly find a flaw of ambiguity in it, but it is generally understood that a standard reference frame is implicitly used in the problem. In fact, when a reference frame is classical, whether or not including it in the physical description of a system is irrelevant. One will get the same prediction by treating the reference frame internally or externally.

To illustrate the point further, a simple system with a ball bouncing off a wall is used. In this system, the wall can be treated either as an external potential or as a dynamical system interacting with the ball. The former involves putting the external potential in the equations of motions of the ball while the latter treats the position of the wall as a dynamical degree of freedom. Both treatments provide the same prediction, and neither is particularly preferred over the other. However, as it will be discussed below, such freedom of choice cease to exist when the system is quantum mechanical.