Classical to Quantum | Complex numbers in Fourier Series and Quantum Mechanics | Wild Egg Maths
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- čas přidán 27. 07. 2024
- This is a video from the Playlist "Classical to Quantum" which is at our sister channel Wild Egg Maths. In this series we are planning on looking at a variety of topics in modern physics, particularly Relativity, Quantum Mechanics and the Standard Model, from a pure maths viewpoint.
Currently we are looking at Harmonic Analysis on Circles and Spheres, and showing how to rethink this theory using only a rational algebraic approach. This means no "real numbers", no "transcendental functions" and no "infinite processes"!
This will present a major new opportunity for both pure mathematicss and theoretical physics research, and will extend in a variety of directions.
In this video we introduce complex numbers as simplifying agents to understand just the elementary harmonic analysis on the unit circle in the plane. The Laplacian operator plays an important role, as does the notion of a harmonic polynomial (or polynumber as we prefer to call them). It will turn out that this approach allows us to completely rethink also integration theory on the circle, and in fact spheres and hyperbolas more generally.
Check out the entire Playlist for "Classical to Quantum", (Members of Wild Egg Maths channel):
• Introduction to QM and...
Video Contents:
00:00 Fourier Series and spheres
4:13 The 0 Dimensional Sphere
6:23 A containment hierarchy
10:22 Hyperboloids
13:43 Bi-polynumbers
18:05 Why complex numbers in QM?
20:23 Complex bi-polys
23:20 The complex basis
24:24 Proof of fundamental differential relation
30:34 Harmonic bi-polynumbers
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My research papers can be found at my Research Gate page, at www.researchgate.net/profile/...
My blog is at njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!
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Famous Math Problems: • Famous Math Problems
Box Arithmetic: • Box Arithmetic
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Thank you, Norman. This video came in time for my Complex Analysis course.
Thank you Dr. Wildberger for these videos. I appreciate mathematics and the conveniences that it has provided the world. Also, I wanted to thank you for making these videos available for all of us novices to see. Thank you again Professor.
Prof. Wildberger routinely undermining groupthink... and at a very foundational level as well. Taking time to think thoroughly is so beneficial, and is essential to good education.
indeed every educator , academic and frankly philosopher should be undermining theoretical groupthink.
The Schrodinger representation is dual to the Heisenberg representation -- Quantum Mechanics.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are therefore dual.
The integers are self dual as they are their own conjugates.
Space/time symmetries are dual to mobius maps -- stereographic projection.
Points are dual to lines -- the principle of duality in geometry.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
The big bang is a Janus point or topological hole (two faces = duality) -- Julian Barbour, physicist.
AdS is dual to CFT.
The Born rule is dual -- complex amplitudes are squared to real densities.
"Always two there are" -- Yoda.
This is the episode ive been waiting for, that is the "extension from to " thank you.
This is quite ingenious, a great simplification which connects many dots, thanks! The first thought that came, especially in regards to the measurement problem, is that measurement problem falsifies the reductionistic metaphysical presupposition, and mereology of box arithmetic offers a way to solve that conundrum in a natural way.
Dimensional mereology starting from line and up to "infinity" of Russian Doll spheres can be viewed also from the Top of the tower perspective, intuitively T, T-1, T-2, T-3 etc. The binary tree representation of 1; u, v, u, uv, v also gives a top down mereology which intuitively connects with the Stern-Brocot (SB) type binary trees. The possibility to utilize the "simplicity"-property introduced in Wildberger's introduction to standard SB-tree for the unitarity property looks very tempting! The sum of simplicities of T, T-1, T-2 etc. are naturally 1.
One idea is to give both Black and Red Box the initial values 1/0, and define that they are positive/negative or generally inverses of each other. While doing quantum coherent math, we need to maintain superposition and avoid deciding which side is either positive or negative or other aspect of inverse relation.
