Dialogues on the New Science - Excerpt

Excerpt from "Dialogues on the new Science"

“What a strange day” said Lisa W to Lisa C. “The men are slowly learning about the new science, I think Michael faster than everybody. I think I grasped the most, Lisa. What I would need now is a correct formalization of the theory, and I know it may take years if I have to do it by myself... There are three aspects that are in need of clarification: the four equations of the S.T.E.M., in the sense that I don’t know about the practical measurement and expression of the time and space equivalence; the definition and operation of noun-less logic, and the definition of the energy operators. You have said that the covariant derivative is one of the energy operators, but I didn’t understand how, and how it relates to the others.”
“Lisa” answered Lisa C “If a give you a sheet of paper full of formulas, it would be only of limited help. When you will be knowledgeable of the entire theory, you will be the person who will write that sheet. Let’s use this time to discuss the items I don’t want to discuss with the group.
Can you tell me your best definition of space?”
“Extension, considered independently of anything which it may contain; that which makes extended objects conceivable and possible” said Lisa W, an continued: “A set of elements or points satisfying specified geometric postulates. The boundless regions of the infinite.” “Ok, said Lisa C “please stop here. All of your textbook definitions are a bunch of lies. The infinite doesn’t exist. A “set” is a true logical paradox, and the geometry, in any flavour, is a self centric depiction of what you want the space to be. Einstein didn’t proceeded any further for this simple reason, because he used the Riemann geometry to describe his original idea, which I believe was fundamentally correct. Levi-Civita did a great job in setting the independence from a reference system of coordinates, which proved to be working also in absence of a space metrics... but, while the study started in a promising and correct manner, you formal logic fooled the mathematicians once again: the linear combination which represent a vector is a correct description in the new science too; but... when this vector represents a geometrical object in space, and you change the linear combination which describes it, your mathematics suffers the pollution of your inexorable need for separability. The linear transformation keeps the identity of the original vector, effectively centring the space around your needs, not around the associated energy translation. The use of the covariant derivative was the brightest idea of Einstein, which could have brought him to see mathematically the structure of our space of the operators, by renouncing the transformation and keeping the pure mathematical concept of space, intended as a sequence hosting another sequence. The true space is no more, no less.”
Lisa W said: “I will never understand how you come to this conclusion. From a later perspective, it looks of great simplicity, but I cannot view, in my mind, how you may have get to it...” “Lisa” said Lisa C “learn to identify each of your thoughts. Learn how they are born in your mind. Every time, ask yourself why it should be correct. If you can identify something of yourself in it, very likely it is wrong...”
“Thank you for the wise suggestion” answered Lisa C, “but very likely needs some logic refinement... but it will be a later job. The next question is, once you have told me the mathematical nature of the space, which coincides with its true nature, how we can measure this? Angstroms, Meters, Years Light, Parsecs... there is a correspondence?” “Yes, if you use the abstraction layer” answered Lisa C. In the future, once the true structure of the space will be better understood, you may want to measure the space just like you measure the time, with an oscillator. The actual way of measuring the space, either the stick which is stored in Sevres or the best reference based on the speed of the light, is abnormal.” Why you say abnormal, Lisa?” said Lisa W. “Because the space is a flow, just like the time, the matter or the energy, of the same nature and scope which is at the base of the principle of commutability that, in turn, allows the translation from a concentrated description to a distributed one. Instead of solving the great problem of the relation between the energy expression and the number of geometrical dimensions, you circumvented it with the addition of space metrics: while a good idea in principle, it just complicated your life uselessly.”
“I am particularly interested to the relation of the number of geometrical dimensions with the energy expression; as you know, there are specific relationships between the subatomic particles and the dimensions, at the base of the M-theory. You already cut my off on the independence of widely spaced branes in the bulk; now you deplete my M-theory...”
“Let’s say I’m expanding on it” answered Lisa C patiently. “Energy flows from commutable space to your geometric space as first level interaction. You don’t see the interaction, you just see particles in geometric space, and when they congregates in a good number, you also note the gravity. Gravity is linear combination at the same level of the mass itself, but the resulting vector has a visible interaction; it is a rotating vector - if you consider its description in the space of the operators. Therefore, it can address energy in all possible forms and you see it can attract matter, modify the path of a light ray, and so on. Gravity makes energy to flow in and out the commutable space, and since this happen at the first level, you aren’t able to quantise it, nor to determine its nature. The matter, on the other hand, relies on a second space operator, which allows multiple successive interactions, seen the sequenced structure of the space operators. All in all, it seems to you that the matter produces a field of gravity, but that is only a partial effect of the flow. If you take into account the full energy and mass of the Universe, which you can deduce from other effects, your accounting is odd, and you have to invent dark matter and dark energy to square it out.
