StrictlyLogical

Language, concept, number, these we are possessed of and are all part of our grasping of and our relating ourselves to entities. Entities themselves, however, are not in any way possessed of any of language, concept, or number. That we "find" them poetic or majestic, of a kind or a phenotype, or multitudinous or stochastic, although somethings of them, something about their identity, is touchable and accessible by our various abstraction apparatuses, those somethings of them are not themselves linguistic, conceptual, or mathematical. For those, are only us, as they only ever could and should be.

Given the specific formalization of QM as accepted I would suspect other kinds of numbers as operators or coefficients to the so-called “states” would be improper somehow. I played around with quaternions, more specifically a sort of Mandelbrot generalization to render fractals on a … believe it or not… Amiga computer back in the day. I believe there are multiple imaginary bases i j and k, each squared is -1 but the product of any two is the other (positive or negative depending on the order of multiplication)

Did you learn anything from this video?

Ah yes, this is the convention of using a complex refractive index in calculations to take into account absorption. Quite a convenient use of complex numbers, relating incident light to absorption of light in the material as a function of depth.

The point was not that real numbers cannot be used for relations... that is obvious and you know it. I'll restrain my snippiness to that comment. The point is that in reality you do not literally have "i" quantity of some entity, as such. "i" can be used to help calculate quantities or to represent/characterize (through its odd mathematical qualities) relationships between real things. Do not get me wrong, I am not guilty of reification of real numbers either, they are abstractions we use to directly quantify things in reality, but they are no more real than "i"... merely the directness of their application, and their function differs.

and that includes relationships between "portions" of the same thing, so to speak. From what I recall in QM probabilities there is never any "absolute" phase, but relative phase is ubiquitous and the foundation for determining interference, and plays much in what the results of the inner product look like (i.e. probabilities). So real numbers we use to quantifying things but "i" and the like are useful for relating things, in particular phase differences.

Real numbers characterize and are abstraction we find useful for quantifying things. Imaginary numbers, as directly as I can relate them to things, characterize and are abstractions we find useful for (some specific) relationships between things.

"Analysis" is a complicated formal construct. That which one builds to surround and is supported by real numbers is not the same as that which one builds to surround and is supported by complex numbers. So no, complex analysis is not "really equivalent" to real analysis. These are two different games played in two different arenas... we can and did make them so. Now the idea of analysis is like the idea of a mathematical expression, how it is used or what symbols are there may be different but what it represents or refers to, is not the same as the expression used. The identity operator helps us understand that although the expressions are not the same what they refer to are one in the same, an identity. So I do not discount the possibility that some form of complicated real number based formalism, using Euler relations etc. cannot create something like an answer, i.e. refer to some quantity, which complex analysis also refers to. When used as a coefficient, a real number can be interpreted as a kind of scaling or "quantity of" operator, so taking i, as operated on by 3 you have three of them, so no 3 times i does not remove the i. i however is like a 90 degree rotation operator in the 2d plane we use to arrange our complicated useful contrivances we call complex numbers. EDIT: The last paragraph seems very important to limit and understand the import of their findings.

No one is proposing that, indeed it would be unnecessary. The potential or purported import of the paper is its implication that *which particular kinds* of abstractions we can choose to use in the context, are somehow actually limited.

