Dimensional Physics
Everything consists of spacetime.
Everything consists of spacetime.
After superposition, entanglement and probability should come next, since both can only be understood through the concept of superposition. That is correct for now. However, we will first discuss the basic elements of QM so that the concept for the entire QM can be recognized early on. The respective further derivations from these basic elements will then come later. It is a popular point of contention in QM whether superposition or uncertainty is the decisive “basic element” of QM. It is often argued that superposition (not local) causes uncertainty, or vice versa. We are pursuing a different structure here. From the perspective of DP, QM has four core elements:
Therefore, we will work through the uncertainty principle. Then we will have already discussed all the basic elements of QM here. The structure of elementary particles and interactions can then be derived with a little additional idea. As always, geometry will guide us.
Let’s start with an explanation of uncertainty. Unlike superposition, uncertainty is not built into the underlying mathematics. It actually arises from the solution approach, e.g., via the Schrödinger equation. Uncertainty was discovered purely mathematically. As a result, the interpretation of uncertainty has changed over time. But let’s take it one step at a time.
The official name is Heisenberg’s uncertainty principle. It was discovered by Werner Heisenberg in 1927. It could only come after the Schrödinger equation (1926) and also had to be discovered. Unlike superposition, the uncertainty principle was not built into the basic mathematical description. It emerged later.
Heisenberg did not find the exact formulation that is commonly used today. His description was still:
\Delta p\space *\space \Delta x\space \approx\space h
To explain the formula, we will explain the official name.
The part “Heisenberg’s” clearly comes from Heisenberg. The current formulation, which we will also use in the following, was found a few months later by Earle Hesse Kennard. However, this is the same principle as in Maxwell’s equations of electrodynamics. He developed the basic idea, and it bears his name, even though he never saw the current formulation with four very short equations during his lifetime. Therefore, the part “Heisenberg’s” fits me.
The part “relation” comes from the mathematical symbol \approx. Heisenberg did not establish an equation, but only a relation. The relationship between x and p is clear, as is the order of magnitude. However, it is not an exact equation. Hence, it is a relation. Later, it became an inequality. The somewhat less precise term “relation” is good enough here; it fits.
The part “uncertainty” is a little more difficult. We need several new sections for that.
Let’s clarify what is meant by uncertainty. The symbol \Delta for p or x is not the measured value for the momentum or position. \Delta p refers to the deviation from the mean value of the momentum from many measurements. In mathematics, this standard deviation is denoted by the symbol \sigma. In fact, Kennard chose this for the exact formulation:
\sigma_p\space * \sigma_x\space \ge\space \cfrac{\hbar}{2}
For whatever reason, the misleading notation with the \Delta has become established. This clarifies the term “uncertainty.” This inequality always refers to the deviation from the mean value and not to the measured value itself. This inequality makes a statement about how close or far away one is from the mean value, i.e., an uncertainty.
In the physical interpretation, they initially backed the wrong horse. Where is the uncertainty supposed to come from? This concept did not exist in classical mechanics. There, it was theoretically possible to carry out a measure with absolute precision. In QM, the end of accuracy is reached at approximately h. That is the statement of this inequality. This led to the idea that this uncertainty has something to do with the interaction during the measurement process. The symbol h stands for an effect. The relation concerns the simultaneous measurement of position and momentum. In order to measure something, we need interaction. If we want to measure the position of a particle, we do so through an interaction with, for example, a photon. However, the photon has a momentum that is transferred to the particle. This changes the value of the momentum in this measurement.
This idea persisted for a long time. Fortunately, it is now only heard of in a historical context. In today’s interpretation, it is assumed that we do not need interaction for uncertainty. It is fundamentally present. It only depends on the possible combination of the measured variables.
Uncertainty is a deviation from a mean value in a measurement. To calculate a mean value, we need at least two measured values. What does uncertainty mean for a single particle or a single measurement? The answer is clear: no statement is possible. In this derivation, uncertainty is a purely statistical statement. In a single measurement, the deviation is exactly zero and the relation is violated.
This is where it gets strange. As soon as I take many measurements, each individual measured value must adhere to the uncertainty. If I take a single measure, this is not relevant. However, many measurements consist of individual measurements. So, what now?
In fact, this is an open question of interpretation in QM. There are two interpretations:
Later, we will get a mixture of both. If you are a supporter of the Copenhagen interpretation, then both positions are not so far apart. A particle only exists there when a measurement is taken. The only problem that remains is whether to take one or many measurements.
Unfortunately, there is also a lot of “folklore” surrounding the topic of uncertainty. These are sayings that everyone is familiar with. Example: “If I determine the location more and more precisely, then the momentum becomes more and more imprecise, and vice versa. If the location is determined exactly, then the momentum is completely indeterminate.” Sayings of this kind haunt some people’s minds. Unfortunately, the statement in the example is wrong. Here, the facts have been simplified to such an extent that the statement becomes false. This is a mistake that I have unfortunately made more than once. Mathematics has clear advantages.
Of course, according to QM, we can measure the momentum and position of a particle simultaneously with 100% accuracy. To do this, we only need to measure the momentum in one direction and the position in another direction. The uncertainty principle only applies to momentum and position if we measure both in the same direction. Here, the uncertainty principle depends on the direction of the measurement.
But that’s not all: there are measured variables where uncertainty is not relevant at all. This only happens with certain combinations. In QM, the mathematical expression of the commutator is used for this purpose. This is written out as follows:
\big[\hat{A},\hat{E}\big]\space =\space \hat{A}\hat{E}\space -\space \hat{E}\hat{A}
With normal numbers, the result is always zero and there is no uncertainty. In QM, however, observables are meant. These are observable quantities. These are not numbers, but “self-adjoint linear operators.” The result may not be zero. This happens when we set \hat{A}\space =\space p and \hat{E}\space =\space x. In fact, it is even more complicated, because the wave function also plays a role. But we don’t need to go into that much detail here.
