Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used for 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.
Contents
- Definitions
- Scalar fields
- Vector fields tensor fields spinor fields
- Action
- Mathematical formalism
- Examples
- Newtonian gravity
- Einstein gravity
- Electromagnetism in special relativity
- Electromagnetism in general relativity
- Electromagnetism using differential forms
- Dirac Lagrangian
- Quantum electrodynamic Lagrangian
- Quantum chromodynamic Lagrangian
- References
This article uses
The Lagrangian mechanics formalism was generalized further to handle field theory. In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a manifold. The dependent variables (q) are replaced by the value of a field at that point in spacetime φ(x, y, z, t) so that the equations of motion are obtained by means of an action principle, written as:
where the action,
and where s = { sα} denotes the set of n independent variables of the system, indexed by α = 1, 2, 3,..., n. Notice L is used in the case of one independent variable (t) and
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 (x, y, z, t) 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
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 and bosons by tensor fields.
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
The spatial volume integral of the Lagrangian density is the Lagrangian, in 3d
Quantum field theories in particle physics, such as quantum electrodynamics, are usually described in terms of
Notice that, in the presence of gravity or when using general curvilinear coordinates, the Lagrangian density
Mathematical formalism
Suppose we have an n-dimensional manifold, M, and a target manifold, T. Let
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
Consider a functional,
called the action.
In order for the action to be local, we need additional restrictions on the action. If
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
The solution is given by the Euler–Lagrange equations (due to the boundary conditions),
The left hand side is the functional derivative of the action with respect to
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 (field) induced by quantities such as mass or charge density at any point
Newtonian gravity
The density
where G in m3·kg−1·s−2 is the gravitational constant. Variation of the integral with respect to Φ gives:
Integrate by parts and discard the total integral. Then divide out by δΦ to get:
and thus
which yields Gauss's law for gravity.
Einstein gravity
The Lagrange density for general relativity in the presence of matter fields is
The last tensor is the energy momentum tensor and is defined by
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
Varying this with respect to ϕ, we get
which yields Gauss' law.
Varying instead with respect to
which yields Ampère's law.
Using tensor notation, we can write all this more compactly. The term
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
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
This Lagrangian is obtained by simply replacing the Minkowski metric in the above flat Lagrangian with a more general (possibly curved) metric
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
One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by Kaluza-Klein theory.
Electromagnetism using differential forms
Using differential forms, the electromagnetic action S in vacuum on a (pseudo-) Riemannian manifold
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 F = dA 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 (annihilation operator),
Quantum electrodynamic Lagrangian
The Lagrangian density for QED is:
where
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