Lagrangian (field theory)

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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

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

$${\displaystyle \frac{\delta \mathcal{S}}{\delta {\phi }_i}=0,}$$

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

$${\displaystyle \mathcal{S}\left[{\phi }_i\right]=\int \mathcal{L}\left({\phi }_i(s),\left\{\frac{\mathrm{\partial }{\phi }_i(s)}{\mathrm{\partial }{s}^{\alpha }}\right\},\{s^{\alpha }\}\right){\mathrm{d}}^{n}s,}$$

where the brackets denote ${\displaystyle \{\cdot \mathrm{\forall }\alpha \}}$; 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, ${\displaystyle \mathcal{L}}$, 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 fiber bundle, wherein the Euler–Lagrange equations can be interpreted as specifying the geodesics on the fiber bundle. Abraham and Marsden's textbook 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 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 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

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 ${\displaystyle \phi }$, the Lagrangian density will take the form:

$${\displaystyle \mathcal{L}(\phi ,\mathbf{\nabla }\phi ,\mathrm{\partial }\phi /\mathrm{\partial }t,\mathbf{x},t)}$$

For many scalar fields

$${\displaystyle \mathcal{L}({\phi }_1,\mathbf{\nabla }{\phi }_1,\mathrm{\partial }{\phi }_1/\mathrm{\partial }t,\dots ,{\phi }_n,\mathbf{\nabla }{\phi }_n,\mathrm{\partial }{\phi }_n/\mathrm{\partial }t,\dots ,\mathbf{x},t)}$$

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

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 ${\displaystyle m}$ real-valued scalar fields, ${\displaystyle {\phi }_1,\dots ,{\phi }_m}$, then the field manifold is ${\displaystyle {\mathbb{R}}^m}$. If the field is a real vector field, then the field manifold is isomorphic to ${\displaystyle {\mathbb{R}}^n}$.

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

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

and the Lagrangian density ${\displaystyle \mathcal{L}}$, which one integrates over all spacetime to get the action:

$${\displaystyle \mathcal{S}[\phi ]=\int \mathcal{L}(\phi ,\mathbf{\nabla }\phi ,\mathrm{\partial }\phi /\mathrm{\partial }t,\mathbf{x},t){\mathrm{d}}^3\mathbf{x}\mathrm{d}t.}$$

The spatial volume integral of the Lagrangian density is the Lagrangian; in 3D,

$${\displaystyle L=\int \mathcal{L}{\mathrm{d}}^3\mathbf{x}.}$$

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 $\sqrt{g}$. 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

$${\displaystyle \mathcal{S}={\int }_M\sqrt{|g|}d{x}^1\wedge \cdots \wedge d{x}^m\mathcal{L}}$$

Here, the ${\displaystyle \wedge }$ is the wedge product and $\sqrt{|g|}$ is the square root of the determinant ${\displaystyle |g|}$ of the metric tensor ${\displaystyle g}$ on ${\displaystyle M}$. For flat spacetime (e.g., Minkowski spacetime), the unit volume is one, i.e. $\sqrt{|g|}=1$ 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 $\sqrt{-g}$ 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 ${\displaystyle \ast (1)}$ where ${\displaystyle \ast }$ is the Hodge star. That is,

$${\displaystyle \ast (1)=\sqrt{|g|}d{x}^1\wedge \cdots \wedge d{x}^m}$$

and so

$${\displaystyle \mathcal{S}={\int }_M\ast (1)\mathcal{L}}$$

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

$${\displaystyle \mathcal{S}={\int }_M\mathcal{L}}$$

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 Euler–Lagrange equations describe the geodesic flow of the field ${\displaystyle \phi }$ as a function of time. Taking the variation with respect to ${\displaystyle \phi }$, one obtains

$${\displaystyle 0=\frac{\delta \mathcal{S}}{\delta \phi }={\int }_M\ast (1)\left(-{\mathrm{\partial }}_{\mu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\phi )}\right)+\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\phi }\right).}$$

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

$${\displaystyle \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\phi }={\mathrm{\partial }}_{\mu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\phi )}\right).}$$

