Classical field theory

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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 (usually in a continuous manner). For example, on weather forecasts, the wind velocity during a day over a country is described by assigning a vector at each point of space (with moving arrows representing the change in wind velocity during the day). 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.

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[edit] 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.

[edit] 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{\vec{r}}

where \hat{\vec{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{\vec{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.

[edit] Electrostatics

Main article: 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{\vec{r}}.

[edit] Magnetism

Main article: Magnetism

[edit] Hydrodynamics

Main article: Hydrodynamics

[edit] Relativistic field theory

Main article: Covariant classical 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.

[edit] Lagrangian dynamics

Main article: Lagrangian

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.

[edit] Relativistic fields

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

[edit] Electromagnetism

Main articles: Electromagnetic field and 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.

[edit] 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.

[edit] 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}.

After some enlightening algebra, this yields

\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.

[edit] Gravitation

Main articles: Gravitation and General Relativity

Newtonian gravitation being found to be inconsistent with special relativity, a new theory of gravitation called general relativity was formulated by Albert Einstein. This treats gravitation as a geometric phenomena ('curved spacetime') caused by masses and the gravitational field is represented 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. 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 EFE:

G_{ab}\, =0

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

[edit] See also

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