It's really a sort of unique but trivial way of progressing from F=r to defining a new fixed point, b. Bifurcation is usually meant to "split" one fixed point into two fixed points. A kind of "Adam's Rib" sort of thing. In this instance, the function in the background (not the function,F, seen here) is a parabolic linear fractional transformation. But enough mathematics. Thanks.

Bifurcation, a transcritical, pitchfork, and Hopf

Nice presentation on traditional bifurcations via DEs. Not relevant to the situation I describe which is a process of growing a second (and repelling) fixed point (FP) as a parabolic LFT (having a single attracting FP) morphs into a non-parabolic LFT (having two FPs). Thanks for posting it.

Yeah, it's different from what you describe. I gave your F a fair amount of thought. If we set the nominator 1 and write F=1/(a-(a-1/r)) (=r), what have we done with r? Say r=10. We can take a=1. This gives 1 and 9/10. Is that the bi in bifurcation? I can't see a parabole, as you write.

It's really a sort of unique but trivial way of progressing from F=r to defining a new fixed point, b. Bifurcation is usually meant to "split" one fixed point into two fixed points. A kind of "Adam's Rib" sort of thing. — jgill

Being familiar with some of my posts, you probably already have a good idea of what I would say about this proposal. The idea of one point becoming two points, without a clearly defined division (division of a point appearing to be impossible), is simple contradiction, in the first place.

I can't see a parabole, as you write. — EugeneW

Here's a link to the short note. This is a technical subject and probably not appropriate for TPF. If there are other questions it's best if they come through messaging.

What's philosophical is the idea of a dimensionless point producing an offspring. This happens as the value of t goes from 0 to 1. Like b(t) = a(1+t).

Love this video about Möbius transformations

Nice imagery isn't it? That's the fairly simple geometric and projective aspects of MTs. The analytic aspect Is what I do. For example.

What's philosophical is the idea of a dimensionless point producing an offspring. — jgill

That's so incoherent it's actually funny.

That's so incoherent it's actually funny. — Metaphysician Undercover

:cool:

(you can see this happening in the first note I linked) :wink:

Cool stuff! Especially the arrow pictures are telling. And the light brown picture is intriguing! All from iterating MT? Are the centers in these arrow fields the two points from one?

The arrow fields are vector fields with arrows showing direction and magnitude when a point is moved by application of the underlying function. When the arrows point to a specific point, that is an attracting fixed point. When they go away from a point, it is a repelling fixed point. Short answer for a sophisticated set of ideas. The brown picture is an image generated by iterating each point a specific number of times and pixel coloring according to magnitude.

I think aliens, actually, prove scientific materialism false. How can someone prove everyone else is not an alien without common sense. You can't get this knowledge from science. Everything has to be thought out by a materialism and no faith can save them unless a more mystical side of reality is considered (Spinoza, Hegel, ect)

What's philosophical is the idea of a dimensionless point producing an offspring. — jgill

In the context of this thread, is a zero-dimension point considered to be Real or Ideal, Physical or non-Physical? As a philosophical or mathematical thought-experiment, the notion of "nothing producing something" might be a valid ideal concept. But as a scientific observation it might be as unrealistic as a vacuum fluctuation popping a particle of matter into existence.

As I understand it, a Virtual Particle is equivalent to a dimensionless point. It works mathematically, but does it exist in reality? And that raises the old conundrum : "is mathematics real or invented?" My question is serious, because the answer could shed some light on the OP topic. :smile:

Is math invented or discovered? :
Mathematics is an intricate fusion of inventions and discoveries. Concepts are generally invented, and even though all the correct relations among them existed before their discovery, humans still chose which ones to study. ___Mario Livio, theoretical astrophysicist
https://www.sfu.ca/~rpyke/cafe/livio.pdf
Q : are "correct relations" equivalent to logical relationships?

Are virtual particles really constantly popping in and out of existence? Or are they merely a mathematical bookkeeping device for quantum mechanics? :
Virtual particles are indeed real particles. Quantum theory predicts that every particle spends some time as a combination of other particles in all possible ways. These predictions are very well understood and tested. ___Gordon Kane, theoretical physicist
https://www.scientificamerican.com/article/are-virtual-particles-rea/

In the context of this thread, is a zero-dimension point considered to be Real or Ideal, Physical or non-Physical? As a philosophical or mathematical thought-experiment, the notion of "nothing producing something" might be a valid ideal concept. But as a scientific observation it might be as unrealistic as a vacuum fluctuation popping a particle of matter into existence.