All the “assertions” of the commutable space can be described as rotating vectors; all the energy of the commutable space cycles in and out, this is why the global effects you can observe doesn’t match the effects produced from visible mass alone. Only a portion, those ten raised at the twenty-tree power solar masses, are rendered as observable mass. The remainder of commutable space cycles in and out, without any second level interaction.”
“Ok” Lisa W answered. “Therefore, based on what you are saying, the space is an expression of dynamic energy, not the blackboard for our static drawings. You have defined “commutable space” the real space, and “geometric space” the one we are accustomed to see, right?”
Lisa C continued: “Yes, my dear. The Einstein field equations describe the equivalence of the space-time, warped from the presence of a mass, to the energy contained within that space-time.
Einstein believed that his description matched the observation, while the “guilty lawyers” of Copenhagen claimed not to be true in the case of quantum mechanics, even if the majority of them didn’t really believed in their own claim.
The relationship “phenomena – observer” was valid, more than ever, for relativity; Einstein could have correctly deducted the real nature of space if he had considered the gravity and the mass as two aspects of the same energy source; instead of establishing a principle of equivalence, who lead him to his ten, and very complicated, field equations...”
“Lisa” said Lisa W “you just said that it is possible to correctly measure the commutable space with an oscillator. If a clock measures comparatively the flow of time, how this oscillator should conduct a comparative measurement of the flow of energy of the commutable space? Space, for our five senses, is perceived as a static living space, while the time is perceived as an evolution...”
“Lisa” answered Lisa C “may be you are a bit tired now. The flow of energy, coming in and getting out of the commutable space, happens as first level interaction, which typically is not perceivable from an individualized point of view. First level interactions doesn’t have space locations, time flow, irradiative emission, and so on.
I want to make you an example of pure fantasy. Let’s say that I want to light the candle that is on this table, or light the bulb in your bedroom, without using a lighter or flipping the switch, respectively. Let’s say that, instead of intervening on the translated perception of the candle or the light bulb, I want scale up my intervention, and act directly on the flow of operators, before the “translation”.
You would “see” me doing totally unrelated things, with specific timings, sequence and intensity, like throwing a stone in a Alaska location, wetting a papyri plant on the Nile, while chanting a song in old Celtic language… and if I repeat the same set of actions a minute later, they will not produce the same effect, very likely they will produce no effect at all.
Any action in the perceived space has a correspondence in the commutable space, and vice versa; basically there is no difference - except the way we construct the mechanisms of cause and effect.
To be back to your original question, a clock to measure the space would not be perceivable. Later and if the circumstances will allow it, I will be happy to provide you a design of a clock which measure the space, by translating inputs from commutative space into the geometric space…”
“Lisa you are making me curious behind any possibility of rescue…” said Lisa W. “How this clock would be described from an human electronic designer?”
“Well” said Lisa C “your designers are familiar with different types of creating a circuit; I refer to the distributed constants and the concentrated constants designs. The reason to use one or another is mainly tied to the frequency of operation of the electronic circuits, as at higher frequencies the components like diodes, capacitors, and so on, start behaving in mixed mode, so a correct concentrated constant design is no longer possible. I am saying this since I need to introduce a distributed criteria of design, which is not of the type commonly intended from circuit designers. Your question regarding the design of such a device expands behind the simple clock to measure the commutable space. Based on the accepted criteria of your designers, it would be composed with a computing unit and some sensors, to measure some parameters tied to the entity you want to measure or control. Trust me, is not going to work this way.
