Heisenberg Uncertainty Principle: Energy-Time Formulation

that we will see in its energetic formulation²⁰. This formulation, although it is the most dangerous and misinterpreted²¹, is the most useful for our subsequent discussions. The Heisenberg uncertainty relation for energy is indeed²²:

$$\Delta E \cdot \Delta t \geq \hbar$$

Where $\hbar$ expresses the value of a constant known as Planck's constant²³. This relation states that an energetic state $\Delta E$ is not well defined if it involves a time $\Delta t$ less than that expressed by the relation. In particular, therefore, the energy of a system can oscillate over distinct values and not be conserved if this oscillation occurs in a time span less than the constraint expressed by the uncertainty principle. In practical terms, the uncertainty principle implicitly states that particles of energy $\Delta E$ can be created and destroyed if their lifetime lasts a time span $\Delta t$ less than the indicated constraint. This continuous creation and annihilation of virtual particles will be the subject of discussion in a subsequent chapter; for the moment, what interests us

is to highlight the philosophical aspect expressed by this principle, which represents the origin of many philosophical debates and discussions. Since, as we have said, energy conservation is closely associated with the temporal invariance of the laws governing the evolution of a physical system, the indeterminacy of energy determines the crumbling, over small temporal intervals, of the rigid causal description by means of physical laws. In practice, the more one wants to restrict the investigation time, the higher the indeterminacy about the system's behavior. A rigid causal description, as we are accustomed to conceive it, would therefore be possible only by taking into consideration sufficiently large times. According to Heisenberg's principle, therefore, for very small times, the causal description becomes increasingly blurred and grainy according to a grain that can be considered of the order of Planck's constant.

In 1927 Werner Heisenberg, in his work on the principle that now bears his name, formulated the situation thus:

"In the plain formulation of the law of causality - if we know exactly the present, we can calculate the future - it is not the conclusion that is wrong, but the premise."
Werner Heisenberg²⁴

From these physical premises, the debate then moved to philosophical ground and currently involves two complementary and experimentally equivalent interpretations of Quantum Mechanics. The first, advocated by Heisenberg himself together with Bohr, wants the uncertainty principle as an essential property of nature that at certain depths would be as if blurred when not observed. The other, first advocated by Einstein and Schrödinger, then continued by Bohm, instead wants Heisenberg's principle as a limit dictated by our perception of nature's essential processes. The first interpretation, often indicated under the name of Copenhagen interpretation, places a limit on the mechanistic and deterministic vision of causal processes; the second, Bohm's interpretation, saves deterministic causality but places a limit on the possibility of human perception. In one case, the quantum of minimal action introduced by Planck represents in a certain way a grain with which nature is woven; in the other, it represents the grain with which the eye that looks at it is woven. Being or perception? Here we see delineated, again, the problem that afflicted Berkeley at the beginning of the XVIII century. Each is called to solve this problem with their own means; we will limit ourselves to using the indicated formulas, leaving the reader the freedom to provide for them the interpretation they prefer.

1.3 Fundamental Units

In the course of the previous paragraphs, the investigation of the mutual relations between space, time, and causality has led us to find two fundamental formulas. The first, coming from Einstein's genius, uses light to link space and time together; the second, fruit of Heisenberg's idealism, seals energy and time in three letters. Uniting these two formulas together, we finally have a relation between space and energy, whose meaning we will see in subsequent paragraphs.

These are three formulas that, united together...