You touch upon a deep issue here, as a matter of fact! It is claimed that symmetries lay at the basis of forces.The SU(2)l×SU(1)ySU(2)l×SU(1)y symmetry for the so-called unified force (splitting in the EM force and weak force after a break of symmetry, namely that of the Higgs potential) the SU(3)SU(3) symmetry for the color force, and a coordinate symmetry for general relativity. You can perform symmetry operations without truly change a system. This is simply done mentally, and by demanding symmetry, forces arise, while in fact it's the other way round. It are forces which give rise to symmetry principles. You can literally force symmetry transformations upon nature, like you do with the squares, and retrospectivelyclaim that forces are the result, but that's indeed putting the horse behind the wagon. You can rotate all points of a square locally and say that because of this forces will appear in the square to let it keep its shape (making it symmetrical wrt to local rotations or gauges), but as you say, you have to pull and push it first for these forces to appear. — AgentTangarine

Thanks for your contribution Agent Tangerine. I must admit that I don't quite grasp what you're talking about here. I'm having a difficult time understanding the concept of gauge symmetry, and especially the role of what is called "internal space", and its relation to space-time. Maybe because it's supposed to be "internal", is the reason for the role reversal which you describe.

Interchangeability of the right and left, which constitutes "invariance", is only possible if the figure is considered to exist free from any context, without a location. Of course nothing really exists without a location, so "symmetry" in this sense is is just a false principle. It cannot be applied to anything real. — Metaphysician Undercover

On the contrary, the second notion of symmetry(17th c) from the quote you provided, ignores the location or context. The left and right are simply equal, or they mimic each other. While the first notion, which is the ancient definition of symmetry, refers to balance. This symmetry, I think, is what's dependent on location. You'll find this a lot in art composition -- paintings for example, around the 15th century.

Good topic, but out of sync, I'm afraid.

I get your point completely and I always wondered about that myself. And you are right. Applying a principle of symmetry to the natural world implies an active manipulation of objects. Symmetry principles reverse the order of things. One can also apply passive operations under which an object stays the same. If you transform coordinates of space and time, an object in it will not change. It will not be seen to have a different motion or shape. If you gauge all clocks and lengths in flat spacetime by a continuous differentiable function then in the new coordinates fictitious forces seem present which make the object move in a way that's related to the gauge function, more specific, by a connection and by demanding that the object stays the same.
By applying an active gauge and demanding the object stays the same (so the object is symmetric under this gauge), real forces are introduced. You start from the position the forces are already there and exactly there lies the subtlety. If you start from non-interacting (free) fields, and apply active local transformations on internal properties of the fields (which require no force though), then demanding that a function of the fields and its derivatives, the Lagrangian (the difference between kinetic and potential energy, which in the case of free fields is only kinetic), stays the same introduces extra fields in it to compensate for the change introduced by the gauge transformations.
In the case of QED, you mentally rotate a real feature of nature. You rotate the complex phase of the wavefunction everywhere in spacetime, but differently at each point.

Sorry, pushed the post button. Not finished yet!

I don't quite get mirror symmetry. Positive and negative numbers are mirror images of each other and yet they do something that other reflection transformations don't do - cancel each other. Maybe our reflections in mirrors are made of antimatter and, for all we know, it could be the other way round. Kinda makes sense, doesn't it? How much does my reflection weigh? I'm about 70 kgs. That means my reflection in a mirror should be -70kgs. Is it 0 kgs? :chin:

On the contrary, the second notion of symmetry(17th c) from the quote you provided, ignores the location or context. The left and right are simply equal, or they mimic each other. While the first notion, which is the ancient definition of symmetry, refers to balance. This symmetry, I think, is what's dependent on location. You'll find this a lot in art composition -- paintings for example, around the 15th century. — Caldwell

Yes I agree, the older notion of symmetry does not involve removing the symmetrical thing from its context. In fact it might be argued that the reason why symmetry is considered to be beautiful is the way that the symmetry is observed to be a fit, within the context. In other words, the beauty of any particular symmetry is given by the context. And one could even take this principle further to argue that symmetry is actually a feature of the context, reducing the internal thing which is supposed to be the symmetry itself, to a simple central point within a balanced environment.

