The Physical Limitations on Mathematical Abstraction, the Representational Effect of Mathematics on Physical Explanation, and the Resulting Expansion of Computability by Steven P Sax

Hi Jon,

You asked an excellent question following through on the half pulse manifestation of the self-referential operator. Let me first clear up some of your questions on the technical infrastructure in the Rubidium setup:

A laser whose output power is constant over time is called a continuous wave laser and this is what was used in the setup. This is opposed to pulsed operation in which the optical power appears in pulses of some duration at some repetition rate (such as in a Q-switching or mode-locking laser). For continuous wave operation it's required for the population inversion of the gain medium to be continually replenished by a steady pump source. Even a laser whose output is normally continuous can be intentionally turned on and off at some rate in order to create pulses of light (the modulation rate is on time scales much slower than the cavity lifetime and the time period over which energy can be stored in the lasing medium or pumping mechanism, and it's still classified as a modulated or pulsed yet nevertheless continuous wave laser) - it is this type of pulse that's being shined on the Rubidium atom in the setup. Pulse of laser light thus refers to a duration of time in which the laser is being shined on the atom, necessary to excite the electron to the excited state. The half pulse refers to shining the laser light for half this duration of time on the atom.

Now, one half pulse (i.e. shining this continuous wave laser on the Rubidium atom for this half-duration time) causes the electron to go into superposition of the ground and excited states. Another half pulse excites it to the definitive excited state. This corresponds to two self referential operators, but that's the key: whatever the physical mechanism is, it is one that manifests the self referential operator. In representing a qubit, the most general state of a quantum two-level system can be written in the form ∣ψ〉 = α ∣0〉 + β∣1〉 where α and β are complex numbers. The state has to be normalized, so ∣α∣^2 + ∣β∣^2 = 1, and an overall phase makes no difference, so either α or β can be chosen to be real. This leads to the parametrization ∣ψ 〉 = cos (θ/2)∣0〉+ (e^iφ) sin (θ/2)∣1〉 in terms of only two real numbers θ and φ, with ranges 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. These are the same as the polar angles in 3-dimensional spherical coordinates, and this leads to the representation of the state as a point on a unit sphere called the Bloch sphere. The Bloch sphere can be traveled with pulses of different lengths and the system driven from the superposition state to the ground or excited states. But this can be done only during a certain amount of time which is called the coherence time. It can vary from several microseconds to several seconds depending on the particular system. When this period is over the system has to be initialized again in the ground state. If the phase of the driving field is fixed, the system will be going in one direction, but if phase is changed by π, the rotation axis in the Bloch representation is changed, and in that case the system might rotate in the reverse direction (for example around -x axis instead of the x-axis). One can also change the phase by π/2 which would result in a mirror image of the state. For example if a π/sqrt(2) pulse is shined and then the axis is changed from x to y, the coefficients of the two states in the superposition are reversed. So the physical manifestation for a self referential operator may not be the same half pulse but rather may change - but this makes sense: there are different physical paths as we see to get back to the ground state; hence there are different physical manifestations for the NOT operator. It then follows there would be different manifestations for the SQR (NOT) and thus for a self referential operator at that particular point. There's lots of research being done on these variations (and part of my own personal research) and I hope this can be shifted to the engineering of self-referential operator gates which would be an extremely fascinating field.

Hi Jon,

You asked an excellent question following through on the half pulse manifestation of the self-referential operator. Let me first clear up some of your questions on the technical infrastructure in the Rubidium setup:

A laser whose output power is constant over time is called a continuous wave laser and this is what was used in the setup. This is opposed to pulsed operation in which the optical power appears in pulses of some duration at some repetition rate (such as in a Q-switching or mode-locking laser). For continuous wave operation it's required for the population inversion of the gain medium to be continually replenished by a steady pump source. Even a laser whose output is normally continuous can be intentionally turned on and off at some rate in order to create pulses of light (the modulation rate is on time scales much slower than the cavity lifetime and the time period over which energy can be stored in the lasing medium or pumping mechanism, and it's still classified as a modulated or pulsed yet nevertheless continuous wave laser) - it is this type of pulse that's being shined on the Rubidium atom in the setup. Pulse of laser light thus refers to a duration of time in which the laser is being shined on the atom, necessary to excite the electron to the excited state. The half pulse refers to shining the laser light for half this duration of time on the atom.

Now, one half pulse (i.e. shining this continuous wave laser on the Rubidium atom for this half-duration time) causes the electron to go into superposition of the ground and excited states. Another half pulse excites it to the definitive excited state. This corresponds to two self referential operators, but that's the key: whatever the physical mechanism is, it is one that manifests the self referential operator. In representing a qubit, the most general state of a quantum two-level system can be written in the form ∣ψ〉 = α ∣0〉 + β∣1〉 where α and β are complex numbers. The state has to be normalized, so ∣α∣^2 + ∣β∣^2 = 1, and an overall phase makes no difference, so either α or β can be chosen to be real. This leads to the parametrization ∣ψ 〉 = cos (θ/2)∣0〉+ (e^iφ) sin (θ/2)∣1〉 in terms of only two real numbers θ and φ, with ranges 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. These are the same as the polar angles in 3-dimensional spherical coordinates, and this leads to the representation of the state as a point on a unit sphere called the Bloch sphere. The Bloch sphere can be traveled with pulses of different lengths and the system driven from the superposition state to the ground or excited states. But this can be done only during a certain amount of time which is called the coherence time. It can vary from several microseconds to several seconds depending on the particular system. When this period is over the system has to be initialized again in the ground state. If the phase of the driving field is fixed, the system will be going in one direction, but if phase is changed by π, the rotation axis in the Bloch representation is changed, and in that case the system might rotate in the reverse direction (for example around the -x axis instead of the x-axis). One can also change the phase by π/2 which would result in a mirror image of the state. For example if a π/sqrt(2) pulse is shined and then the axis is changed from x to y, the coefficients of the two states in the superposition are reversed. So the physical manifestation for a self referential operator may not be the same half pulse but rather may change - but this makes sense: there are different physical paths as we see to get back to the ground state; hence there are different physical manifestations for the NOT operator. It then follows there would be different manifestations for the SQR (NOT) and thus for a self referential operator at that particular point. There's lots of research being done on these variations (and part of my own personal research) and I hope this can be shifted to the engineering of self-referential operator gates which would be an extremely fascinating field.