The device you ask me would not be manufactured with the process of assembling concentrated or distributed components. The solution is not of technological, but of epistemological nature. You already have the technology for controlled crystal growth, used to manufacture complex semiconductor structures. You can already control the growth at single atom level: you just don’t care or don’t know what happen inside the crystal you make grow, you just care of using the properties of the free electrons to built miniature switches, and use them with your flawed logic. It is a very coarse use of the technology which you have been able to develop, a clear indication of a failure of your science, not your minds. Your creativity, when is not tightly embroidered with your science, as when it is the case of building machines, can make miracles. The crystals you already are able to build are the microprocessors for logic calculations, or to produce electricity from the light, or to produce light from electricity; to produce electricity from the heat flow; to detect electric or magnetic fields, to convert mechanical into electrical form of energy and so on. If you note, your crystals can make only what can be addressed with your quantum mechanics science. With the existing manufacturing plants, you could do much more, only if you had a correct addressing. Now, your clock to measure the space would be no different from an actual GPS device, except the fact it wouldn’t need a constellation of artificial satellites to work, because the crystal inside it would be tuned on the correct parameters to make itself able to detect its own position in the geometrical space. It may sound science fiction but it isn’t. It is based on the principle that, if there is a geometrical space where you can be in, therefore there is an energetic flow, a first level interaction, in the commutative space which can be detected with the proper means.”
“Lisa” said Lisa W, “you promised me the macros describing the most important sequences of space operators which translate into the most important second level events and phenomena. Do you think I will be ever able to grasp the full reasoning behind the sequencing by myself?”
“I was scared from this very question of yours, my dear” said Lisa C. Your mind is not designed to be away from itself for too long, a necessary precondition to “see” the non-translated operators at work. In addition, your mind cannot rationally solve, and sometime not even grasp, situations where there are more than six or seven simultaneous variables.
Let’s say you are driving on the motorway. It is raining, you are on a slope, which is also curving. Parallel to your motorway lane, there is an entry lane, and you see another car which is about to enter. You don’t want to hit the car obviously, and automatically you start to make an estimation of a possible collision point, in order to avoid it.
In your school, you have been taught just basic mathematics. You don’t know about mathematical functions, differential equations, or other procedures where the incognita is a function… but your mind start estimating the velocity function of your car, the other’s car velocity function, the effect of the slope, the tangential acceleration produced from the curve, the effect of the rain, the power of the engines and the effects of the brakes, effectively setting and solving a complex differential equation.
Analogically, there are already some functions of your brain which can unknowingly elaborate the space’s operator sequencing. You name them “gut feelings” and your science has already misrecognised them. There are documented cases where scientific discoveries have been produced only from such feelings; these discoveries were made following an hunch. The philosopher Immanuel Kant suggested the existence of galaxies one hundred fifty years before their effective discovery, but nobody ever credited him for his prediction. Other scientists were able to prove the result of their hunches, and were credited for their discovery. An “hunch” is not repeatable and therefore is not important for your science. I’ll give you a scientific method to correctly interpret your hunches and this has to suffice at the beginning, until you learn how to build a conscious process in your mind. The only prerequisite to build such a process is to study and understand the new noun-less logic.
Yes, time will come my dear when you will be able to follow, up to a given level of precision, the sequencing of the space’s operators.”
The door bell rang, and Lisa W stood up to open. Michael, Andrew, Mariah and Daniel were back from the kitchens with a cart full of thermal boxes. As soon they stepped in, Lisa C looked at Michael with a spirited look. Michael understood her without a word being said. “Can you help me to unload the cart? I need to bring it back otherwise the boys tomorrow will be upset with me!” “Sure” said Daniel, who understood the situation, too.
The pizzas and the drinks were quickly removed from the boxes. Michael said: “Andrew, please, can you bring the cart back to the kitchens? Tomorrow we will all very busy and I don’t want the boys to be after me…”
“Sure…” Was his answer. “I will be back in few minutes, don’t eat all the pizza please…”
When Andrew was out of the door, and the door was shut, Michael waited few more seconds before relaxing, while Mariah said: “Again, the spies?”
“Good idea” said Lisa W. And we all sat around the table. Michael was serving the pizza, potato chips, fried mozzarella, and rice balls. “Here are some beers if you want some” Michael said, while placing various cans of soda and bottles of water on the table. The pizza was still hot, thanks to the very good thermal container, and the drinks were very cold for the same reason. Armed of forks and knifes, we all started eating the well deserved foods… not the healthiest in my opinion; we could have easily drove to a good restaurant, for which our area was famous, but we liked to spend all the available time together.