We’d better stop here with the explanation of the uncertainty principle. We can see that there are different combinations of measured variables with different behaviors:
As always in QM, there is no physical/logical explanation for this. It is embedded in mathematics with commutators. In fact, the uncertainty principle arises from various mathematical considerations. Therefore, we can consider this to be very well established mathematically. That is not enough for us. Let’s see if we can find a logical explanation.
Let’s try to apply what we have to the DP. We’ll stick with the example of momentum and position. We repeat Heisenberg’s formula
\Delta p\space *\space \Delta x\space \approx\space h
In this form, the connection is not yet clear. So, let’s just write the units of measurement on the left side:
\Delta p[m\space *\space v]\space *\space \Delta x[l]\space \approx\space h
It says [mass\space *\space speed]\space *\space [lenght]\space \approx\space h. The magic word to solve the puzzle is Planck. Uncertainty still has to do with a measurement in our spacetime. Let’s insert the characteristic Planck values for our spacetime into the relation:
(m_P\space *\space c)\space *\space l_P\space \approx\space h
Then we could even turn the \approx into a =. That is our definition of h across the low-dimensional boundary.
This leads us to the following requirement: there must be no uncertainty in gravity. Uncertainty causes interaction. This means that it is linked to interaction across the low-dimensional boundary. The interaction causes uncertainty, not the state. With our model, we no longer have a problem with uncertainty in a 3D spacetime density with an infinite number of 2D manifestations. Gravity and uncertainty are not mutually dependent, as there is no uncertainty in a state in 3D.
This leads us to the interpretation that uncertainty only arises in the measurement process. The state is irrelevant.
Now we must not allow ourselves to be distracted by the formula for defining h. On the left-hand side, we have two separate deviations from two measured values and not one deviation from h. We check whether the Planck values make any sense here at all.
The smallest effect in our spacetime is h. This means that a deviation which, when combined, results in an effect, should make the formula look more like this:
(m_P\space *\space c)\space *\space l_P\space \ge\space h
We cannot get below h. In the exact formula, however, there is a 4\pi in the denominator for h. This brings us below h in the effect. This looks like an error. If we follow pure statistical interpretation, we have no problem. Purely mathematically, there may be values for a deviation from a mean value that does not exist in the individual measurement. Example: The mean number of dots on a dice is 3.5. We do not find this value on the dice. Here, the mean value is already impossible in a single measurement. The same applies to the deviation. However, we also want to have a logic for each individual measurement. Then we cannot proceed with the purely mathematical argument.
We always need a length to map a spacetime density. An effect changes the spacetime density. This means that a length must always be involved. A length cannot be measured precisely to l_P in an interaction. If we want to arrive at a Planck length in a measurement process and thus also in the deviation, we create a black hole in 3D, which in turn has no 2D representation and therefore has nothing to do with QM. In fact, we therefore have to multiply the Planck length by 2. This gives us the smallest possible deviation. We take this to the other side and then have:
(m_P\space *\space c)\space *\space l_P\space \ge\space \cfrac{1}{2}h
That looks better. These \frac{1}{2} must therefore always occur. We are still missing 2\pi. We already know the problem. With the proportionality constant k in Einstein’s field equation, there were still 8\pi. There it was 4\space *\space 2\pi. Per spacetime direction 2\pi. Here we only have one direction and 2\pi are missing.
However, there is still no real reason for the uncertainty. In particular, there is no reason why it only exists to a limited extent and is not always valid. According to the argument so far, it should always be valid.
The uncertainty can be explained very simply in logical terms. We must return to our initial approach. All forms of energy are represented as spacetime density in the geometry of spacetime. The uncertainty itself is already defined in the spacetime density.
The fundamental problem with a measurement is that we combine measured variables that measure the identical quantity in the spacetime density. Let us stick with the example of location and momentum.
If we want to measure the location in the x-direction precisely, we thereby limit the possible length in this direction. A location measurement is nothing else. However, if we also want to measure momentum, we need a lot of length. Momentum is a spacetime density relative to its surroundings. When I measure the length of the spacetime density, it has no momentum in the spacetime density itself. We only obtain this if we can compare this definition of spacetime density with the surrounding spacetime density. If we measure the location precisely, we no longer have any “surrounding” spacetime to determine the momentum. If we want to determine the momentum exactly, we need the entire length of the x-direction for the most accurate value. Then the location is undefined. Since we are measuring the identical object, one specific measurement excludes the basis for the other measurement. Therefore, we have no problem measuring location and momentum in different directions or from different objects.
Uncertainty also applies when we measure the identical object using different methods. As an example, let’s take the uncertainty of time and energy. However, this uncertainty then takes on a slightly different character. This is because one measurement (time) does not affect the other measurement (energy), but rather a longer time measurement is identical to a more accurate energy measurement. Energy is the geometry of spacetime. However, this already indicates the “distance” to the spacetime boundary via length contraction. Time is exactly the same measurement. When we measure time, we measure energy.
The fact that uncertainty cannot be linked solely to quantum mechanics also comes from mathematics. We can switch between the momentum and position representations of the description. This is done using a Fourier transformation. This transformation already contains a form of uncertainty between the representations. Therefore, the uncertainty must originate from the joint observation of an identical object.
This describes the uncertainty precisely enough for our purposes. What is important for us is that the uncertainty is already inherent in the definition of spacetime density and is only connected to QM via the spacetime boundary, with the effect h. However, the uncertainty itself does not result from QM.