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:

$${\displaystyle \mathcal{L}(\mathbf{x},t)=-\frac{1}{8\pi G}(\mathrm{\nabla }\mathrm{\Phi }(\mathbf{x},t){)}^2-\rho (\mathbf{x},t)\mathrm{\Phi }(\mathbf{x},t)}$$

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

$${\displaystyle \delta \mathcal{L}(\mathbf{x},t)=-\rho (\mathbf{x},t)\delta \mathrm{\Phi }(\mathbf{x},t)-\frac{2}{8\pi G}(\mathrm{\nabla }\mathrm{\Phi }(\mathbf{x},t))\cdot (\mathrm{\nabla }\delta \mathrm{\Phi }(\mathbf{x},t)).}$$

After integrating by parts, discarding the total integral, and dividing out by δΦ the formula becomes:

$${\displaystyle 0=-\rho (\mathbf{x},t)+\frac{1}{4\pi G}\mathrm{\nabla }\cdot \mathrm{\nabla }\mathrm{\Phi }(\mathbf{x},t)}$$

which is equivalent to:

$${\displaystyle 4\pi G\rho (\mathbf{x},t)={\mathrm{\nabla }}^2\mathrm{\Phi }(\mathbf{x},t)}$$

which yields Gauss's law for gravity.

Scalar field theory

Main article: Scalar field theory

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

$${\displaystyle \mathcal{L}=\frac{1}{2}{\mathrm{\partial }}^{\mu }\varphi {\mathrm{\partial }}_{\mu }\varphi -V(\varphi )=\frac{1}{2}{\mathrm{\partial }}^{\mu }\varphi {\mathrm{\partial }}_{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2-\sum _{n=3}^{\mathrm{\infty }}\frac{1}{n!}{g}_n{\varphi }^n}$$

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 ${\displaystyle T=m{v}^2/2}$. The scalar theory is the field-theory generalization of a particle moving in a potential. When the ${\displaystyle V(\varphi )}$ is the Mexican hat potential, the resulting fields are termed the Higgs fields.

Sigma model Lagrangian

Main article: sigma model

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:

$${\displaystyle \mathcal{L}=\frac{1}{2}\mathrm{d}\varphi \wedge \ast \mathrm{d}\varphi }$$

where the ${\displaystyle \mathrm{d}}$ is the differential. An equivalent expression is

$${\displaystyle \mathcal{L}=\frac{1}{2}\sum _{i=1}^n\sum _{j=1}^n{g}_{ij}(\varphi ){\mathrm{\partial }}^{\mu }{\varphi }_i{\mathrm{\partial }}_{\mu }{\varphi }_j}$$

with ${\displaystyle g_{ij}}$ the Riemannian metric on the manifold of the field; i.e. the fields ${\displaystyle {\varphi }_i}$ are just local coordinates on the coordinate chart of the manifold. A third common form is

$${\displaystyle \mathcal{L}=\frac{1}{2}\mathrm{t}\mathrm{r}\left(L_{\mu }{L}^{\mu }\right)}$$

with

$${\displaystyle L_{\mu }=U^{-1}{\mathrm{\partial }}_{\mu }U}$$

and ${\displaystyle U\in \mathrm{S}\mathrm{U}(N)}$, the Lie group SU(N). This group can be replaced by any Lie group, or, more generally, by a symmetric space. The trace is just the Killing form in hiding; the Killing form provides a quadratic form on the field manifold, the lagrangian is then just the pullback of this form. Alternately, the Lagrangian can also be seen as the pullback of the Maurer–Cartan form to the base spacetime.

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

Main article: Covariant formulation of classical electromagnetism

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

$${\displaystyle -q\varphi (\mathbf{x}(t),t)+q\dot{\mathbf{x}}(t)\cdot \mathbf{A}(\mathbf{x}(t),t)}$$

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:

$${\displaystyle \mathcal{L}(\mathbf{x},t)=-\rho (\mathbf{x},t)\varphi (\mathbf{x},t)+\mathbf{j}(\mathbf{x},t)\cdot \mathbf{A}(\mathbf{x},t)+\frac{{ϵ}_0}{2}{E}^2(\mathbf{x},t)-\frac{1}{2{\mu }_0}{B}^2(\mathbf{x},t).}$$

Varying this with respect to ϕ, we get

$${\displaystyle 0=-\rho (\mathbf{x},t)+{ϵ}_0\mathrm{\nabla }\cdot \mathbf{E}(\mathbf{x},t)}$$

which yields Gauss' law.