After five minutes of silence, where we all appreciated in full the delicious taste of the foods, it was clear that everybody was thinking about the explanations Lisa C was providing to us… her ability to simplify, sometime excessively, the most important theory known to man, was of continuous stimulus for all of us. Our cultural backgrounds are different... myself being just a self taught amateur logician. In a different environment and situation, I would have kept listening without any further enquiry, since any question of mine would have uncovered the big knowledge gap between myself and the rest of the participants.
The way Lisa C was teaching us the new theory was, on the contrary, encouraging our curiosity at the expense of our personal pride, so I asked her:
“Lisa I have a question for you.” “Go ahead Tony” was her calm reply. “You know I don’t have any formal preparation in the theoretical physics, but I kept reading whatever explanation didn’t had mathematics inside it, hoping to understand what is behind it. I have read about the wave function, the probability it has to materialise, and about the example of the cat in the box that Schrodinger used to explain it to non-elect people. If I understand it correctly, when you open the box to see if the cat is dead or alive as result of the release of a poison caused from the possible atomic decay, you start the game of chance... in his words, there is a collapse into reality of the wave function. I don’t like it, but this is what the mathematics say. I connected this knowledge of mine about the cat to what you said about the creation of the Universe-observer relationship: the creation of this relationship makes the wave function collapse. Am I wrong, or there is a similarity between this concept, and what you said about the connection of perceived Universe, the abstraction layer, and the Universe described with the space of operators?
“Tony” answered Lisa C “sorry to say this, but you are still confused... but just a bit. If you could ask Schrodinger if the cat was dead or alive before you opened the box, he doesn’t have an answer. In the quantum world, the description doesn’t match the reality, and you don’t need to know the reality, mainly because you cannot know it. In our new theory, there is an exact match between the two sides of the abstraction layer; there is the same cat, but “here” it is described in terms of atoms, molecules, proteins and all the subsystems that produces life. On the other side of the layer, the cat IS MADE of a sequence of mathematical operators which you can describe as a “schema”. At the same, you can have the Schrodinger setting properly translated, and it wouldn’t be any wave function. The emission of the subatomic particle that would kill the cat at random wouldn’t be random anymore. You can know about its state if your computing capability, trough the abstraction layer, is powerful enough to calculate with enough precision the evolution of the involved operators. But... there is a caveat. Forget for a moment the abstraction layer. We haven’t talk yet about what a “schema” is in the new theory, and its relation with the priorities of the operators. It is just an anticipation, but the life, intended as a phenomena, is a schema of high priority. It wasn’t certainly in the intention of Schrodinger, but tying a life to the quantum behaviour of a particle is a very particular case, which is not a good demonstration for our purpose...”
“Lisa” said Lisa W “are you saying that the real Universe has a purpose?” “Not here, Lisa” answered Lisa C” “but a definite behaviour, yes. The quantity of energy flowing in and out the commutative space has specific energetic constraints. The mathematical space operators tends mainly to organise the energy in order to maximise its flow, and this happens at all energetic levels. The principle of the minimum state of energy of a system, which you know very well, is a direct consequence of this behaviour. Life is another consequence. Life is an universal schema, repeated wherever is possible, intended to maximise the flow from the commutative space; note that I said the flow, not the dissipation... I’m not stating the anthropic principle, I am just making a statement of coherence, valid in a “place” where the energy is the order...”
“Lisa” Andrew intervened with anxiety. “I was thinking of what you said of the first level of interaction, which we cannot detect, so all the particles and phenomena we observe are a second level interaction. I want to cite Gödel, what he said about theories which encloses other theories.
It is possible that you just wrapped another layer of theory around the existing one, just to encompass it, and at a more careful observation your theory is incomplete by definition, again?”
“Good question” said Lisa C. The answer is short and simple: no. The incompleteness, which was predicated from Gödel, and recognised but not accepted from the scientific community, is a gentle way to exclude logical paradoxes caused from mixing verbs and nouns into the same paradigms. When I put together the new theory, I have excluded a priori any illogical consequence, of which the incompleteness is one of the most obvious; to me it sounds so strange, but so strange, that all of your theorists never complete their proposals. It seems to me they accept incompleteness as a fact of life... at a logical observation, your theories, those which approaches better the reality of facts, are almost purely conceptual. This should be the hint driving your scientists toward the understanding that individualisations are creating paradoxes, instead they prove a strange feeling in making their theories totally abstract, as much as they can... a good step toward finding the correct theory of Universe, but as long as there will be only one, I say only one noun or a reference to a noun in their statements, there will be unsolvable paradoxes. This is why I preferred a radical approach, with the use of the abstraction layer, where I separate totally the two worlds... there is no risk of mistakes or incompleteness or paradoxes... and I should say... it is the only way.”
