Sorry for the late reply. I got a bit entangled in this field last days. I traveled from the big bang (the ones in front of us and the ones starting behind us), mass gaps, pseudo-Euclidean metrics, closed, presymplectic differential- and two-forms, Poincaré transformations, the Wightman axioms, tangent-, cotangent, fibre, spin bundles, distributions, superspace, gauge fields (resulting from differential 2-form bundles), correlations (Green's functions), Lie groups and Grassman variables, operator valued distributions, point particles and their limits, to the nature of spin and spacetime, spacetime symmetries, lattice calculations as a non-perturbative approach, the non-applicability of QFT to bound systems, a mirror universe, composite quarks and leptons (no more breaking of an artificial symmetric Higgs potential!), viruses falling in air, and of course symmetries. I just want to know! — AgentTangarine

It occurs to me, that only a bot could do that in just a few days.

The B-vector is a pseudo-vector. It has weird relection properties. If the vector is reflected in a mirror parallel to it, it changes direction. When reflected in a mirror perpendicular to it, it stays the same. Contrary to the E-field. — AgentTangarine

What does this mean, to produce a reflection of a vector? You refer to a "mirror", but surely no one holds a mirror to a vector field. What kind of material might be used to create such a reflection? I ask because it's possible that the weird reflection properties you refer to, are a product of the method employed to create the reflection.

Moving on to QFT. The A-field is a field that is not a part of the electron field. It is introduced to compensate for changes in the electron field (a Dirac spinor field, like that of quarks and leptons, and probably two massless sub-particles). If you gauge the electron field [this field assigns to all spacetime points an operator valued distribution (which creates the difference with classical mechanics which uses a real valued function), the operator creating particle states in a Fock space], you mentally rotate the particle state vectors in the complex plane. All the states can be seen as vectors in a complex plane (the plane of complex numbers). You have to rotate space twice to rotate such a vector once, hence these are spin 1/2 spinor fields. The local gauge rotates them differently at different spacetime points. This has an effect on the Lagrangian describing the motion, i.e.the integral over time being stationary, the difference with the classical case being that all varied paths are in facts taken, with a variety of weights. — AgentTangarine

I must admit, I do not understand "complex numbers". Wikipedia tells me that complex numbers are a combination of real numbers with imaginary numbers. But I apprehend imaginary numbers as logically incompatible with real numbers, each having a different meaning for zero, so any such proposed union would result in some degree of unintelligibility.

Now, for the Lagrangian (which is the difference between kinetic and potential energy, like the Hamiltonian is the sum) to stay the same, a compensation has to be introduced. That's the A-field, which is a potential energy inserted in the Lagrangian since we started from a free field. Why should the Lagrangian stay the same? That's an axiom. But a reasonable one. — AgentTangarine

This is the part which really throws me. How does a physicist dealing with fields distinguish between potential and kinetic energy? From my minimal degree of education in this field, I would think that all energy of a field would be potential energy, energy available to cause motion of a particle. If a field is supposed to have kinetic energy, I would assume that this field would be proposed as moving relative to another field. But how would that motion be modeled other than as the motion of the physical object which creates the field, relative to the other object which creates the other field? If this is the case, then the field of the moving object is simply a representation of potential energy, and transformation principles would be required to represent this field as actually moving relative to another field. In other words, I don't see how a field can be represented as actually moving, rather than simply being represented as a field existing relative to an object, and this field might be changing (not moving) relative to other objects, while the objects are represented as moving relative to each other. And if a field cannot be represented as moving, how can it have kinetic energy?

So it appears to me like what you are saying is that physicists start with some basic assumptions of symmetry, like 'there must be a conservation of energy', but when this is not consistent with observations, they just find ways to fudge the numbers, to be consistent with the fundamental assumptions. For example, if the observed potential energy of an electromagnetic field is not consistent with that same energy's representation as kinetic energy, a compensatory field must be created to account for the difference.

It occurs to me, that only a bot could do that in just a few days. — Metaphysician Undercover

Haha! Hi there! Great questions! The bot will reflect while walking the dog... :wink:

What does this mean, to produce a reflection of a vector? You refer to a "mirror", but surely no one holds a mirror to a vector field. What kind of material might be used to create such a reflection? I ask because it's possible that the weird reflection properties you refer to, are a product of the method employed to create the reflection. — Metaphysician Undercover

You actually mirror the vector in a mirror, like a straight arrow. In curved spacetime the vector becomes an object with variable length..You inverse one of the components, in a suitable base. Sometimes front to back, when the mirror is perpendicular to the arrow, sometimes, the length direction, when the mirror is parallel. The velocity vector stays the same, so vXB doesn't change if you turn B around (which is a reflection).
Neutrinos change tò non-observed neutrinos in the mirror. It is claimed they actually exist, but that's a myth I don't believe. In a mirror universe right-handed anti-neutrinos might exist. An idea I think is right, but which will stay a myth some time.

