Dear professors, researchers,
After some models for the Riemann Xi and zeta function, I was able to get a curious equation which is based on two parameters in function of one final independent parameter that rules the entire equation and of course the complex variable s or z which is the variable for the domain of the Riemann Zeta function and the Riemann Xi. Although the model has not been revised at all, due to I need to compute the parameter I have mentioned, which demands transcendental equations in some parts, not difficult to compute but their analysis must be done, I have noticed that I have arrived to some similarities described in important articles regarding the conjecture of a hypothetical Hamiltonian, there are a lot of similarities like the fact of involving Bernoulli summations in some operators in some references and hyperbolic trigonometric functions mentioned in some works, and similarities like that seem to appear in my own model purely related to a mathematical methodology ( I am not defining physical terms like position, time or potential functions). Yesterday, after checking old articles and new ones regarding the Hamiltonian and the Polya's conjecture, Berry and other authors, I have noticed that I achieved some components as a resemblance to the "H*i about the i*H that is PT symmetric with a broken PTsymmetry" or at least what is understood in Article Hamiltonian for the Zeros of the Riemann Zeta Function
,with the imaginary unit i =sqrt(-1) , and the term 1/2 involved and the possibility to factor the structures of my model for the expected eingevalues and potential functions within a physic model. However, the work is not concluded and I have just wanted to be instructed by physicists or other experts to know the brief concrete mathematical and physical characteristics of an operator like H quantized and how to quantize it or describe it from my own results, how to understand properly the self-adjoint property and if it is obligatory to look for an Hamiltonian or it could be other operator that involve the eingevalues or in this case the imaginary part of the non trivial zeros within a physical context.I would like to have a serious contact with physicists and mathematicians who are interested in to resolve this part of the mathematical model, since I am convince that hyperbolic trigonometric functions and Bernoulli numbers are enrolled in this path!
Carlos
Carlos
The Riemann-Zeta function is used in analytic number theory and such things as the distribution of primes.
It has nothing direct to do with Hamiltonians in the usual quantum mechanics.
I just think here that the main interest is just mathematical.
PT symmetry and pseudo hermitian QM are in mathematical physics, mainly examples of non hermitian operators who still have real eigenvalues (at some cost). Not yet very relevant to actual physics.
How to quantize a Classical Hamiltonian for use in QM is a well known proceedure in physics. ie. consult modern physics ,book by Beiser for example.
It means translating dynamical variables into operators.
I suggest you follow the following related Lectures :: https://www.youtube.com/watch?v=v9nyNBLCPks
https://www.youtube.com/watch?v=way0jAWpjZA
https://www.quantamagazine.org/how-i-learned-to-love-and-fear-the-riemann-hypothesis-20210104/
Hyperbolic algebra is similar to complex algebra, except that you use
x+ty where tt=1 instead of tt=-1
Then you can have series expansions of known trig functions, for example,
in terms of this new variable.
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