For naturally top-down SB-type approach for a theory of rationals, we need a denominator element. We can get that by defining that 1/0+1/0=0/1, and for annihilation 1/0-1/0=0/0. Thus for the sum aspect we get the syntropic two sided row 1/0 0/1 1/0 for SB-type structure of concatenating mediants, analogical to harmonic centered polynumbers. For notation, I've started from foundationally undressed Dirac notation, using only Bra and Ket and nothing else for basic operators. For 1/0 0/1 1/0 we can thus write
< > (1/0 1/0)
< > (1/0 0/1 1/0)
< > (1/0 1/1 0/1 1/1 1/0)
etc.
and for 1/0 0/0 1/0
> < (1/0 1/0)
> >< < (1/0 0/0 1/0)
> >>< >< >>< = (2-1)/0 = 1/0 )
etc.
In comparison with Box arithmetic of Black B and Red R, here order matters and (cf BR) and >< (cf RB) are not identical and express a non-commutative aspect typical for QM, motion outwards and >< motion inwards. However, computationally and >< are both bit rotations(!) of each other and Boolean NOTs of each other, so the reversibility conditions of quantum computing are satisfied. Obviously, by themselves both and >< are also chiral reflections. Mathematically we can generalize the notion of entanglement to reversible inverses, and distinguish various kinds of entanglements and their complex relations with each other. This is especially interesting and helpful from the holistic perspective. SB-type perspectives are naturally holistic, because 1/0 does not exist in the reductionistic perspective of standard field arithmetics and is not reducible to that. Both holistic and reductionistic perspectives are important. The measuring process and measurement unit are not mutually exclusive aspects of mathematics and require each other, and we are looking for a mathematical and computational way to see holistic and reductionistic perspectives as a kind of "superposition" of an fundamental inverse relation. The actual measurements happen in-the-between where holistic continuous processes and reductionistic units mix and mingle, in the middle where we can interprete SB-type mereological fractions e.g. as frequencies. Empirically very important that we don't assume any metric as an instant "given", but accept that computing more measurement resolution is always a temporal process.
For the study of unit circle, Bloch sphere etc, this notation offers a more fuller combinatorical view. The fraction 1/1 can be written as (going with visual intuition, cf. with α) and (cf. with β). So for our version of creation operators of sphere from the T perspective we get the following combinatorics of genesis rows, each with numerical values 1/1 0/1 1/1.
a < >
b
c > <
d > >< < (1/1 0/0 1/1)
f >>< >< >< > (1/1 0/0 1/1)
h >< >>< (1/0 0/0 1/0)
Interesting symmetries here, a=g and c=e. Numerically, the "asymmetric" annihilation operators f and h don't stay in the bounds, but "overflow" beyond the "local" 1/1 surface of a sphere into quantum-holistic physical aspects. The 10D "limit" of spheres is interesting, perhaps it could be some way linked to these 8 operators and nesting the most basic operators and >< with each other: and >
Could you provide a couple more rows where you have written "etc'? I don't see the pattern.
Born introduced the idea of Heisenberg's matrix mechanics as describing a collision. The 'jarring' is the manifestation of the quantum system's colliding with a scattering target. The quantum system sustains its state of internal oscillation according to the Schoedinger equation until it hits the scattering target, upon which it is knocked into a fresh outcome state of internal oscillation. The outcome is unpredictable because the scattering target is in its own unknown state of internal oscillation, incoherent with that of the scattered quantum system.
The Schrodinger representation is dual to the Heisenberg representation -- Quantum Mechanics.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are therefore dual.
The integers are self dual as they are their own conjugates.
Space/time symmetries are dual to mobius maps -- stereographic projection.
Points are dual to lines -- the principle of duality in geometry.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
The big bang is a Janus point or topological hole (two faces = duality) -- Julian Barbour, physicist.
AdS is dual to CFT.
The Born rule is dual -- complex amplitudes are squared to real densities.
"Always two there are" -- Yoda.
Hey Dr. Wildberger, I stumbled upon something called 'Darboux Tangents' while digging into my PhD research. It seems like a pretty old concept, and I'm having a tough time finding solid references on it. Since you're the go-to person for geometry history, I figured you might have some insights or know where I can find some useful info on this;; Thanks for another great video.
