# My E&M textbook claims that fields are a form of matter

I'm studying out of Classical Electrodynamics by Ohanian and in chapter 2 (Electrostatics) he makes the following claim while discussing the electric field:

As we will see later, fields have energy. They therefore are a form of matter, and they can be regarded as the fifth state of matter (solid, liquid, gas, and plasma are the other four states of matter).

I'm a little confused by this, and I can't seem to find any sources that share this view. I'm even more confused because in the preceding paragraph he says:

The electric field is a useful concept because it lets us think of electric effects as being due to local action, or action-by-contact. This means that we can think of the force on a charge as being due to the altered state of space surrounding the the charge, not as due to action-at-a-distance of the other charge.

So I'm just not sure what he's trying to tell me. Is the electric field literally a state of matter, analogous to solid, liquid, and gas? If so, then which matter is taking on that state? Or is it, as claimed in the previous paragraph, a property of space at each point that is altered due to the presence and location of electric charges? Would this be true of scalar fields as well as vector fields? If it's defined for scalar fields, then how does this square with the fact that a potential field is only defined up to an arbitrary constant such that at two points x and y we could for instance have V(x) = φ(x) + 3 and V(y) = φ(y) + 8?

If we assume that the electric potential field is a form of matter then this would seem to suggest that there is a form of matter whose intrinsic physical properties could take on different values at different locations (I would not expect a 5 kg mass to suddenly have a mass of 2 kg if I walk to the other end of the room). If we assume that the electric potential field is a property of space, then this would seem to suggest that space isn't homogeneous since space would be physically different at two points. Or would this be avoided by saying that the potential field is not matter or a property of space, in which case why do we allow the electric field to be?

Or am I just completely overthinking this and this was just a little attempt at philosophy, or there's something he's going to explain later in the book that would make this clear?

EDIT: The title of this post is meant to say "My E&M textbook claims that fields are a form of matter", how do I fix that?

• Delta2

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I do not think it is helpful to think of a field as a "form of matter" in the same sense as liquids or solids (which, in the continuum case, can be described by several fields themselves). By definition, a field is an object that takes a particular type of value at each point in space (and time). In that sense, the electromagnetic field is something that is present everywhere and its value at a particular point describes a state of the field at that point.

Would this be true of scalar fields as well as vector fields?
Yes.

If it's defined for scalar fields, then how does this square with the fact that a potential field is only defined up to an arbitrary constant such that at two points x and y we could for instance have V(x) = φ(x) + 3 and V(y) = φ(y) + 8?
Generally, potentials are not physical, only potential differences are. Essentially, what you are doing here is ##V(\vec x) = \phi(\vec x) + f(\vec x)##, where ##f## is some function. As long as you are not including ##\phi## in ##f##, you will not change the equations of motion for ##\phi## by doing this.

If we assume that the electric potential field is a form of matter then this would seem to suggest that there is a form of matter whose intrinsic physical properties could take on different values at different locations (I would not expect a 5 kg mass to suddenly have a mass of 2 kg if I walk to the other end of the room).
The electric potential is not physical. It is defined only up to gauge transformations (which also involve the vector potential). The physical field is the electromagnetic field itself.

If we assume that the electric potential field is a property of space, then this would seem to suggest that space isn't homogeneous since space would be physically different at two points.
You have to separate the theory being invariant under translations from the field being able to take on different values at different points (which depends on the field sources, etc). If you ignore the dynamics of the field (e.g., a static field) and consider it only as a background to some moving charge, then the field may indeed break translational invariance, which boils down to momentum not being conserved for a particle moving in the field.

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Thanks for the explanation, but I'm still confused. What's throwing me is that the claim that "Electromagnetic fields are a form of matter" seems to be so at odds with what I've already learned in my E&M courses, which was that fields are essentially just convenient mathematical tools that allow us to state problems in electromagnetism as problems in vector analysis and that in reality electric and magnetic forces arise due to quantum mechanical interactions where the charges exchange photons.

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I think you are getting into philosophical territory here. You could say exactly the same thing about point particles. Do they exist or are they just a useful mathematical tool? This is actually irrelevant for the physics, which makes predictions for observations.

However, if you consider photons or electrons ”real” then you really have no choice but to consider the EM field as ”real” as well.

atyy
It is quite uncontroversial. In one sense of the word - when one says "spacetime and matter" in classical general relativity, matter is anything with localizable energy. So the electromagnetic field is matter.

Other definitions of matter are possible. One just has to figure out which one the author is using.

For example, in the context of special relativistic quantum field theory, fermions (electrons) are matter, whereas the gauge bosons (electromagentic field) are force carriers. In this context, the electromagnetic field is not matter.

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EDIT: The title of this post is meant to say "My E&M textbook claims that fields are a form of matter", how do I fix that?
You report your own post, with the reason being that you want the title changed, and one of the mentors will fix it.

CWatters
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Perhaps this is just a poor reference to mass energy equivalence. Charge up a capacitor or inductor or wind up a spring and gets very slightly more massive.

I think the bit about "altered space" rather than "action at a distance" is just trying to say that it doesn't really matter what creates the field around the point of interest. The field close to a point can be created many different ways. You don't have to analyse every possible case just the effect of the field on the particle.

robphy
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Since Ohanian is a relativist (he authored "Gravitation and Spacetime" ISBN 1107012945 https://www.amazon.com/dp/1107012945/?tag=pfamazon01-20 ), I would concur with @atyy 's interpretation. The electromagnetic field and other such "matter fields" will show up in the stress-energy tensor ##T_{ab}## in Einstein's field equation ##G_{ab}=\kappa T_{ab}##.

In my first-edition copy of the electrodynamics text, your quote is from Ch 2.
Looking ahead on the first page of Ch 5, he writes:
Ohanian (Classical Electrodynamics-Ch 5) said:
In the interpretation of the electric potential energy, we are faced
with a question: Is the potential energy located at the point charges or is it
located in the electric fields? This question has no answer in the context of
electrostatics. The outcome of experiments in electrostatics is independent of
where the energy is located. In principle, we could decide this question with
a gravitational experiment—all forms of energy exert gravitational
attractions; and if the electrostatic energy is distributed over the electric field, then
the electric field should exert gravitational attractions. Unfortunately, these
gravitational effects are so small that a direct measurement is impossible. To
decide where the electric energy is located, we must go beyond electrostatics.
We must look at electromagnetic waves, such as light or radio waves. These
waves carry energy from one place to another; thus, there can be no doubt
that the electric fields of these waves contain energy. For the sake of
consistency, we will then suppose that the electric fields of a static distribution of
charges also contain energy. This means that fields are a form of matter. If
we reckon solids, liquids, gases, and plasmas as the first four states of matter,
then fields must be reckoned as the fifth state of matter.

Khashishi
It's just arbitrary classification, but I don't think that electromagnetic radiation should be considered matter. It has energy, but not rest-energy.

Orodruin
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It's just arbitrary classification, but I don't think that electromagnetic radiation should be considered matter. It has energy, but not rest-energy.
An EM field does not need to be a radiation field to carry energy.

Khashishi
An EM field does not need to be a radiation field to carry energy.
That's why I chose the term radiation here. The near field that surrounds charges has energy, but this energy can be considered to be part of the mass energy of the charges itself, and so it can be considered matter. So I would say some part of the EM field is matter and some part is not.

• Ken Morcom
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