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

$${\displaystyle 0=\mathbf{j}(\mathbf{x},t)+{ϵ}_0\dot{\mathbf{E}}(\mathbf{x},t)-\frac{1}{{\mu }_0}\mathrm{\nabla }\times \mathbf{B}(\mathbf{x},t)}$$

which yields Ampère's law.

Using tensor notation, we can write all this more compactly. The term ${\displaystyle -\rho \varphi (\mathbf{x},t)+\mathbf{j}\cdot \mathbf{A}}$ 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

$${\displaystyle j^{\mu }=(\rho ,\mathbf{j})\text{and}{A}_{\mu }=(-\varphi ,\mathbf{A})}$$

We can then write the interaction term as

$${\displaystyle -\rho \varphi +\mathbf{j}\cdot \mathbf{A}=j^{\mu }{A}_{\mu }}$$

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

$${\displaystyle F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }}$$

The term we are looking out for turns out to be

$${\displaystyle \frac{{ϵ}_0}{2}{E}^2-\frac{1}{2{\mu }_0}{B}^2=-\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu }=-\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}_{\rho \sigma }{\eta }^{\mu \rho }{\eta }^{\nu \sigma }}$$

We have made use of the Minkowski metric to raise the indices on the EMF tensor. In this notation, Maxwell's equations are

$${\displaystyle {\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }\text{and}{ϵ}^{\mu \nu \lambda \sigma }{\mathrm{\partial }}_{\nu }{F}_{\lambda \sigma }=0}$$

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

$${\displaystyle \mathcal{L}(x)=j^{\mu }(x)A_{\mu }(x)-\frac{1}{4{\mu }_0}{F}_{\mu \nu }(x)F^{\mu \nu }(x)}$$

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 and the Yang–Mills equations

Using differential forms, the electromagnetic action S in vacuum on a (pseudo-) Riemannian manifold ${\displaystyle \mathcal{M}}$ can be written (using natural units, c = ε₀ = 1) as

$${\displaystyle \mathcal{S}[\mathbf{A}]=-{\int }_{\mathcal{M}}\left(\frac{1}{2}\mathbf{F}\wedge \ast \mathbf{F}-\mathbf{A}\wedge \ast \mathbf{J}\right).}$$

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

$${\displaystyle \mathrm{d}\ast \mathbf{F}=\ast \mathbf{J}.}$$

These are Maxwell's equations for the electromagnetic potential. Substituting F = dA immediately yields the equation for the fields,

$${\displaystyle \mathrm{d}\mathbf{F}=0}$$

because 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 ${\displaystyle \mathrm{S}\mathrm{U}(3)\times \mathrm{S}\mathrm{U}(2)\times \mathrm{U}(1)}$ 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.

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

$${\displaystyle \mathcal{S}[\mathbf{A}]={\int }_{\mathcal{M}}\mathrm{t}\mathrm{r}\left(\mathbf{A}\wedge d\mathbf{A}+\frac{2}{3}\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\right).}$$

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

Main article: Ginzburg–Landau theory

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:

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

where ${\displaystyle \psi }$ is a section of a vector bundle with fiber ${\displaystyle {\mathbb{C}}^n}$. The ${\displaystyle \psi }$ corresponds to the order parameter in a superconductor; equivalently, it corresponds to the Higgs field, after noting that the second term is the famous "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

$${\displaystyle D\star D\psi =\frac{1}{2}\left(\sigma -|\psi {|}^2\right)\psi }$$

and

$${\displaystyle D\star F=-\mathrm{Re}⟨D\psi ,\psi ⟩}$$

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

Main article: Dirac equation

The Lagrangian density for a Dirac field is:

