Merged Relativity+ / Farsight

Craig B · Jun 19, 2013

Tim Thompson said:
... So I wonder if Mr. Farsight might enlighten me with the answer to a couple of questions.

1) Please define what a field is, as rigorously or non-rigorously as you think necessary.

2) Please explain, in light of the answer to (1) how the electromagnetic field is, in reality, not a field.

Merci Beaucoup.

Maybe it's like the Holy Roman Empire: Neither electric nor magnetic, nor a field. (See Voltaire at http://www.newworldencyclopedia.org/entry/voltaire.)

sol invictus · Jun 19, 2013

Farsight said:
As Simon pointed out, I'm afraid you've misunderstood what I was saying here. I said the electromagnetic field is a field, but that the electric field and the magnetic field aren't. Instead E and B denote the forces that result from electromagnetic field interactions. See section 11.10 of Jackson's Classical Electrodynamics where he says "one should properly speak of the electromagnetic field Fuv rather than E or B separately".

As usual, garbled nonsense. E and B obviously cannot and do not denote forces - they have the wrong units. And what Jackson is saying is that it is simpler to group the components of E and B into a second-rank spacetime tensor than to regard each as an independent vector. That is a tensor field, and its components that correspond to E are a vector field, while the components that correspond to B are a pseudo-vector field.

W.D.Clinger · Jun 19, 2013

fields vs Farsight: a summary

Tim Thompson said:
Well, I confess the language here has me a bit baffled.

Farsight said:
A field is typically a spatial disposition or structure. It isn't something separate from space. It's a "state of space". When that state is uniform and homogeneous, we usually say there's no field present.

As ctamblyn noted, that is "not the definition of a field, but your personal mental image of certain types of field, which is of questionable applicability to nature." ctamblyn went on to observe "You have, however, confirmed my belief that you don't use the same definition of "field" as the physics community, so at least we've cleared something up."

As edd pointed out in that other thread and again in this thread, uniformity and homogeneity do not imply a zero field. This gets back to Farsight's stubbornly false claim that uniformity and homogeneity imply flat spacetime:

http://www.internationalskeptics.com/forums/showthread.php?postid=9286413#post9286413
http://www.internationalskeptics.com/forums/showthread.php?postid=9286434#post9286434
http://www.internationalskeptics.com/forums/showthread.php?postid=9286463#post9286463
http://www.internationalskeptics.com/forums/showthread.php?postid=9287484#post9287484
http://www.internationalskeptics.com/forums/showthread.php?postid=9287576#post9287576
http://www.internationalskeptics.com/forums/showthread.php?postid=9287650#post9287650
http://www.internationalskeptics.com/forums/showthread.php?postid=9287661#post9287661

Farsight said:
I said the electromagnetic field is a field, but that the electric field and the magnetic field aren't. Instead E and B denote the forces that result from electromagnetic field interactions.

In the other thread, Farsight offered an even more forceful statement of the mistake I've highlighted here when he wrote:

Farsight said:
E and B aren't fields! They're forces!

That's wrong on both counts. E and B are fields, and they aren't forces. E is a ~~scalar~~ vector field giving the force per unit charge, and B is a (pseudo)vector field whose relationship to forces is given by the Lorentz force law, which involves charge and velocity as well as the fields E and B.

The rest of Farsight's recent post just repeats mistakes he's been making since the very first page of this thread.

edd · Jun 19, 2013

E is a vector field not a scalar. The electric potential is scalar though, maybe that's the source of the confusion. edit: Now fixed of course

Farsight · Jun 19, 2013

I'd prefer it if our JREF moderator had not interfered with the thread that Tim started, deleting posts and merging it so that Tim's OP is no longer apparent. I'm afraid it comes across as an attempt to bury a sincere physics discussion because it challenges "the experts" via robust supporting references.

Sol: see Jackson section 1.2 where he says "Although the thing that eventually gets measured is a force" and "At the moment the electric field can be defined as the force per unit charge acting at a given point". E does denote force. It is not a field in its own right. The field concerned is the electromagnetic field. The interaction of two electromagnetic fields results in force. Also see the Minkowski quote from Space and Time:

"In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect."

godless dave · Jun 19, 2013

So if E is a force, you can just plug it into F=ma, right?

