Fields and dimensions

[RichIsnang]

Is the electromagnetic (field) a field or a dimension or neither? And what is the difference between a field and a dimension? Does QCD say that photons are like ripples in the electromagnetic dimension/field?

[pmb]

Is the electromagnetic (field) a field or a dimension or neither?

It's a field To be more precise its a vector field. In physics a vector field is a vector function of the position vector. That means that for every point in space represented by the position vector R = (x, y, z) you can associate a vector. A scalar field is a scalar function of the position function, i.e. at every point in space you can associate a number. The electric field is a vector field. In relativity the electromagnetic field is a second rank tensor field which is a tensor function of the 4-position X = (ct, x, y, z)

 

And what is the difference between a field and a dimension? Does QCD say that photons are like ripples in the electromagnetic dimension/field?

A field is described above. A dimension is a number which serves to determine uniquely the configuation of a system. E.g the position vector R = (x, y, z) is three dimensional, i.e. it has three dimensions. Each dimension helps to determine a location in space. Each of the components, x, y, z are each a dimension. The 4-position X = (ct, x, y, z) is 4 dimensional, i.e. it has four dimensions. Each dimension serves to determine a point in spacetime.

 

I don't know QED so you'll have to get the answer on that from someone else.

[studiot]

Addendum to pmb's description.

 

A field need not extend to all space, just some region of space. However there must be a scalar or vector assigned to each and every point in that region.

 

The region need not have the same number of dimensions as the space. So in 3D space the region may be only 2D ie a surface. This is very common. For instance the surface of a shell or sphere is often used.

[timo]

Does QCD say that photons are like ripples in the electromagnetic dimension/field?

You presumably meant "QED", not "QCD". The simplified answer to your question is "yes".

[pmb]

Addendum to pmb's description.

 

A field need not extend to all space, just some region of space. However there must be a scalar or vector assigned to each and every point in that region.

I can't think of a counter example. Its quite reasonable to assign a vector to every point in space, even if the vector is the null vector.

 

The region need not have the same number of dimensions as the space. So in 3D space the region may be only 2D ie a surface. This is very common. For instance the surface of a shell or sphere is often used.

The 2-d space is a subspace of 3-d. All you do is evaluate the field over the surface in question. Each point on the surface is still in 3-d. But its kinf of unnatural to restrict an electric field to a 2-d surface. I can't even think of an example except for things like a gaussian surface. But that surface is embeded in 3-d space.

[juanrga]

Is the electromagnetic (field) a field or a dimension or neither? And what is the difference between a field and a dimension? Does QCD say that photons are like ripples in the electromagnetic dimension/field?

 

The electromagnetic field is a field. A field is a physical system and carries energy and momentum. A dimension of a space or object is the minimum number of coordinates needed to specify any point within it. Electromagnetic field is defined in a four-dimensional 'space' named spacetime.

 

The quantum field theory of photons is QED not QCD and yes QED say that photons are excitations of the electromagnetic field.

[studiot]

Actually, pmb, the distinction is worth making.

 

One property of a field (from the mathematical definition not physics) that is useful (vital even) in physics is that the field is closed under addition.

 

This is important if you wish to postulate a finite yet unbounded 3D universe.

 

Juanrga

 

A dimension of a space or object is the minimum number of coordinates needed to specify any point within it.

 

This is a good concise definition I couldn't quite remember before. Thanks

Edited July 2, 2012 by studiot

[pmb]

Actually, pmb, the distinction is worth making.

 

One property of a field (from the mathematical definition not physics) that is useful (vital even) in physics is that the field is closed under addition.

As an undergraduate one of my majors was mathematics. One of the required courses was abstract algebra. While I'm sure that group theory and all the stuff about fileds and rings is useful somewhere in physics, I myself have never had cause to use it need it in my studies an research.

[studiot]

'Closed under addition' is used all the time.

 

It is (part of) the justification for saying that any two plane vectors add to a third plane vector in the same plane.

 

In 3D it is the justification for saying that any two the vectors on a surface such as a sphere add to form another surface vector on that spherical surface.

As a result the dimension of that surface is two not three, as Juan pointed out, even though it is embedded in 3D space.

This is significant.

Edited July 3, 2012 by studiot

[pmb]

'Closed under addition' is used all the time.

That's also part of linear algebra, which I use quite often.

[studiot]

Yes except that the algebra doesn't have to be linear.

[RichIsnang]

Ok so the electromagnetic field is a field, what is the relation(if any) to the spatial dimension(could they be described as fields?) and what about time? Is the Higgs field the proposed idea about the spatial dimensions with the Higgs boson working as the photon? And is that QFT?

[studiot]

I do believe a Higgs field is something totally different, but I will leave that to the particle physics specialists.

 

Did you catch the implications of Juan's definition of dimension, your last post would suggest not.

[juanrga]

Ok so the electromagnetic field is a field, what is the relation(if any) to the spatial dimension(could they be described as fields?) and what about time?