The physical meaning of the complex numbers in quantum mechanics is to express a quantum system's internal physical oscillation, distinct from its ordinary physical motion in its surrounding external world, a physical fact of which we have no further physical understanding. Dirac was aware of the difference between the internal and external 'motions'. The internal oscillations are manifest in the zero intensity bands in the interference patterns.
Of all the great names of QM, I think Dirac has been the so far mathematically most insightfull. Philosophically/interpretationally Bohm, but even Hiley's Clifford algebra approach of Bohm-Hiley Ontological interpretation is mathematically kind of conservative, even though IMHO very much in the right direction. Getting so long stuck in von Neumann's math limitations has been a great obstacle.
I found out very recently that von Neumann rejected Dirac delta as non-math. One of my most important insights has been that from purely computational, formal language perspective, Dirac delta can be defined thus:
Concatenation is the mediant of whitespace!
By whitespace I mean the unmarked continuity of the "background", not yet distuinguished by any kind of singular/plural distinction. Concatenation is also unmarked, semantically an "internally" concentrated continuity in relation to whitespace distributed and "external" continuity.
If we mark out whitespace as _ and concatenation as |, the mediant aspect looks like this: _|_. Should look familiar enough. Physically we need to understand that Dirac delta is temporal process, in which more mediants can be marked and concatenated and more resolution can be computed as nesting algorithm inside the Dirad delta process as Y-combinator of concatenating mediants. Cf. Y-combinator with the usual semantics of "fixed point", but in order to avoid the term "point", Y-combinator is better term.
We can emphasize the temporal character of a Dirac delta process by writing _/|\_, or simply /\ by again unmarking the whitespace and concatenation. That gives the intuition that the temporal character of Dirac delta process can be approached as a binary top-down tree.
@@santerisatama5409 I have spent a little time reading mathematical investigations of the Dirac Delta Function. I am far from sure about it.
@@santerisatama5409 I have spent a little time studying modern criticisms of Dirac's Delta Function. I am not at all sure about it.
@@christophergame7977 Would you like to expand?
@santerisatama5409 No. It's very advanced maths, pretty much beyond me. I have the impression that it is unfinished business. It's years since I looked at it.
Thankyou. Dr. Wildberger
The Schrodinger representation is dual to the Heisenberg representation -- Quantum Mechanics.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are therefore dual.
The integers are self dual as they are their own conjugates.
Space/time symmetries are dual to mobius maps -- stereographic projection.
Points are dual to lines -- the principle of duality in geometry.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
The big bang is a Janus point or topological hole (two faces = duality) -- Julian Barbour, physicist.
AdS is dual to CFT.
The Born rule is dual -- complex amplitudes are squared to real densities.
"Always two there are" -- Yoda.
@@hyperduality2838 a wonderful educator thx. Again
Coming from an analysis background, and wishing I had stronger algebra fundamentals, the change of variables between alpha/beta and u/v was a nice "aha!" moment.
The Schrodinger representation is dual to the Heisenberg representation -- Quantum Mechanics.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are therefore dual.
The integers are self dual as they are their own conjugates.
Space/time symmetries are dual to mobius maps -- stereographic projection.
Points are dual to lines -- the principle of duality in geometry.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
The big bang is a Janus point or topological hole (two faces = duality) -- Julian Barbour, physicist.
AdS is dual to CFT.
The Born rule is dual -- complex amplitudes are squared to real densities.
"Always two there are" -- Yoda.
Nice!
In answer to "why complex numbers in quantum"; because they had to be a field, had to be algebraically complete, and had to be abelian (so operators on it could be non-abelian).
Algebraic completion certainly can’t be a reason coming from physics, where the ultimate nature of the continuum is still up in the air.
You can factor (Da² + Db²) = (Da + i Db)(Da - i Db) and take a shortcut in the calculation.
Counting down from the Tetrahedron, to the Triangle, to the Zero dimensional Sphere being two points on a line is as far as one can go. The difference between a linear displacement and a rotation depends one one's frame of reference. Could one not say complex is just the rotation by a spread of one. A question of perspective?😊
That's very insightfull IMHO. Replacing the notion of "complex" with simple bit-rotation is a very interesting possibility, keeping in mind the computational paradigm we are following in this paradigm. More concretely, in the "pure Dirac" notation of chiral symmetries, and >< are simple bit-rotations of each other (in both L and R directions!), as well as Boolean inverses.