Andrew intervened again. “With the latest attempt in looking for particles based on the supersimmetry, which cannot be observed in the Standard Model, every particle should have a partner, therefore...”
“Andy” said Lisa C with decision “First of all, this is not the completeness I was talking about. You are speaking of the completeness of the supersimmetry model, which is good from one side, since it would explain dark matter, but is bad since it is sensible to the level of energy. The spin of the particles, on which the supersimmetry is based upon, is a relational behaviour of a particle, and logically incorrect. Supersimmetry relates particles that makes up matter to the particles that transmit forces, and this is a rare point of union between your physics and our new theory of Universe, as it would support the structure of space as we described it. When you decompose the particle with a spin trough the abstraction layer, the result from the other side is the sequence of the operators “harmony” and the signalling operator “no return”, repeated to create all the wanted instances of the particle, with matter and forces all together... The “social” behaviour of the particle, like the spin, comes afterward.”
“Lisa!” screamed Lisa W “it is the first time in absolute you name the space operators. Do you intend to disclose it now?”
“Sorry” said Lisa C. “No, it is not my intention yet to disclose the theory in full. Even if you name the operators, their name will just be an indication, from which you cannot elaborate much. As I promised you, at a certain point I will provide you with some macros; will be a later job, when the necessary mathematics will be ready, to disclose the full theory. At the moment, we are proceeding step by step. To answer Andy’s question, the supersimmetry is a much better model than the Standard Model, but is full of incoherencies because the way you describe your particles makes them to appear or not appear based on the level of energy you conduct your observations. Particles are materialised from the flow of energy, which goes in and out the commutable space, and they present themselves with all the properties you correctly observe. In reality, the particles appear or not following a balance of energy with a different logic, where the level of energy represent an operators priority shift, which only indirectly influences the materialisation of the particles you see. I mean you could observe the Higgs’ boson above one teraelectronvolt, and confirm a model or another; your technical calculations will be always be based on the principle of quantum mechanics, therefore the observation will not add any knowledge. You will be looking for a cause effect mechanism, making a relation between the presence of the Higgs particle and the energy, based on a principle which cannot confirm its existence but only a description of a range of existence. There are at least a dozen of logical errors in it, and we already discussed them...”
“Lisa” said Lisa W “I think we can postpone this discussion to the moment you will decide to disclose the nature of the space operators...” “Yes, maybe” replied Lisa C. “For the time being,” continued Lisa W, “would you be so kind to tell us what do you think of the theory of the strings? You never mentioned it in your lessons and discussions...”
“Because strings haven’t changed the history” was the straight answer from Lisa C. In the attempt to describe the graviton with a string, string theorists hoped to unify quantum with relativity, but the interaction of strings didn’t produced the expected results.
You know that strings are floating in space-time, as they are a direct replacement for particles: their “vibrations” represents the mass and the energy. You know what we think of the real nature of space, as we just finished to talk about it. I don’t want to dismiss strings by just saying that they are an embellishment of the particles theory: there is something good in strings, and I want to note it. The bad side of it is the fact that the string, at the distances you are interested in, where you would observe the gravity interaction with particles, have necessarily to be ten raised to minus thirty-three centimetres, or the Planck length: this rules out their direct observation. The good think is the fact that, based on the new theory of the commutative space, high energy supersimmetry is definitively true: for every fermion there is a boson; there is no need to build powerful particle accelerators to prove it, it is enough our noun-less logic and the theory of commutative space which, among other things, makes particles theory blend with gravity. Back to the strings... the supersimmetry confirms the strings model for very small scale. It would not be neat as our commutative space theory, but it would be a step in the correct direction. So good for the string theory.
The string theory, more than any other modern theory, suffer from incompleteness, and the self dual point is its most prominent “demonstration”... I bet you can’t find the paradox in higher dimensions, with nested string theories, and the disappearance of massive particles and strange symmetries.