Thanks again,

Steve Sax

Thanks Michel for your very insightful comments and references. It's amazing how the self-referential operator can take on many different forms. I want to just clarify a couple important distinctions that I sense you were aiming to bringing out: The macroscale "classical" coin flip CF that randomizes the input is indeed idempotent, as any number N of CF operations in sequence are equivalent to a single one: CF^N = CF like you exemplified. But it is not a self-referential operation per se. Idempotence is related to referential transparency with regard to computation in that an expression can be replaced by it's value without changing the behavior of the program and this might be what you meant. But the self-referential operation (SF) results in undecidability; in the representation of qubits this would take the system to a superposition of states. SF^2 = NOT, and would not be equivalent to the original SF operator. In this sense the QCF (quantum coin flip) of Hayes would indeed be a form of the self-referential operator - it's a fascinating distinction. That's very interesting how you combined the Hadamard and Pauli Z gates - I want to make sure we're using the same terminology on the shift gate but I very much like your ingenuity to engineer these gate combinations; it's really quite grand how this can be built up. Just to review for posterity: the Hadamard operation is similar to a SF (and thus SQR(NOT) ) operation in that both take the system to superposition, but a second Hadamard operation returns the system to the original state while a second SF as we know takes it to NOT of the original state. For example,

H(0>) = superposition 0> 1>, H.H(0>) = 0>.

H(1>) = superposition 0> - 1>, H.H(1>) = 1>.

SF(0>) = superposition 0> 1>, SF.SF(0>) = 1>.

SF(1>) = superposition 1> - 0>, SF.SF(1>) = -0>.

That's absolutely stimulating Michel that you connected these concepts to the time entanglement from my other essay and thanks for engaging that. Time entanglement is quite profound and your connection here of that to the CNOT gate in view the SF operator, undecidability, and incompleteness as explained in my current essay is an excellent observation - thanks very much.

I very much enjoyed your comments, and look forward to reading and rating your essay very soon.

Steve Sax

Thanks again Torsten, and as I elaborated on your page, your essay has many great points about which we would agree - including our discussions of mathematical abstraction and the relationship to physical explanation - and a lot of our ideas would support each other.

One point I just want to clarify: my essay did not dismiss the infinite universe per se, much as it did reign it in a little. Utilizing MUH to devise mathematically a universe that wouldn't allow for the physical explanation of an infinite multiverse really explores moreso the fundamentality of superposition and the role of the self-referential operator, as I explained further in the thread just above this.

Best regards, and thanks again,

Steve Sax

Thank you Mohammed for your kind words about my essay, and I'm glad you enjoyed it. I look forward to soon reading yours as well.

Best regards,

Steve Sax

Hi Sara,

Thanks for your very fascinating questions. That's an excellent distinction regarding requiring the self-reference to be manifest at a given instant versus allowing it to dynamically distribute through time. Since a system can be analysed from the perspective of other observables besides time, perhaps a way to generalize this distinction then is to consider whether the self-reference applies to a specific state, or instead manifests as an emergent phenomena throughout a distribution of states and their paths. The former scenario can be more active; that is a self referential 'operator' can be applied to a given state, and this physically can be accomplished by a specific physical event like the half pulse of laser light shining on the rubidium atom. Looking at it that way - from instantiating self reference at a given state - quantum superposition lends itself as a compelling description because it can explain efffects that we observe. The latter scenario (emerging through a distribution of states and paths) seems more passive, but need not necessarily be the case. In a path dependent multi-state system like that described in your paper, perhaps the particular state path (which may be nonlinear, as your paper suggests) in which a self reference emerges can be looked at as one macrostate M. The key would be what defines the self reference for such a macrostate, and what physical phenomena led to it. Did it evolve and if so is there any way to predict or assign probability to it happening? Could it be 'pushed' or 'tweeked' to emerge that way from an added physical event? That macrostate M applied to itself would be a NOT function of itself, but there would be many possibilities in a path dependent system that could satisfy NOT [M]. It may be that if such a self reference were interpreted to yield a quantum superposition, or a system of superpositions of the particular microstates involved, then it would have to be so complex that the observable effect yielded could perhaps be inferred as many discernible effects. This could be difficult if they in some way interfere with each other or are not fully addressed depending on the measuring environment. Even more revealing is if together they could indeed be perceived as one phenomena or concept. I'm thinking out loud at this point and would need to consider this much more, but needless to say your question has been very inspiring! It's an intriguing suggestion that self reference is a defining feature of living systems. I approached it from the idea of consciousness and its connection to causality, and this is a good example where the specific consideration of time (i.e. self reference manifest at an instant in time versus distributed in time) would especially apply. The analysis in this discussion here adds yet another dimension; perhaps there are environments in which the complex macrostate superposition mechanism described would yield phenomena that yet triggers other self referencing. If so, a modifed defining feature of life could be not just self reference, but a framework in which self reference is constantly or iteratively being invoked in a coordinated manner, (for example such as by these complex mechanisms).

Thanks again Sara for some very thought provoking questions; these are just some of my thoughts on it, and I definitely want to explore this distinction of instant versus emergent self reference much more!

Kindly, and best regards,

Steve Sax