As I understand it, a Virtual Particle is equivalent to a dimensionless point — Gnomon

I would say a zdp is both real and ideal, but not physically real. As for vps, they are excitations of the underlying quantum fields, so perhaps zdps are as well.

It's all Greek to me. I just try to do the math. :cool:

But as a scientific observation it might be as unrealistic as a vacuum fluctuation popping a particle of matter into existence. — Gnomon

A vacuum fluctuation can be seen as an eternal presence of a particle in the vacuum. It oscillates in time with a non-fixed energy-momentum relation ($E^2-p^2=m^2$). Only a real particle or superstrong gravity with an horizon can promote it to a real status. So it's not the vacuum that creates but the non-vacuum or its curvature.

Virtual particles are indeed real particles. Quantum theory predicts that every particle spends some time as a combination of other particles in all possible ways — Gnomon

A particle doesn't change continuously into other particles. Rather, it couples with the omnipresent field of virtual particles, by means of it charge. So the electron interacts with other electrons by interacting with the virtual photon field between them. And this virtual photon field can, on its turn, interact with virtual electrons which again can couple to photons. Zillions of couplings are involved in an interaction between two electrons. Can you imagine? All at the same time.

A vacuum fluctuation can be seen as an eternal presence of a particle in the vacuum. — EugeneW

What is this concept of "a particle in the vacuum"? Is it a vacuum, which is not really a vacuum because there is something there, which must be a particle, but it's not really a particle because it has no location? So it's not really a vacuum, nor is there a particle, just convenient terms.

I would say a zdp is both real and ideal, but not physically real. As for vps, they are excitations of the underlying quantum fields, so perhaps zdps are as well.
It's all Greek to me. I just try to do the math. — jgill

That's coming close to what I was getting at. The linked quote below indicates that a Virtual Particle is treated as-if it's "Real", but it doesn't add your qualification : "but not physically real". So my question, is "in what sense is a non-physical object Real?. Is that faux reality an equivocation?

Since Quantum Fields consist of dimensionless-points-in-space, they are "real" only in the sense that they have the Potential to produce physical particles. So, it seems that a mathematically-defined Field is deemed capable of creating mathematical (virtual ; un-real?) particles from nothing-but-numbers. Yet, Aristotle contrasted "Potential" with "Actual". So, you could say that the statistical-possibility-for-future-existence "exists" only as the hypothetical power to create Actual (now) particles from mathematical Probability (predictability). Ouch! I grok what you mean by "it's Greek to me".

I understand "as-if" to mean "hypothetically" or "metaphorically". As you noted, and as the VP definition below indicates, such conceptual objects are "not physically real". But isn't it misleading to label such abstract notions as "real". When I use the philosophical terms "Ideal" or "Meta-Physical" for such common conceptual abstractions as "Zero" & "infinity", I get protests for employing a religious term, even though I'm not using it in a strictly philosophical context. [edited to strike out "not"]

Therefore, I've been searching for a viable alternative term to mean "non-physical reality". "Virtual" and "Essential" do indeed refer to abstractions, such as " "excellence, potency, efficacy". Even "Mathematical" or "Statistical" refer to non-physical or not-yet-actual abstractions. But they have traditional non-philosophical scientific currency. So, I guess the "real" question could be expressed as, for example : A> "Is Mathematics Real, or is it Metaphysical, in the sense defined below?" or B> "does a Virtual Particle have physical form?" How can we distinguish between Physical Reality and Virtual Reality? Is it OK for philosophers to postulate in terms of "non-physical Ideality", or "beyond-physical-reality"? :brow:


Is a Virtual Particle Real ? :
Compared to actual particles — It is not. "Real particles" are better understood to be excitations of the underlying quantum fields. . . . .
Since it is possible to perform quantum field theory calculations completely absent virtual particles being referenced in the math used, as seen in lattice field theory, then it is believed virtual particles are simply a mathematical tool.
https://en.wikipedia.org/wiki/Virtual_particle

What keeps space empty? :
Perfectly "empty" space will always have vacuum energy, the Higgs field, and spacetime curvature.
https://www.wtamu.edu/~cbaird/sq/2012/12/20/what-keeps-space-empty/
Q : do those mathematical entities actually occupy space?