Lisa C, for a moment, stopped her vehement flow of words, which had the same impetuosity of a river who wants to get to the sea. Mariah was listening with unexpected interest, Michael had abandoned his inborn anxiety and looked very calm. Andrew, which in the group was the least indoctrinated at theoretical level, was releasing a previously unseen self-esteem. Daniel was taking some notes from time to time, and myself... I was listening and learning, with a speed which I thought impossible for my mind. Lisa C, I am sure, was using some subliminal trickery to speed up our learning. Lisa, from time to time, was looking fuzzy, then very enthusiast.
Lisa C continued: “Let’s tackle one argument at time. First of all there is the need for more than four dimensions for the string theory. As I said, the string theory is an halfway theory suffering of incompleteness; if you add to this what we have studied about the equivalence of S.T.E.M., it is easy to understand that the need for more dimensions is caused from the inability to manage particles or phenomena with an elevated time content. One day, you will not need dimensions at all; for the time being, to manage the patterns produced from the shift of energy from observed mass and energy toward time and space, since you don’t understand the true nature of the latest, you must work with more geometrical dimensions. There is nothing bad, since you understand you are working on a model, but... why then hook up yourself with brane worlds in higher dimensions? I already told you that more dimensions pops up as required, they are not a static geometry of your perceived space. They exists in the layout of the geometric space, but their manifestation is evident only when dealing with the organisation of higher energies. Higher dimensions are crossed from gravity force, but to get the full extent of it, we need to refer to the modality energy cycles in and out of commutative space. The best your theorists have been able to achieve here is the supersimmetry; in the commutative space it describes the commutability of different manifestations which are the matter and the gravity. Remember this cycle is not related to any geometrical manifestation; energy gets out - and in again, “pushing” - in terms of “harmony” - as hard as it can, and is limited only from “organisational” factors: it is the true law of thermodynamics, which we will discuss later. And all of this happens below your threshold of perception, as you see the particles and the gravity force only after they have interacted at first level, in the act of “appearing”. If you could look under this “cover”, and slowly you will, you would see that things are much simpler than what you theorise. An high level of conceptualisation allow the elimination of the need of individualisation; therefore your better theories may look transcendentally complicate; I want to cite as an example the heterotic strings. As you eliminate the majority of nouns from your logic reasoning, the results look more solid; your scientists, almost unconsciously, are pursuing this road, driven from the promising solutions, but... unless somebody removes the geometric components from the reasoning, they are fated to run in circle forever. It is a pity that the most fundamental argument on Earth is resulting to be a childish hide-and-seek game, and even if embellished with academic importance and sustained from a scientific cooperation never seen before in history, will never get to the point – unless somebody will superimpose the necessary paradigm shift; and for this, we are here.
To conclude my answer to your question about the string theory, I want to add just few more notes. The dimensionless commutable space looks like more information exchange, rather than an energy exchange. Instead, seen from the geometrical space, information can “travel” without any energetic contribution, therefore without any speed limit: it is an intrinsic property of the commutative space.”
“Yes” said Lisa W “it was clear from the beginning, even if you didn’t mentioned explicitly yet, that in the world of space’s operators information do not require a carrier. It remain tough to understand the functioning of the noun-less logic in the commutative space, but I’m getting in it - as you said, slowly.

Michael intervened. “Lisa, I think after this fruitful day, and a quick but tasteful pizza based lunch, we all want to rest... or, at least, to relax listening some music or playing cards. Before going to bed, we need to have a clear agenda for the coming days.

“We have to” answered Lisa C. “There is another fundamental aspect of quantum physics that I have left uncovered, and it may take half hour to clarify... Than we can relax with some gossip. Lisa?” Lisa W was tapping on her palm computer, and didn’t realised the request from Lisa C. I looked at the small screen, trying to understand what was attracting all of her attention. Well... she was playing the jewel game. From time to time, I was playing the jewel game to, both on my palm and on my net top, but... Lisa was showing an unexpected level of skill, effectively pre-empting the computer in finding the combinations of coloured jewels. “Yes” said Lisa W, give me few more seconds and I finish this level. I was listening you, Lisa, even if I didn’t responded...”
“Lisa” continued Lisa C. “What is the last part of quantum mechanics I still need to put in perspective?”
