Dear Juan;
Thank you for your detailed comments and appreciated criticism. I am not surprised at all that the Duality of Time is extremely difficult to be digested and appreciated, even by experienced physicists such as yourself, although it would have been much easier if you look at the positive sides before criticizing the new concept that has never been introduced before. I have been studying physics and philosophy, at the most profound levels, for more than thirty years, and I spent the last decade on this new concept, yet if I leave it away for few days I find it again difficult to accept. Nevertheless, the theory has now become mature enough to be challenged. There is a dedicated website: http://smonad.com, and the Duality of Time book has been published recently and started to make excellent impressions.
Let me briefly answer your comments, especially those which are related to the main concept of the Duality of Time:
Your Comment>>Quantum mechanics is not restricted to discrete spectrum, we have continuous spectra in quantum mechanics. Also fields aren't discrete objects.
My Reply= QM and QFT are based on the quantization of energy and fields.
Your Comment>>The cosmological constant problem is unrelated to the incompatibility between relativity and quantum mechanics. The problem is backed to incorrectly interpretting the lambda as a vaccum term and then using QFT to calculate it. A solution to this problem is available, but the fundamental incompatibilities between relativity and quantum mechanics remain, just as they remain between relativity and classical mechanics.
My Reply= The informativity between Relativity and Classical is well understood, and the correspondence principle is fulfilled here since CM is a special case of GR. This is not the case between QM and GR because they are conceptually incompatible in situations such as entanglement and tunneling. The Duality of Time solves this fundamental problem of nonlocality without breaking the speed of light limit. Moreover, the speed of light constancy and invariance, as well as the other two principles of relativity, are theoretical postulates or axioms, while they follow directly from the DoT postulate, or the re-creation in the inner levels of time.
Your Comment>>LQG community tries to quantize spacetime because it confounds physical gravity with geometrical gravity. There is no reason to quantize spacetime in a physical description of gravity, and so any theory that predicts "discretuum structure of space-time" is very far from a TOE.
My Reply= You have to distinguish between the physical space-time and the abstract one, the latter is that defined by classical mechanics as an absolute reference frame (of space-time continuum or Euclidean space), and the first is introduced by General Relativity which has to be quantized in order for gravity to be quantized. The Duality of Time combines between these two as vacuum and void respectively.
Your Comment>>Spacetime isn't needed to describe physics. Example? Newtonian theory, where spacetime doesn't exist. In fact, the concept of spacetime only arises on a superficial field-like description of Nature.
My Reply= See the previous answer.
Your Comment>>Eq. (2) is dimensionally and mathematically incorrect.
My Reply= This equation is correct, but there is a minor typo in the second line, which is corrected afterwards.
Your Comment>>Things aren't being "created from nothing" in this model, but from pre-existent structures.
My Reply= There are many details here about how to define "nothing" in philosophy. In physical terms, "nothing" is the lack of any objects or motion, which corresponds to a homogeneous flat space. This is the space that split into two time arrows in the Duality of Time Theory.
Your Comment>>Imaginary time concepts helps to simplify some computations but makes other more complex. The trick is used only in the formulas where this simplification happens.
My Reply= Here the outer time is genuinely imaginary, which is the first time in physics and mathematics that imaginary numbers have genuine reality. If imaginary numbers make some problems more complex, this is a minor issue not related to the fundamental realities we are discussing here.
Your Comment>>Time is the evolution parameter that synchronizes correlations. It has to be unique; otherwise this role is lost. So theories of dual time either are incorrect or are redundant. In fact theories of dual time usually differentiate both times according to the evolution criteria, with only one of them playing the role of evolution parameter whereas the other participates in some geometrical spacetime structure. Such theories are redundant; and moreover they have additional difficulties to make direct connection with experiments.
My Reply= Please note that in the Duality of Time Theory, there is only one unique time that is either flowing inwardly to (metaphysically) create space and matter, or outwardly in which the kinetic evolution of matter is observed. The quantization of space is a result of interrupting the inner flow of time, which makes a new dimension of space or direction of time that is the outer time. So time is still fundamental to the extent that everything else is a result of its unique flow and fluctuation between the inner and outer dimensions.