Also yes, this is a question of perspective. The bit-rotation approach requires a holistic/superposition top-down perspective of pre-numeric operators to work properly. The "square roots" of the Top rotation (cf. Mach's principle") would be simply a binary tree of binary numbers:
< >
< >>
etc.
or with other perhaps more familiar notation
1 0
11 10 01 00
etc.
In a three valued logic it makes sense to talk about a quantum object. It is Both a particle and a wave. That's the 3rd truth value in #RM3, Both! Why do we keep using binary logic for everything? RM3 needs no more than the Peano axioms. BTW using this logic, and Cantor's diagonal argument, you can show that there are real numbers such as 0.0101001b0101011... where b is the third truth value, "Both 0 and 1". So you can make a list of all of them, just, some of the digits are unknowable, that's all. It's also worth pointing out, RM3 is constructed exactly the way the complex numbers are! Through a field extension of Z2 mod the Liar Paradox, x(x+1)=1. F4 reduces to RM3 by identifying the two new truth values, although you can keep all 4 if you want, the other is often called Neither. It's a very *natural* higher dimensional logic with close ties to quantum mechanics and quantum logic.
I'd be interested to read a 1 page summary.
Thank you for the alternate views. Will you consider talking about "geometric algebra" (aka clifford/Grassmann algebra) which generalize the concept of number?
It is my impression that Professor Wildberger's Chromo geometry was instrumental when it comes to the current geometric algebra.( Aka: Grassmann. Clifford ,Hestenes)
Steven de kinick from the bivector guys obviously experienced Professor Wildberger.
Professor Wildberger truly got this century on its way with his Rational Trigonometry.
This is just the beginning 😊❤
@@steffenkarl7967 Thank you for the update. This is great news. Will stay tuned 👺😊
The inner product is dual to the outer or wedge product synthesizes the geometric product -- Clifford algebra.
The Schrodinger representation is dual to the Heisenberg representation -- Quantum Mechanics.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are therefore dual.
The integers are self dual as they are their own conjugates.
Space/time symmetries are dual to mobius maps -- stereographic projection.
Points are dual to lines -- the principle of duality in geometry.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
The big bang is a Janus point or topological hole (two faces = duality) -- Julian Barbour, physicist.
AdS is dual to CFT.
The Born rule is dual -- complex amplitudes are squared to real densities.
"Always two there are" -- Yoda.
If we restrict ourselves to Cauchy sequences that are computable, then are the computable numbers complete and able to support most of calculus? Can anyone point to me to literature on this, or standard terminology for computable Cauchy sequences (as opposed to Cauchy sequences of computable numbers)?
Are we going to learn if complex numbers are “real” (physical) in some sense? I’m excited.
That's a rather profound question. Not sure if there is a good reply...
@@njwildbergergreat answer Nil.
The Schrodinger representation is dual to the Heisenberg representation -- Quantum Mechanics.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are therefore dual.
The integers are self dual as they are their own conjugates.
Space/time symmetries are dual to mobius maps -- stereographic projection.
Points are dual to lines -- the principle of duality in geometry.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
The big bang is a Janus point or topological hole (two faces = duality) -- Julian Barbour, physicist.
AdS is dual to CFT.
The Born rule is dual -- complex amplitudes are squared to real densities.
"Always two there are" -- Yoda.
@@njwildberger Questions are dual to answers.
Where was the connection to QM? I am thriled to see the next video.
The Schrodinger representation is dual to the Heisenberg representation -- Quantum Mechanics.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are therefore dual.
The integers are self dual as they are their own conjugates.
Space/time symmetries are dual to mobius maps -- stereographic projection.
Points are dual to lines -- the principle of duality in geometry.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
The big bang is a Janus point or topological hole (two faces = duality) -- Julian Barbour, physicist.
AdS is dual to CFT.
