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 clear 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

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 ${\displaystyle \varphi (x,y,z,t)}$ so that the equations of motion are obtained by means of an action principle, written as:

where the action, ${\displaystyle {\mathcal {S}}}$, is a functional of the dependent variables ${\displaystyle \varphi _{i}(s)}$, their derivatives and s itself

where the brackets denote ${\displaystyle \{\cdot ~\forall \alpha \}}$; and s = {s^α} denotes the

of the system, including the time variable, and is indexed by α = 1, 2, 3, ..., n. The calligraphic typeface, ${\displaystyle {\mathcal {L}}}$, is used to denote the density, and ${\displaystyle \mathrm {d} ^{n}s}$ 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

.)

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 ${\displaystyle \varphi }$, the Lagrangian density will take the form:^{[nb 1][4]}

For many scalar fields

In mathematical formulations, the scalar fields are understood to be

.

Vector fields, tensor fields, spinor fields

The above can be generalized for

are described by tensor fields, which include scalar and vector fields as special cases.

For example, if there are ${\displaystyle m}$ real-valued scalar fields, ${\displaystyle \varphi _{1},\dots ,\varphi _{m}}$, then the field manifold is ${\displaystyle \mathbb {R} ^{m}}$. If the field is a real

to ${\displaystyle \mathbb {R} ^{n}}$.

Action

The

denoted by S. In field theory, a distinction is occasionally made between the Lagrangian L, of which the time integral is the action

$${\displaystyle {\mathcal {S}}=\int L\,\mathrm {d} t\,,}$$

and the Lagrangian density ${\displaystyle {\mathcal {L}}}$, 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

In the presence of gravity or when using general curvilinear coordinates, the Lagrangian density ${\displaystyle {\mathcal {L}}}$ will include a factor of ${\textstyle {\sqrt {g}}}$. This ensures that the action is invariant under general coordinate transformations. In mathematical literature, spacetime is taken to be a Riemannian manifold ${\displaystyle M}$ and the integral then becomes the volume form

Here, the ${\displaystyle \wedge }$ is the

and ${\textstyle {\sqrt {|g|}}}$ is the square root of the determinant ${\displaystyle |g|}$ of the metric tensor ${\displaystyle g}$ on ${\displaystyle M}$. For flat spacetime (e.g., ), the unit volume is one, i.e. ${\textstyle {\sqrt {|g|}}=1}$ 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 ${\textstyle {\sqrt {-g}}}$ 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 ${\displaystyle *(1)}$ where ${\displaystyle *}$ is the . That is, and so

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

is frequently seen. Do not be misled: the volume form is implicitly present in the integral above, even if it is not explicitly written.

Euler–Lagrange equations

The

of the field ${\displaystyle \varphi }$ as a function of time. Taking the variation with respect to ${\displaystyle \varphi }$, one obtains

Solving, with respect to the

:

Examples

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

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 ${\displaystyle {\mathcal {L}}}$ 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 ${\displaystyle {\mathcal {L}}=T-V}$, with the ${\displaystyle T=-(\nabla \Phi )^{2}/8\pi G}$ providing a kinetic term, and the interaction ${\displaystyle V=\rho \Phi }$ 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 δΦ the formula becomes:

which is equivalent to: which yields Gauss's law for gravity.

Scalar field theory

The Lagrangian for a scalar field moving in a potential ${\displaystyle V(\phi )}$ can be written as

It is not at all an accident that the scalar theory resembles the undergraduate textbook Lagrangian ${\displaystyle L=T-V}$ for the kinetic term of a free point particle written as ${\displaystyle T=mv^{2}/2}$. The scalar theory is the field-theory generalization of a particle moving in a potential. When the ${\displaystyle V(\phi )}$ is the .

Sigma model Lagrangian

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:

where the ${\displaystyle \mathrm {d} }$ is the differential. An equivalent expression is with ${\displaystyle g_{ij}}$ the on the manifold of the field; i.e. the fields ${\displaystyle \phi _{i}}$ are just of the manifold. A third common form is with and ${\displaystyle U\in \mathrm {SU} (N)}$, the to the base spacetime.

In general, sigma models exhibit

that has withstood the test of time.

Electromagnetism in special relativity

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

are replaced by terms involving a continuous charge density ρ in A·s·m⁻³ and current density ${\displaystyle \mathbf {j} }$ in A·m⁻². The resulting Lagrangian density for the electromagnetic field is:

Varying this with respect to ϕ, we get

which yields .

Varying instead with respect to ${\displaystyle \mathbf {A} }$, we get

which yields .

