Mathematical motivations for application of QM formalism outside physics

0:00 If we integrate the square of the norm with respect to this measure, and this is nothing more than dispersion of measure. If average is zero, then, for example, for usual measure, if this is on finite dimensional space, if this integral is zero, then dispersion of this measure is just second moment. The same we have in infinite dimensional space, if we denote this dispersion, but we can calculate this integral and we can prove that this trace of correlation appears. The calculation is quite simple, you just expand this norm with respect to coordinates and integrate for each coordinate this one-dimensional integral. And you will get this sum, it will be trace of the covariation operator. So, in our model, we consider it measures which have dispersion alpha. So in our model, this trace is alpha. So the only difference from connoisseur density operator is that trace is not alpha. Therefore, we are practically about to get connoisseur density operator by dividing by alpha, more or less. So, we see that each measure, in fact, contains four normal density operators and its covariation operator with some scale. So, our map would be in future that we put just measure into scaling of its covariation operator. So in this approach, full normal density operators are not appeared by some postulating for some mystical reasons, they are just covariation operators of usual classical measures, but on this infinite dimensional space. So this is the idea of how we shall proceed.

2:30 And now we can formulate mathematical results. Which belong to this functional class. I have not defined it, but I can say that it consists of analytic functionals on this space. So analytic is the same as unusual. So, of course, from the first point of view it's a kind of problem because function is an infinite dimension of space. But we can define analyticity in usual way. We expand function into power series. And we can say that the function is analytic if this power theory is converged on this space. In recent mathematics, this theory was well developed. These are analytical functions on banks, spaces, and so on and so on. So we can consider analytic functions. And we can consider these measures that we considered with small dispersion. The dispersion is alpha. And then this is mathematical result. Then, this classical average, which is integral, is equal to the trace, this just correlation operator, we take this measure, which was here, we make it scaling with respect to square root of alpha, and we denote this by dc. So, this is more or less quantum term plus some perturbation. And now I would like to make some mathematics, and they have an estimation of this term, which is below the alpha square, with respect to this intensity of fluctuations of quantum. Now I would like maybe to make some mathematics to prove this equality and show what is in fact quantum mechanics from this point of view. From this point of view, quantum mechanics is just expansion in the Taylor series. So I shall try to show this mathematical exercise.

5:00 Sorry, just in the notation, what's the C there, the small c? This is small c, it's just complexification of this operator. Because in quantum mechanics we work in complex Hilbert space. Up to now my model was real. And to get complex Hilbert space I just made complexification of this space. I write it in this way. Just consider the second with I. So if I had the sign in this way. And if before I had some operator from H to B, I would like to consider it in the complex Hilbert space and I denote this Hc. So in some sense this is complexification. Because quantum mechanics work in complex Hilbert space, and if I would like, on this side should be quantum mechanical objects. So therefore I shall take a covariation operator, and then I consider this covariation operator not here, but in this complexification. Because usually a normal density operator is complex. It's set in complex Hilbert space. Therefore I should also make complexification. But if you forget about complex structure, you can just take away this and it would be the usual covariation operator of this scale. Now we prove this equality. In some sense this is the essence of this approach. And in some sense it demonstrates, if this model is correct, what should really happen in quantum mechanics. So we have this classical average. Classical average is nothing else than the usual integral with respect to this omicron. Omicron is this direct product of two Hilbert spaces. Now, my measure was such zero, and measure was concentrated very, very narrow around zero, so it was measure all.

7:30 Now I would like to make such transformation of variables, as people do in probability theory, in the law of large numbers and so on, to transform this measure into measure with unit dispersion. It's more or less standard trick, for example, when people formulate central linear theorem, they take the sum of variables and divide by square root of dispersion. So I do the same. And this is such change of variables. So I denote psi-smoke, square root of alpha, and here is psi-large, and u-variable. Just this change of variables in this integral. So, and the result would be integral over omega, here, alpha, psi-large. And here is new measure, which I called scaling of my previous measure. I made the change of variable and I shall get new measure. By computing Fourier transform, we can directly prove that the covariation operator of this scaling, you know, for example, this d, the covariation operator of this scale, is nothing else than the covariation operator of rho divided by r. This is more or less general fact of measure theory which has nothing to do with physics. If we make such change of variables, we get new covariation operator which is scaled in this way. Therefore Gaussian measure is clear because it's quadratic form with covariance operator. And we scale by r. So it's important to remember. So we know the covariation operator of this scaled measure.

