Near physical exhaustion point, the other night I was listening all night to YouTube videos of some Oxford professor type lecturing on the math of quantum mechanics. And in that he addresses core topics – including energy. In my delirium state, I’m half dreaming – about his quantum states and his argument in which he builds out wave functions FROM such state-representation algebra.
At some point, he goes on about these energy states. They certainly-defined (i.e. non probabilistic) energy levels – which the dream immediately concludes are really the orbitals of the hydrogen atom. One has energy amplitudes, the dream reasons, much as one has other amplitudes. Much as one might be conserving energy, one can be abstractly conserving probabilities – in many forms (since its so abstract). Remember, his early lectures are about pure kets and their representation in some coordinate system founded on a choice of basis, and moving between one basis and a next using matrix representation of (functional) operators. He is transitions from this core reasoning framework and starting to introduce wave functions, built out of the quantum states and their representations. Before or after ( I cannot recall right), he is basically introducing the notion that bras, matrix operators and commutators are ALL types of functional. One has within this abstract new-new-math all the elements of school-boy math recast into a neater formalism (i.e. counting coordinates, matrix rows as simultaneous equations, and differential equations). In particular, the commutator (as the generalization within conjugation-capable algebras of the basic calculus differentiation operator) encapsulates the need to “create” specifically-small probability differences – as the process for manufacturing commutators of ever lesser dimension goes along its merry way.
In particular, the semi-dream cuts to me conjecturing that the stuff about which he is talking can be
“intuitively” projected onto DES. I’m a student (in his classroom, along with others) and Im day-dreaming (in the dream, recall). I see, in the dream’s day-dream, that the processes of DES are the same as his equations.
First, and don’t laugh, the dream-me equates the core energy states with the subkeys of DES. In dream-time, just before that idea comes to the dream’s dream-mind, I actually realize that the differential Schrodinger equation (that he is postulating and motivating) is a process-differentiating process identical with the rather similarly process-diffusing-process that lies at the heart of DES. I figure that just as DES uses rounds to build dynamically a future-predicting process that ensures that differential probabilities get small and smaller, so the Schrodinger equation is the generalization of this. Or, rather, DES is one specialized representation of the whole wave-function elaboration being built out from raw quantum states (of the electron orbitals!) The Fiestel round mixes energy, deviating from the pure state representing of the subkey!
Clearly my dream is taking some of my favorite ideas and mixing them up with memories of his lecture, which may even be ongoing. In particular, it sees the way in which the DES expander works to select which of the sboxes are active, having duplicated input bits in a non-linear manner, and then having used the output within the permutation to work with the rules of the expander and the design of sboxes to implement a particular future-path-deciding hamiltonian-characterized energy transducer, we see how bit and key dependency is used to create a stream of bras, which act on the state to progressively reduce the differentials uniformity.
Later, this dream state continues – but has shifted century. Now I see that Turing’s use of inner products in a normed vector space was, all along, a veiled statement – breaking secrecy – that his cipher designs were themselves REALLY expressions of quantum mechanical processes, and said process is a function-generating process manufacturing bras (functionals) that when acting on the kets of the current differences found in current round-text further diffuse the probabilities encountered between bits of input (the dream actually had characters in an enigma-break) , till reaching the uniform probability space. AS the projection of the applied bra upon the ket-difference gets smaller and smaller, so one is approaching the uniformity required as the difference in any historical or future trajectory/projection within the local function is within a certain small bound – above or below 1/2.
Yes it’s a dream – improperly mixing up linear and differential cryptanalysis techniques! But the point was made that to design such a hamiltonian one has to consider the particular spectrum of just the kind of “energy operator” that can ensure a correspondence between reducing differentials in probability space and reducing commutators in quantum space. Stepping ouf of the dream, of course, it was considering the topic of how to set the rules of the engine, so that, given an initial condition, not only is their a unique output but a guaranteed and complete set of ever-decreasing commutators.