Is Every Physics Equation a Field?

[Isaac0427]

Hi all!
I'm only 12 but I am very interested in quantum physics. I do not have any resources at school to ask these questions, so I have to do independent research. As I didn't have a good background in physics, I had trouble understanding everything.
So first of all I think I understand the concept of a field, but I was wondering if any physics equation is a field, like e=mc^2. Would that be a classical field?
Also, I know that when a classical field is quantized it is made to work with subatomic particles. Do the variables or the coefficients change when a field is quantized? If not, what makes the field different? A few examples would be nice.
Thanks,
Isaac Smith

 

[Nugatory]

A few examples would be nice.

You'll find some examples of fields in the first paragrapg or so of the thie wikipedia article: http://en.wikipedia.org/wiki/Field_(physics)

 

[Isaac0427]

You'll find some examples of fields in the first paragrapg or so of the thie wikipedia article: http://en.wikipedia.org/wiki/Field_(physics)

I have already looked there. I mean examples of quantization.

 

[robphy]

A scalar field is a scalar (number) at every point in space. Example: temperature map.
A vector field is a vector at every point in space. Example: wind-velocity map.
Physics equations tell us how those fields change with time.

 

[Isaac0427]

A scalar field is a scalar (number) at every point in space. Example: temperature map.
A vector field is a vector at every point in space. Example: wind-velocity map.
Physics equations tell us how those fields change with time.

Ok, so if I am getting this right, an equation is not a field, but they explain how the fields function. So would quantization just be different equations to explain the system on a quantum scale?

 

[lonely_nucleus]

Fields can represent many effects like forces experienced by a charge particle; fields can represent "lines of force". Fields go on forever and usually decrease as the square of the distance. For example gravitation field force decreases by a factor of 4 if you move 2 units away from the origin of the field; F=1/d^2. A field is invisible and a field surrounds its origin.

If you really want to learn quantum science this is what you have to do. First you have to learn some algebra then you can go straight to pre-calculus, after pre-calculus you can learn calculus. Take it one step at a time, get an algebra book and do 1 problem a day until you finish the algebra book then you can get a pre-calculus book and read 1 lesson a day until you finish the book. After that you can get a calculus book and do 1 problem a day. After calculus you can get a small ordinary differential equation's book and do 1 problem a day. If each book has 91 lessons then you should be able to finish all the math you will need in 12 months with plenty of leisure time. The key is to simply be consistant and devote 1 hour a day to this, it will not take more than an hour to do a lesson. If you do this as a young boy then you will have a solid foundation for any science course or book. Science is based on math, this would really help you.

Sorry if I am a too straightforward; I like to see kids actually learning science through books

 

[Nugatory]

I have already looked there. I mean examples of quantization.

You won't get very far with examples of a quantized field if you haven't understood what a classical field is, and the way you've asked your question suggests that you haven't.

However, there is one fairly easy example, the electromagnetic field.

At every point in space, at any moment, there are electrical and magnetic effects. I can observe these effects by measuring the force on a charged test particle with charge ##q## and velocity ##\vec{v}## at that place at that time, and it turns out that the force can always be written as ##\vec{F}=q(\vec{E}(x,y,z,t)+\vec{v}\times\vec{B}(x,y,z,t))## where ##\vec{B}## and ##\vec{E}## are vectors and ##\vec{B}(x,y,z,t)## and ##\vec{E}(x,y,z,t)## are the values of those vectors at time ##t## at the point in space specified by ##(x,y,z)##. (If you're not clear on how I can use three numbers to specify a point in space, google for "Cartesian coordinates" - I didn't meet this concept until I was thirteen).

OK, that makes ##\vec{E}## and ##\vec{B}## fields (vector fields, to be specific) according to the definition in wikipedia. For every point in space, at a given time, there's a vector ##\vec{B}## which we call the "magnetic field" and a vector ##\vec{E}## which we call the "electrical field".

Now, let's ask what it means to say that a field is "quantized". The classical, non-quantized equation for the effects of these fields on a charged particle is, as I said, ##\vec{F}=q(\vec{E}(x,y,z,t)+\vec{v}\times\vec{B}(x,y,z,t))##. This equation says that if ##\vec{E}## or ##\vec{B}## change by a very small amount, the force on the particle will change by a corresponding small amount; arbitrarily small changes in one imply arbitrarily small changes in the other. However, sensitive experiments show that we don't get arbitrarily small changes in either; instead, when the electromagnetic forces act on a particle, they change its energy by tiny steps not continuously. That's a quantized field, one whose effects come in tiny but distinct steps.
(Warning - This last paragraph is hugely oversimplified. To get a sense of just how much it's oversimplified, you might take a look at http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf to see what the real mathematical treatment of field quantization looks like. But you're doing self-study at age twelve so you don't need to worry about this now... A year or so of college math and it won't look so hard).