10:00 And somewhere here we roll this t, this covariation operator, divided by r. We know that hr is a small parameter of our model. And we have this function. It was a lesson from the school, from my teacher, the QC function. Small parameter. Use Taylor formula. So it was more or less standard. So we use Taylor formula. Here on the integral we have f at 2.0 plus square root of alpha divided by 2. Second derivative of this function into Taylor series with respect to this small parameter. I make restrictions to the class of functions. I consider only functions which have zero is zero. This is quite natural restriction. This is vacuum. And I consider all such variables which from vacuum produce vacuum. So there is no effect of vacuum. Or you can just say mathematical restriction to the class of functions. We have vacuum. It would be strange if we get some non-zero value from vacuum. So this disappears by definition. So we consider only such variables.

12:30 This integral is in fact zero because mean value of measure is zero. Watch the proof. This is linear functional. We can take it from integral. And then it would be finally such stuff, f prime 0, apply to integral, psi, 0, psi, integral is linear, functional is linear, we can take functional from integral, but this is just mean value of measure, this is 0, so for measures this is 0 mean value, and for functions which do not produce from vacuum anything, The first non-trivial contribution is given by second derivative. So, our classical integral can be represented in such a way. There is not alpha square, there is alpha because it was square root. Square root means square, so it would be just alpha. And here is integral, infinite dimensional integral, of second derivative, psi, psi, below scaling, higher than alpha. And this integral is possible to calculate. This is quadratic form. Now we expand this psi with respect to orthonormal basis. And it will be just the sum of two dimensional quadratic forms. There are two types of two-dimensional integrals. They can be calculated, and the answer would be the following. After some mathematical computations, we shall get such an answer. This integral is equal to alpha divided by 2, and here we get the trace of this covariation operator t of this measure. And then there is a remark, since measure is symmetric, the third moment of this measure also disappears.

15:00 The first which would be non-zero would be with derivative of degree four. So the additional term which is here is of the order plus something of the order alpha squared. So this expansion. This is the correspondence between classical, model, and quant. Function is mapped in its Hessian, it's always self-adjoined. And measure is mapped into D, which is the variation operator, scaling. This is the correspondence between classical, model, and quant. This is huge projection. Huge class of functions has the same second derivative as zero. So, more or less, quantum mechanics pay attention only to second derivative and vector. This huge class of measures has the same covariation operators. So, quantum mechanics could not distinguish classical measures. So, what you call quantum state, it could be huge ensemble of different classical states. But then you have the same image, the same covariation matrix, the same von Neumann density operator. And then again, if we go to these relative intensities, we shall get such, if we take classical average of this relative intensity, it would be quantum plus something of the order alpha. So it would be, you can say, I don't know, maybe better. So, how we can now interpret quantum mechanics?

17:30 Quantum mechanics is special mechanism of approximation of classical averages for relative intensities, which is based on Taylor formula. So we take classical function, which is function of classical fields, expand it into Taylor series up to the second term, and take the first non-zero contribution. If we forget about these additional terms, it will be quantum mechanical description. But this model tells us that in general it's not precisely the average. It tells us that there could be additional terms. So if we, for example, make quantum predictions that the average of energy in this state is such, then I say that starting with some level of precision, There should be a deviation between quantum prediction and experimental which is given by this real integral. The main question is what is the magnitude of this deviation? What is the physical meaning of this parameter alpha? Because I already gave you talks at laboratories and people said OK, but we never check just that one quantity is different from another. You should show how many digits we should measure. They do not check inequality, it's impossible. They should know how many digits you expect it would be consistent and then different. Because in other way they could not compute the effect of other perturbations and so on. So the main question is how to find alpha. And if you are not tired, it's a different story which is related to Schrödinger non-linear equation. Or you can come to the very first time here. I'm not tired. I mean, I was about to ask about the dynamics. Okay, so then I go to dynamics. Yeah, so good. So the inconsistence of power is true. Go to dynamics. Yes, good question. Can you please give a precise example where you write a function in your space h cross h which corresponds to a simple quantum state, for example, a clear state?