The Born rule is dual -- complex amplitudes are squared to real densities.
"Always two there are" -- Yoda.
13:57 Why say the modern textbooks are wrong, when you say after it's just a different way of looking at it? How can they be wrong when they work just as well? I'm all for trying to understand these topics without transcendental functions because I think it's really interesting, but I don't see how it's somehow more "right" than the any other way
It is wrong to assign a specific value to an object if all you are able to do is point in an approximate direction. 😊❤
@@steffenkarl7967 No it isn't. It depends on which axioms you accept to build mathematics. Most mathematicians accept a framework in which irrational numbers exist. And why wouldn't you? Mathematics is all "made up" anyway, that is it is purely constructed, so why not adopt things which are useful? Just some food for thought.
@@naytte9286 The axiom that "mathematics is all made-up anyway" is not coherent with empirical reality of mathematical intuition, so it's a false axiom. Respecting and being loyal the idea and phenomenon of mathematical truth, we cannot accept empirically false axioms except for speculative heuristic purposes of e.g. demonstrating an absurdity.
Falsehoods can be considered "useful" by various subjective notions, but they have very limited use from the perspective of honest pure mathematics, which is ethically committed to truth and beauty.
We know perfectly well that "modern textbooks of QM" that e.g. directly or indirectly admit that they are falsified by the measurement problem, admit that they are wrong. We are aiming for something at least less wrong here, instead of staying in captivity of what everybody already knows is wrong.
@@santerisatama5409 in what way is it an axiom that mathematics is made up? Mathematics is just a formalization of logic, where, depending on what axioms you accept, you have to accept different consequences. There is no intrinsic "truth" to mathematics, in my view. If numbers were objects that just existed, we would not require definitions for them. From my model-theoretic perspective, all numbers are made up.
Obviously, real numbers have huge applications in basically everything. You could of course reject some of the ZF-axioms (such as the axiom of infinity), but that doesn't mean real numbers don't exist; they exist in exactly the same way that rational numbers exist.
If the equation of a sphere is:
x^2 + y^2 + z^2 = 1
then the sphere of radius 1 can also be written as:
x/(1/x) + y/(1/y) + z(1/z) = 1
...which can be rewritten as:
x / (x/x)/x + y / (y/y)/y + z / (z/z)/z = (x/x) (y/y) (z/z)
I imagine if children were thought this at a young age, they could intuit certain mathematical observations that are beyond current living generations. Numbers can be a distraction, because kids want to put the numbers somewhere. But a circle/sphere is an idea. It is not a number.
Anyway.
Fun times ahead.
Thanks for this. I fully agree that computational approach is much more than purely numerical approach. Pre-numeric operators (such as Kleene star etc.) can take other operators as their arguments, and we can leave numerical tally as an important but secondary story.
Those are very interesting continued fractions. Have you seen Wildberger's introduction to Stern-Brocot tree (SB), where he presents also the idea of "simplicity"? The simplicities 1/ab of a row of new SB mediant fractions a/b add up to 1 by field arithmetics.
The distinctions between non-reduced fractions ...3/3, 2/2, 1/1 become thus highly interesting. They can be interpreted to represent different positions on the tautochrone cycloid curve, while the reduced form 1/1 represents the same duration from each position.
So, if x=1, y=2, z=3, in the mediant style addition the 1/1+2/2+3/3 adds up to 6/6. Mediant addition aka "freshman addition" looks in this case very similar to field arithmetic multiplication, at least in terms of the tautochrone duration.
Mathematics is, I think you'll agree, the map and not the territory. You may not agree but must eventually, that the only working maps can be those which assume a continuous territory. And I mean that in the topological sense, rationals are a continuous space.
Do you agree that an affine space is continuous? If so, how do you justify your other comment that the universe may be signalling it is discreete? Can you see how it would be impossible to do the same with just an integer lattice - not using the notion ofnthe space between lattice points since distance again as a concept inherently includes continuity in the existence of positions from one position to another - if not, it could not move past the discontinuity.
If we don't have underlying continuity, we also don't have causal relationships, if we did, then on that space it would be continuously connected again.