That understanding would be incorrect. Particularly that last sentence. Anything characterizable in complex numbers must be characterizable in real numbers, because every complex number is characterizable in real numbers. As for things, any complex attribute of a thing can be represented by two non-complex attributes associated with the thing. In fact the necessity of characterizing an attribute as (specifically) complex is nothing more and nothing less than having to use more than one (specifically two) real number to characterize that attribute. Every operator and mathematical calculation in the complex plane deals simultaneously with two real quantities, phase and magnitude or alternatively real and imaginary parts. A complex number IS two things, and of a necessity is b0th reducible analytically into those two things and cannot be constituted by less than those two things. It can never be a "simple" number, after all it is a complex number, and it has two absolutely independent components, and cannot be thought of as having any less than two components, and therefore it IS two things. So what is the error? Reification. The fact that an attribute (in a particular framework or theory of reality), to reflect reality has two things associated with it, and that the operators must take into account both, say phase and magnitude, means that a thing has a two-attribute attribute. The reification is an erroneous identification of this two attribute attribute WITH the abstractions we use to work with them... it comes from the way we understand and express and work with this two attribute attribute, namely with complex numbers. Moreover, the simplicity with which we deal with the calculations of the two attribute attribute as if it were one attribute (because they always go together)... every phase must have a magnitude... leads one to believe the formalism as expressed is the only way to express that formalism. That when you change the expressions one has changed the formalism... and that "QM based on real numbers" means somehow trying to use real numbers in a framework built for using complex numbers... The idea of a hypothetical "real quantum state" is nonsensical. Why? Because the referent of the modifier term "real" is purely mathematical or abstract, and the referent of the term "quantum state" is supposed to be an entity of reality. This should be a huge clue to how the authors are thinking...or how they are not being careful about what they are talking about, i.e. what refers to abstractions and what refers to reality. The foundation of QM on the idea of representing reality as states in a vector space, and whose inner product corresponds with probabilities of outcomes. We assign states in the same direction when probability is 1 and assign states as orthogonal vectors when probabilities are 0. We have operators to rotate those vectors modeled on causation and interaction. A nice little game no? It turns out that correspondence between these vectors we have concocted to real world outcomes, requires the use of complex coefficients and operators... but what has that done to the formalism? All that has cone is doubled the degrees of freedom. Sure one could not write QM in the standard formulation, the standard way with real numbers, the correspondence between it and reality requires complex numbers but that does not mean one could not rewrite the entire thing, vector spaces and all using real numbers.

How familiar are you with Objectivism as a philosophy? As a former academic of physics, I highly recommend it. Interpretations of physics and in particular of QM are interesting but until we have a complete understanding of the mechanics of measurement (rather than a formalism and assumptions) I think they will remain rather fanciful and mystical… I have found THAT is where the mind goes when it encounters something it does not understand… not merely an acknowledgement of the unsolved but a kind of ecstasy in the “mysterious”. I would suggest you hold onto as much of the solid foundations of thought as possible when you encounter the myriad flights of fancy of both your common and uncommon physicist. Be rigorous and disappointingly real about the distinctions between our abstractions and entities referenced by them. Good luck on your journey!

@Bill Hobba I note a great many physicists completely dispense with philosophy and philosophical theory and the rigours which may be attained therewith… do you bring any philosophical scrutiny to the physics you cite or do you simply take them and the interpretations therein, as true?

Please excuse me I am unfamiliar with the details of Noether’s theories re. fields. If the electric field of an electron has a separate energy which you imply has a mass, then when you expend energy to accelerate the mass of an electron do you need to expend energy to accelerate the electric field of the electron which you imply also has mass? In fact, when you accelerate an electron are you accelerating i.e. moving, its electric field as well? Are you moving two things are one thing... are there two things or one thing? As student of philosophy what is your understanding of an attribute or property of an entity as regards its nature and relationship to the entity? Some would say they certainly are real but are not independently real, they are only always “of” entity or entities.

A funny thing about how we tend to use language, and funnier when we are talking about physicists, “Quantum Mechanics” is sometimes interpreted as referring to what reality does, when it is far more accurate to say QM is something we do, which to the extent it corresponds with observables of what reality does do, is valid and useful. That paper is more about how we process what reality does, not what attributes and properties which are possessed by entities. The third person would point out that a complex number is nothing more than a complicated (not very) combination of real values. They are absolutely and always reducible to real values. We happen to call them phase and magnitude. But again this is mere characterization of the abstraction which is QM, merely interpretations of the abstractions as more or less complex … when in fact it is all the same and beside the point. The processing abstraction is not the referent to which its predictions are directed. Observe there is no absolute phase in the complex coefficients, and also observe that statistical in nature they are not strictly speaking possessed by any single entity, and of course are never observable properties possessed by any single entity. As such, any assumption about complex numbers, I put to you, is more of an assumption about our abstractions referring to physical reality than an assumption about physical reality itself.

@Bill Hobba I should let you know that hearing your interest in these subjects is refreshing. So many mathematicians, physicists, and philosophers stay in their own lanes far too much or hold on to what they have been told by authorities in their field with far too little independent scrutiny. That said, as an analogy I would like to introduce the statement “abstractions are founded in reality” as a generalization which is subject to the same problem. When abstractions are used in a context of referring to reality, with any language, mathematical or not, the system of abstractions should be founded in reality. These sorts of considerations were never really investigated in my training in physics. I would argue the BEST professors admonished us to look at the equations SOLELY as a tool for predicting and quantifying reality… implicitly our way of dealing with reality is not reality… a nice warning about the clear distinction between our abstractions and their referents in reality, without sophisticated explication.