# 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.

One motivation for the development of the Lagrangian formalism on fields, and more generally, for classical field theory, is to provide a clean mathematical foundation for quantum field theory, which is infamously beset by formal difficulties that make it unacceptable as a mathematical theory. The Lagrangians presented here are identical to their quantum equivalents, but, in treating the fields as classical fields, instead of being quantized, one can provide definitions and obtain solutions with properties compatible with the conventional formal approach to the mathematics of partial differential equations. This enables the formulation of solutions on spaces with well-characterized properties, such as Sobolev spaces. It enables various theorems to be provided, ranging from proofs of existence to the uniform convergence of formal series to the general settings of potential theory. In addition, insight and clarity is obtained by generalizations to Riemannian manifolds and fiber bundles, allowing the geometric structure to be clearly discerned and disentangled from the corresponding equations of motion. A clearer view of the geometric structure has in turn allowed highly abstract theorems from geometry to be used to gain insight, ranging from the Chern–Gauss–Bonnet theorem and the Riemann–Roch theorem to the Atiyah–Singer index theorem and Chern–Simons theory.

## Overview[edit]

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 Riemannian 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:

*action*, , is a functional of the dependent variables , 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*. The calligraphic typeface, , is used to denote the density, and is the volume form of the field function, i.e., the measure of the domain of the field function.

In mathematical formulations, it is common to express the Lagrangian as a function on a fiber bundle, wherein the Euler–Lagrange equations can be interpreted as specifying the geodesics on the fiber bundle. Abraham and Marsden's textbook^{[1]} provided the first comprehensive description of classical mechanics in terms of modern geometrical ideas, i.e., in terms of tangent manifolds, symplectic manifolds and contact geometry. Bleecker's textbook^{[2]} provided a comprehensive presentation of field theories in physics in terms of gauge invariant fiber bundles. Such formulations were known or suspected long before. Jost^{[3]} continues with a geometric presentation, clarifying the relation between Hamiltonian and Lagrangian forms, describing spin manifolds from first principles, etc. Current research focuses on non-rigid affine structures, (sometimes called "quantum structures") wherein one replaces occurrences of vector spaces by tensor algebras. This research is motivated by the breakthrough understanding of quantum groups as affine Lie algebras (Lie groups are, in a sense "rigid", as they are determined by their Lie algebra. When reformulated on a tensor algebra, they become "floppy", having infinite degrees of freedom; see e.g. Virasoro algebra.)

## 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]}^{[4]}

For many scalar fields

In mathematical formulations, the scalar fields are understood to be coordinates on a fiber bundle, and the derivatives of the field are understood to be sections of the jet bundle.

### 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.

For example, if there are real-valued scalar fields, , then the field manifold is . If the field is a real vector field, then the field manifold is isomorphic to .

### 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

**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,

The action is often referred to as the "action functional", in that it is a function of the fields (and their derivatives).

### Volume form[edit]

In the presence of gravity or when using general curvilinear coordinates, the Lagrangian density will include a factor of . This ensures that the action is invariant under general coordinate transformations. In mathematical literature, spacetime is taken to be a Riemannian manifold and the integral then becomes the volume form

Here, the is the wedge product and is the square root of the determinant of the metric tensor on . For flat spacetime (e.g., Minkowski spacetime), the unit volume is one, i.e. and so it is commonly omitted, when discussing field theory in flat spacetime. Likewise, the use of the wedge-product symbols offers no additional insight over the ordinary concept of a volume in multivariate calculus, and so these are likewise dropped. Some older textbooks, e.g., Landau and Lifschitz write for the volume form, since the minus sign is appropriate for metric tensors with signature (+−−−) or (−+++) (since the determinant is negative, in either case). When discussing field theory on general Riemannian manifolds, the volume form is usually written in the abbreviated notation where is the Hodge star. That is,

Not infrequently, the notation above is considered to be entirely superfluous, and

### Euler–Lagrange equations[edit]

The Euler–Lagrange equations describe the geodesic flow of the field as a function of time. Taking the variation with respect to , one obtains

Solving, with respect to the boundary conditions, one obtains the Euler–Lagrange equations:

## Examples[edit]

A large variety of physical systems have been formulated in terms of Lagrangians over fields. Below is a sampling of some of the most common ones found in physics textbooks on field theory.