$${\displaystyle \mathcal{L}=\overline{\psi }(i\hslash c\mathrm{\partial }/-m{c}^2)\psi }$$

where ${\displaystyle \psi }$ is a Dirac spinor, ${\displaystyle \overline{\psi }={\psi }^{†}{\gamma }^0}$ is its Dirac adjoint, and ${\displaystyle \mathrm{\partial }/}$ is Feynman slash notation for ${\displaystyle {\gamma }^{\sigma }{\mathrm{\partial }}_{\sigma }}$. There is no particular need to focus on Dirac spinors in the classical theory. The Weyl spinors provide a more general foundation; they can be constructed directly from the Clifford algebra of spacetime; the construction works in any number of dimensions, 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

Main article: Quantum electrodynamics

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:

$${\displaystyle {\mathcal{L}}_{\mathrm{Q}\mathrm{E}\mathrm{D}}=\overline{\psi }(i\hslash cD/-m{c}^2)\psi -\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu }}$$

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 ${\displaystyle D_{\sigma }={\mathrm{\partial }}_{\sigma }-ie{A}_{\sigma }}$ where ${\displaystyle A_{\sigma }}$ 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. The full gauge-invariant classical formulation is given in Bleecker.

Quantum chromodynamic Lagrangian

Main article: quantum chromodynamics

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:

$${\displaystyle {\mathcal{L}}_{\mathrm{Q}\mathrm{C}\mathrm{D}}=\sum _n{\overline{\psi }}_n\left(i\hslash cD/-m_n{c}^2\right){\psi }_n-\frac{1}{4}{G}^{\alpha }{}_{\mu \nu }{G}_{\alpha }{}^{\mu \nu }}$$

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.

Einstein gravity

Further information: Einstein–Hilbert action

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

$${\displaystyle {\mathcal{L}}_{\text{GR}}={\mathcal{L}}_{\text{EH}}+{\mathcal{L}}_{\text{matter}}=\frac{c^4}{16\pi G}\left(R-2\mathrm{\Lambda }\right)+{\mathcal{L}}_{\text{matter}}}$$

where ${\displaystyle \mathrm{\Lambda }}$ is the cosmological constant, ${\displaystyle R}$ 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 ${\displaystyle {\mathcal{L}}_{\text{EH}}}$ 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 define the metric connection on spacetime. The gravitational field itself was historically ascribed to the metric tensor; the modern view is that the connection is "more fundamental". This is due to the understanding that one can write connections with non-zero torsion. These alter the metric without altering the geometry one bit. As to the actual "direction in which gravity points" (e.g. on the surface of the Earth, it points down), this comes from the Riemann tensor: it is the thing that describes the "gravitational force field" that moving bodies feel and react to. (This last statement must be qualified: there is no "force field" 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.

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

$${\displaystyle R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+g_{\mu \nu }\mathrm{\Lambda }=\frac{8\pi G}{c^4}{T}_{\mu \nu }.}$$

${\displaystyle T_{\mu \nu }}$ is the energy momentum tensor and is defined by

$${\displaystyle T_{\mu \nu }\equiv \frac{-2}{\sqrt{-g}}\frac{\delta ({\mathcal{L}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}\sqrt{-g})}{\delta g^{\mu \nu }}=-2\frac{\delta {\mathcal{L}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}}{\delta g^{\mu \nu }}+g_{\mu \nu }{\mathcal{L}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}.}$$

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 $\sqrt{-g}{d}^{4}x$. 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). This is an example of the volume form, previously discussed, becoming manifest in non-flat spacetime.