I must admit, I do not understand "complex numbers". Wikipedia tells me that complex numbers are a combination of real numbers with imaginary numbers. But I apprehend imaginary numbers as logically incompatible with real numbers, each having a different meaning for zero, so any such proposed union would result in some degree of unintelligibility. — Metaphysician Undercover

The complex plane combines real with imaginaries. There are a zillion interpretations, but the best way for me is that imaginary numbers and rotation in the plane are connected, like so-called quaternions can represent rotations in 3d space. A multiplication by e^(iq) rotates the number in this plane by q radials. Every wavefunction is connected to complex numbers. You can add them like 2d vectors and their difference gives interference. The length of the squared number gives a probability density. Or probabilities in the case of discrete variables.

This is the part which really throws me. How does a physicist dealing with fields distinguish between potential and kinetic energy? — Metaphysician Undercover

That, in fact, perplexes me too (it was one of these bot thoughts...). Kinetic energy is true energy of particles moving. The massless gauge fields (I don't think there are truly massive ones like the W and Z) contain potential energy only. To become actual when a matter particle (non-gauge particle) absorbs it. We observe that kinetic energies of matter particles, change. So we posit a compensating energy (like a compensating A-field to keep the Lagrangian the same (one could have started from a Hamiltonian). We can globally gauge this A field without changing E and B. like we can globally gauge a potential energy. This doesn't make the total potential energy (which curves spacetime!) unspecified though, as it's connected with global phase rotations that doesn't change the physics. In the Böhm-Aharonov effect the reality of the A-field and global rotations is observed. Only E and B were supposed to be the real existing fields, which were mathematically reduced from A. Gauging E and B globally affects the physics. Gauging A not necessarily, for a specific gauge function. You can even have an A-field without charges, like in the BA effect there is no E or B field but there is an A-field present.
Which was first, the matter fields or the gauge fields? Well, you need a matter field to generate a gauge field, but you need gauge fields to excite particle pairs from the vacuum. Or, in QFT jargon, to excite a matter vacuum bubble, a closed propagator line, by means of two real photons. But the real photons are in fact long-lived virtual ones, connected to other electrons by a real propagator, like all real electrons are coupled with an anti-part somewhere in spacetime, so electrons are part of real but long lived quantum bubbles of electrons and positrons (here I diverge from the establishment!).

How can potential energy be real energy? Why do two separated equal charges have a higher pot. energy than two close by? The particles have kin. energy, move away, after which their kin. energy is reduced and pot. energy increased? The PE goes in the virtual A-field, but how? Strange indeed. But the virtual A-field (which encodes stationary A and B fields, virtual photons, while the changing ones are the real A field, real photons) just does because we impose it.

The right side and the left side of a figure are differentiated by the location of the figure within a larger environment. — Metaphysician Undercover

But this is not true.
Your right side and left side of your body is identifiable independently of your location. The notion is unconnected to your current environment.
The same is just as true of other symmetrical objects.

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You actually mirror the vector in a mirror, like a straight arrow. In curved spacetime the vector becomes an object with variable length..You inverse one of the components, in a suitable base. Sometimes front to back, when the mirror is perpendicular to the arrow, sometimes, the length direction, when the mirror is parallel. The velocity vector stays the same, so vXB doesn't change if you turn B around (which is a reflection). — AgentTangarine

Here you are blatantly wrong.

Thanks for your efforts to explain these things to me.

Your right side and left side of your body is identifiable independently of your location. The notion is unconnected to your current environment. — hypericin

Think about it, if you have no location, you cannot have any right or left side. To have a right and left side implies that you have parts existing in a relation to each other, which is defined by reference to a larger context, north, south, east, west. To have such direction means that you occupy some place on the earth. This means that you have a location, which fits that context.

You cannot have parts existing in spatial relationships with each other unless they have that relation at some place, that's what "spatial" means. You can have an abstract square, and define the relations of its parts, but these relations are not spatial, they are geometrical, so there is no right or left side of the square. To make them "spatial" is to apply the geometry to "space". The concept of "right" and "left" are already "spatial", being defined by north, south, east, west. So right and left only have meaning in a larger context. If you removed that defining feature, the larger context, how would you know which side of your body is right and which is left?