“Hilbert spaces. You have mentioned them only in a scattered manner during the discussions. Few more seconds...ok.” The palm computer emitted a nice winning melody. “I topped my previous record of one hundred thousand points in the untimed game.”
“Very good” said Lisa C. “Do you have any specific question on the argument?”
Andrew intervened in the discussion. What exactly are you talking about Lisa? Why these spaces are needed?”
“Good question, as usual Andrew.” Answered Lisa C. And continued: “The Hilbert spaces were introduced from Hilbert more than a century ago, and used from Von Neumann for his formalisation of the quantum mechanics. There is a variety of these spaces described in mathematics. They are used as a container to ease the calculations of the variations; the mathematics call them “space of functions”. You can define a “vector” which describes a force, or any other physical manifestation; in the function spaces these vectors are interoperated with an operation called “scalar product”, which relates two vectors and the result is a number, which is also called “scalar”. In a finite dimensions, these products are used for very important calculations in your physics: solution of some differential equations, differentiable functions around a steady point, for the conical functions, and so on. In infinite dimensions, the scalar product defines the most important applications of the Hilbert spaces. The scalar products are called “Hermitians” when they include complex numbers.
In the quantum mechanics, it is common practice to define a “physical state” as a vector in the Hilbert space. When you want to explicit the properties of this vector, it is enough to project it on the auto state of an observable. This produces a new element in a new Hilbert space; they are called dual space, and the new element is called a wave function.”
“Wow!” said Andrew. I almost understood it!”
“Andrew” said Lisa C “the others knows these things very well already, as they are the basic notions of quantum physics. I’m repeating them in a very short form for your benefit, but also with a personal purpose, which will emerge later.”
“I’m happy” said Andrew. “Please, Lisa go ahead.”
“Well” continued Lisa C “When you observe the state of the observable, you make its state to collapse into the reality: there is a superimposition of all possible states, but when you watch the wave function, which describes all of these states, it will collapse into one state. I mean, you can see if the cat is dead or alive...
...in the Hilbert spaces, the observable is described by a linear operator, often an Hermitian operator, since it will likely contain complex numbers.”
“Very good” intervened Daniel. This is the shortest, yet effective, description of the Hilbert spaces I ever heard...”
“It is not so” replied Lisa C, and speaking to Lisa W, she said: “Lisa, you will pardon me if this explanation is not up to the fine level of description which is your standard, but this is not a lesson, and my aim now is to explain something really important.”
Lisa W replied: “It is OK Lisa, should I ever explain the Hilbert spaces in an elementary school, I will be proud of using your words. I may just want to say better what complex number are, since the kids don’t know what an imaginary number is... They don’t even know about square roots... I don’t know how I would explain complex numbers to kids.” Lisa C said: “Just tell them that complex numbers are numbers that contain an angle. They will understand.”
Lisa remained in silence, and her face was looking fuzzy again. After ten or fifteen seconds, her face sprung back to life, and with a smile she said: “I understand it too, now!”
Lisa C continued: “I have said that Hilbert spaces were one step from being the correct explanation of our new Space of Operators. The duality of Hilbert spaces is what was necessary to translate sequence of operators into observables... but it doesn’t work, sorry. Von Neumann did his magic, but he wasn’t able to beat what he had chosen to ignore. The fault doesn’t lie into the definition of Hilbert spaces. Their application, yes, it is faulty.”
“It has something to do with the logic of the application, I can guess.” Said Lisa W. “You are right Lisa, but do you know why?” replied Lisa C, including the question.
“Well” my Lisa replied “it has to do with our faulty logic, due to the inclusion of numbers?”
“You can lawfully use numbers at this side of the abstraction layer, Lisa. It is unlawful to include individualised vectors into a set before the duality.”
Michael jumped off the sofa. Lisa W, if I can say so, yelled in silence.
The entire concept of a logic set is correct, if you refer to a set that contains concepts and abstracted entities. It fails when you consider a set of numbers, vectors, or other “objects with a name”.
“Wow.” Lisa W pronounced. “We are at a standstill...” “Lisa, continued Lisa C “do you know that the masterpiece of human logic, the theorem of Hahn-Banach, is false? In your functional analysis, vector subspaces are a false pretence, by definition. This is only a consequence of the fundamental failure present in your science. We have always been talking of the fact that the inclusion of individualisations make any logic reasoning to fail, and you accepted it by faith, I would say.”