# 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 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.

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 (*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 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 with their derivatives and *s* itself

- ,

where the brackets denote ;
and *s* = {*s ^{α}*} 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. .

## Contents

## Definitions[edit]

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[edit]

For one scalar field , the Lagrangian density will take the form:^{[nb 1]}^{[1]}

For many scalar fields

### Vector fields, tensor fields, spinor fields[edit]

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[edit]

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 √*g*, making it a scalar density. This procedure ensures that the action is invariant under general coordinate transformations.

### Mathematical formalism[edit]

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^{[which?]} 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* → ∞ (this will help in doing integration by parts), 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 (due to the boundary conditions):

## Examples[edit]

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 . 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 (**1**), 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[edit]

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 resulting Lagrangian for the classical gravitational field is:

where *G* in m^{3}·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[edit]

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. Plugging this Lagrangian into the Euler-Lagrange equation and taking the metric tensor as the field, we obtain the Einstein field equations

The last tensor is the energy momentum tensor and is defined by

is the determinant of the metric tensor when regarded as a matrix. is the cosmological constant. 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 (the determinant by itself is negative).^{[2]}

### Electromagnetism in special relativity[edit]

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.^{[3]}^{[4]}

### Electromagnetism in general relativity[edit]

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 (possibly curved) 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 (written in natural units and with charge Q):^{[5]}

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[edit]

Using differential forms, the electromagnetic action *S* in vacuum on a (pseudo-) Riemannian manifold can be written (using natural units, *c* = *ε*_{0} = 1) 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 **F** = d**A** immediately yields the equation for the fields,

because **F** is an exact form.

### Dirac Lagrangian[edit]

The Lagrangian density for a Dirac field is:^{[6]}

where *ψ* is a Dirac spinor (annihilation operator), is its Dirac adjoint (creation operator), and is Feynman slash notation for .

### Quantum electrodynamic Lagrangian[edit]

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[edit]

The Lagrangian density for quantum chromodynamics is:^{[7]}^{[8]}^{[9]}

where *D* is the QCD gauge covariant derivative,
*n* = 1, 2, ...6 counts the quark types, and is the gluon field strength tensor.

## See also[edit]

- Calculus of variations
- Covariant classical field theory
- Einstein–Maxwell–Dirac equations
- Euler–Lagrange equation
- Functional derivative
- Functional integral
- Generalized coordinates
- Hamiltonian mechanics
- Hamiltonian field theory
- Kinetic term
- Lagrangian and Eulerian coordinates
- Lagrangian mechanics
- Lagrangian point
- Lagrangian system
- Noether's theorem
- Onsager–Machlup function
- Principle of least action
- Scalar field theory

## Notes[edit]

**^**It is a standard abuse of notation to abbreviate all the derivatives and coordinates in the Lagrangian density as follows:*μ*is an index which takes values 0 (for the time coordinate), and 1, 2, 3 (for the spatial coordinates), so strictly only one derivative or coordinate would be present. In general, all the spatial and time derivatives will appear in the Lagrangian density, for example in Cartesian coordinates, the Lagrangian density has the full form:

## Citations[edit]

**^**Mandl, F.; Shaw, G. (2010). "Lagrangian Field Theory".*Quantum Field Theory*(2nd ed.). Wiley. p. 25–38. ISBN 978-0-471-49684-7.**^**Zee, A. (2013).*Einstein gravity in a nutshell*. Princeton: Princeton University Press. pp. 344–390. ISBN 9780691145587.**^**Zee, A. (2013).*Einstein gravity in a nutshell*. Princeton: Princeton University Press. pp. 244–253. ISBN 9780691145587.**^**Mexico, Kevin Cahill, University of New (2013).*Physical mathematics*(Repr. ed.). Cambridge: Cambridge University Press. ISBN 9781107005211.**^**Zee, A. (2013).*Einstein gravity in a nutshell*. Princeton: Princeton University Press. pp. 381–383, 477–478. ISBN 9780691145587.**^**Itzykson-Zuber, eq. 3-152**^**"Quantum Chromodynamics (QCD)".*www.fuw.edu.pl*. Retrieved 12 April 2018.**^**Hilf, E. R. "Semiclassical QCD-Lagrangian for Nuclear Physics" (PDF).**^**Sluka, Volker (January 10, 2005). "Talk" (PDF). Archived from the original (PDF) on June 26, 2007.