Farsight · Jun 19, 2013

No, you use the Coulomb force expression, see hyperphysics or Coulomb's Law on wikipedia. Note that the force result from the interaction of two charged particles, each with their electromagnetic field. The force is the result of the interaction of two electromagnetic fields, it is not a field in its own right.

Tim Thompson · Jun 19, 2013

W.D.Clinger said:
This started in another thread …

Not surprised, though when I started my own thread last night I didn't realize this one existed and was already devoted to the same idea.

Tim Thompson said:
(1) Please define what a field is, as rigorously or non-rigorously as you think necessary.

Farsight said:

A field is typically a spatial disposition or structure. It isn't something separate from space. It's a "state of space". When that state is uniform and homogeneous, we usually say there's no field present. …

Click to expand...

Well, that is an answer to the question, and certainly is of the non-rigorous variety. And that I think is the real problem. Surely you realize by now that you are supremely alone in your claim that the electric "field" is actually a "force" and not a "field". I am quite convinced that this claim of yours is factually incorrect, and based on a poor understanding of what constitutes a "field", the poor understanding being vested in a highly non-rigorous statement. "Field" is a concept in physics that is borrowed from mathematics, and I think a rigorous definition of a field in mathematics is required.

Consider this:

"A ring is a set X together with two internal operations (x,y) -> xy and (x,y) -> x+y, called respectively multiplication and addition, such that X is an Abelian group under addition, and that multiplication is associative & distributive with respect to addition … if there exists an element e such that ex = xe = x, for all x in the set X, then the ring is a ring with identity. … A ring with identity is called a field if all of its elements except zero (neutral element of addition) are regular (invertible & non-singular)."
{Slightly edited for form by me; Analysis, Manifolds & Physics; Yvonne Choquet-Bruhat, Cecile DeWitt-Morette & Margaret Dillard-Bleick, North Holland Publishing, revised edition 1982, page 8}

It is not much of an exercise to see that the set of complex numbers is a mathematical field, and therefore so are real numbers. A field in physics is simply a mathematical field with a physically defined quantity assigned to every element of the mathematical field (there are likely more rigorous physical definitions, but this should do for now). The real numbers used to define the coordinates in 3-dimensional space constitute a field, and the electric field strength & direction assigned to every point in that space constitute a valid physical vector field.

Furthermore, the electric field & electric force are easily separable: The electric field vector E is assigned to each and every point in space, but it does not become a force until we have a charge (q), and the force is qE (which is also a vector field by the way; there is no reason I can see that prevents a force from being a field too).

So, where is my mistake? What have I said that is factually incorrect? And if I have made a mistake, how does it affect the argument that the electric "field" is or is not a vaild "field"?

ctamblyn · Jun 19, 2013

Tim Thompson said:
A field in physics is simply a mathematical field with a physically defined quantity assigned to every element of the mathematical field (there are likely more rigorous physical definitions, but this should do for now). The real numbers used to define the coordinates in 3-dimensional space constitute a field, and the electric field strength & direction assigned to every point in that space constitute a valid physical vector field.

I don't think that's quite right. In physics a field is just a (often continuous, but not always) mapping from space (or spacetime, depending on context) to some codomain representing possible values of a physical quantity of interest (e.g. a 3D vector space in the case of the electric field, or the reals in the case of the e/m scalar potential). The mathematical concept of "field" you cited isn't really related to the physical one, except by an unfortunate common choice of name.

For example: in classical electromagnetism you would usually model the electromagnetic field as a mapping from spacetime - a (pseudo)Riemannian manifold in GR or Minkowski spacetime in SR - to the 2-forms in four dimensions, with not a field in the mathematical sense in sight (at least, in no direct way).

Aside: I believe (but am too lazy to check) that the origin of the term in physics is the phrase "field of influence", while for mathematics the term (according to here) was introduced by Eliakim Hastings Moore as an English equivalent of the older German name for the same concept, which was Körper (body, corpus).

