Lagrangian (field theory)


Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
This article uses for the Lagrangian density, and L for the Lagrangian.
The Lagrangian mechanics formalism was generalized further to handle field theory. In field theory, the independent variable is replaced by an event in spacetime, or more generally still by a point s on a manifold. The dependent variables are replaced by the value of a field at that point in spacetime so that the equations of motion are obtained by means of an action principle, written as:
where the action,, is a functional of the dependent variables, their derivatives and s itself
where the brackets denote ;
and s = denotes the set of n independent variables of the system, including the time variable, and is indexed by α = 1, 2, 3,..., n. Notice that the calligraphic typeface,, is used to denote volume density, where volume is the integral measure of the domain of the field function, i.e..

Definitions

In Lagrangian field theory, the Lagrangian as a function of generalized coordinates is replaced by a Lagrangian density, a function of the fields in the system and their derivatives, and possibly the space and time coordinates themselves. In field theory, the independent variable t is replaced by an event in spacetime or still more generally by a point s on a manifold.
Often, a "Lagrangian density" is simply referred to as a "Lagrangian".

Scalar fields

For one scalar field, the Lagrangian density will take the form:
For many scalar fields

Vector fields, tensor fields, spinor fields

The above can be generalized for vector fields, tensor fields, and spinor fields. In physics, fermions are described by spinor fields. Bosons are described by tensor fields, which include scalar and vector fields as special cases.

Action

The time integral of the Lagrangian is called the action denoted by S. In field theory, a distinction is occasionally made between the Lagrangian L, of which the time integral is the action
and the Lagrangian density, which one integrates over all spacetime to get the action:
The spatial volume integral of the Lagrangian density is the Lagrangian, in 3d
Note, in the presence of gravity or when using general curvilinear coordinates, the Lagrangian density will include a factor of, making it a scalar density. This procedure ensures that the action is invariant under general coordinate transformations.

Mathematical formalism

Suppose we have an n-dimensional manifold, M, and a target manifold, T. Let be the configuration space of smooth functions from M to T.
In field theory, M is the spacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there are real-valued scalar fields,, then the target manifold is. If the field is a real vector field, then the target manifold is isomorphic to. Note that there is also an elegant formalism for this, using tangent bundles over M.
Consider a functional,
called the action.
In order for the action to be local, we need additional restrictions on the action. If, we assume is the integral over M of a function of, its derivatives and the position called the Lagrangian,. In other words,
It is assumed below, in addition, that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives.
Given boundary conditions, basically a specification of the value of at the boundary if M is compact or some limit on as x → ∞, the subspace of consisting of functions,, such that all functional derivatives of S at are zero and satisfies the given boundary conditions is the subspace of on shell solutions.
From this we get:
The left hand side is the functional derivative of the action with respect to.
Hence we get the Euler–Lagrange equations :

Examples

To go with the section on test particles above, here are the equations for the fields in which they move. The equations below pertain to the fields in which the test particles described above move and allow the calculation of those fields. The equations below will not give you the equations of motion of a test particle in the field but will instead give you the potential induced by quantities such as mass or charge density at any point. For example, in the case of Newtonian gravity, the Lagrangian density integrated over spacetime gives you an equation which, if solved, would yield. This, when substituted back in equation, the Lagrangian equation for the test particle in a Newtonian gravitational field, provides the information needed to calculate the acceleration of the particle.

Newtonian gravity

The Lagrangian density for Newtonian gravity is:
where Φ is the gravitational potential, ρ is the mass density, and G in m3·kg−1·s−2 is the gravitational constant. The density has units of J·m−3. The interaction term mΦ is replaced by a term involving a continuous mass density ρ in kg·m−3. This is necessary because using a point source for a field would result in mathematical difficulties. The variation of the integral with respect to Φ is:
After integrating by parts, discarding the total integral, and dividing out by δΦ the formula becomes:
which is equivalent to:
which yields Gauss's law for gravity.

