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.

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 is Newtonian gravitation, which describes the gravitational force as a mutual interaction between two masses.

Any massive body M has a gravitational field g which describes its influence on other massive bodies. The gravitational field of M at a point r in space is found by determining the force F that M exerts on a small test mass m located at r, and then dividing by m:[1]

$\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m}.$

Stipulating that m is much smaller than M ensures that the presence of m has a negligible influence on the behavior of M.

According to Newton's law of universal gravitation, F(r) is given by[1]

$\mathbf{F}(\mathbf{r}) = -\frac{G M m}{r^2}\hat{\mathbf{r}},$

where $\hat{\mathbf{r}}$ is a unit vector lying along the line joining M and m and pointing from m to M. Therefore, the gravitational field of M is[1]

$\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m} = -\frac{G M}{r^2}\hat{\mathbf{r}}.$

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.

Because the gravitational force F is conservative, the gravitational field g can be rewritten in terms of the gradient of a gravitational potential Φ(r):

$\mathbf{g}(\mathbf{r}) = -\nabla \Phi(\mathbf{r}).$

Electromagnetism

Electrostatics

Main article: Electrostatics

A charged test particle with 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 the electric field due to a single charged particle as

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

The electric field is conservative, and hence can be described by a scalar potential, V(r):

$\mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r}).$

Magnetostatics

Main article: Magnetostatics

A steady current I flowing along a path will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by I on a nearby charge q with velocity v is

$\mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}),$

where B(r) is the magnetic field, which is determined from I by the Biot–Savart law:

$\mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\boldsymbol{\ell} \times d\hat{\mathbf{r}}}{r^2}.$

The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a vector potential, A(r):

$\mathbf{B}(\mathbf{r}) = \boldsymbol{\nabla} \times \mathbf{A}(\mathbf{r})$

Electrodynamics

Main article: Electrodynamics

In general, in the presence of both a charge density ρ(r, t) and current density J(r, t), there will be both an electric and a magnetic field, and both will vary in time. They are determined by Maxwell's equations, a set of differential equations which directly relate E and B to ρ and J.[2]

Alternatively, one can describe the system in terms of its scalar and vector potentials V and A. A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J,[note 1] and from there the electric and magnetic fields are determined via the relations[3]

$\mathbf{E} = -\boldsymbol{\nabla} V - \frac{\partial \mathbf{A}}{\partial t}$
$\mathbf{B} = \boldsymbol{\nabla} \times \mathbf{A}.$

Hydrodynamics

Main article: Hydrodynamics

Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation and Newton's laws connect the density, pressure, and velocity fields:

$\dot \bold{u} = \mathbf{F} - {\nabla p \over \rho}$${\displaystyle \ \dot{\rho} + \nabla \cdot (\rho \bold u) = 0 }$

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

Main article: Lagrangian

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

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

$\mathcal{S} = \int{\mathcal{L} \mathrm{d}^4x}.$

Therefore the Lagrangian itself is equal to the integral of the Lagrangian Density over all space.

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

$\frac{\delta \mathcal{S}}{\delta\phi}=\frac{\partial\mathcal{L}}{\partial\phi} -\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)+.~.~.+(-1)^m\partial_{\mu_1} \partial_{\mu_2}.~.~.\partial_{\mu_{m-1}} \partial_{\mu_m} \left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu_1} \partial_{\mu_2}...\partial_{\mu_{m-1}}\partial_{\mu_m} \phi)}\right)=0.$

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 $F^{ab}$, 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

Main articles: Gravitation and General Relativity

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.