Classical field theory

Classical field theory

A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics (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. As the day progresses, the directions in which the vectors point change as the directions of the wind change. From the mathematical viewpoint, classical fields are described by sections of fiber bundles (covariant classical field theory). 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.

Descriptions of physical fields were given before the advent of relativity theory and then revised in light of this theory. Consequently, classical field theories are usually categorised as non-relativistic and relativistic.

Contents

Non-relativistic field theories

Some of the simplest physical fields are vector force fields. Historically, the first time fields were taken seriously was with Faraday's lines of force when describing the electric field. The gravitational field was then similarly described.

Newtonian gravitation

A classical field theory describing gravity was Newtonian gravitation, which describes the gravitational force as a mutual interaction between two masses.

In a gravitational field, if a test particle of gravitational mass m experiences a force F, then the gravitational field strength 'g' is defined by "g = F/m", where it is required that the test mass, m, be so small that its presence effectively does not disturb the gravitational field. Newton's law of gravitation says that two masses separated by a distance, r, experience a force

\vec{F}=-\frac{Gm_1m_2}{r^2}\hat{r}

where \hat{r} is a unit vector pointing away from the other object. Using Newton's 2nd law (for constant inertial mass), F=ma leads to a definition of the gravitational field strength due to a mass m as

\vec{g}=-G\frac{m}{r^2}\hat{r}.

The experimental observation that inertial mass and gravitational mass are equal to unprecedented levels of accuracy leads to the identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the equivalence principle, which leads to general relativity.

Electrostatics

A charged test particle, charge q, experiences a force, F, based solely on its charge. We can similarly describe the electric field, E, so that F=qE. Using this and Coulomb's law tells us that, we define the electric field due to a single charged particle as

\vec{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}.

Magnetism

Hydrodynamics

Relativistic field theory

Modern formulations of classical field theories generally require Lorentz covariance as this is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using Lagrangians. This is a function that, when subjected to an action principle, gives rise to the field equations and a conservation law for the theory.

We use units where c=1 throughout.

Lagrangian dynamics

Given a field tensor ϕ, a scalar called the Lagrangian density \mathcal{L}(\phi,\partial\phi,\partial\partial\phi, ...,x) can be constructed from ϕ and its derivatives.

From this density, the functional action can be constructed by integrating over spacetime

\mathcal{S} [\phi] = \int{\mathcal{L} [\phi (x)]\, \mathrm{d}^4x}.

Then by enforcing the action principle, the Euler-Lagrange equations are obtained

\frac{\delta \mathcal{S}}{\delta\phi}=\frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu  \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)=0.

Relativistic fields

Two of the most well-known Lorentz covariant classical field theories are now described.

Electromagnetism

Historically, the first (classical) field theories were those describing the electric and magnetic fields (separately). After numerous experiments, it was found that these two fields were related, or, in fact, two aspects of the same field: the electromagnetic field. Maxwell's theory of electromagnetism describes the interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the electric and magnetic fields. With the advent of special relativity, a better (and more consistent with mechanics) formulation using tensor fields was found. Instead of using two vector fields describing the electric and magnetic fields, a tensor field representing these two fields together is used.

We have the electromagnetic potential, A_a=\left(-\phi, \vec{A} \right), and the electromagnetic four-current j_a=\left(-\rho, \vec{j}\right). The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor

F_{ab} = \partial_a A_b - \partial_b A_a.

The Lagrangian

To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have \mathcal{L} = \frac{-1}{4\mu_0}F^{ab}F_{ab}. We can use gauge field theory to get the interaction term, and this gives us

\mathcal{L} = \frac{-1}{4\mu_0}F^{ab}F_{ab} + j^aA_a.

The Equations

This, coupled with the Euler-Lagrange equations, gives us the desired result, since the E-L equations say that

\partial_b\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_b A_a\right)}\right)=\frac{\partial\mathcal{L}}{\partial A_a}.

It is easy to see that \partial\mathcal{L}/\partial A_a = \mu_0 j^a. The left hand side is trickier. Being careful with factors of Fab, however, the calculation gives \partial\mathcal{L}/\partial(\partial_b A_a) = F^{ab}. Together, then, the equations of motion are:

\partial_b F^{ab}=\mu_0j^a.

This gives us a vector equation, which are Maxwell's equations in vacuum. The other two are obtained from the fact that F is the 4-curl of A:

6F_{[ab,c]} \, = F_{ab,c} + F_{ca,b} + F_{bc,a} = 0.

where the comma indicates a partial derivative.

Gravitation

After Newtonian gravitation was found to be inconsistent with special relativity, Albert Einstein formulated a new theory of gravitation called general relativity. This treats gravitation as a geometric phenomenon ('curved spacetime') caused by masses and represents the gravitational field mathematically by a tensor field called the metric tensor. The Einstein field equations describe how this curvature is produced. The field equations may be derived by using the Einstein-Hilbert action. Varying the Lagrangian

\mathcal{L} = \, R \sqrt{-g},

where R \, =R_{ab}g^{ab} is the Ricci scalar written in terms of the Ricci tensor \, R_{ab} and the metric tensor \, g_{ab}, will yield the vacuum field equations:

G_{ab}\, =0,

where G_{ab} \, =R_{ab}-\frac{R}{2}g_{ab} is the Einstein tensor.

See also

References

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • classical field theory — klasikinė lauko teorija statusas T sritis fizika atitikmenys: angl. classical field theory vok. klassische Feldtheorie, f rus. классическая теория поля, f pranc. théorie classique du champ, f …   Fizikos terminų žodynas

  • Covariant classical field theory — In recent years, there has been renewed interest in covariant classical field theory. Here, classical fields are represented by sections of fiber bundles and their dynamics is phrased in the context of a finite dimensional space of fields.… …   Wikipedia

  • Field theory (physics) — There are two types of field theory in physics:*Classical field theory, the theory and dynamics of classical fields. *Quantum field theory, the theory of quantum mechanical fields.ee also*Field (physics)External links* [http://www dick.chemie.uni …   Wikipedia

  • Quantum field theory — In quantum field theory (QFT) the forces between particles are mediated by other particles. For instance, the electromagnetic force between two electrons is caused by an exchange of photons. But quantum field theory applies to all fundamental… …   Wikipedia

  • Covariant Hamiltonian field theory — Applied to classical field theory, the familiar symplectic Hamiltonian formalism takes the form of instantaneous Hamiltonian formalism on an infinite dimensional phase space, where canonical coordinates are field functions at some instant of time …   Wikipedia

  • Scalar field theory — In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a scalar , in contrast to a vector or tensor field. The quanta of the… …   Wikipedia

  • Unified field theory — In physics, a unified field theory is a type of field theory that allows all of the fundamental forces between elementary particles to be written in terms of a single field. There is no accepted unified field theory yet, and this remains an open… …   Wikipedia

  • Polymer field theory — A polymer field theory within the framework of statistical mechanics is a statistical field theory, describing the statistical behavior of a neutral or charged polymer system within the field theoretic approach.It can be derived by transforming… …   Wikipedia

  • Class field theory — In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields. Most of the central results in this area were proved in the period between 1900 and 1950. The theory takes its name… …   Wikipedia

  • Common integrals in quantum field theory — There are common integrals in quantum field theory that appear repeatedly.[1] These integrals are all variations and generalizations of gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”