Aristotle describes potentiality and actuality, or potency and action, as one of several distinctions between things that exist or do not exist. In a sense, a thing that exists potentially does not exist, but the potential does exist.
https://en.wikipedia.org/wiki/Potentiality_and_actuality

As-If Fallacy :
Offering a poorly supported claim about what might have happened in the past or future, if (the hypothetical part) circumstances or conditions were different. The fallacy also entails treating future hypothetical situations as if they are fact.
https://www.logicallyfallacious.com/logicalfallacies/Hypothesis-Contrary-to-Fact

Virtual :
The meaning "being something in essence or effect, though not actually or in fact"
https://www.etymonline.com/word/virtual

Metaphysical : adjective. without material form or substance.
https://www.vocabulary.com/dictionary/metaphysical

Since Quantum Fields consist of dimensionless-points-in-space, they are "real" only in the sense that they have the Potential to produce physical particles. — Gnomon

You could say that the point signifies something. So it's similar to when some one draws a map, and marks a point to signify a city. The point signifies something, but what it signifies does not at all resemble a point. The spacetime is mapped and the designated points signify something, an aspect of that which is being mapped

What is this concept of "a particle in the vacuum"? Is it a vacuum, which is not really a vacuum because there is something there, which must be a particle, but it's not really a particle because it has no location? So it's not really a vacuum, nor is there a particle, just convenient terms. — Metaphysician Undercover

A virtual particle just runs around in time with disentangled energy and momentum (off-shell). On all paths at once (or oscillating between all fast). Time does not go forwards nor backwards for them. They are time, they are vacuum. No, correction. The can serve as timekeepers. As clocks that themselves have no temporal direction.

Point particles are a myth. A fairy tale image, leading to infinities, regularization, renormalization, and other ghost stories.

Real particles are just as virtual as virtual particles. Virtual particles are potential in the sense that they can get promoted from fluctuation to excitation. The quantum field is not more basic than a particle though, as is understandably assumed by quantum field theoreticians.

If you understand what the fields stand for, it turns out that the particle shows up again. Quantum fields are
operator valued distributions. These operators operate on a product of Hilbert spaces, that is Fock space. They create or destroy particle wavefunctions (or wavefunctionals), which add up to represent all particle states with definite energy and momentum, meaning that free particles are not moving on one fixed path but many at once, or shifting between them rapidly.

The vision of a point particle moving in space is a popular image of such excitation, ad is the image of a particle pair appearing and disappearing, which in reality is one particle always being there.

Is math invented or discovered? :
Mathematics is an intricate fusion of inventions and discoveries. Concepts are generally invented, and even though all the correct relations among them existed before their discovery, humans still chose which ones to study. ___Mario Livio, theoretical astrophysicist — Gnomon

I would alter this slightly by saying the correct relations among them come into existence when the concepts do. Then these are sought out.

You could say that the point signifies something. — Metaphysician Undercover

Yes. An immaterial grid-point in empty space is like the symbol "X" (unknown) which serves as a stand-in for a real number, that is un-realized until calculated. I assume that, for a physicist, defining a Virtual Particle as a mathematical point, is essentially an ellipsis, a blank to be filled-in at a later date. So it points toward something imaginary, that could be realized, but not yet physically real.

Using Terrence Deacon's Incomplete Nature terminology, a non-dimensional point on a Quantum Field grid is "Absential". But would I be understood, if instead of labeling a non-physical concept as "Meta-Physical", I called it "Absential"? For example, "the general, non-specific philosophical notion of 'Being' is Absential" ; a potential existence at a point in space to be specified later". Perhaps, "non-being" would be, not only non-physical, but also non-potential, or impossible.

I'm sorry, but I'm confusing myself with the limitations of common language, which is necessarily Materialistic, and forces abstract concepts, like "LOGOS" and "Logic", to be expressed in physical metaphors or analogies. That's why the term "Meta-Physical" seems to me more concise & intuitive : implying something non-physical, yet meaningful. It's too bad, such a useful comparative term has acquired a negative association with flesh-less Spirits & Ghosts, instead of with matter-less Ideas & Abstractions. :smile:

Ellipsis :
1. the omission from speech or writing of a word or words that are superfluous or able to be understood from contextual clues
2. a set of dots indicating an omission..