20:00 Any pure state. Here how we can write measure corresponding to this psi. Good question, right? Would you be satisfied? We take any quantum pure state psi. If you like tolerable state or it's not important. Just pure state. Psi, normalizing vector. Now you want classical measure that will produce this pure state. So one of the strange features of this model is that pure state is not pure in my model. It's mixture. It's a measure which is not concentrated in one form. So a measure is produced in the following way, which I'll go over in my construction. First I know, and this I know from quantum mechanics, that this pure state is nothing else than von Neumann density, operator of such a form. Extend the product of these two wave functions. This is for Neumann density operator, for this pure state. Because if you take trace dA for any self-adjoined operator, you know this dPsi, then this trace is nothing less than dPsi, Psi. So these are the same. So take the trace with respect to pure state. It is this one. Or to take a trace with respect to this von Neumann density operator. So this is density operator corresponding to pure state.

22:30 Now we would like to write a measure. So now we would like to write its covariation operator of rho. And by this procedure it is alpha. Alpha is any parameter in our model unknown. Denote this by b. And then I can write just Gaussian measure on this infinite dimensional space with this covariation operator. So it would produce this Hilbert state. It would be special because this is projector. This is very special density operator. It's projector to this one dimensional space. This Gaussian measure also would be special. It would be just Gaussian measure concentrated on this space, on this one dimensional space. But nevertheless, it's not concentrated in one point, but in some distribution on this space. So this is classical measure that produces what we call in quantum mechanics pure state. So more or less even pure state is not pure from the point of view of this model. So we ignore these degrees of freedom and then we tell that she's pure. But if we not ignore them, it's not pure. Then make example how to write function, a sign which would produce for us, for example, quantum Hamiltonian. So it is also not a problem if we take quantum Hamiltonian. Then it should be just second derivative of my function. It's very easy to construct. Many functions which has a second derivative of this operator, but the simplest is just quadratic form. So if you just write this quadratic form, this function will produce this operator. So to produce just quantum mechanics, it's enough to consider quadratic forms.

25:00 But if you think that quantum mechanics is not complete, Then the second derivative would be the same. So quantum mechanics would not feel that in reality this operator is essentially more complicated, because this is small and we ignore this. But if we, at least my model tells maybe reality, maybe model is not adequate to reality, that if we go deeper to this precision then we should see this effect. And in fact this would be the origin of non-linear Schrodinger equation. But now I shall proceed to dynamics in more, in more detailed way. So dynamics. Dynamics is if you consider usual classical mechanics of usual classical phase space, R3, which kind of dynamical equations we have? We have Hamiltonian equations. So dynamics of classical particle in this phase space is described by Hamiltonian equations. Therefore, we do the same on our infinite dimensional phase space. So we consider on this infinite dimensional phase space such Hamiltonian equations. But now this is point of Hilbert space. Now these derivatives are not usual derivatives. Sometimes they call it variational derivatives. People use an inoptimal control and so on. This is H is functional. And this is variation of this functional with respect to this field. So because this is field, this is field and we make variation of this functional with respect to this field. But more or less definition of derivative is the same. So if we have functional infinite dimensional space, then we define derivative precisely the same way.

27:30 It should be represented in such a way which is linear functional delta x and here is osmol delta x and osmol is such terms which limit norm delta x goes to zero, module goes equal to zero. To define derivative on Banach space is the same as in real life but everywhere you write instead of absolute value the norm. So if everywhere in all analysis on the real life you write instead of absolute value the norm, you will get differential calculus on Banach space, on Hilbert space and so on. So there is Taylor formula, there is everything, so there is norm. So we can write such equations. So these are Hamiltonian equations on this infinite dimension of space. Now, I would like to get Schrodinger equation in some way. And I restrict my consideration only to quadratic Hamilton functions. Now I consider not arbitrary function h of q and p, but I consider only quadratic function. Some linear... So, and then I can write this equation and this form if this operator H is represented, because it's an operator in the indirect order, so it's represented by a matrix from H1 to H1, this is from this into this, this is from this to this, and so on, so this is more or less if you consider operator in R.

