Time and space

Space and time are philosophically loaded concepts. Let us try to delineate what they are in physics.

In QED (the most accurate theory we have), space and time are parameters ranging continuously in R^4, coordinatizing the fields that contain the physical information. These coordinates have no absolute meaning since changing them by means of a Poincare transformation (a combination of translation + rotation + Lorentz boost) does not alter the physics.

The resulting affine pseudo-metric space, called Minkowski space, is (in QED) absolute and physically meaningful: All Poincare invariants expressible in terms of the fields can (in principle) be determined objectively. In particular, this holds for the Minkowski distance between space-time points that can be defined in terms of the fields. Such space-time points include for example all positions of stars, which are local maxima of field intensities in the backward light cone of an observer at a particular time, singled out objectively by appropriate observables.

According to established physics, a real observer is a macroscopic object with the capacity to record information. The recording process is described by means of irreversible thermodynamics. In particular, observers can be described to good accuracy classically, in terms of their associated macroscopic observables. These are expectation values of corresponding aggregated microscopic variables, behaving essentially classically according to Ehrenfest's theorem. Large objects such as stars can similarly be described by their associated macroscopic observables. The position of an observer and the objects it observes changes in time, defining their trajectories = world lines (apart from a global Poincare transformation). This change is (on the macroscopic description level appropriate for observers) continuous. (The world lines get more and more fuzzy as one focusses on smaller and smaller details. When the scale is reached where quantum effects dominate they become thick tubes within which positions are undetermined in principle. Indeed, the Heisenberg uncertainty principle forbids well-defined trajectories of arbitrary accuracy.)

For example: The observer might be the Mount Palomar observatory, at a fixed time t. (This time may be defined locally, say, ''one year after it was built'', a property that may be encoded in terms of QED using known physics.) The observer's past light cone cuts out from 4-space a 3-dimensional manifold, which intersects the world lines of the objects observed at definite points (within the accuracy of the whole construction) - the positions x(t) of the visible stars at time t. This is consistent with how astronomical positions are determined.

On the level of QED - and even one level below, in the standard model - space and time are not quantized in any sense. What is quantized are the observable fields, in dependence on time and position.

The situation changes slightly if we consider quantum gravity, thought to be relevant on the smallest significant scales of our universe. It is tentatively explored by current physicists, but without any definitive experimental results so far. Judging from general relativity (which must be the classical limit of any meaningful quantum gravity), Minkowski space is now replaced by a pseudo-Riemannian manifold, the translation subgroup of the Poincare group is extended to the diffeomorphism group (or, because of anomalies, perhaps only to the volume-preserving subgroup). Therefore, only the invariants under this bigger group are physical. This still includes geodesic distance, so that, for any particular observer, the picture painted above remains valid in the observer's Riemann normal coordinates.

There is an intrinsic asymmetry between observed position and time - even in the classical relativistic case!

Whether measured or not, a state is _always_ a state at a particular time t. Thats why we write state densities as rho(t) and wave functions as psi(t), and have a dynamical equation to tell how the state changes with time.

Note that in a relativistic theory, position becomes like time - rather than time like position: Instead of trajectories depending on time we have fields depending on space and time. Note that absolutely precise particle positions for multiparticle systems don't make relativistic sense - position becomes an intrinsically smeared concept, even classically!

What one can have consistently in relativity is only relative positions of one particle with respect to a frame attached to a particular particle and its world line - in the case of the GPS this ''particle'' is the earth. These positions are again time-dependent.

In nonrelativistic quantum mechanics, a unique position operator is defined only for a system consisting of a single particle alone in the universe, in an observer-dependent coordinate system. (The observer must be outside this mini universe.)

For an N-particle system, one has N position operators. If time were like position, each particle would have its own time, which would make the concept of time meaningless. But time has meaning; so this is not an option.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