# How to define a field? [duplicate]

Possible Duplicate:
What is a field, really?
What are electromagnetic fields made of?

What is a field ? What is magnetic field or other fields made of or what it is, How do u define it (To my knowledge field is thought to exist without a proof and theories are built over it) ?

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Thanks, The answers in the post tell me that a thing could be defined only if its not fundamental but field is a fundamental so it can't de defined –  Abhay Kumar Jul 6 '12 at 12:31
A field isn't a fundamental object from a mathematical viewpoint. You can define riggidly how fields work/look like –  Michael Jul 6 '12 at 12:42
I didn't get you Micheal –  Abhay Kumar Jul 6 '12 at 12:45
Perhaps you mean something different when talking about fields. But in QFT, a field itself is just some function of space-time (see my answer below). The really interesting part is, how these fields transform. Or more precisely, what type of elements are assigned to each point in space. This is how we define the particle associated to that field –  Michael Jul 6 '12 at 12:50

## marked as duplicate by Qmechanic♦, David Z♦Jul 6 '12 at 17:21

A field is simple a function of space(-time) that assigns some value, vector or pretty much anything, to a certain point in space(-time).

Nothing fancy really. A normal function like $f(x) = x^2$ can be viewed as a field that assigns a real value to the space $\mathbb R$.

We call the field by the type of element it assigns. So a Vector field assigns a Vector to each point in space.

You can define your fields over whatever elements you wish. Heck, you could create an apple field, by assigning a certain number of apples to each point in space.

It's pretty much the same as the mathematical answer. The only difference lies in how you interpret these fields.

So a Vector field, that assigns a vector to each point in space can be viewed as a magnetic field.

For example, think of a ball (the earth). Now, think of a vector field on this ball, i.e. some function that assigns a vector to each point. We say this vector field is smooth, if the vectors of two nearby points only differ by some small $\epsilon$-vector. Think of the vectors as hairs on the ball. Smoothness then just means that the hair looks tidy and two neighbouring hairs have almost the same direction.

You can reconstruct the earth magnetic field by choosing an appropriate function. Interestingly, if you do this, we have a mathematical theorem that says, that such a vector field must essentially have (at least) 2 poles (or a pole of multiplicity 2). In our case, the North and South pole. Or in short "You can't comb the hair on a coconut". This theorem is called the Hairy ball theorem.

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Thanks, I get your point. Can you tell me why does a field exist. What makes magnet have magnetic field around it ? –  Abhay Kumar Jul 6 '12 at 12:47
Well, Ferromagnets have the property that they consist of alot of tiny elementary magnets, all aligned in one direction. Each of these elementary magnets carries a Magnetic Moment: en.wikipedia.org/wiki/Magnetic_moment –  Michael Jul 6 '12 at 13:00

I am starting with your argument:

To my knowledge field is thought to exist without a proof and theories are built over it.

If you've problem with this mathematical modelling of physical problem, ask yourself, "Do you know what Energy is?" Man, its all that number to describe how things happen in the world around us.

The field is actually distribution of physical quantity on each point of spacetime (you can say only Space in classical sense, but it'd pose major problems while describing today's complex fields).

But, why field when we can measure physical quantities on any point without it?
Well, it simplifies things with graphical model. With one field equation (or, family of equations), you can tell the physical reality of spacetime in your problem domain. Its can be seen as attribute of spacetime. With it, you can tell anything related to physical quantity at once in entire region of Spacetime.

Quantum Field
Here it gets real value. Unlike classical field, quantum fields aren't continuous. Its quantized with discrete values in spacetime. To explain it, quantum of fields were introduced. They are basically boson particles also known as messenger particles or force carriers. In an electromagnetic field, electromagnetic force is exchanged by photon bosons which are quantum (or, messenger particles; or, force carriers) of electromagnetic field. Similarly, Graviton Bosons are quantum of gravitational field. And, popular Higgs Bosons are quantum of Higgs field. There's always gap between them so that the field isn't continuous. Higher field intensity means higher density of force carriers (Think, why not the model became easier)!

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Quantum fields are not "quantized with discrete values in spacetime". Please don't just make up random stuff when you feel like answering a question. –  user1504 Jul 6 '12 at 13:17
@user1504 Tell me where am I wrong? Can you find a physical quantity continuously over a quantum field? –  Sachin Shekhar Jul 6 '12 at 13:24
@Sachin: Yes, you can. It is highly unusual for a quantum field to take only discrete values. The spectrum of a field operator is continuous in all of the standard examples. –  user1504 Jul 6 '12 at 14:43
@user1504 So.. is that the only amplitude which is quantum of field? –  Sachin Shekhar Jul 6 '12 at 15:03