### Newtonian gravity[edit]

The Lagrangian density for Newtonian gravity is:

*G*in m

^{3}·kg

^{−1}·s

^{−2}is the gravitational constant. The density has units of J·m

^{−3}. Here the interaction term involves 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.

This Lagrangian can be written in the form of , with the providing a kinetic term, and the interaction the potential term. See also Nordström's theory of gravitation for how this could be modified to deal with changes over time. This form is reprised in the next example of a scalar field theory.

The variation of the integral with respect to Φ is:

After integrating by parts, discarding the total integral, and dividing out by *δ*Φ the formula becomes:

### Scalar field theory[edit]

The Lagrangian for a scalar field moving in a potential can be written as

### Sigma model Lagrangian[edit]

The sigma model describes the motion of a scalar point particle constrained to move on a Riemannian manifold, such as a circle or a sphere. It generalizes the case of scalar and vector fields, that is, fields constrained to move on a flat manifold. The Lagrangian is commonly written in one of three equivalent forms:

In general, sigma models exhibit topological soliton solutions. The most famous and well-studied of these is the Skyrmion, which serves as a model of the nucleon that has withstood the test of time.

### Electromagnetism in special relativity[edit]

Consider a point particle, a charged particle, interacting with the electromagnetic field. The interaction terms

^{−3}and current density in A·m

^{−2}. The resulting Lagrangian density for the electromagnetic field is:

Varying this with respect to ϕ, we get

Varying instead with respect to , we get

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

^{[5]}

^{[6]}

### Electromagnetism and the Yang–Mills equations[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

**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

**F**= d

**A**immediately yields the equation for the fields,

**F**is an exact form.

The **A** field can be understood to be the affine connection on a U(1)-fiber bundle. That is, classical electrodynamics, all of its effects and equations, can be *completely* understood in terms of a circle bundle over Minkowski spacetime.

The Yang–Mills equations can be written in exactly the same form as above, by replacing the Lie group U(1) of electromagnetism by an arbitrary Lie group. In the Standard model, it is conventionally taken to be although the general case is of general interest. In all cases, there is no need for any quantization to be performed. Although the Yang–Mills equations are historically rooted in quantum field theory, the above equations are purely classical.^{[2]}^{[3]}

### Chern–Simons functional[edit]

In the same vein as the above, one can consider the action in one dimension less, i.e. in a contact geometry setting. This gives the Chern–Simons functional. It is written as

Chern–Simons theory was deeply explored in physics, as a toy model for a broad range of geometric phenomena that one might expect to find in a grand unified theory.

### Ginzburg–Landau Lagrangian[edit]

The Lagrangian density for Ginzburg–Landau theory combines together the Lagrangian for the scalar field theory with the Lagrangian for the Yang–Mills action. It may be written as:^{[7]}

### Dirac Lagrangian[edit]

The Lagrangian density for a Dirac field is:^{[8]}

^{[3]}and the Dirac spinors appear as a special case. Weyl spinors have the additional advantage that they can be used in a vielbein for the metric on a Riemannian manifold; this enables the concept of a spin structure, which, roughly speaking, is a way of formulating spinors consistently in a curved spacetime.

### Quantum electrodynamic Lagrangian[edit]

The Lagrangian density for QED combines the Lagrangian for the Dirac field together with the Lagrangian for electrodynamics in a gauge-invariant way. It is:

*D*is the gauge covariant derivative, and is Feynman notation for with where is the electromagnetic four-potential. Although the word "quantum" appears in the above, this is a historical artifact. The definition of the Dirac field requires no quantization whatsoever, it can be written as a purely classical field of anti-commuting Weyl spinors constructed from first principles from a Clifford algebra.