So the point I tried to make in the op is that the modern use of "symmetry" as it is used in pop metaphysics, in the sense of symmetry-breaking and similar concepts, derived from the application of mathematics in physics, is what we might call a perverted sense of "symmetry". It places "symmetry" as a feature of an object rather than as an arrangement of objects. We could say that it abstracts "a symmetry" as an object, from "symmetry" which is necessarily an arrangement, therefore a plurality of things. In essence, a true symmetry requires an arrangement of parts, whereas a modern symmetry is considered to be an invariant whole, thereby denying the possibility of parts.

Good topic, but out of sync, I'm afraid. — Caldwell

I don't get you. Out of sync with what? Out of sync with the modern sense?

I'm tempted to say that Symmetry isn't true by correspondence, rather symmetry means "truth by correspondence".

This is by considering symmetry to be the same thing as an isomorphism, i.e an invertible mapping, where an invertible map is a property of a description or acts of description.

Let me continue. In the case of QED, you locally apply gauge transformations. At every point in space and time you rotate an internal vector in the complex plane. These internal complex vectors belong to electron field. They are associated with probability densities. Their squared length represents the probability density which give you the probability of finding an electron in some volume if you integrate over the volume. If you globally perform such a gauge (everywhere the same) then there will be no change in the physics. The probability density stays the same everywhere. Now you might expect this to be the case also when you rotate the vector locally (differently at each point in spacetime). After all, the length of the vector in the complex plane doesn't change if you multiply it by $e^{iq(x)}$. The transformation is an element of the unitary abelian Lie group U(1). So the operations are not completely random but continuous and differentiable.
The point is that the Lagrangian changes if you apply the gauges locally. So a compensating field is needed to keep the Lagrangian invariant, or make it symmetric under the gauge. This field is the A field, which is a 4-vector field comprised of the electric potential in the time component and the vector potential A in the space components.

Now when one states that the force or interaction appears because of a symmetry principle (the Lagrangian staying the same) one turns the world upside down. It only looks as if. When you apply the gauges you mentally change the electron field and this induces an A-field to compensate for the extra terms appearing in the Lagrangian. In the real world, the A-field induces gauge transformations in the electron field which make themselves noticed exactly when electrons interfere with each other.

Let me explain the last point. In the Bohm-Aharonov effect, an A-field is introduced between two slits and a screen. The A-field has no corresponding electric or magnetic fields. Before the advent of QM the A-field was thought to be a mathematical object only. One could apply gauge transformations to it without changing the corresponding E and B fields. QFT changed that image as an A-field without E- and B-fields can introduce (global) gauge transformations in the electron field. Before the introduction, an electron field passing through the slits forms an interference pattern on the screen. When the A-field is inserted, the field introduces global rotations in the two fields coming from the slits, and this difference in rotations shows up as a shift in the interference pattern. So it's the change in the two parts which actually translates the interference pattern. The individual parts would have shown no difference when projected on the screen. If we would have introduced an A-field with corresponding E- and B-fields the fields would experience local transformations which would have shown itself in an interference pattern on the screen that suggests the electrons have interacted with a real E- or B-field, or both, if the A-field varies in time.

The A-field in QED is caused by the electrons themselves and they induce local gauge transformations on the electron field, precisely in such a way that the Lagrangian of the conserved. The gauge changes introduced cause similar shifts in interference patterns as in the BA effect. This causes electron fields to get shifted like the interference pattern is shifted in the effect above-mentioned. The difference is that the shift is not the same everywhere (global) but rather varies from place to place. The induced local gauge transformations show themselves as interference effects (which is the only way to observe rotations of internal vectors in the complex plane).