Your Comment>>On the local level the arena for dynamics is the dual structure (r;tau). This structure can then be approximated by (r,ct). Dividing this last expression by c, will produce multiple time coordinates associated to the "time-time frame". However, no one of them is true time tau. Moreover, dividing by c to create a time coordinates doesn't introduce any new physics, since it is simply a change of labels.
My Reply= Dividing by c to introduce the time-time frame, and treating the outer time as hyperbolic, lead to the same Lorentz transformations (when you calculate the complex time or velocity), which is so far nothing new, but equivalent to Minkowski space-time, or the principles of Special Relativity. However, when you apply the same logic to calculate the complex momentum or energy you arrive to the equivalence principle necessary for General Relativity, and also the equivalence between mass and energy (E=mc^2). These principles were introduced by Einstein in his famous elevator experiment but there is not rigorous derivation of this equivalence, as Einstein repeatedly admitted. This was the missing puzzle which the Duality of Time solved. Please refer to the Preprint: https://www.preprints.org/manuscript/201708.0050/v1 for details about this important issue, which is also explained further in the book: https://www.amazon.com/dp/B07645RMZG .
Your Comment>>Space and time aren't treated equally in Minkowski spacetime, as the metric shows.
My Reply= That was a general expression indicating that they are both treated as dimensions, but of course they are not equal.
Your Comment>>The explanation of negative modulus for t_c and of the direction of time aren't convincing.
My Reply= This is well known in hyperbolic numbers.
Your Comment>>Inertial mass doesn't increase with speed. What increases with mass is the old and useless concept of relativistic mass.
My Reply= Effectively it does.
Your Comment>>The problem with the vacuum and the void concepts is not that both are abstract concepts not related to observation, and thus useless, but they have unwelcome properties such as infinite size.
My Reply= It is true that vacuum and void are two extreme theoretical states, but the Duality of Time explains how everything is introduced by mixing these two states in the real and imaginary flow of time.
Your Comment>>It is possible to build a model of nature with "real flow of time" without requiring continuous re-creation of space and matter. What is more, what matter is not created or destroyed is a basic principle of science. No observation invalidates it. No, pair 'creation' doesn't invalidate the principle.
My Reply= The conservation of energy/matter is a correct universal observation, and it is not violated in the Duality of Time Theory. Actually, this observation is also equivalent to causality or also the equivalence principle of GR, and it is rigorously derived in the DoT theory as explained in the above sources. The re-creation referred to here is perfumed in the inner levels of time, so there is not creation or destruction on the outer normal level that we encounter.
Your Comment>>Motion is not discrete, but a result of infinitesimal change. Zeno paradoxes confounded his contemporaries, but they is not longer paradoxes.
My Reply= This is not correct, although physicists and many philosophers like to believe that axiomatic mathematics has habitually dispelled Zeno's paradoxes. While the new mathematics can calculate where and when the moving Achilles will overtake the Tortoise, it does not really address the central point in Zeno's infinity argument that is indispensable if we want to solve his paradoxes.
Your Comment>>Ancient expressions as E=mc2 are better forgotten and replaced by modern ones: E0=mc2.
My Reply= Please refer to my prolonged discussion the in the above sources (https://www.preprints.org/manuscript/201708.0050/v1 , https://www.amazon.com/dp/B07645RMZG) which clearly show that this famous equation cannot be rigorously derived without the Duality of Time postulate. Einstein had already admitted that this relation cannot be derived within the frame of Relativity.
Your Comment>>Checked as well the external references. I find additional meaningless claims as such that photons move from rest to c in zero time in pair production or when emitted or absorbed. Besides the speed of photons being constrained to c due to being massless particles, the author seems to believe that photons are localized and at rest inside emitters before being emitted and then instantaneously accelerated to speed c during the emission process. Of course Nature doesn't behave in this way.
My Reply= Well, I think it is enough to remind you that I have studying and teaching physics for decades. I would not say that photons (themselves) are localized at rest and that they move from zero to c.
I urge you to have a deeper thoughtful look at my essay and my other cited sources, and please try to look this time at the positive side, because it is going to open many doorways to whole new kinds of sciences.