W.D.Clinger · Jun 19, 2013

ctamblyn posted while I was composing this. What I'm about to say agrees with ctamblyn.

Tim Thompson said:
"Field" is a concept in physics that is borrowed from mathematics, and I think a rigorous definition of a field in mathematics is required.

True, but mathematicians use the word "field" with two entirely different meanings. You stated the meaning of field in algebra, but that's not the meaning that's relevant here.

Farsight's mistakes have to do with the meaning of "field" in physics and mathematical physics: a function that assigns some mathematical object to each point of a space. (That function is usually assumed to be continuous, and is often assumed to be differentiable; for those concepts to make sense, the mathematical objects in the range of the field must themselves come from a topological space. This gets even more complicated in general relativity, where the domain of the field is a manifold instead of a topological vector space.)

Tim Thompson said:
Furthermore, the electric field & electric force are easily separable: The electric field vector E is assigned to each and every point in space, but it does not become a force until we have a charge (q), and the force is qE (which is also a vector field by the way; there is no reason I can see that prevents a force from being a field too).

So, where is my mistake? What have I said that is factually incorrect? And if I have made a mistake, how does it affect the argument that the electric "field" is or is not a vaild "field"?

I think your only mistakes here were in quoting the wrong definition of field and speculating about the etymology. Farsight will disagree.

Tim Thompson · Jun 19, 2013

Interesting. I would have thought the two were necessarily related. So what would be an example of a "field" in physics that is not a "field" in the mathematical sense that I portrayed?

Perpetual Student · Jun 19, 2013

It is clear that the term field is a mathematical construction describing a physical quantity that has a value (scalar or vector) for each point in space. It's not that hard. So, we can regard electricity as a vector field with the property F = qE. Similarly, B can also be regarded as a vector field, such that F = qvxB. Because of their mutual relationship, these fields have been unified and can be regarded as the electromagnetic field, which I believe is best described as a tensor field. The point is that there is no basis for which to reject a mathematical analysis of E or B as separate vector fields, when such treatment is useful. It appears that simply because they can be unified into a tensor field, Farsight has concluded that their treatment independently as fields is somehow not proper. He seems not to understand that this is a mathematical construction, intended for the purpose of describing and understanding physical phenomena that are quite real as either the vector fields E or B, or as the unified electromagnetic field described in Maxwell's equations.
This reminds me of Mozina and the word "discharge."

W.D.Clinger · Jun 19, 2013

Tim Thompson said:
Interesting. I would have thought the two were necessarily related. So what would be an example of a "field" in physics that is not a "field" in the mathematical sense that I portrayed?

As defined by Jackson's Classical Electrodynamics (and indeed by everyone except Farsight), the electric field E is a function from 3-dimensional space (isomorphic to the vector space R³) to the vector space R³.

E is a field in the sense of physics, but isn't a field in the sense of algebra.

R³ isn't a field in the sense of algebra either, because it isn't even an integral domain: multiplication of two vectors in R³ isn't commutative, and there is no identity element for that multiplication operation.

ETA: Actually, the electric field E must often be treated as a function of both position and time. Jackson starts out by defining it as a function of position only, because he starts with electrostatics instead of proceeding directly to electrodynamics.

ben m · Jun 19, 2013

Tim Thompson said:
Interesting. I would have thought the two were necessarily related. So what would be an example of a "field" in physics that is not a "field" in the mathematical sense that I portrayed?

It's really sort of apples and oranges. A mathematical "field" is the domain over which an algebra operates. It has elements, it has two distinct binary operations on elements (one generalizing "addition" and another generalizing "multiplication"), and there are a list of requirements about identities and inverses to these operations.

For a physics "field", the important thing is taking some physical quantity and writing it as a function of coordinates. If you're looking at those physical quantities and saying, "hey, electric potential is a real number, and the reals are the elements of a (mathematical) field." But the reals are not a (math) field by themselves---they're a field when combined with the addition and multiplication operators. There's nothing about the (physical) field that insists on the existence of these operators in any general sense---for example, I'm not aware of any physics in the world of vector-valued (physical) fields that requires a multiplication-like operator. (Neither the dot product nor the cross product can serve as "multiplication" to make a field whose elements are the vectors.)