It's actually the charge that is used for the local gauge transformations, the function $q(x)$ in $e^{iq(x)}$. Charge is the generator of the transformations. Together with the invariancy of the Lagrangian under the action of the gauge, this entails the conservation of charge
Likewise, energy, momentum, and angular momentum are generators of translations in time and space, and rotations respectively. And because the translations and rotations leave the Lagrangian invariant (like the gauge transformations generated by charge), energy, linear momentum, and angular momentum are conserved, and are also referred to as charges.

The conservation laws are not a consequence of symmetry, like the symmetry of the Lagrangian doesn't cause the A field. It are rather the conservation laws that are cause of the symmetry. So again, the world put upside down or the cart put in front of the horse.

You can rotate a square locally with your mind and demand that the because of an invariant shape of the square forces are induced to keep the shape intact. But in reality you apply forces to the square (albeit mentally) which induces contra forces that keep it in shape. The invariant shape is not the cause of the forces but a consequence. So it's force in the first place that induces force to keep it in shape, like it is the A-field in the first place that gives rise to gauge transformations and keeps the Lagrangian invariant when the gauge transformations are applied on it. But it's a matter of taste. If you hold the symmetry fundamental, the symmetry is the cause. If you hold the fields fundamental the symmetry follows.


It's possible to make the symmetry of the shape (by which I mean that it stays the same, so any form will do, also the asymmetrical ones) of the square (or any object) the cause of forces appearing in it. As I already wrote, this is turning the world upside down. You need to apply forces first which have as a consequence that shape remains fixed.
You can rotate the square (or any form) globally, in the external space, or translate it in space or time. This will involve forces obviously. But these are not internal transformations and the square being translated or rotated globally, will be related to energy, momentum, or angular momentum conservation.

@AgentTangarine is a bot. Not a real member posting. Can we put a restraining feature on this bot?

And one could even take this principle further to argue that symmetry is actually a feature of the context, reducing the internal thing which is supposed to be the symmetry itself, to a simple central point within a balanced environment. — Metaphysician Undercover

Nice observation. In fact, it was the attitude at the time to showcase symmetry in visual arts. So in essence, you're admiring a painting because it has symmetry (balance), but you're not supposed to notice it.

So the point I tried to make in the op is that the modern use of "symmetry" as it is used in pop metaphysics, in the sense of symmetry-breaking and similar concepts, derived from the application of mathematics in physics, is what we might call a perverted sense of "symmetry". It places "symmetry" as a feature of an object rather than as an arrangement of objects. We could say that it abstracts "a symmetry" as an object, from "symmetry" which is necessarily an arrangement, therefore a plurality of things. In essence, a true symmetry requires an arrangement of parts, whereas a modern symmetry is considered to be an invariant whole, thereby denying the possibility of parts. — Metaphysician Undercover

This is almost similar to what I'm saying above. Symmetry becomes the object itself, and the main event becomes the background -- a supporting role to symmetry. Is this close to what you're thinking here?

I don't get you. Out of sync with what? Out of sync with the modern sense? — Metaphysician Undercover

Ignore this. I get your point now. But to clarify what I meant when I posted it, I meant your OP was out of sync.

AgentTangarine is a bot. Not a real member posting. Can we put a restraining feature on this bot? — Caldwell

Then what? Return to quantum babble? Be thankful there are a couple of physicists on board. :roll:

Then what? Return to quantum babble? Be thankful there are a couple of physicists on board. :roll: — jgill

If you say so.

AgentTangarine is a bot. Not a real member posting. Can we put a restraining feature on this bot? — Caldwell

How do you know this?


