Time, Phenomenologically.

Phenomenological philosophy claims that time is not the place, the scene, the container or the medium for events (changes), nor a dimension along which everything flows. According to Jean-Toussaint Desanti, a French scholar of Husserl, we should forget “the ordinary meaning of the preposition “in” we spontaneously use when we talk about our experience of time. It is even this use, so ancient that should be the subject of our review. Really it would be strange that what we have learned to call “time” can contain anything. And yet we say without anxiety: “It is time that everything goes.”

But what is happening “in” time does not remain as a place. In fact, this is the major objection of Bergson against Einstein’s Special Relativity, that he has dimensioned time, something immeasurable in the same way as space, which is, of course (in everyday life), measurable. This kind of reasoning in phenomenology is not that far from the one in modern physics.

As Smolin says,

There is a deeper problem, perhaps going back to the origin of physics… time is frozen as if it were another dimension of space. Motion is frozen, and a whole history of constant motion and change is presented to us as something static and unchanging… We have to find a way to unfreeze time — to represent time without turning it into space.

In the words of Carlos Rovelli,

Today, the novelty that comes from quantum gravity is that space does not exist. … But combining this idea with relativity, one must conclude that the non-existence of space also implies the non-existence of time. Indeed, this is exactly what happens in quantum gravity: the variable t does not appear in the Wheeler-DeWitt equation, or elsewhere in the basic structure of the theory. … Time does not exist.

The claim about the imaginary, surreal, even exotic nature of time is not new in philosophy and physics. Of course, there have always been, too, physicists defending the real existence of time, even so real to define such a quantum variable as the chronon with the idea in mind to reconcile special and general relativity with quantum field theory. This “atom” of time was supposed to be the duration for light to travel the distance of the classical (non-quantum) radius of an electron. This model implies a lowest level of actuality, as asserted in the Planck scale.

In his book “Time Reborn” Smolin argues that physicists have inappropriately banned the reality of time because they confuse their timeless mathematical models with reality. His claim was that time is both real (which means external to him) and fundamental, hypothesizing that the very laws of physics are not fixed, but evolve over time. This stance is not really a new one. But it means again an absolute external reference axis and a direction for placing events in a sequence, which phenomenologists decline as the only option. Some of them, partly inspired by the late works of Heidegger and Merleau-Ponty, approach time neither from the standpoint of simultaneity alone, nor from that of succession. For instance, the dualism of these two concepts is surpassed in favor of a temporal dialectic in which simultaneity and succession are entwined, without denying their separate meanings. Heidegger’s concept of “true time” speaks to this approach to phenomenology.

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Vector Representations and Why Would They Deviate From Projective Geometry? Note Quote.

There is, of course, a definite reason why von Neumann used the mathematical structure of a complex Hilbert space for the formalization of quantum mechanics, but this reason is much less profound than it is for Riemann geometry and general relativity. The reason is that Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics turned out to be equivalent, the first being a formalization of the new mechanics making use of l₂, the set of all square summable complex sequences, and the second making use of L₂(R³), the set of all square integrable complex functions of three real variables. The two spaces l₂ and L₂(R³) are canonical examples of a complex Hilbert space. This means that Heisenberg and Schrödinger were working already in a complex Hilbert space, when they formulated matrix mechanics and wave mechanics, without being aware of it. This made it a straightforward choice for von Neumann to propose a formulation of quantum mechanics in an abstract complex Hilbert space, reducing matrix mechanics and wave mechanics to two possible specific representations.

One problem with the Hilbert space representation was known from the start. A (pure) state of a quantum entity is represented by a unit vector or ray of the complex Hilbert space, and not by a vector. Indeed vectors contained in the same ray represent the same state or one has to renormalize the vector that represents the state after it has been changed in one way or another. It is well known that if rays of a vector space are called points and two dimensional subspaces of this vector space are called lines, the set of points and lines corresponding in this way to a vector space, form a projective geometry. What we just remarked about the unit vector or ray representing the state of the quantum entity means that in some way the projective geometry corresponding to the complex Hilbert space represents more intrinsically the physics of the quantum world as does the Hilbert space itself. This state of affairs is revealed explicitly in the dynamics of quantum entities, that is built by using group representations, and one has to consider projective representations, which are representations in the corresponding projective geometry, and not vector representations.