And there's nothing in the (math) field that specifies the "spatialness" of (physical) fields. A particle might have, e.g., a kinetic energy that changes over time---"hey, it's a real number that takes on a variety of values! Electric potential is also a real number that takes on a variety of values!"---but that doesn't make it a (physics) field.

Reality Check · Jun 19, 2013

Farsight said:
Racklever: it's worth having a read of Matt Strassler's blog about virtual particles.

Farsight: Perhaps you should read Matt Strassler's blog about virtual particles where he writes that electrons do exchange virtual photons. The problem with that picture is as he states (as anyone who knows physics): it leads to the confusion that actual photons are being exchanged.

This is basically a issue with historical usage. Virtual particles are in fact not particles - Duh! As Matt Strassler says - they are not even "ripples in the field". What they are is a nice and simple concept to describe the electromagnetic field.

Arguing about semantics is just ridiculous. Scientists call the virtual photons that make up the electromagnetic field virtual particles - live with it Farsight!

ben m · Jun 20, 2013

It's amazing that Farsight can spend multiple pages worth of denying that QFT---the theory that, e.g. uses wave equations to describe point like particles with magnetic moments, all of which Farsight thinks is gibberish---makes any sense, and insulting anyone who cites QFT in any way ... and half a breath later try to shoot down a different argument by citing the (excellent) "QFT for dummies" page written by noted QFT expert Matt Strassler.

By the way, Farsight, do you still think that the 1/q^2 dependence of electron-electron scattering is consistent with a whirlpool-like electron?

tusenfem · Jun 20, 2013

Farsight said:
Sol: see Jackson section 1.2 where he says "Although the thing that eventually gets measured is a force" and "At the moment the electric field can be defined as the force per unit charge acting at a given point". E does denote force. It is not a field in its own right. The field concerned is the electromagnetic field. The interaction of two electromagnetic fields results in force. Also see the Minkowski quote from Space and Time:

My red. It seems that Farsight cannot actually read. Jackson does not say that the electric field (he uses field throughout section I.1 in my red version of the book) is a force but:

Jackson said:
Section I.1 page 3
The electric and magnetic fields E and B (I.1 <- Maxwell equations) were originally introduced by means of the force equation (I.3 <- Lorentz force equation). In Coulomb's experiments forces acting between localized distributions of charge were observed. There it is found useful (see Section 1.2) to introduce the electric field E as the force per unit charge. Similarly, in Ampère's experiments the mutual forces of current carrying loops were studied (in Section 5.2). With the identification of NAqv as a current in a conductor or cross-sectional area A with N charge carriers per unit volume moving at velocity v, we see that B in (I.3) is defined in magnitude as the force per unit current. Although E and B thus first appear just as convenient replacements for forces produced by distributions of charge and current, they have other important aspects. First, their introduction decouples conceptually the sources from the test bodies experiencing electromagnetic forces. If the fields E and B from tow source distributions are the same at a given point in space, the force acting on a test charge or current at that point will be the same, regardless of how different the source distributions are. This gives the E and B in (I.3) meaning in their own right, independent of the sources.

So, clearly Jackson does not say that the electric field is a force, it is a force per unit charge, which would have a unit of Newton-per-Coulomb.

So either Jackson is a dimwit who keeps on mistakenly writing E and B field, or Farsight does not understand what he is reading. I know what my choice is, I leave it up to you to make your own.

Farsight · Jun 20, 2013

Tim Thompson said:
Well, that is an answer to the question, and certainly is of the non-rigorous variety. And that I think is the real problem. Surely you realize by now that you are supremely alone in your claim that the electric "field" is actually a "force" and not a "field".

That answer was backed up by an Einstein reference. And whilst I'm "supremely alone" here on JREF, I'm not elsewhere. And I've got Minkowski and Maxwell on my side too.

Tim Thompson said:
I am quite convinced that this claim of yours is factually incorrect, and based on a poor understanding of what constitutes a "field", the poor understanding being vested in a highly non-rigorous statement.