Thanks for the detailed and informative post. Regardless of whether you are a bot or not, I find your posting to be both interesting and coherent. Perhaps, in relation to subjects like this, it's better to be a bot, because human beings tend to be emotional

There's a couple questions I have. The first, concerns the nature of the described A field. You said:

The A-field in QED is caused by the electrons themselves and they induce local gauge transformations on the electron field, precisely in such a way that the Lagrangian of the conserved. The gauge changes introduced cause similar shifts in interference patterns as in the BA effect. This causes electron fields to get shifted like the interference pattern is shifted in the effect above-mentioned. The difference is that the shift is not the same everywhere (global) but rather varies from place to place. The induced local gauge transformations show themselves as interference effects (which is the only way to observe rotations of internal vectors in the complex plane). — AgentTangarine

If I understand correctly, the classical "electromagnetic field" which is a property of electrons, can be represented as two distinct fields, electric field and magnetic field. I understand the electric field (E) to be spatial, representing a spatial relation to the position of the electron. The magnetic field (B) I understand as temporal, representing the changing position of the electron. If I understand you correctly, you are saying that the relationship between the E field and the B field, which ought to represent "electromagnetism", is not strictly invariant, so there is a need to introduce an A field to compensate. I would conclude that the relationship between space and time is not invariant. It is made to appear as invariant through the use of the A field. If you can explain where this interpretation is misunderstanding, or deficient, I'd be grateful.

On the other hand, here's another question. It has to do with the use of "planes". I understand the E field to be represented as a different plane from the B field. I don't understand why this principle is employed in the first place. But there is evidently a problem, because there is a fundamental difficulty in relating two planes, known as the irrational nature of the square root of two. Further, when the relationships is presented as curved lines, circles or arcs, there is the irrational nature of pi to deal with. So distinct planes is a problematic concept to me. Can you tell me what is meant by "the complex plane"?

This is almost similar to what I'm saying above. Symmetry becomes the object itself, and the main event becomes the background -- a supporting role to symmetry. Is this close to what you're thinking here? — Caldwell

Yes, I think this is close. The problem being that the symmetry cannot actually be the object, so this is where the notion of falsity, or in the case of your example of artwork, maybe even a form of deception, is involved. "Symmetry" is a descriptive term referring to the relationship between things, thereby implying a multitude of things. If the symmetry is the object, we'd say that the multitude of things are parts of a whole, and the whole is "the symmetry". The issue is that a whole never really is a symmetry, so that is a misrepresentation. So when we see "the main event" as a symmetry, we are not seeing the whole, we are seeing the parts involved in an event which is seen as symmetrical. And the whole is something completely different from a symmetry, so if we see the whole work of art, we are not seeing a symmetry, and it would be wrong to describe it this way.

In other words, to describe a whole as a symmetry is to lose track of the essence of what a whole really is.

Can you tell me what is meant by "the complex plane"? — Metaphysician Undercover

Complex Plane

The issue is that a whole never really is a symmetry, so that is a misrepresentation — Metaphysician Undercover

I tend to think this too. But I'm tolerant of math and physics.

Reflection symmetry, is a relation between two halves of an object. I don't think there's a need for any third party involvment. Here: A, B, D, E, T, U, W, X, C possess reflection symmetry with no reference/dependence to/on a "background".

A reflection, or mirror image, is not identical to the thing reflected. Notice that when you look into the mirror, the features on the right side of your face are reflected as the features of the left side of your face.

That, I confess, is something I haven't studied carefully enough (lateral inversion it's called). However, reflection symmetry is defined in such a way that lateral inversion is ignored.

There's a difference though but I trust mathematicians - there must've been a very good reason lateral inversion has been swept under the rug. Can you figure out why?

There's a difference though but I trust mathematicians - there must've been a very good reason lateral inversion has been swept under the rug. Can you figure out why? — Agent Smith

Simplicity, I suppose. The object and the image are on opposites sides of the plane. And the reality of the turning required such that they face each other (what it really consists of), is ignored for simplicity sake.