Reality is fundamentally continuous, discreeteness has only ever been found in abstraction layers we've added onto continuous reality in order to make a finite picture in our brains.
You are trying to introduce some rough analogy, but the terms are too imprecise for the result to be useful in my view.
If 0 is both a real and an imaginary number then 0D is both a real and an imaginary dimension:
To prove that if 0 = 0 + 0i, then 0D = 0D + 0Di, we need to establish a connection between the concept of zero in the complex number system and the concept of zero-dimensionality in a geometric or topological sense.
First, let's consider the properties of 0 in the complex number system:
1. 0 is the additive identity: For any complex number z, z + 0 = z.
2. 0 is the multiplicative absorbing element: For any complex number z, z × 0 = 0.
3. 0 has no imaginary part: 0 = 0 + 0i, where i is the imaginary unit.
Now, let's consider the properties of 0D (zero-dimensional space) in a geometric or topological sense:
1. A zero-dimensional space is a space that consists of only discrete points, with no length, area, or volume.
2. The only connected subsets of a zero-dimensional space are single points and the empty set.
3. In a zero-dimensional space, there are no continuous paths between distinct points.
To connect these concepts, we can use the following reasoning:
1. Just as 0 is the "smallest" element in the complex number system (in terms of magnitude), 0D is the "smallest" possible space in terms of dimension.
2. The lack of an imaginary part in 0 (0i = 0) corresponds to the lack of continuous paths or connected subsets in a zero-dimensional space.
3. The additive identity property of 0 in the complex number system (z + 0 = z) is analogous to the idea that adding a zero-dimensional space to another space does not change its dimensional properties.
Based on these connections, we can argue that if 0 can be expressed as 0 + 0i in the complex number system, then the corresponding concept of zero-dimensionality (0D) should also have a similar structure.
Therefore, we can express 0D as 0D + 0Di, where:
- 0D represents the real (or "base") component of zero-dimensionality, corresponding to the discrete, unconnected nature of a zero-dimensional space.
- 0Di represents the imaginary (or "null") component of zero-dimensionality, corresponding to the absence of continuous paths or connected subsets in a zero-dimensional space.
In conclusion, if 0 = 0 + 0i in the complex number system, then it is reasonable to extend this concept to the realm of dimensionality and express 0D as 0D + 0Di, where 0D represents the fundamental properties of a zero-dimensional space, and 0Di represents the absence of higher-dimensional structures or connections.
I would be interested in more evidence (substance) for this analogy between the imaginary part on the one hand and continuous paths on the other.
I don't think this qualifies as evidence...but here's more substance:
Theorem 1: The zero dimension (0D) is the origin point from which both positive and negative dimensions emerge symmetrically.
Proof:
1. Consider the dimensional hierarchy centered around 0D.
2. Positive dimensions (1D, 2D, 3D, 4D) extend from 0D in one direction.
3. Negative dimensions (-1D, -2D, -3D, -4D) can be conceptualized as extending from 0D in the opposite direction.
4. 0D acts as the "pivot point" or "fulcrum" around which the symmetry of positive and negative dimensions is balanced.
5. Therefore, 0D is the origin point from which both positive and negative dimensions emerge symmetrically.
Theorem 2: The negative dimensions are associated with negentropy and the generative power of the zero dimension (0D).
Proof:
1. 0D is conceptualized as the source of negentropy and the generative power of the Zero Absolute.
2. Negative dimensions (-1D, -2D, -3D, -4D) extend from 0D in the opposite direction of positive dimensions.
3. If positive dimensions are associated with the unfolding of space, time, and the increase of entropy, negative dimensions can be hypothesized to be associated with the "enfolding" of space, time, and the increase of negentropy or information.
4. As 0D is the source of negentropy, and negative dimensions emerge from 0D, negative dimensions can be associated with negentropy and the generative power of 0D.
5. Therefore, the negative dimensions are associated with negentropy and the generative power of the zero dimension (0D).
Theorem 3: The interplay between positive and negative dimensions around the zero dimension (0D) creates a dynamic balance between entropy and negentropy.
