The Schrödinger equation is integrable for completely unitary systems. This hold for Hamiltonians that are complex or nonlinear, where perturbation methods are often used. My thought is that all possible quantum eigenvalues are computable with Diophantine equations, where some Gödel numbering scheme from the solutions to the actual numbers can exist.
LC On Sunday, June 17, 2018 at 3:42:56 PM UTC-5, Jason wrote: > > Hi Lawrence, > > Is the evolution of states of the wave function computable? If so then the > result of MRDP implies it is Diophantine. > > Jason > > On Sunday, June 17, 2018, Lawrence Crowell <[email protected] > <javascript:>> wrote: > >> I have Matiyasevich's paper on the MRDP theorem. I have not as yet read >> it. I have had this idea that a general scheme for quantum eigenvalues >> could by Diophantine. This would then be a sort of universal dovetailer of >> all possible physical states. Unfortunately this is an area I have thought >> about some, but as yet have never endeavored to pursue. >> >> LC >> >> On Saturday, June 16, 2018 at 7:18:41 PM UTC-5, Jason wrote: >>> >>> In solving Hilbert's 10th problem >>> <https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem> in the >>> negative, the work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and >>> Julia Robinson culminated in 1970 with the MRDP theorem >>> <https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's_theorem> >>> which concludes: >>> >>> *Every computably enumerable set has a representation as a Diophantine >>> equation <https://en.wikipedia.org/wiki/Diophantine_equation> (an equation >>> involving only integer coefficients and variables).* >>> >>> This shocked number theorists, because it meant simple equations >>> involving nothing more than a few integer variables have the full power of >>> Turing machines. In fact, it was shown by Yuri Matiyasevich that a >>> universal Diophantine equation can be made with as few as 9 unknowns. >>> >>> Some examples: >>> >>> - k is even if there exists a solution to: k - 2x = 0 >>> - k is a perfect square if there exists a solution to: k - x^2 = 0 >>> - k is a Fibonacci number if there exists a solution to: k^4 - >>> k^2*x^2 - x^4 - 1 = 0 >>> - (k+2) is a prime number if there exists a solution to the sum of: >>> (these >>> 14 equations >>> <http://mathworld.wolfram.com/PrimeDiophantineEquations.html>) >>> - k is a LISP program having output n, if the equation having >>> variables: k, n, x1, x2, x3 ... x20000 (a polynomial having ~20,000 >>> variables <https://arxiv.org/pdf/math/0404335.pdf>) has a solution. >>> >>> The universality of Diophantine equations means there are polynomial >>> equations that compute things quite surprising, such as polynomials that >>> have solutions of 0, IFF: >>> >>> - One of the variables "k" is a valid MP3 file. >>> - One of the variables "k" is a JPEG image containing the image of a >>> cat (where the equation implements the same computation as a neural >>> network >>> trained to recognize images of cats) >>> - For two of the variables "y" and "x", "y" equals a state of a >>> chess board after deep blue makes a move given a chess board with a >>> state >>> of "x". >>> - For two of the variables "y" and "x", "y" equals the state of the >>> Universal Dovetailer after performing "n" steps of execution. >>> >>> >>> The last example seems to suggest to me, that pure arithmetical truth, >>> concerning the solutions to equations, is identical to computation. That >>> is to say, certain mathematical statements carry with them (effectively) >>> Turing machines, and their executions. >>> >>> Just as all solutions to the deep-blue implementing equation is >>> equivalent to the computations that Deep blue makes when evaluating the >>> board, and all solutions to the cat recognizing equation are equivalent to >>> the processing done by the trained neural network, all solutions to the >>> LISP equation are equivalent to the execution of every possible LISP >>> program (including the UD). >>> >>> Does this our conscious experience might be a direct consequence of >>> Diophantine equations? >>> >>> Can Diophantine equations for a single set of parameters model >>> non-halting programs like the UD, or one must consider the set of of all >>> possible parameters? >>> >>> Jason >>> >>> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected] <javascript:>. >> To post to this group, send email to [email protected] >> <javascript:>. >> Visit this group at https://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