Simplicity, I suppose. The object and the image are on opposites sides of the plane. And the reality of the turning required such that they face each other (what it really consists of), is ignored for simplicity sake — Metaphysician Undercover

What's the other word now? "Immaterial" or "inconsequential"?

Anyway, that they (mathematicians/physicists) have a name for the phenomenon - lateral inversion - implies it hasn't escaped their notice.

We might need to look into the concept of chirality/handedness to settle this matter. Chirality requires two conditions

1. A point of reference which is usually a(n) imaginary line that passes through the object (the line of symmetry)

2. A front (anterior) and a back (posterior). I guess this is another (imaginary) line perpendicular/orthogonal to the line of symmetry.

In short, if there's an object, we have two lines perpendicular to each other passing through this object with their point of intersection somewhere inside that object. This framework then provides us with chirality/handedness.

When I look in the mirror, I see myself looking back at me (the reflection). Based on the above system of lines, my left becomes my image's right and my right becomes the image's left (lateral inversion).

Lateral inversion is crucial to the Chinese yin-yang symmetry as it maps onto yin (-) and yang (+), opposing forces that interact with each other, this eventually leading to some kind of an equilibrium.

That's all I have to say for now.

@Metaphysician Undercover

Lateral inversion seems to be a feature of 1D, 2D and 3D objects. A 0 dimensional object, a point, doesn't (seem to) undergo/experience lateral inversion (a point doesn't have a left & right side).

A lateral inversion (reflection transformation) can be achieved via a combo of translation (slide) and rotation (turn). What should we make of this?

Deleted..

If there's a giant mirror (does the mirror have to be big) that can hold an image of the entire earth, in one detailed high resolution image, the mirror-world would be dominated by southpaws and not right-handed folks. So, lefties, if there be some among us, if you're feeling the blues because there are so few of us, just take a mirror and see the world's chirality reversed in your favor. Cold comfort though since you're right handed to mirror-folks.

n short, if there's an object, we have two lines perpendicular to each other passing through this object with their point of intersection somewhere inside that object. This framework then provides us with chirality/handedness.

When I look in the mirror, I see myself looking back at me (the reflection). Based on the above system of lines, my left becomes my image's right and my right becomes the image's left (lateral inversion). — Agent Smith

These two are very different. The imaginary plane and lines you describe are imaginary and may be positioned arbitrarily. The mirror is a real (though artificial) object with a spatial separation between it an the object whose image is reflected by the mirror. So there is a reason for the so-called lateral inversion which the mirror produces, it's due to the spatial separation between the object and the reflecting plane.. In the case of the arbitrary plane, or arbitrary point, within a supposed object there is no medium between the thing and its reflection (one side of the plane and the other), so the lateral inversion of this object is completely fictitious and not an adequate representation of a mirror reflection. It is lacking a key element, which is the medium between the object and the reflecting plane.

So there is a reason for the so-called lateral inversion which the mirror produces, it's due to the spatial separation between the object and the reflecting plane.. — Metaphysician Undercover

Not all the blame falls on the mirror then, huh? Suppose a line's on the line of symmetry (flush with the mirror's surface), this line, as per you, doesn't undergo lateral inversion then. However, such a line (remember only 2D objects can achieve 0 distance between itself and the mirror's surface/line of reflection) and the line of reflection/the mirror surface would be indistinguishable i.e. we're no longer talking about an object at all but the mirror itself.

What you say makes sense but it's something like a degenerate triangle which isn't actually a triangle but a line.

Not all the blame falls on the mirror then, huh? Suppose a line's on the line of symmetry (flush with the mirror's surface), this line, as per you, doesn't undergo lateral inversion then. However, such a line (remember only 2D objects can achieve 0 distance between itself and the mirror's surface/line of reflection) and the line of reflection/the mirror surface would be indistinguishable i.e. we're no longer talking about an object at all but the mirror itself. — Agent Smith

I don't get your point. But a mirror's surface is not a true 2D plane. Look at it under a microscope, and you'll see this. So I don't see how you can propose to reduce a mirror's surface to a plane in this way. There is no such thing as a "2D object", things don't exist as planes, and such 2D things are imaginary fictions.