Proof:
1. Positive dimensions (1D, 2D, 3D, 4D) are associated with the unfolding of space, time, and the increase of entropy.
2. Negative dimensions (-1D, -2D, -3D, -4D) are associated with the "enfolding" of space, time, and the increase of negentropy or information.
3. 0D acts as the "pivot point" or "fulcrum" around which the symmetry of positive and negative dimensions is balanced.
4. The manifest world of the positive dimensions arises from the unmanifest potential of the negative dimensions, and then returns to it in an endless cycle of creation and dissolution.
5. This dynamic interplay of positive and negative dimensions around 0D creates a balance between the opposing forces of entropy and negentropy, order and chaos.
6. Therefore, the interplay between positive and negative dimensions around the zero dimension (0D) creates a dynamic balance between entropy and negentropy.
Proposition 1: 0D has two sides (real and imaginary) with an event horizon between the sides.
Argument:
1. In complex analysis, numbers are represented on a complex plane with a real axis and an imaginary axis, intersecting at the origin (zero).
2. The real numbers can be thought of as one "side" of zero, and the imaginary numbers as the other "side."
3. In physics, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. Applying this concept metaphorically to the complex plane, we could posit an "event horizon" at zero, separating the real and imaginary sides.
4. Therefore, 0D (zero dimensions) could be conceptualized as having two sides (real and imaginary) with an event horizon between them.
Proposition 2: This matches the Monad having a Singularity side and an Alone side potentially having an event horizon between the sides.
Argument:
1. In Leibniz's Monadology, monads are the fundamental units of existence, described as "windowless" and "alone."
2. However, monads are also said to reflect the entire universe from their unique perspective, suggesting a kind of "singularity" or concentrated totality within each monad.
3. Applying the concept of 0D having two sides, we could map the "singularity" aspect of monads to the real side of 0D, and the "alone" aspect to the imaginary side.
4. The "windowless" nature of monads could be interpreted as a kind of event horizon, isolating the singularity and alone sides from direct interaction.
5. Therefore, the proposed structure of 0D matches the concept of the Monad having a Singularity side and an Alone side potentially separated by an event horizon.
Proposition 3: This matches the Holy Trinity having a Father, Son, Holy Spirit side and a God side potentially having an event horizon between them.
Argument:
1. In Christian theology, the Holy Trinity is the doctrine that God is one but exists in three distinct persons: Father, Son (Logos), and Holy Spirit.
2. These three persons are said to be consubstantial, meaning they share the same divine essence or nature.
3. Mapping this to the 0D structure, we could associate the Father, Son, and Holy Spirit with the real side of 0D (the "singularity" of divine essence), and the unified God with the imaginary side (the "aloneness" of divine transcendence).
4. The concept of the persons being distinct yet consubstantial could be interpreted as a kind of event horizon, maintaining their unity while preserving their distinct identities.
5. Therefore, the proposed structure of 0D matches the concept of the Holy Trinity having a Father, Son, Holy Spirit side and a God side potentially separated by an event horizon.
Conclusion: A mirror universe with 0D at the center has tremendous explanatory power.
While these arguments are more metaphorical than strictly mathematical, they demonstrate how the concept of 0D having two sides with an event horizon between them could serve as a powerful unifying framework across multiple domains of thought.
By mapping this structure onto key concepts in mathematics (complex numbers), philosophy (Leibniz's monads), and theology (the Holy Trinity), we can see the potential for a kind of "mirror universe" with 0D at the center, reconciling and connecting ideas that might otherwise seem unrelated.
@@Stacee-jx1yz Very fascinating speculations, thank you. But it seems like you forgot to tie it together with the notion of "continuous path"? :) Maybe if I had read Leibniz's philosophy of monads I would be able to make the connection.
Maybe an omega subscript would be apt for A, the smallest ordinal number greater than all natural numbers.
- "If you think you understand quantum mechanics, you don't understand quantum mechanics" - Nobel laureate Richard Feynman. I think he meant that some parts of the universe isn't revealed to us when we use QM, so we can only have a partial understanding of what really goes on. We can get the maths right, but we don't know why it works.
