The modern mathematical description of physics focuses on symmetries. Physical theories are considered beautiful because of their symmetries. What are these symmetries and where do they come from?
The brilliant mathematician Emmy Noether (1882-1935) made a major contribution to the understanding of symmetries in physics. Noether's theorem, named after her, states that every symmetry in a mathematical function space leads to a conservation variable. Translation invariance in time leads to conservation of energy, conservation of momentum is a consequence of translation invariance along one dimension of space, and conservation of angular momentum is a consequence of rotation invariance in three-dimensional space. Translational invariance of time means that the mathematical equations describing physical events are always the same, regardless of the specific time of observation. Translational invariance of space means that the laws of physics are the same at every point in space. Rotational invariance of space means that the laws of physics are the same in every direction of space.
But where do these symmetries come from?
Regarding the translational invariance of the laws of physics with respect to time and space, and the rotational invariance of space, one could argue that these symmetry properties are a compelling consequence of our man-made claim that scientific knowledge is valid regardless of our respective locations in space and time. The laws of physics should not change as time passes or as an observer moves in space. These knowledge requirements lead to mathematically imperative conservation quantities that we call energy, momentum, and angular momentum. With Immanuel Kant, we could say that these conservation quantities are a priori certain because they are a direct consequence of our knowledge conditions. The law of conservation of energy is therefore not a surprising insight into the nature of nature, but a simple consequence of the claim that our physical theories are equally valid regardless of the point in time. Conservation of energy follows from the requirement of translational invariance of time.
What is remarkable, however, is that we actually manage to build measuring devices that can measure these conservation laws. If we were to build our measuring devices "incorrectly", we would not be able to empirically confirm the theoretically postulated conservation laws. For example, if we were to divide time into hours as the ancient Egyptians did, dividing the light day into 12 hours and the dark night into 12 hours, an hour would have a different length depending on the time of year or the geographical location of the place. Measured with such a clock, neither the conservation of energy nor the conservation of momentum would be empirically observable in pendulum movements, for example. In the practice of the natural sciences, empirical methods of measurement and theoretical considerations are successively reconciled in such a way that they fit together. Thus, empiricism is not a neutral judge of theory. Rather, theory determines what is worth measuring.
Our man-made demands on scientific knowledge alone cannot ensure that nature behaves in our measuring apparatus as we expect it to. This becomes even clearer with other symmetries and conservation quantities: For example, from what condition of knowledge should it logically necessarily follow that we observe conservation of electric charge in all experiments? The fact that the sum of positive and negative electric charges is invariant cannot be a result of our knowledge alone, but must have something to do with the nature of matter. This brings another possible explanation into view: We determine the existence of a certain elementary particle, among other things, by the fact that it carries a certain electric charge. The conservation variables could therefore be the defining properties for the basic elements that make up the world. We had previously postulated that all known elementary particles could be traced back to a single universal particle whose behavior could be described as that of an autonomous agent.
This universal particle would therefore have to be defined by its internal configuration, from which the quantum numbers for the electric, weak and strong charge, as well as its energy, momentum and angular momentum result. The symmetry properties of space and time would then not be the cause of the conservation laws for energy, momentum and angular momentum, but the translational invariance of space and time and the rotational invariance of space would follow from the defining properties of the universal particles. Interestingly, location is not a defining characteristic of the universal particles, because the location of an object is variable in time, and the postulated universal particles do not have a fixed location in space, but fill space completely.
This conclusion is consistent with the conviction expressed at the beginning that space and time do not set the stage for the physical world, but result from the behavior of the basic building blocks of the universe.
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