 

All of this was answered above. The electromagnetic field is defined in a four-dimensional spacetime, with time being one of the dimensions.

 

Did you catch the implications of Juan's definition of dimension, your last post would suggest not.

 

That definition is not from mine. It is a standard mathematicians definition. I do not know who invented it.

Edited July 3, 2012 by juanrga

[studiot]

I do not know who invented it.

 

I believe it was Lagrange

[pmb]

Ok so the electromagnetic field is a field, what is the relation(if any) to the spatial dimension(could they be described as fields?) and what about time?

To evaluate the field one needs to specialize the location in space r = (x, y, z) which to evaluate the field and the time t which it is evaluated. That mean we need to know the position vector of the location and that requires three numbers to uniquely specify it. The field is then expressed as

 

E = E(r, t)

 

This is what it means to speak of three spatial dimensions. This can be expressed as

 

E = E(x, y, z, t)

 

The element [v]X[/b] = (ct, x, y, z) is called a point in spacetime. Its also a position 4-vector. It requires 4 numbers to specify a point in spacetime. In relativity one uses the Faraday Tensor tensor to specify the electromagnetic field. Such a tensor is called a second rank 4-tensor.

 

Is the Higgs field the proposed idea about the spatial dimensions with the Higgs boson working as the photon? And is that QFT?

I don't know about all that Higgs stuff. I do know that QFT refers to Quantum Field Theory.

[timo]

Is the Higgs field the proposed idea about the spatial dimensions

That doesn't make any sense to me. The answer is probably "no". I think you should delete the term "dimension" from your vocabulary and re-learn it with a proper definition (e.g. the above-mentioned "number of independent degrees of freedom").

 

Is the Higgs field [something weird] with the Higgs boson working as the photon?

Higgs bosons are something like "ripples in the Higgs field", so in that sense "yes". I'd like to point out that the translation (Higgs field, Higgs boson)<->(Electromagnetic field, photon) is not completely one-to-one in all aspects, though.

[pmb]

I think you should delete the term "dimension" from your vocabulary and re-learn it with a proper definition (e.g. the above-mentioned "number of independent degrees of freedom").

Well said. Let's start from scratch.

 

We call a set of values x¹, x², ... , x^N a point. The variables x¹, x², ... , x^N are called coordinates. The totality of points corresponding to all values of the coordinate with certain ranges constitute a space of N dimensions.

[RichIsnang]

OK, ill try to be careful when I write this

 

 

we have space-time, which requires 4 co-ordinates to specify any point in it.

 

then on top of that we have the electromagnetic field, which is a vector field, and I guess we have the Higgs field on top of it too?

i say on top of because i don't really know the proper wording, I guess any point in space-time has many fields with certain vectors in it? on it? lol

 

how many fields are there?

 

and space-time and the fields are pretty much two separate things?

 

[juanrga]

OK, ill try to be careful when I write this

 

we have space-time, which requires 4 co-ordinates to specify any point in it.

 

then on top of that we have the electromagnetic field, which is a vector field, and I guess we have the Higgs field on top of it too?

i say on top of because i don't really know the proper wording, I guess any point in space-time has many fields with certain vectors in it? on it? lol

 

The electromagnetic field is not a vector field, but is given by a tensor [math]F^{\mu\nu}[/math].

 

The Higgs field is not on "top of" another field. As explained above fields are defined over spacetime. The Higgs field is another field.

 

A point in spacetime cannot have a field nor many fields. A field is a physical system extended over the whole spacetime. At any given spacetime point the field takes a value. By virtue of the superposition principle several fields can be existing over the same spacetime.

[studiot]

By virtue of the superposition principle several fields can be existing over the same spacetime.

 

Superposition has nothing to do with whether several fields (or other effect) may coexist in the same system.

 

It does however have a great deal to do with the result of their interaction if they do.

[pmb]

..then on top of that we have the electromagnetic field' date=' which is a vector field, ...

[/quote']

The electromagnetic field is a second rank 4-tensor field. The electric field and the magnetic field are both 4-vector fields. They are defined in General Relativity by Wald on page 64

For an observer with 4-velocity v^a, the quantity

 

[math]E_a = F_{ab}v^b[/math]

 

is interpreted as the electric field measured by that observr, while

 

[math]B_a = -\frac{1}{2}\epsilon_{ab}^{cd}F_{cd}v^b[/math]

 

is interpreted as the magnetic field,

...

 

and I guess we have the Higgs field on top of it too?

You got me my friend. I have no idea about all that Higgs stuff. I hope to learn about it someday.

[MigL]

Higgs' field is a scalar field, see PMB's definition of a scalar field in one of his previous posts.

That being said, QFT dictates that there are field 'excitations' we call Higgs bosons.

 

The only connection between fields ( of infinite extent ) and dimensionality is that field intensity ( strength ) drops off with r^(n-1) where n is the number of spatial dimensions. This has some interesting effects on gravity in 10D ( spatial ) M theory.

[RichIsnang]

Ooooooo cool I didnt know that