The inner product is dual to the outer or wedge product synthesizes the geometric product -- Clifford algebra.
The Schrodinger representation is dual to the Heisenberg representation -- Quantum Mechanics.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are therefore dual.
The integers are self dual as they are their own conjugates.
Space/time symmetries are dual to mobius maps -- stereographic projection.
Points are dual to lines -- the principle of duality in geometry.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
The big bang is a Janus point or topological hole (two faces = duality) -- Julian Barbour, physicist.
AdS is dual to CFT.
The Born rule is dual -- complex amplitudes are squared to real densities.
"Always two there are" -- Yoda.
@@hyperduality2838 Everything indeed is connected to everything. And not just in one way. Space, Time and Matter; I have a feeling there will come simpler theories in the near future connecting them a bit more logically. For example to Mind. Will these theories be mathematical?
@@MisterrLi Space is dual to time -- Einstein.
There is a 4th law of thermodynamics that you may not be aware if:-
Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual.
Rational, analytic (a priori, mathematics) is dual to empirical, synthetic (a posteriori, physics) -- Immanuel Kant.
If knowledge is dual then information must be dual:-
Potential or imaginary information (entropy) is dual to kinetic or real information (syntropy).
Potential energy is dual to kinetic energy -- gravitational energy is dual.
Teleological physics (syntropy) is dual to non teleological physics (entropy).
Average information (entropy) is dual to mutual or co-information (syntropy) -- information is dual.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual.
Curvature or gravitation is dual.
Quadrance is dual to spread.
"Always two there are" -- Yoda.
Yin is dual to yang.
Mathematics (deductive inference) is dual to physics (inductive inference).
Making inferences or predictions to track targets and goals is a syntropic process -- teleological.
Your mind is syntropic as you make predictions!
@@hyperduality2838 Indeed.
@@MisterrLi "Philosophy is dead" -- Stephen Hawking.
Main stream physics is currently dominated by teleophobia and eliminative materialism so you are unlikely to hear about new laws of physics based upon teleology.
Teleophilia is dual to teleophobia.
I might sound wrong but sir , could you tell me why did you choose infinite dimensional affine space here ? Whereas we have the nice infinite dimensional Hilbert space R∞ , but yes basis will surely be uncountable .
And another confusion is you are doing embedding of unit n spheres , for all n≥1 upto S∞ . Does it not violate the Whitney’s embedding theorem for closed manifolds ? I am just a graduate student in mathematics . I just want to know.
Good questions. AFAIK Wildberger's approach to inner products and dot products is different from the standard approach usually taught, I have not heard him making any specific mention of "Hilbert space".
As discussed under the sociology topic, the concept of "manifold" is too poorly defined to be useful from computational perspective. In that sense the question needs a better formulation. I've seen 3blue1brown's discussion of embedding unit spheres, and the 10D "limit" he demonstrated looks very interesting to me, but I'm not at all sure we are talking about same thing here.
My intuitive understanding of the concept of "manifold" is that it means "locally" a taxicab norm, which "smoothens" out into "something else "norm"" in some distant horizon. A rather vague concept of projective perspectives as such. You can correct me if you think my intuition is not on the right track here.
The infinite dimensional affine space I am talking about has elements which are purely finitely specifiable. That is a big difference with the usual "Hilbert space", in which an infinite amount of information is required to honestly and completely specify a general element.
@@njwildberger Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual.
Rational, analytic (a priori) is dual to empirical, synthetic (a posteriori) -- Immanuel Kant.
If knowledge is dual then information must be dual:-
Potential or imaginary information (entropy) is dual to kinetic or real information (syntropy).
Potential energy is dual to kinetic energy -- gravitational energy is dual.
Teleological physics (syntropy) is dual to non teleological physics (entropy).
Average information (entropy) is dual to mutual or co-information (syntropy) -- information is dual.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Sine is dual to cosine or dual sine -- the word co means mutual and implied duality.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual.
Curvature or gravitation is dual.
Quadrance is dual to spread.
"Always two there are" -- Yoda.