 

[rootone]

Your interest in that at 12 years old is remarkable.
Look up a guy called Richard Feynman

 

[bhobba]

I have already looked there. I mean examples of quantization.

Quantum Field theory is very advanced - even college graduates that are math and/or physics majors find it rather difficult and challenging.

But before going into that you need to understand what a field is classically.

Basically, classically, fields are usually defined in terms of effects like the force on charged particles. The reason they are considered real was sorted out by some no go theorems a very great mathematical physicist called Wigner came up with - if you don't know about Wigner check him out:
http://en.wikipedia.org/wiki/Eugene_Wigner

What those no go theorems show is in interactions momentum and energy are not conserved. But there is a very powerful theorem called Noethers theorem that says they should be conserved. Something fishy here. The solution is the missing momentum and energy is stored in the fields. If they have energy and momentum then physicists are inclined to think them real.

Thanks
Bill

 

[lonely_nucleus]

Your interest in that at 12 years old is remarkable.
Look up a guy called Richard Feynman

It is only remarkable if he puts the effort necessary to understand that type of science. Many kids are interested in qm but they do not try to study it.

 

[sophiecentaur]

A 'Field' can be regarded as just a bit of Maths, in the same way that the numbers we use when doing calculations about our money or the algebra we use when solving physical problems. It's quite a problem, conceptually, and has been the subject of much argument over the years.
Consider two masses M₁ and M₂, at a distance r apart. We say that the force between them will be M₁M₂G/r². That doesn't involve Fields at all. On the other hand, we say that a mass M₁ experiences a force F in a field g. So the Mass M₂ can be regarded as producing a Field g = M₂G/r². But the same can be said of the other Mass producing it own field. Which one to choose? Do we choose your personal g field and ask what effect it will have on the Earth? You should get the same answer - i.e. our weight..
This all involves vectors, of course, because the forces have a magnitude and direction but I left that out, for simplicity.

 

[Isaac0427]

Thanks guys! This really helps me understand better.

 

[robphy]

Related to sophiecentaur's post about the Maths and the gravitational force and the associated gravitational field...
is the story of the Electric Field (at least the way I tell the story to my students).
Yes, it is introduced as a mathematical tool for Coulomb's Law for the Electrostatic Force.
Later, however, the Electric Field is seen as an entity of its own, without any source charges, when Faraday's Law is introduced.
In fact, Faraday is usually credited with the development of the "field" concept: http://en.wikipedia.org/wiki/Line_of_force

 

[lonely_nucleus]

Too add more to what @robphy said about electric fields here are some of the differences between electric and magnetic fields
Electric Fields unique characteristcs:Exhists if a charge is present, imaginary field lines are emited perpendicular from surface of charge, imaginary field lines go from positve charge to negative charge by convention.
Magnetic Fields unique characteristics:Exhists when there is a charge in motion, has a north pole and a south pole that can never be separated, imaginary field lines go from north pole to south pole, north pole is defined as the pole that field lines come out, south pole defined as pole that receives the field lines(yes that means Earth's northern hemisphere is where south magnetic pole is located)

here is a site that shows a more organized list of the differences http://www.diffen.com/difference/Electric_Field_vs_Magnetic_Field

 

[tkjtkj]

Too add more to what @robphy said about electric fields here are some of the differences between electric and magnetic fields
Electric Fields unique characteristcs:Exhists if a charge is present, imaginary field lines are emited perpendicular from surface of charge, imaginary field lines go from positve charge to negative charge by convention.
Magnetic Fields unique characteristics:Exhists when there is a charge in motion, has a north pole and a south pole that can never be separated, imaginary field lines go from north pole to south pole, north pole is defined as the pole that field lines come out, south pole defined as pole that receives the field lines(yes that means Earth's northern hemisphere is where south magnetic pole is located)

here is a site that shows a more organized list of the differences http://www.diffen.com/difference/Electric_Field_vs_Magnetic_Field

This might interest you (synthetic, yes, but perhaps a pathway to the nature of nature) :
http://phys.org/news/2014-01-physicists-synthetic-magnetic-monopole-years.html

 

[yoron]

How would one define a field relative observer dependencies? A Lorentz contraction for example? One problem for me is whether to see it as a 'global representation', possibly dynamically changing, or just to stop at the observer level, defining it locally. It can be seen as a field both ways naturally but it doesn't seem to matter what definition I have of the 'forces', acting there, scalar or vector? None of them makes it any simpler to me, thinking of it that way.