SpaceTime and detection

A continuous spacetime is assumed, homogenous and isotropic with an attached metric (see material for the first term students).


But to detect a particle (that is asking for 'where and when' it is) is a different question.


Space measurement (in one dimension): Opperationally, following the advices of measuring theory we build a set of 10 finite size detectors of length, say 1[cm]. With them we plaster (covering an area with a single leakless layer) an area of, say 10[cm]. If we detect the particle then we know the position up to an accuracy of one centimeter. Next we construct more precise detectors with half a centimeter length. More effort (more detectors) gained a better precision in the result. Iterating we might assume to get to 'infinitely precise' space measurements. However, this limes does not exist in real physics because the technical effort (number of counters) goes to infinity. This phenonemon has gotten many names, one of which is coarse graining of space measurements.


A good outcome is: we should never need integrals in quantummechanics, e.g. here: the probability of finding the particle somewhere on the grid is just the sum of the probabilities to be found in one of the detectors, $W=\sum_1^N \Vert \psi(x_n,t)\Vert^2$. A single experiment (one particle in) gives the particle still integer at one of the many counters and simultaneously at none of the others. We can give the information on where it was actually found by giving an ordered set of answers: is it in the first counter, is it in the second counter, is it in.... We will call the result of the particle being found in counter, say number n as the n-th state of the possible states of the particle. Clearly, all the possible states form a countable (orderable) set of states. The limit to infinitesimally small counters in this way would lead us to infinitely many but countable states (just like the rational numbers in contrast to real numbers) not to a larger (unorderable) set of states ¹¹.


Heisenberg realized that these mathematical results are well known in wave physics (acoustics, ocean waves, optics,..), saying that it is impossible to prepare a finite length wave train better than the length of the train divided by its main wavelength smaller than $2\pi$, that is

$$\Delta x/ \Delta \lambda \geq 2\pi \quad , \quad \Delta \omega \Delta t \geq 2 \pi \quad .$$ (1)

 Musicians know that the shorter the duration of a tone, the more lousely is allowed to hit the frequency without being spotted (see G. Suessmann). By putting into the equation the relations between momentum and energy, and wave number and frequency of the respective probability amplitude wave $\psi(x_i, \Delta x,t)$ we gain the famous Heisenberg-uncertainty relations.


The coarse graining of time has to be equivalently described.


To ease calculations and to simplify notations well spread is a short hand writing for the sums over plastered finite but small detectors by replacing them with integrals over space (and time). Since the probability to find a particle in an infinitesimally small detector is zero, but the sum over all detectors is still 1 (particle must be somewhere), the notation uses densities, just as the material density of a infinitesimally small piece of lead stays constant even though the volume goes to zero.

$$\psi(x_i, \Delta x, t) = \int_{\mathrm{counter}} \mathrm{d} x\psi(x,t) /\sqrt{\Delta x}$$ (2)

The function $\psi(x,t)$ of the real spacetime variables is thus a 'probability-amplitude-density'. We will see in more detail, how the use of integrals simplifies calculations, but needs some precautions, since the sets of possible positions are now real numbers and thus over-accountable, causing lots of subtle mathematical problems. However, the reader, whenever in doubt, should replace real spacetime variables by a plastering of the spacetime with finite but small counters.


Immediately he will gain two benefits: 1) no hayyling with integrals any more, 2) numerical results become stationary, when the detectors get as small as theleading wave length of the probability amplitude density function, well short of the mathematical limit.


Same procedure for time measurements.