30:00 And then it would be written in such a block way. One is made from this into this, one from this into this, and this diagonal made this into this. So, in our case, we write in this way. Comptonian equation is just a linear equation. So I just compute derivative of this quadratic form. So I write the previous equation in this particular case. And now this equation, and it's well known in usual classical mechanics, we can write in such vector form that G is so-called symplectic operator. The symplectic operator is operator of such a form. You need not refer to some theory or so on, you can just put it. If you take this here and multiply this matrix by this, you will get these two equations. So you can check this formula directly. You put symplectic operator here, put this representation of H, and you will get this. So you are all of this formula in vector form. And we know that g square is minus 1. So it's nothing else. If we would like to write this in complex form, we get the complex Schrodinger equation. So this is Schrodinger equation. I just did manipulation. So we have such equation. Multiply both sides by g. g square. And g square is minus 1. And I send it here.

32:30 And now I make correspondence I, I realize by this operator it's square still minus one, and it's nothing else than Schrodinger equation. So the lesson, the Schrodinger equation is very special form of Hamiltonian equation of this infinite dimension of phase, space, when we restrict Hamiltonian functions to quadratic forms. Linear, Hamiltonian equations which can be written in this complex form, but in my model there is no reason to restrict yourself to quadratic forms. I can consider any function on this infinite dimensional space, and therefore in general my evolution equation has additional terms, which are non-linear terms. It is non-linear terms. So the basic equation is non-linear Schrodinger equation. But now you should understand why we do not observe this effect of non-linear terms. The measures that correspond to quantum space are probability distributions over these signs, and since these signs obey these Hamiltonian equations then we get an evolution that is induced on the probability measures And that turns out, if we look at the projection onto the quantum state, it then turns out that it's just, again, the Euclidean equation. Yeah, yeah. This is one step missing, but it's pretty obvious.

35:00 Yeah, yeah, yeah. So we should make scaling from this size to this. I would not like to discuss this more than scaling. But I would like to say that if we, this was equation with respect, with already psi large, more or less this is after scaling. If I write some equation before scaling and then make scaling, it would be after scaling of such form, for example. If I consider additional term of the degree 4, then my equation, which I get, this is just linear term. And here will be this small parameter with this non-linear term. So, either I can work on the z-level that is before scaling, and then this term is large, and I should take into account all non-linear effects. But I tell I do not want to work on this level. I would like to consider quantum state after this renormalization. Then additional terms would appear with this small parameter. So after scaling, if I make scaling in my Hamiltonian function, then they appear this parameter. But if I work with psi small, which is before scaling, there is no this alpha, and here on this level before scaling nonlinear effects are very strong. But I make scaling and I went to our... And now there is the idea how to estimate this alpha through theory of non-linear Schrodinger equation, because it was completely separate activity that people just told. We do not believe that fundamental equation of quantum mechanics is linear. They tell it is non-linear, just axiomatically. Then let us try to estimate what should be the effect of this contribution, that we still do not observe these measurements, that in experiments we still see all experiments confirm linear equations.

37:30 Let us estimate this parameter from this experiment. How small should be this alpha, that in all existing experiments we still do not see this non-linear effect. And it was a huge activity. It was a lot of papers and many experiments. So there are a few papers where this idea was discussed. So the theoretician was, one of the theoreticians was this guy, one called Bilansky, Birula. And they have theoretical fundamental paper and also physics. But then many, many people, they did experiments. Just to check if there is nonlinear contribution or not. And they put such an estimate, they tell that at the moment, this alpha, if we use energy scale, it should be less than 10 to the degree minus 15 electron volts. So this is in fact the level that is approachable now. So until this level, people do not see nonlinear effects in quantum mechanics. As I spoke recently in Berlin, soon they hope to approach 10-90. So maybe at this level such non-linear effects would appear and then this can even have consequences for my model. So this is the coincidence point because non-linear Schrodinger equations appear very naturally in my model. But of course there maybe should be another independent estimation of alpha. For me, I see the main problem that I could not get estimates of this small parameter inside my model. So I predict that quantum average gives only approximation, but I cannot say of which order. Therefore, the conversation with experimenters, they never speak about such thing.