Einstein gravity

The Lagrange density for general relativity in the presence of matter fields is
is the curvature scalar, which is the Ricci tensor contracted with the metric tensor, and the Ricci tensor is the Riemann tensor contracted with a Kronecker delta. The integral of is known as the Einstein-Hilbert action. The Riemann tensor is the tidal force tensor, and is constructed out of Christoffel symbols and derivatives of Christoffel symbols, which are the gravitational force field. is the cosmological constant. Substituting this Lagrangian into the Euler-Lagrange equation and taking the metric tensor as the field, we obtain the Einstein field equations
is the energy momentum tensor and is defined by
is the determinant of the metric tensor when regarded as a matrix. Generally, in general relativity, the integration measure of the action of Lagrange density is. This makes the integral coordinate independent, as the root of the metric determinant is equivalent to the Jacobian determinant. The minus sign is a consequence of the metric signature.

Electromagnetism in special relativity

The interaction terms
are replaced by terms involving a continuous charge density ρ in A·s·m−3 and current density in A·m−2. The resulting Lagrangian for the electromagnetic field is:
Varying this with respect to ϕ, we get
which yields Gauss' law.
Varying instead with respect to, we get
which yields Ampère's law.
Using tensor notation, we can write all this more compactly. The term is actually the inner product of two four-vectors. We package the charge density into the current 4-vector and the potential into the potential 4-vector. These two new vectors are
We can then write the interaction term as
Additionally, we can package the E and B fields into what is known as the electromagnetic tensor.
We define this tensor as
The term we are looking out for turns out to be
We have made use of the Minkowski metric to raise the indices on the EMF tensor. In this notation, Maxwell's equations are
where ε is the Levi-Civita tensor. So the Lagrange density for electromagnetism in special relativity written in terms of Lorentz vectors and tensors is
In this notation it is apparent that classical electromagnetism is a Lorentz-invariant theory. By the equivalence principle, it becomes simple to extend the notion of electromagnetism to curved spacetime.

Electromagnetism in general relativity

The Lagrange density of electromagnetism in general relativity also contains the Einstein-Hilbert action from above. The pure electromagnetic Lagrangian is precisely a matter Lagrangian. The Lagrangian is
This Lagrangian is obtained by simply replacing the Minkowski metric in the above flat Lagrangian with a more general metric. We can generate the Einstein Field Equations in the presence of an EM field using this lagrangian. The energy-momentum tensor is
It can be shown that this energy momentum tensor is traceless, i.e. that
If we take the trace of both sides of the Einstein Field Equations, we obtain
So the tracelessness of the energy momentum tensor implies that the curvature scalar in an electromagnetic field vanishes. The Einstein equations are then
Additionally, Maxwell's equations are
where is the covariant derivative. For free space, we can set the current tensor equal to zero,. Solving both Einstein and Maxwell's equations around a spherically symmetric mass distribution in free space leads to the Reissner–Nordström charged black hole, with the defining line element :
One possible way of unifying the electromagnetic and gravitational Lagrangians is given by Kaluza-Klein theory.

Electromagnetism using differential forms

Using differential forms, the electromagnetic action S in vacuum on a Riemannian manifold can be written as
Here, A stands for the electromagnetic potential 1-form, J is the current 1-form, F is the field strength 2-form and the star denotes the Hodge star operator. This is exactly the same Lagrangian as in the section above, except that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression. Note that with forms, an additional integration measure is not necessary because forms have coordinate differentials built in. Variation of the action leads to
These are Maxwell's equations for the electromagnetic potential. Substituting immediately yields the equation for the fields,
because F is an exact form.

Dirac Lagrangian

The Lagrangian density for a Dirac field is:
where ψ is a Dirac spinor, is its Dirac adjoint, and is Feynman slash notation for.

Quantum electrodynamic Lagrangian

The Lagrangian density for QED is:
where is the electromagnetic tensor, D is the gauge covariant derivative, and is Feynman notation for with where is the electromagnetic four-potential.

Quantum chromodynamic Lagrangian

The Lagrangian density for quantum chromodynamics is:
where D is the QCD gauge covariant derivative,
n = 1, 2,...6 counts the quark types, and is the gluon field strength tensor.

Citations