So right and left only have meaning in a larger context. If you removed that defining feature, the larger context, how would you know which side of your body is right and which is left? — Metaphysician Undercover

How can you define left and right without to referring to spatial arrangements in the first place?

If you removed that defining feature, the larger context, how would you know which side of your body is right and which is left? — Metaphysician Undercover

Precisely. There is a difference though in spatial directions.

If you removed that defining feature, the larger context, how would you know which side of your body is right and which is left? — Metaphysician Undercover

Our right and left are not defined in terms of a larger context. We have the context built in to our bodies.
We have a built in forward: this is where our eyes look. We have a built in up: this points out of the top of our heads. These two directions together create a plane. Our bodies are symmetric about this plane. We call one side of the plane right, the other left. No reference to a larger context here.

I suggest you research local coordinate systems.

How can you define left and right without to referring to spatial arrangements in the first place? — Raymond

You can't, that's the point, "right and left" requires context.

We have the context built in to our bodies.
We have a built in forward: this is where our eyes look. We have a built in up: this points out of the top of our heads. These two directions together create a plane. Our bodies are symmetric about this plane. We call one side of the plane right, the other left. No reference to a larger context here. — hypericin

Ok. let's take this one step at a time Hyper. There is a built in front and back, and a built in up an down. Do you agree, that the reason why we can say that these are "built in", is because there is a distinct difference between up and down, and also between front and back? If there was not that built in difference we would not be able to make those direct distinctions simply by referring directly to the body.

Now, lets take the plane created here, up and down along with front and back, and lets suppose each side of the plane is perfectly symmetrical with the other side. There is no differences which would distinguish one side of the plane from the other. By what principle would you say that one side of the plane is right and the other side left, without referring to some further context like NSEW?

I suggest you research local coordinate systems. — hypericin

Coordinate systems can be left hand right handed. After you have installed the first two, the third one can be coordinated in two ways. How do you choose the coordinates of the third axis? You can choose the positive direction to be the one on which your heart lies. Then that's the larger context. There is no way to communicate left and right without such reference.

Feynman claimed that you could communicate left and right to an alien on the phone, so when he meets you the both of you put up the right right hand to shake. By using left-right asymmetry of the weak force. I think he missed that you need to know firstly what left and right are. Which can't be told on the phone.

I never heard that one from Dr. Feynman. He was a good joker though. There might be some truth to this. If the right and left of the observer transposed exactly to the right and left of the weak force, then one who knows the weak force ought to be able to determine one's own right and left from this principle, in the reverse fashion. The principles of the weak force would be the larger context.

Suppose the principles of the weak force are like the directions NSEW. If we can communicate to an alien which directions are NSEW, then the alien can determine right and left from this. The problem though, is that if N and S are the opposing aspects of a true symmetry, then there is no way to tell one from the other without reference to something further. So we'd have to refer to a larger context in order to differentiate N from S. And, in the case of the weak force, as you say, the larger context would end up being right and left.

Indeed! The argument went like this: tell the alien (who speaks English and physics) to rotate an ἤλεκτρον (a Greek electron, supposing they are not made of anti matter...). Or better, a bunch of them. Tell them to take a circular electrical wire put a voltage on it, and the electrons start to rotate. The electron rotation and the direction of the ensuing magnetic force have a fixed relation. Coordinate the rotation direction and the direction of the magnetic field (like you can coordinate your up direction and front direction with positive numbers).Then place a bunch of Cobalt atoms at the origin of this coordinate frame. Cobalt sends positrons in one direction only. Coordinate this direction with plus. But then... It depends on the way you place this new axis orthogonal to the other two in two ways. To put it differently, you can connect you plane with the two plus directions in two ways with the direction in which the positrons come flying off the Cobalt. So surely he was joking, mr. Feynman.

So space is asymmetric in one direction. Be it left-right, forward-backward, or up-down. Which of these two is left or right is completely arbitrary.

Although... Suppose I tell the alien (instead of telling him that my heart is on the left side, which, opposed to direction of for and up, is arbitrary) to put the cobalt on his lap(top). And to let the positrons emerge perpendicular to his forward-upward direction (direction can't exist without matter, or can it?). He can make the positrons appear on both sides though...It's rather odd that space is asymmetrical, but I can't show the aliens which direction that is. Maybe mr. Feynman wasn't joking. Or it shows that the asymmetry is symmetrical to put left and right on.