40:00 It's a more or less standard question. I wanted to have a class and I did maybe 50 talks during the last two months in Moscow, in Berlin and in other places. It's a standard question. More or less all people ask me where is the place of non-commutativity. I think in my model no place in fact. So, more or less, I think people overestimate the role of non-commutativity. In my model, operators non-commutative just because second derivative is Hessian, and if you have two classical functions, and if you take two classical Hessians, they do not commute. Nothing more. And I consider one function, take its second derivative. And I consider another function, I take its second derivative. Then sometimes they commute, sometimes non-commute. It may be strange for, I think, standard quantum ideology, but here it's no fundamental thinking, no convocativity. And it's clear because the model is ontic and on ontic level I can define simultaneously everything, so for me it's not important do they commute or not commute, everything is defined but I do not claim such measures because there is no theory of measurement. There is this ontic form on this space. It's well-defined at the same time for this operator and this simultaneously, but I don't know how to measure them. Maybe it's impossible. Do you have a counter-path of non-commutativity? Yeah, it explains non-commutativity, but... You mean that it works the same result when you... Yeah, you are right. Yeah, so this is a counter-path of non-commutativity, but I do not see fundamental role.

42:30 So it's not something fundamental that, for example, for two functions, the hessians do not commute. Heisenberg uncertainty relations in my model is precise. There is an additional term for Heisenberg uncertainty relations. In principle, I could define everything simultaneously, but there is some restriction. That is a good question. I mean, what you do have are these averages. So, you are going to have an average of something that corresponds to the quantum average of The commutator of two operators. In the model it just corresponds to function which has Hessian which is commutator of these operators. It's not defined internally through two classical variables. Because the second derivative of the product is not... The commutator is not internally defined in the classical terms. The question is, is there anything that corresponds to the algebraic structure of the operators in a natural way? No, for co-edition it corresponds, because the sum is going into sum and linear combination is going into linear combination. So because if you take two functions and the second derivative of sum, you have this formula of this.

45:00 Linear structure is preserved. As linear spaces, they are in correspondence. But composition is not preserved. But the iteration of composition of two quantum observables is not derivative, it is not competitive, it is not more self-adjunctive. In some ways, we always know that from a mathematical point of view, if you have two operators, you can write these things. But what is physical meaning? It is not more self-adjunctive, they are not competitive. But it is a productive model, so I was even criticized in some places. Precisely for this, there is no correspondence for composition operation. The operation of composition of quantum operators has nothing natural on classical level. In some sense it's totally quantum operation which is not present in my theory. In some sense it's interesting that it's also axiom of von Neumann. In his novel Celerium, he used this. Even for non-computative, he used this. And this was very strongly criticized by Bell and Ballantyne. But nevertheless, in my model, it's also dissectionist-satisfied. I don't know if I have. I mean, even in quantum mechanics, it's so that it doesn't matter. No, no, no, not very clear, yes, not very clear. Because some people think that this just means first we measure this, then we measure this. But it's not, it's not the case, yes, because at least it's not self-adjoined operator, it's not observable at all, yes. Yeah, if they know that you're doing this addition, it's a problem, yes.

47:30 What is the composition of the operators in quantum mathematics? By myself, I was showing this and I observed this maybe two years ago when I gave this to my PhD student and then I started to think, well, but this is not self-adjointed if they are not committed to what you are talking about and then I was a little bit surprised. Roughly speaking, I do not know how in general case, how or why. Okay, we have these two observators, A and B. They represent a class of adjoined operators. Then I define such a measurement procedure. First I measure B and then A. Which is the operator which represents this measurement procedure? I don't know. It's a basement operator. This is a positive operator value measure. This is positive operator value measure, yeah. Have you got something in your model that corresponds to positive operator value? This I did not think about. I did not think about this. Yeah. Yeah, it's a good question, sir, because not only for Neumann observables exist, but exist, of course, positive variables and greater measures. Maybe some. I do not know. But it's a good question.