There is No Unified Equation
Are physicists looking for a unified equation? This is one of the big lies told to young people. There is no unified equation. There cannot be one.
Physics is full of wonderful equations and full of stories about how they were discovered.
Today, all knowledge about nature has been summarized into two equations.
The first equation is the Hilbert Lagrangian of general relativity. It is a compact way to write Einstein’s field equations of general relativity.
The second equation is the Lagrangian of the standard model, with massive Dirac neutrinos.
Everything is described by those two equations. It might be that the effects attributed to ‘dark matter’ are not described by either, but the issue is not settled. (I think that they are included, but in a tricky way that still needs to be understood.)
But now comes the surprise. The unified theory that describes all of nature cannot be written as an equation. There are many reasons, but four are essential.
First, as the experiments show, there is not even a need for a different equation from the two existing ones. (Except if ‘dark matter’ surprises us…) The existing equations in general relativity and particle physics describe all observations in nature. In short,
1. There is no reason to look for a unified equation.
The existing equations already describe nature. Exactly. Not approximately. The equations are exact. Experiments do ask us to understand where the existing equations come from. But there is no need, from experiments and observations, to find a different equation or even a unified equation. This is the amazing result of all research completed in 1973, fifty years ago, when the standard model of particle physics with massive Dirac neutrinos was completed, in all its details. (General relativity was already completed earlier on.)
Second, nature has a minimum length. It is also the smallest possible length measurement error. In fact, in relativistic quantum gravity, there is a measurement error for every physical quantity. The error is not due to human limits, it is given by nature. Nature does not allow exact values of observables. This means:
2. No unified equation can be valid in nature.
There is no way, in principle, to prove that two expressions, the left and the right side of an equation, are equal to each other. This only happens in relativistic quantum gravity. In general relativity alone, or in quantum theory alone, errors can be made as small as desired. Only in relativistic quantum gravity, at the foundations of nature, is this impossible.
Third, equations and Lagrangians are about observables. A unified equation must define the observables before using them in equations of motion. This is logically impossible.
3. No unified equation of motion holds in relativistic quantum gravity.
A unified equation of motion, or a unified Lagrangian is based on fields. Examples are the metric, or the photon field, etc. These fields have to be defined and to be measurable. But no unified equation can explain their origin. In other terms, explaining the origin of particles and fields is not possible with an equation of motion that assumes their existence.
Fourth, equations of motion contain derivatives with respect to time and space. These quantities are undefined at the smallest scales.
4. Space and time are not continuous.
There is a minimum length, given by twice the Planck length, and a minimum length measurement error. Therefore, one cannot state that length is continuous. (Or that it is discrete, for the same reason.) Space cannot be made of points. Nor can time be made of instants. As a consequence, derivatives cannot be defined, and equations of motion do not exist at the fundamental scale.
Summing up all these statements: there is no unified equation. This might be the most striking consequence of relativistic quantum gravity. Unification cannot be done with equations. For example, rumours say that Einstein was working on unified equations even on his deathbed. However, the minimum length implies that he had no chance to succeed. There is no “God equation”. The existence of such a unified equation is a lie. This lie or a similar one is regularly told to young people. It is impossible that such an equation exists.
It is terrible that well-known physicists appear in the media and say that the big task of physics is to find a set of unified equations. This is wrong. It is more than wrong. It is a lie.
In reality, the remaining task is to find out where the known equations — which describe nature perfectly — come from. This is the task of fundamental physics that is left. (So, if you are searching for unified equations, you have been put on the wrong track.)
And there are many questions open! Above all: Where do the masses of the elementary particles and their coupling constants come from? This is the main open issue of fundamental physics.
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