Absential :
1. the state of a thing not yet realized
2. un-realized potential

Virtual particles are as real as real particles. They have space-pervading presence, and are not moving forward nor backward in time. They are just circulating, oscillating, fluctuating in time. They do this with a momentum and energy that are independent of one another, so they can have energy with zero momentum, momentum without energy, or, in short, any momentum value combined with any energy. Its mathematically described by a propagator propagating a combination of all possible free particles with all possible momenta and energies (or positions and times). It's this "glue", this "condensate" charged particles couple to when interacting. So an interaction is not by an the exchange, the emission and absorption of virtual particles, as often read in popular treatments (and causing a lot of confusion), but by coupling to them. The same popular writings say that the vacuum is filled with particle pairs popping in and out of existence. There are no pairs involved. Only single particles. :smile:

Virtual particles are as real as real particles. — EugeneW

If so, why give one of them an un-real name? I'm aware that mathematical theorists treat "Virtual" Particles as-if they are real. But the differentiating name they pinned on them belies their reality from a common sense perspective. This is just one of many paradoxes emanating from the Pandora's Box of quantum science. They make our world seem to be an mirage of many delusions, as contrasted with the mundane Actual Reality of classical science. Biologists & Chemists are still mostly working in the old-fashioned Real world. But Physicists seem to be exploring a sci-fi fantasy realm of parallel Realities.

Ironically, Aristotle made the same real/unreal distinction 2500 years ago, when he defined the meanings of "Actual" and "Potential". So, which is it : is the "Real" world Potential or Actual or both? As a non-mathematical layman, I have been forced to punt on that either/or question, and to view paradoxical reality in terms of the BothAnd Principle. Reality is a system of systems, but only the more familiar material aspects seem really Real, and the immaterial, non-physical parts seem weird, or surreal, or imaginary. :gasp:

Virtual :
1. almost or nearly as described, but not completely or according to strict definition.
2. not physically existing as such . . .

Sir Roger Penrose details three different kinds of reality :
The three levels are Proto Energy which is energy as pure vibration, Proto TimeSpace where space and time become a single point looping with no direction, and Proto Matter where matter is infinite potentialities.
https://www.vice.com/en/article/kbnkqx/explore-the-subatomic-world-of-energy-fields-in-virtual-reality

The Paradox of Reality :
It was interesting to read great scientific minds like Prof Roger Penrose grapple with the apparent paradox of reality. . . . . Objective, impersonal reality, untainted by private sensation, must ever remain an inference and construct, however successfully these may seem to approximate to that reality. . . . We mortals are heir to illusion...but at least we may strive to avoid delusion.
https://ronaldtkwong.com/news/TheParadoxofReality.html

Both/And Principle :
My coinage for the holistic principle of Complementarity, as illustrated in the Yin/Yang symbol. Opposing or contrasting concepts are always part of a greater whole. Conflicts between parts can be reconciled or harmonized by putting them into the context of a whole system.
BothAnd Blog Glossary
Note -- Apparently, our world is neither Actual nor Potential, but both simultaneously. However, the Actual stuff of reality is physical, while the Potential essence of reality of non-physical (mathematical, ideal, meta-physical, etc.). So, Physical Reality is merely the sensory part of the whole world, and Virtual Reality is the imaginary or ideal aspect of the world system.

energy with zero momentum — EugeneW

Yes, but energy-without-momentum is what we know as Potential energy, as contrasted with the Kinetic (or Actual) energy of moving matter. For example a typical car battery has 12 volts of Potential, but when no current is flowing there is no Actual physical work being done. That reminds me of the storage box used by the Ghost Hunters to trap poltergeists so they can do no harm. :joke:

Can you have energy but no momentum?
Yes, something can have energy without having momentum. Momentum is defined as the mass of an object times its velocity. Even in a rest state, when momentum is zero, Body still has potential energy, U=mgh (where, m= mass of body). Hence, it is possible to have energy without having momentum.
https://www.quora.com/Can-something-have-energy-without-having-momentum

If so, why give one of them an un-real name? — Gnomon

Well, they have unreal properties. Or rather, non-intuitive properties. Like energy (not potential) without momentum and momentum without energy. Off-shell, that is. But beside that, they are as real as real particles (which are on-shell). Iif a virtual particle had fixed a momentum energy relation it would be a real particle. The vacuum is filled with these particles. Why should a field fluctuation be less real than an excitation? If one wants to call virtual particles unreal, then so are real particles.

That is the problem with "mass". It's just not at all understood by physicists. Momentum is the product of mass and velocity. But then we have "energy" which could be velocity without mass, hence velocity with no momentum. If there is no mass which is moving, then what is the velocity attributed to? What a mess physicists have found themselves in, due to the adaptation of speculative theories which are not grounded in sound ontology.

Maybe all particles are basically massless. Maybe them interacting renders mass. Massless virtual photons (a closed propagator line in a Feynman diagram, a "bubble") can have energy without momentum and momentum without energy, and in between. They can even have negative energies and momenta. Charged real fields interact by coupling to the virtual photon field (which can interact with virtual charged fields, which can couple to virtual photons, etc...), which adjusts its energies and momenta to the interaction.