This is going to be a fun post to write, furthering ideas in https://yorkporc.wordpress.com/2013/12/31/turings-argument-on-limiting-distributions/. We have enough confidence now to relate Turing’s thinking to the harmonic oscillator, now. Strangely enough, its Susskind “off the cuff” comments that make the difference – when he gets out of standard-university teaching modes and gets back to what he is good at: motivating intuition, and conveying mental models (that are not orthodox and examination material).
Susskind builds his formal model starting with the Hamiltonian of a system. The theory model chosen is the world of spin-half particles from which one obtains the raising and lower operators a+ and a-. He models Dirac ket states functionally, using the wave function PSI(x). TO this wave function – in its position vs momentum basis, operators such as P and X are applied. He distinguishes this representation from the abstract equivalent of P and X acting on dirac kets. In the concrete representation, suitable for integral calculus, his wave function of function is acted upon by either P or X. In the resulting functional expansion, such applicions of P and X operators induce a particular computational expression: multiply PSI(x) by the value x for the X operator, and differentiate PSI(x) wit respect to X for the P operator. Of course, a wave function (in positional basis) in dirac notation is just the inner product of the bra ‘X’ with the ket “Psi”, where Psi is a superposition of position and momentum terms). The functional expression PSI(x) is the projection of the PSI superposition onto the X axis. The functional expression when applying the operator X to PSI(x) becomes, in Dirac notation the product of x with the wave function now as the inner product expressed in dirac notation, as above.
When one wants to calculate the generated wave functions of the harmonic oscillator – the violin string or the atomic system – one needs to know what distinguishes each such function. So one now analyzes the Hamiltonian as a constrained energy system, given its computational form and given the computational operators for X and P. As one adds “levels l” of angular momentum to the system (in phase space one goes around the orbit an integer l times), one “generates” a new wave function. The number of poles (zero points) is a function of l.
Remember, that we must not forget that we have baseline equilibrium for level = 0- which is modeled as that simplest of all wave functions – the simple Gaussian about zero. All other generated wave functions (for l=1,…) oscillate, being fixed always at zero, where the number of zero crossings is the integer value of l.
since Susskind has solved the second order differential equation of the hamiltonian in an discrete energy world, he gets us to the “calculation form” of a- and a+. cast as an interaction between P (in its integral calculation form) and X (also in its integration calculation form).
Remembering now how a+ or a- are defined, in calculation form, as an interaction of P and X
he points how that a product expression of the form a-.a+ or a-.a, or … – will be inducing +x and –x terms in the calculation form, the resultant sign of x being due to the signs of a+ or a- as they interact, in a parity and anti-symmetric sense. Thus, the anti-symmetry will induce the change-impetus to cancel, inducing a lowering of energy level by one, or they will cohere inducing a raising of by 1 quantum number level. In wave function terms, we now have a world in which the function being generated looks like a “Gaussian times x” – which generates all the different wave functions of the oscillator … as x varies.
From this we get to the “Gaussian about zero” – associated with the ground state, |g>. And one can have a first oscillator wave function – associated with the first excited state, |e1>. And moreover, if one is in |e2> one can lower to |e1> , going backwards.
The point is that a number of a+ and a- operators once put in sequence (think Turing’s paper on words) define a particular wave function. As Susskind points out, the action of these a+/a- terms is to generate polynomial terms. When the number of a+ (add 1) and a- (subtract 1) sums to zero, we have the ground state – or the Gaussian about zero.
Given this better portrayal of the harmonic oscillator and its relevance to physics (compare it to the crap Binny gave, bored at talking math to undergrads) , we finally understood Turing’s quantum computing model better. For Turing, the Octonian generator +i is playing role of a+, and +j is a-. Powers of I and j become multiples of a+ and a-. That different generators seem to playing the ‘a’ role is not a problem. That, in an expression a+.a- we then have the interaction of two generators is again NOT a problem since the whole octonian model underlying the 7 point geometry and the representation of that as a suitable non-exceptional cipher wheel wiring can be recast in terms of powers of a single generator, k, anyways (allow for a co-cycle of 2)
Each couplet of wheels, nominally inducing generator I and then generator J, generates a new “level of” wave function. 4 couplets gives l= 4, which under l(l+1) gives 20 states in the expanded wave function. So we see how the complexity model works, now. And we see the constraining and concentrating model too, if you go in reverse – with a contra-variant orientation.
Now, it also makes perfect sense to think in terms of norms, series of diminishing norms and the limits of norms, where for Turing changes in the norms due to averaging/diffusing/variance-concentrating effect of a mean =1 wheel density are what the physicists would call the energy eigenvalues. As the wheel couplets average , the variance in a series of plaintext characters through the wheel reduces to an simpler expression in terms of only a+/a-, which are surrogates for generators, which are always inducing a move towards the ground state, since the powers of the generators sum to zero.
This process is dividing the differences that is, acting as normed division algebra, in a compact space that induces an accumulation point as the dividing reaches a limit. In energy level and norm terms, the norm levels l are decreasing, as the reverse sequence of generated wave function concentrate the original energy/entropy of the plaintext back into a single Gaussian, having mixed it properly and till it reaches the shannon capacity of the channel – the ground state, or the minimal ground state energy level of that particular “plaintext channel” – or the decibannage of the run (to use a colossus era terminology).
So why does the wheel induce change in l, given the generators power sum to zero – and would seem to be having a null/equilibrium effect therefore? One has to remember that one is rotating the 7point geometry, as each plaintext character is entered. Unlike an enigma world in which one literally steps the a wheel inducing change of rods, stepping is a operator in Turing’s pure functional-lambda world. If one has say 4 couplets –where each couple is a 2 (i with j) wheel pair, it is the *overall* sum of the powers of generators i and j that must sum to zero – not the I and j offsets of a particular couplet. Since any two neighboring (identically wired) wheels may be (statically) set by the instruction of an indicator key in initial offset positions, the math of rod formation induced by the resultant conjugation is due to this local coupling of the wheels. Turing shows this, in his first pages. So. one is rotating the geometry, three turns of of which gets you back to identity, of course.
In his example, Turing shows that after just two rotations of his geometry, which performs “quantum-stepping” of the wheels in phase space, his norms have converged to the decibannage of the plaintext reflected in the fact that is no predictability of the ciphertext output by the wheel set. We recall
re rotations hamiltonians and group-level symmetry operators.
Of course, one has generated a virtual wheel, since input a still produce the same output, regardless of where a is found in the plaintext sequence. So Turing is using wheel wiring to generate a composite virtual wheel – which one can think of a a giant stecker.
ok I see how the “differencing mechanism” works. One has a custom division algebra, associated with the octonions. One takes the positions of the poles of a higher-level wave function and forces the choice of pole in a lower-level wave function (remembering these are fewer than the world one just came from). Thus one divides distance – by averaging. We can imagine that Turing’s point about the wheel “function” having mean 1 is meant to signal that the pole choice, during wave function mapping, is unbiased. We infer that this citation of mean 1 is ‘’Turing equivalent” to modern thinking about unbiased coin-tossing operators in the quantum random walk. As we learned, one coin-operator, the Hadamard transform, may be subtly biased; inducing a skewed output distribution that other operator designs may avoid in favor of constant output distribution.