^{[3]}The full gauge-invariant classical formulation is given in Bleecker.

^{[2]}

### Quantum chromodynamic Lagrangian[edit]

The Lagrangian density for quantum chromodynamics combines together the Lagrangian for one or more massive Dirac spinors with the Lagrangian for the Yang–Mills action, which describes the dynamics of a gauge field; the combined Lagrangian is gauge invariant. It may be written as:^{[9]}

*D*is the QCD gauge covariant derivative,

*n*= 1, 2, ...6 counts the quark types, and is the gluon field strength tensor. As for the electrodynamics case above, the appearance of the word "quantum" above only acknowledges its historical development. The Lagrangian and its gauge invariance can be formulated and treated in a purely classical fashion.

^{[2]}

^{[3]}

### Einstein gravity[edit]

The Lagrange density for general relativity in the presence of matter fields is

*per se*; moving bodies follow geodesics on the manifold described by the connection. They move in a "straight line".)

The Lagrangian for general relativity can also be written in a form that makes it manifestly similar to the Yang–Mills equations. This is called the Einstein–Yang–Mills action principle. This is done by noting that most of differential geometry works "just fine" on bundles with an affine connection and arbitrary Lie group. Then, plugging in SO(3,1) for that symmetry group, i.e. for the frame fields, one obtains the equations above.^{[2]}^{[3]}

Substituting this Lagrangian into the Euler–Lagrange equation and taking the metric tensor as the field, we obtain the Einstein field equations

^{[5]}This is an example of the volume form, previously discussed, becoming manifest in non-flat spacetime.

### 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

^{[5]}

One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by Kaluza–Klein theory.^{[2]} Effectively, one constructs an affine bundle, just as for the Yang–Mills equations given earlier, and then considers the action separately on the 4-dimensional and the 1-dimensional parts. Such factorizations, such as the fact that the 7-sphere can be written as a product of the 4-sphere and the 3-sphere, or that the 11-sphere is a product of the 4-sphere and the 7-sphere, accounted for much of the early excitement that a theory of everything had been found. Unfortunately, the 7-sphere proved not large enough to enclose all of the Standard model, dashing these hopes.

### Additional examples[edit]

- The BF model Lagrangian, short for "Background Field", describes a system with trivial dynamics, when written on a flat spacetime manifold. On a topologically non-trivial spacetime, the system will have non-trivial classical solutions, which may be interpreted as solitons or instantons. A variety of extensions exist, forming the foundations for topological field theories.

## See also[edit]

- Calculus of variations
- Covariant classical field theory
- 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:see four-gradient. The*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.

## Citations[edit]

**^**Ralph Abraham and Jerrold E. Marsden, (1967) "Foundations of Mechanics"- ^
^{a}^{b}^{c}^{d}^{e}^{f}David Bleecker, (1981) "Gauge Theory and Variational Principles" Addison-Wesley - ^
^{a}^{b}^{c}^{d}^{e}^{f}Jurgen Jost, (1995) "Riemannian Geometry and Geometric Analysis", Springer **^**Mandl, F.; Shaw, G. (2010). "Lagrangian Field Theory".*Quantum Field Theory*(2nd ed.). Wiley. p. 25–38. ISBN 978-0-471-49684-7.- ^
^{a}^{b}^{c}Zee, Anthony (2013).*Einstein gravity in a nutshell*. Princeton: Princeton University Press. pp. 344–390. ISBN 9780691145587. **^**Cahill, Kevin (2013).*Physical mathematics*. Cambridge: Cambridge University Press. ISBN 9781107005211.**^**Jost, Jürgen (2002). "The Ginzburg–Landau Functional".*Riemannian Geometry and Geometric Analysis*(Third ed.). Springer-Verlag. pp. 373–381. ISBN 3-540-42627-2.**^**Itzykson-Zuber, eq. 3-152**^**Claude Itykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory"