What is Classical field theory: Definition and 31 Discussions
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations. The term 'classical field theory' is commonly reserved for describing those physical theories that describe electromagnetism and gravitation, two of the fundamental forces of nature. Theories that incorporate quantum mechanics are called quantum field theories.
A physical field can be thought of as the assignment of a physical quantity at each point of space and time. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space. Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a vector field. As the day progresses, the directions in which the vectors point change as the directions of the wind change.
The first field theories, Newtonian gravitation and Maxwell's equations of electromagnetic fields were developed in classical physics before the advent of relativity theory in 1905, and had to be revised to be consistent with that theory. Consequently, classical field theories are usually categorized as non-relativistic and relativistic. Modern field theories are usually expressed using the mathematics of tensor calculus. A more recent alternative mathematical formalism describes classical fields as sections of mathematical objects called fiber bundles.
In 1839, James MacCullagh presented field equations to describe reflection and refraction in "An essay toward a dynamical theory of crystalline reflection and refraction".
Let us consider an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. The solution of the field ##c## is given by the expression ##f(a,b)##. On taking into account the relations obtained from the solutions for ##a## and ##b##, we find that ##f(a,b)=0##. If the...
If a Lagrangian has the fields ##a##, ##b## and ##c## whose equations of motion are denoted by ##E_a, E_b## and ##E_c## respectively, then if
\begin{align}
E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c
\end{align}
where ##f_1## and ##f_2## are some functions of the fields, if ##E_b## and ##E_c## are...
Hi,
I am trying to learn relativistic classical field theory as a preparation for studying quantum field theory.
I am currently reading chapter 13 i Herbert Goldstein's Classical Mechanics edition 3, but I think that this book is a bit too brief and does not fully derive and explain the...
I am trying to understand how one derives the dilaton monopole interaction. In "Black holes and membranes in higher-dimensional theories with dilaton fields", Gibbons and Maeda mentioned that one could obtain the dilaton monopole interaction as such:
where the action is given by
However, I...
From this post-gradient energy in classical field theory, one identifies the term ##E\equiv\frac{1}{2}\left(\partial_x\phi\right)^2## as the gradient energy which can be interpreted as elastic potential energy.
Can one then say that $$F\equiv -\frac{\partial...
1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##.
The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...
On examining Maxwell's third equation which is about time varying magnetic fields (Faraday's electromagnetic induction) we find that time varying magnetic fields produce loops of electric fields in space irrespective of whether a coil is present or not, if any coil is present then these loops of...
First I found the equations of motion for both fields:
$$\partial_\mu \partial^\mu \psi = -\frac{\partial V(\psi^* \psi)}{\psi^*}$$
The eq. of motion with the other field is simply found by ##\psi \rightarrow \psi^*## and ##\psi^* \rightarrow \psi## due to the symmetry between the two fields...
hi, I'm currently taking a classical field theory class (electromagnetism in the language of tensors and actions and etc) and we have just encountered the gauge symmetry, that is for the 4 vector potential we can add a gradient of some smooth function and get the same physics (if we take Aμ →...
When a classical field is varied so that ##\phi ^{'}=\phi +\delta \phi## the spatial partial derivatives of the field is often written $$\partial _{\mu }\phi ^{'}=\partial _{\mu }(\phi +\delta \phi )=\partial _{\mu }\phi +\partial _{\mu }\delta \phi $$. Often times the next step is to switch...
Couldn't really fit the precise question in the title due to the character limit. I want to know what are some sufficient conditions for a model in classical field theory to possesses infinitely many conserved quantities. The sine-Gordon and KdV equations are examples of such systems. Now...
I'm not sure this is the best place for this question, and apologize if it isn't. I'm studying the classical field solutions on the first few chapter of Rajaraman's Solitons and instantons : an introduction to solitons and instantons in quantum field theory. Well, my question is about one of...