I don't get your point. — Metaphysician Undercover

It's simply me trying to parse your claims given what I know. That's all.

I'm gonna steer this discussion in a slightly different direction. The major class of higher animals (aves, pisces, mammalia, amphibia, and reptilia) exhibit bilateral symmetry (reflection symmetry). It's as if there's a(n) (imaginary) mirror going right through what's known as the body's sagittal plane. So, when I look in a mirror, what's really happening:

1. My left/right side being reflected along my sagittal plane and the composite, my whole body (left side + right side) being

2. Reflected on a mirror.

It's all good then, no? One side of my body is actually a reflection of my other side and the mirror proves the point by lateral inversion (flipping my left and right sides) as no one (usually) reports anything amiss in our reflections.

It's all good then, no? One side of my body is actually a reflection of my other side and the mirror proves the point by lateral inversion (flipping my left and right sides) as no one (usually) reports anything amiss in our reflections. — Agent Smith

The point though, is that it's not really true that the right side of your body is a duplication of the left. A look at a representation of a brain will show this to you. However, it may be the case, that to produce a balanced, stable body within a turbulent environment, something similar to symmetry facilitates this. But "similar to symmetry" is not symmetry, and the capacity to act is a fundamental feature of the human body. So I propose that it is the non-symmetrical features which enable this capacity to act, and this is likely more essential to the existence of a living being than the symmetrical features. And once we see the non-symmetrical as more essential than the symmetrical within living beings, we can move to inanimate objects, and see that the non-symmetrical is more essential to such objects as well. From here we can understand that representing objects as symmetries is a mistaken adventure.

:ok:

Crucially, the parts are interchangeable with respect to the whole

Well, try to change the left and right hands in a human body! :grin:
(Just a joke, but still, it defeats the above statement.)

BTW, referring to the title of your topic, "symmetry" is not either a true or a false principle. Because it is a quality or attribute, not a principle! :smile:

I would think that a quality or attribute which is impossible for a thing to have, is a false principle.

I would think that a quality or attribute which is impossible for a thing to have, is a false principle. — Metaphysician Undercover

From Oxford LEXICO:
A principle is "a fundamental truth or proposition that serves as the foundation for a system of belief or behaviour or for a chain of reasoning"
A quality is "a distinctive attribute or characteristic possessed by someone or something."

So, no. A quality cannot be true or false. A principle, on the other hand, can.
They are totally different things.

If I understand correctly, the classical "electromagnetic field" which is a property of electrons, can be represented as two distinct fields, electric field and magnetic field. I understand the electric field (E) to be spatial, representing a spatial relation to the position of the electron. The magnetic field (B) I understand as temporal, representing the changing position of the electron. If I understand you correctly, you are saying that the relationship between the E field and the B field, which ought to represent "electromagnetism", is not strictly invariant, so there is a need to introduce an A field to compensate. I would conclude that the relationship between space and time is not invariant. It is made to appear as invariant through the use of the A field. If you can explain where this interpretation is misunderstanding, or deficient, I'd be grateful. — Metaphysician Undercover

I think it was a female bot that called me a bot. They tend to react quite emotionally. Especially when they don't understand WTF their male fellow bots are talking about.No, seriously... 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! Consequence of the story of science we are told already at young age. Even obliged to learn at our schools... It's a nice story though. My wife had to suffer from my apparent absence. With Christmas even... Well, I'm out of it, luckily. We saw a nice movie ("Don't look up", which somehow reminded me of this Corona era, the look-uppers and don't-look-uppers being the vaxers and non-vaxers; there was a nice quote from a Jack Handey: "My grandfather died in his sleep. While the passengers in his bus screamed in agony". I'm not sure what the connection with the movie was. Which was about an astronomy professor with his assistent who discover a meteorite on collision course with Earth and all ensuing madness; funny and serious at the same time. A symmetry?), and after I have given you this answer, there is the relief of closure.