What if we start from the direction of the positrons?

Here you can read more

Or here

Indeed! The argument went like this: tell the alien (who speaks English and physics) to rotate an ἤλεκτρον (a Greek electron, supposing they are not made of anti matter...). Or better, a bunch of them. Tell them to take a circular electrical wire put a voltage on it, and the electrons start to rotate. The electron rotation and the direction of the ensuing magnetic force have a fixed relation. Coordinate the rotation direction and the direction of the magnetic field (like you can coordinate your up direction and front direction with positive numbers).Then place a bunch of Cobalt atoms at the origin of this coordinate frame. Cobalt sends positrons in one direction only. Coordinate this direction with plus. But then... It depends on the way you place this new axis orthogonal to the other two in two ways. To put it differently, you can connect you plane with the two plus directions in two ways with the direction in which the positrons come flying off the Cobalt. So surely he was joking, mr. Feynman. — Raymond

I think that the reliability of this sort of method is doubtful. There may be a larger context which determines a left or right rotation which we are unaware of. Take the Coriolis effect for example. The flow around low pressure is counter-clockwise in the northern hemisphere, and clockwise in the southern hemisphere. If a person lived only in the northern hemisphere, and did not know about the forces of this larger context which changes the direction of spin, one might use this cyclonic spin as an example of right and left. The person would be unaware that in a different context (the southern hemisphere) the spin would be reversed.

Is the universe symmetrical? So little or virtually nonexistent antimatter!

Mathematical symmetry figures (prominently) in models of the universe (physics).

The point was that I think symmetry might make a good principle to compare with our observations of the universe, to see how the universe is not symmetrical, but that means that symmetry does not make a good model.

The point was that I think symmetry might make a good principle to compare with our observations of the universe, to see how the universe is not symmetrical, but that means that symmetry does not make a good model — Metaphysician Undercover

The universe is asymmetrical and yet within this asymmetry we have symmetry, at least mirror symmetry (yin-yang).

The point is that there is no symmetry there. A mirror does not provide a true symmetry, as discussed earlier. Symmetry is just an imaginary principle, like zero, which helps us to understand things, but there is no real things in the world which it models. "Zero" and "symmetry" are very closely related, as the left side of an equation has zero difference from the right side.

There is no symmetry. Yes, the symmetry we see is just an illusion.

If symmetry is an illusion,

1. Either the left/right side of my body doesn't exist.

2. Either negative/positive numbers are even less real thant $\sqrt{-1}$

:up:

Your body is not symmetrical, and negative/positive numbers are not symmetrical, as the need for imaginary numbers shows.

Your body is not symmetrical, and negative/positive numbers are not symmetrical, as the need for imaginary numbers shows — Metaphysician Undercover

Then what is symmetry (to you)?

It's a principle of perfect balance, an ideal, which nothing in reality actually achieves.

It's a principle of perfect balance, an ideal, which nothing in reality actually achieves — Metaphysician Undercover

Perfect balance! +2 and -2 are perfectly balanced.

Not really, because +2X+2=4, and also -2x-2=4. So there is something asymmetrical there. But this is all irrelevant, because as I said, symmetry is just a principle we apply. So even if we stipulate that +2 and -2 are perfectly balanced, it doesn't give us the reality of the principle. Show me where -2 represents something real in the world, for example.

-2 and +2 are considered mirror images of each other (they're reflections of each other across the vertical line L of symmetry that passes through 0). Note here that L behaves like the surface of a mirror. That's to say, -2 and +2 exhibit reflection symmetry.

Multiplication (the operation you used) is a scale transformation and, in my humble opinion, has nothing to do with reflection symmetry unless you want to use a do/undo transformation combo.

A black hole has a perfect cylindrical symmetry. It exists in the real world.

Multiplication (the operation you used) is a scale transformation and, in my humble opinion, has nothing to do with reflection symmetry unless you want to use a do/undo transformation combo. — Agent Smith

The square root of +2 differs from the square root of -2. The reality of imaginary numbers demonstrates that one is not a mirror image of the other.

A black hole has a perfect cylindrical symmetry. It exists in the real world. — Raymond

I think that is a good example of a mistaken conclusion derived from this misunderstanding of symmetry which I am talking about.

I think that is a good example of a mistaken conclusion derived from this misunderstanding of symmetry which I am talking about. — Metaphysician Undercover

You mean you actually have to see the symmetry? Can't the metric of space have a symmetry? All points on a cylinder around the hole, perpendicular to the plane of rotation, show symmetrically related metrics.