Hi everyone,
I have a question that, when came to me, sounded a bit silly to me as well, but then I realized, I myself maybe don't understand the logic behind this 100%, so why not discussing with you about it.
So my question is the following. Usually we are used to do quantum field theory...
In field theory a typical Lagrangian (density) for a "free (scalar) field" ##\phi(x)## is of the form $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi -\frac{1}{2}m^{2}\phi^{2}$$ where ##m## is a parameter that we identify with the mass of the field ##\phi(x)##.
My question is...
In classical field theory, translational (in space and time) symmetry leads the derivation of the energy-momentum tensor using Noether's theorem.
From this it is possible to derive four conserved charges. The first turns out to be the Hamiltonian, and thus we have energy conservation.
The...
Hey,
I am about to do my bachelor project in physics and I really want to dive into the classical theory of fields, this could be General Relativity (GR) and/or Electrodynamics (ED). I have some books on the subject: Barut, ED and classical theory of fields and particles, and Landau&Lifshitz...
In classical field theory, the field, φ, is usually constructed from a very large number of coupled harmonic oscillators. Let's say our φ consists of just electrons.
What does φ best represent physically, a very large number of electrons or can it represent just a few electrons? Which is the...
Homework Statement
In this problem, you will calculate the perihelion shift of Mercury simply by dimensional analysis.
(a) The interactions in gravity have
##\mathcal{L}=M^{2}_{Pl}\Big(-\frac{1}{2}h_{\mu\nu}\Box...
Homework Statement
A class of interesting theories are invariant under the scaling of all lengths by ##x^{\mu} \rightarrow (x')^{\mu}=\lambda x^{\mu}## and ##\phi(x) \rightarrow \phi'(x) = \lambda^{-D}\phi(\lambda^{-1}x)##.
Here ##D## is called the scaling dimension of the field.
Consider...
Hello, I am taking an introductory class on non relativistic classical field theory and right now we are doing the more mathematical aspect of things right now. The types of differential equations in the function ##f(\vec{r},t)## that are considered in this course are linear in the following...
One page 24 of his book on classical field theory (4th edition), Landau derives the relativistic equation of motion for a uniformly accelarated particle. How does he get the differential equation that leads him to his result?
There's very few problems in Landau's books. I'm the kind of guy that properly learns material by doing tons of problems. Of course I can pull from other textbooks but there's the issue of different notation, extra material within chapters, etc...
Does anyone know of a good resource that can...
Hey everyone,
I posted this a while back in General Physics without any reply, and it looks like this is actually the germane forum (despite the fact that I'm explicitly NOT looking for QFT) -- but I couldn't find the "move thread" option...
Anyway:
I'm looking for some books that...
Hey everyone,
I'm looking for some books that really dig into the topic of classical field theory -- and not necessarily just the fields that were known during the heyday of classical physics (electromagnetic / gravitational), but not necessarily all about Yang-Mills and Dirac fields, either...
Homework Statement
Could someone please explain what is meant by the term:
\partial_{[ \mu}F_{\nu \rho ]}
Homework Equations
I have come across this in the context of Maxwells equations where F^{\mu \nu} is the field strength tensor and apparently:
\partial_{[ \mu}F_{\nu \rho...
Hi,
I have a problem in classical field theory.
I have a Lagrangian density \mathcal{L}=\frac{1}{2}\partial_\lambda \phi \partial^\lambda \phi + \frac{1}{3}\sigma\phi^3 . Upon solving the Euler-Lagrange equation for this density, I get an equation of motion for my scalar field \phi (x), where...
Hello,
I'm looking to get a book on classical field theory at a beginner level, so please don't recommend anything that a first year grad student wouldn't understand!
Anyways I was look into getting Landau and Lifgarbagez's book any other suggestions? I don't really have any idea of which...
Hello folks,
I would like to know more about the standard books in Classical Field Theory which I am not really familiar with.
I would be grateful if you suggest something (be it a book/lecture notes etc...) in line with the 2nd chapter of the following lecture notes...