The story of the A-field. Classically, the A-field is a contravariant four-vector, with the electric potential as time component, and the magnetic vector potential as the space part. Contravariant just means that the component values get bigger/smaller if the base vectors get smaller/bigger (derivatives, the change per length, get smaller if you go to smaller base: 10 per meter becomes 0.1 per centimeter, hence derivatives are covariant).

You can apply gauges to this A-field without changing the E-field (not the electron field!) and B-field. The magnetic field is an electric field also, but it is seen only when charge moves, and its effect is only felt by charge moving in it. It's a "relativistic E-field" in the sense that relative velocity causes the electric field to compress in the direction of motion. Hence its connection with being the space part of the A-field. It's a pseudo-vector and only gives a force when a charge has velocity in the vector field. The exterior product with velocity (and charge) gives the force, which is just an electric force. In an EM wave the are perpendicular, like you envisioned with the two planes. A charge feels the E-field and the magnetic gives an electric (caused moving charges elsewhere) which lies in a plane perpendicularly to it, with a direction dependent on the velocity of the charge.

If two charges move parallel in space, with equal velocity, they experience different fields as when standing still. What is an electric field only in the frame of non-moving charges, becomes a combination of E and B in a frame in which the both move. The time part of the A vector aquires a non-zero spatial part (in a fixed gauge) when the charges move. The E field gets smaller while the B-field increases (the length of the A vector is Lorentz invariant. So the decreased E-field is compensated by the appearing B-field, so the total force is the same in both frames. I think this is the compensation you refer to.

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.

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.

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.

Now you can say the A-field is caused by the symmetry of the Lagrangian under the U(1) gauge. But... You can just as well say that the gauge field comes in the first place, and that it causes the Lagrangian to stay the same. There are interactions (by means of an A-field), and these give a gauge symmetric Lagrangian. The symmetry runs behind the facts, so to speak. Symmetry can indeed only be es
tablished after manipulations, like that of an equilateral triangle. Some parts of it have to be compared with other parts, and there is no pre-existing thing like symmetry to which the parts have to obey. Of course, afte arranging themselves in a certain way, there can be symmetry. Even if you draw a triangle when you see one in your mind, the image arises from three equal parts. Like the A-field induces a symmetry by keeping the Lagrangian the same.

Can a symmetry exist on its own? Well, symmetry means that aspects stay the same. Like the combination of the kinetic and potential energy, or like the distribution of particles on the corners of the triangle. Do these aspects conform themselves to a symmetry? The corners of the triangle can be created in similar circumstances. They have to be compared to know if they are the same, like potential and kinetic energy after a gauge. Are there symmetry principles lying at base of nature? If things stay the same, symmetry follows, but to say symmetry lays at the base? I don't think so, and the present-day urge in physics to symmetrize is dangerous, because it projects sameness on stuff that's not the same. As I already briefly mentioned, I think there is no symmetry on the basic level, after which a breaking of this symmetry gives rise to difference. It is said that the symmetry of the electroweak interaction at high energy (meaning that both forces are the same, stemming from the same gauge, which, by locally varying it gives rise to the EW force like the A-field in the EM case) is a unique force, but carried by four massless particles, like the photon for the A-field. For low energies the Higgs field falls into the rim of the potential energy form, thereby creating a weird vacuum with finite field values (normally, for a vacuum the fields are zero particle fields). I think the desire for symmetry got the upper hand, which made Higgs create his strange field. Well, actually to account for massive gauge bosons, which can be addressed in a more natural, less artificial way. The mechanism was used to artificially unify the weak and EM force (which is a completey different unification from the unification of E and B, which actually are the same (under spacetime Lorenz rotations).

Nice thread! You got me thinking...