Using

, we can write all this more compactly. The term ${\displaystyle -\rho \phi (\mathbf {x} ,t)+\mathbf {j} \cdot \mathbf {A} }$ 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 ${\displaystyle F_{\mu \nu }}$. We define this tensor as The term we are looking out for turns out to be We have made use of the to raise the indices on the EMF tensor. In this notation, Maxwell's equations are where ε is the . 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.^[5][6]

Electromagnetism and the Yang–Mills equations

Using

, the electromagnetic action S in vacuum on a (pseudo-) Riemannian manifold ${\displaystyle {\mathcal {M}}}$ can be written (using natural units, c = ε₀ = 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 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 .

The A field can be understood to be the

.

The

, it is conventionally taken to be ${\displaystyle \mathrm {SU} (3)\times \mathrm {SU} (2)\times \mathrm {U} (1)}$ 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

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

.

Ginzburg–Landau Lagrangian

The Lagrangian density for

$${\displaystyle {\mathcal {L}}(\psi ,A)=\vert F\vert ^{2}+\vert D\psi \vert ^{2}+{\frac {1}{4}}\left(\sigma -\vert \psi \vert ^{2}\right)^{2}}$$

where ${\displaystyle \psi }$ is a with fiber ${\displaystyle \mathbb {C} ^{n}}$. The ${\displaystyle \psi }$ corresponds to the order parameter in a

"Sombrero hat" potential

. The field ${\displaystyle A}$ is the (non-Abelian) gauge field, i.e. the

Yang–Mills field

and ${\displaystyle F}$ is its field-strength. The Euler–Lagrange equations for the Ginzburg–Landau functional are the Yang–Mills equations and where ${\displaystyle {\star }}$ is the Hodge star operator, i.e. the fully antisymmetric tensor. These equations are closely related to the Yang–Mills–Higgs equations. Another closely related Lagrangian is found in Seiberg–Witten theory.

Dirac Lagrangian

The Lagrangian density for a

$${\displaystyle {\mathcal {L}}={\bar {\psi }}(i\hbar c{\partial }\!\!\!/\ -mc^{2})\psi }$$

where ${\displaystyle \psi }$ is a Dirac spinor, ${\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}}$ is its Dirac adjoint, and ${\displaystyle {\partial }\!\!\!/}$ is Feynman slash notation for ${\displaystyle \gamma ^{\sigma }\partial _{\sigma }}$. There is no particular need to focus on Dirac spinors in the classical theory. The , which, roughly speaking, is a way of formulating spinors consistently in a curved spacetime.

Quantum electrodynamic Lagrangian

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:

where ${\displaystyle F^{\mu \nu }}$ is the electromagnetic tensor, D is the gauge covariant derivative, and ${\displaystyle {D}\!\!\!\!/}$ is Feynman notation for ${\displaystyle \gamma ^{\sigma }D_{\sigma }}$ with ${\displaystyle D_{\sigma }=\partial _{\sigma }-ieA_{\sigma }}$ where ${\displaystyle A_{\sigma }}$ is the

Quantum chromodynamic Lagrangian

The Lagrangian density for

$${\displaystyle {\mathcal {L}}_{\mathrm {QCD} }=\sum _{n}{\bar {\psi }}_{n}\left(i\hbar c{D}\!\!\!\!/\ -m_{n}c^{2}\right)\psi _{n}-{1 \over 4}G^{\alpha }{}_{\mu \nu }G_{\alpha }{}^{\mu \nu }}$$

where D is the QCD gauge covariant derivative, n = 1, 2, ...6 counts the quark types, and ${\displaystyle G^{\alpha }{}_{\mu \nu }\!}$ 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

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

where ${\displaystyle \Lambda }$ is the cosmological constant, ${\displaystyle R}$ is the . The integral of ${\displaystyle {\mathcal {L}}_{\text{EH}}}$ is known as the ".)

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

Substituting this Lagrangian into the Euler–Lagrange equation and taking the metric tensor ${\displaystyle g_{\mu \nu }}$ as the field, we obtain the Einstein field equations

${\displaystyle T_{\mu \nu }}$ is the and is defined by where ${\displaystyle g}$ 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 ${\textstyle {\sqrt {-g}}\,d^{4}x}$. This makes the integral coordinate independent, as the root of the metric determinant is equivalent to the , previously discussed, becoming manifest in non-flat 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 ${\displaystyle {\mathcal {L}}_{\text{matter}}}$. 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 ${\displaystyle g_{\mu \nu }(x)}$. 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 ${\displaystyle D_{\mu }}$ is the covariant derivative. For free space, we can set the current tensor equal to zero, ${\displaystyle j^{\mu }=0}$. Solving both Einstein and Maxwell's equations around a spherically symmetric mass distribution in free space leads to the

$${\displaystyle \mathrm {d} s^{2}=\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)\mathrm {d} t^{2}-\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)^{-1}\mathrm {d} r^{2}-r^{2}\mathrm {d} \Omega ^{2}}$$

One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by

Standard model

, dashing these hopes.

Additional examples

• 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

Notes

Citations