# Lagrangian (field theory)

# 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 usesfor the Lagrangian density, and`

*L*for the Lagrangian.`The Lagrangian mechanics formalism was generalized further to handlefield theory. In field theory, the independent variable is replaced by an event inspacetime(`

*x*,*y*,*z*,*t*), or more generally still by a point*s*on amanifold. The dependent variables (*q*) are replaced by the value of a field at that point in spacetimeso that theequations of motionare obtained by means of anactionprinciple, written as:`where the`

*action*,, is afunctionalof the dependent variableswith their derivatives and*s*itself- ,

`where the brackets denote; and`

*s*= {*sα*} denotes thesetof*n*independent variablesof 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 (*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, the Lagrangian density will take the form:`

^{[1]}^{[2]}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 allspacetimeto 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 densitywill include a factor of√`

*g*, making it ascalar density. This procedure ensures that the actionis invariant under general coordinate transformations.Mathematical formalism

`Suppose we have an`

*n*-dimensionalmanifold,*M*, and a target manifold,*T*. Letbe the configuration space ofsmooth functionsfrom*M*to*T*.`In field theory,`

*M*is thespacetimemanifold and the target space is the set of values the fields can take at any given point. For example, if there arereal-valuedscalar fields,, then the target manifold is. If the field is a realvector field, then the target manifold isisomorphicto. Note that there is also an elegant formalism for this, usingtangent bundlesover*M*.Consider a functional,

- ,

called the action.

`In order for the action to belocal, we need additional restrictions on theaction. If, we assumeis theintegralover`

*M*of a function of, itsderivativesand 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.

`Givenboundary conditions, basically a specification of the value ofat theboundaryif`

*M*iscompactor some limit onas*x*→ ∞ (this will help in doingintegration by parts), thesubspaceofconsisting of functions,, such that allfunctional derivativesof*S*atare zero andsatisfies the given boundary conditions is the subspace ofon shellsolutions.From this we get:

`The left hand side is thefunctional derivativeof theactionwith respect to.`

Hence we get the Euler–Lagrange equations (due to the boundary conditions):

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. 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

`The densityhas 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 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

`is thecurvature scalar, which is theRicci tensorcontracted with themetric tensor, and theRicci tensoris theRiemann tensorcontracted with aKronecker delta. The integral ofis known as theEinstein-Hilbert action. The Riemann tensor is thetidal forcetensor, and is constructed out ofChristoffel symbolsand derivatives of Christoffel symbols, which are the gravitational force field. Plugging this Lagrangian into the Euler-Lagrange equation and taking the metric tensoras the field, we obtain theEinstein 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 thecosmological 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 theJacobian determinant. The minus sign is a consequence of the metric signature (the determinant by itself is negative).`

^{[3]}Electromagnetism in special relativity

The interaction terms

`are replaced by terms involving a continuous charge density ρ in A·s·m−3and current densityin 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.

`Usingtensor notation, we can write all this more compactly. The termis actually the inner product of twofour-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 theelectromagnetic 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.^{[4]}^{[5]}

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 (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

`whereis thecovariant 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 theReissner–Nordström charged black hole, with the defining line element (written innatural unitsand with charge Q):`

^{[6]}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

`Usingdifferential forms, the electromagnetic action`

*S*in vacuum on a (pseudo-) Riemannian manifoldcan be written (usingnatural units,*c*=*ε*_{0}= 1) asHere, **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

The Lagrangian density for a Dirac field is:^{[7]}

`where`

*ψ*is aDirac spinor(annihilation operator),is itsDirac adjoint(creation operator), andisFeynman slash notationfor.Quantum electrodynamic Lagrangian

The Lagrangian density for QED is:

`whereis theelectromagnetic tensor,`

*D*is thegauge covariant derivative, andisFeynman notationforwithwhereis theelectromagnetic four-potential.Quantum chromodynamic Lagrangian

The Lagrangian density for quantum chromodynamics is:^{[8]}^{[9]}^{[10]}

`where`

*D*is the QCDgauge covariant derivative,*n*= 1, 2, ...6 counts the quark types, andis thegluon field strength tensor.See also

Calculus of variations

Covariant classical field theory

Einstein–Maxwell–Dirac equations

Functional derivative

Functional integral

Generalized coordinates

Hamiltonian field theory

Kinetic term

Lagrangian and Eulerian coordinates

Lagrangian mechanics

Lagrangian point

Lagrangian system

Onsager–Machlup function

Principle of least action

Scalar field theory

## References

*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: Here we write the same thing, but using ∇ to abbreviate all spatial derivatives as a vector.

*Quantum Field Theory*(2nd ed.). Wiley. p. 25–38. ISBN 978-0-471-49684-7.

*Einstein gravity in a nutshell*. Princeton: Princeton University Press. pp. 344–390. ISBN 9780691145587.

*Einstein gravity in a nutshell*. Princeton: Princeton University Press. pp. 244–253. ISBN 9780691145587.

*Physical mathematics*(Repr. ed.). Cambridge: Cambridge University Press. ISBN 9781107005211.

*Einstein gravity in a nutshell*. Princeton: Princeton University Press. pp. 381–383, 477–478. ISBN 9780691145587.

*www.fuw.edu.pl*. Retrieved 12 April 2018.