I don't have the poor understanding Tim. You do. Hopefully I can correct that.

Tim Thompson said:
"Field" is a concept in physics that is borrowed from mathematics, and I think a rigorous definition of a field in mathematics is required.

Consider this:

"A ring is a set X together with two internal operations (x,y) -> xy and (x,y) -> x+y, called respectively multiplication and addition, such that X is an Abelian group under addition, and that multiplication is associative & distributive with respect to addition … if there exists an element e such that ex = xe = x, for all x in the set X, then the ring is a ring with identity. … A ring with identity is called a field if all of its elements except zero (neutral element of addition) are regular (invertible & non-singular)."
{Slightly edited for form by me; Analysis, Manifolds & Physics; Yvonne Choquet-Bruhat, Cecile DeWitt-Morette & Margaret Dillard-Bleick, North Holland Publishing, revised edition 1982, page 8}

It is not much of an exercise to see that the set of complex numbers is a mathematical field, and therefore so are real numbers. A field in physics is simply a mathematical field with a physically defined quantity assigned to every element of the mathematical field (there are likely more rigorous physical definitions, but this should do for now). The real numbers used to define the coordinates in 3-dimensional space constitute a field, and the electric field strength & direction assigned to every point in that space constitute a valid physical vector field.

I know all that Tim. As ctamblyn said, you can use this definition to claim that there's a field in your bathtub.

Tim Thompson said:
Furthermore, the electric field & electric force are easily separable: The electric field vector E is assigned to each and every point in space, but it does not become a force until we have a charge (q), and the force is qE (which is also a vector field by the way; there is no reason I can see that prevents a force from being a field too).

So, where is my mistake? What have I said that is factually incorrect? And if I have made a mistake, how does it affect the argument that the electric "field" is or is not a vaild "field"?

Your mistake is that you're elevating an abstract mathematical definition to reality. A charged particle q1 doesn't have an electric field, it has an electromagnetic field. It doesn't matter how you arrange some collection of charged particles, those particles each have their electromagnetic field.

A single charged particle q1 has an electromagnetic field, another single charged particle q2 has an electromagnetic field. This field has a "screw" nature, so we depict the particles like this:

Image credit: Jim Bumgardner, see http://jbum.com/pixmagic/

If those particles are initially stationary, they move apart with a force:

$mimetex.cgi$

This force is only present when the two particles are present, because it's the result of electromagnetic field interaction. Neither of those particles ever had an electric field as opposed to an electromagnetic field. See this physics classroom page which describes the electric field as per your understanding. It even refers to a field in a swimming pool! Page down a little and see image A which shows radial "electric field lines". Those aren't field lines. They're lines of force.

Have a read of this. Note the reference to a guy call Thompson. No relation I hope. ETA: I do believe it's Thomson, not Thompson.

Farsight · Jun 20, 2013

ben m said:
It's amazing that Farsight can spend multiple pages worth of denying that QFT---the theory that, e.g. uses wave equations to describe point like particles with magnetic moments, all of which Farsight thinks is gibberish---makes any sense, and insulting anyone who cites QFT in any way ... and half a breath later try to shoot down a different argument by citing the (excellent) "QFT for dummies" page written by noted QFT expert Matt Strassler.

I've never said I think QFT or wave equations are gibberish, and no way have I been "insulting anyone who cites QFT in any way". Please try to stick to the physics instead of casting aspersions.

ben m said:
By the way, Farsight, do you still think that the 1/q^2 dependence of electron-electron scattering is consistent with a whirlpool-like electron?

Yes. Note though that a Falaco soliton is a better analogy than a whirlpool. It's like half a smoke ring, but in water. Thomson and Tait experimented with smoke rings. And it was they who introduced the phrase "spherical harmonics".

edd · Jun 20, 2013

Farsight said:
A single charged particle q1 has an electromagnetic field, another single charged particle q2 has an electromagnetic field. This field has a "screw" nature, so we depict the particles like this:

I can think of no reasonable way in which that depiction is correct.

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