Electromagnetism in general relativity

Main article: Maxwell's equations in curved spacetime

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

$${\displaystyle \begin{array}{rl}\mathcal{L}(x)& =j^{\mu }(x)A_{\mu }(x)-\frac{1}{4{\mu }_0}{F}_{\mu \nu }(x)F_{\rho \sigma }(x)g^{\mu \rho }(x)g^{\nu \sigma }(x)+\frac{c^4}{16\pi G}R(x)\\ & ={\mathcal{L}}_{\text{Maxwell}}+{\mathcal{L}}_{\text{Einstein–Hilbert}}.\end{array}}$$

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

$${\displaystyle T^{\mu \nu }(x)=\frac{2}{\sqrt{-g(x)}}\frac{\delta }{\delta g_{\mu \nu }(x)}{\mathcal{S}}_{\text{Maxwell}}=\frac{1}{{\mu }_0}\left(F_{\lambda }^{\mu }(x)F^{\nu \lambda }(x)-\frac{1}{4}{g}^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right)}$$

It can be shown that this energy momentum tensor is traceless, i.e. that

$${\displaystyle T=g_{\mu \nu }{T}^{\mu \nu }=0}$$

If we take the trace of both sides of the Einstein Field Equations, we obtain

$${\displaystyle R=-\frac{8\pi G}{c^4}T}$$

So the tracelessness of the energy momentum tensor implies that the curvature scalar in an electromagnetic field vanishes. The Einstein equations are then

$${\displaystyle R^{\mu \nu }=\frac{8\pi G}{c^4}\frac{1}{{\mu }_0}\left({F^{\mu }}_{\lambda }(x)F^{\nu \lambda }(x)-\frac{1}{4}{g}^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right)}$$

Additionally, Maxwell's equations are

$${\displaystyle D_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }}$$

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 Reissner–Nordström charged black hole, with the defining line element (written in natural units and with charge Q):

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

One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by Kaluza–Klein theory. 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

• 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.

Rainforest

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Rainforest

A thick rainforest in Chiapas, Mexico

 

Olympic rainforest located at Olympic Peninsula, Washington state

 

Rainforest at Kinabalu Park, Borneo

 

A paranomic view of the Tropical rainforest, Nilgiri mountains, India

 

Canopy of Khao Sok tropical rainforest

 

Primitive tropical rainforest in Palawan

Rainforests in Pacific Rim National Park Reserve

Rainforests are forests characterized by a closed and continuous tree canopy, moisture-dependent vegetation, the presence of epiphytes and lianas and the absence of wildfire. Rainforests can be generally classified as tropical rainforests or temperate rainforests, but other types have been described.

Estimates vary from 40% to 75% of all biotic species being indigenous to the rainforests. There may be many millions of species of plants, insects and microorganisms still undiscovered in tropical rainforests. Tropical rainforests have been called the "jewels of the Earth" and the "world's largest pharmacy", because over one quarter of natural medicines have been discovered there.

Rainforests as well as endemic rainforest species are rapidly disappearing due to deforestation, the resulting habitat loss and pollution of the atmosphere.

Definition

Rainforests are characterized by a closed and continuous tree canopy, high humidity, the presence of moisture-dependent vegetation, a moist layer of leaf litter, the presence of epiphytes and lianas and the absence of wildfire. The largest areas of rainforest are tropical or temperate rainforests, but other vegetation associations including subtropical rainforest, littoral rainforest, cloud forest, vine thicket and even dry rainforest have been described.

Tropical rainforest

Worldwide tropical rainforest climate zones.

Main article: Tropical rainforest

Tropical rainforests are characterized by a warm and wet climate with no substantial dry season: typically found within 10 degrees north and south of the equator. Mean monthly temperatures exceed 18 °C (64 °F) during all months of the year. Average annual rainfall is no less than 168 cm (66 in) and can exceed 1,000 cm (390 in) although it typically lies between 175 cm (69 in) and 200 cm (79 in).

Many of the world's tropical forests are associated with the location of the monsoon trough, also known as the intertropical convergence zone. The broader category of tropical moist forests are located in the equatorial zone between the Tropic of Cancer and Tropic of Capricorn. Tropical rainforests exist in Southeast Asia (from Myanmar (Burma)) to the Philippines, Malaysia, Indonesia, Papua New Guinea and Sri Lanka; also in Sub-Saharan Africa from the Cameroon to the Congo (Congo Rainforest), South America (e.g. the Amazon rainforest), Central America (e.g. Bosawás, the southern Yucatán Peninsula-El Peten-Belize-Calakmul), Australia, and on Pacific Islands (such as Hawaiʻi). Tropical forests have been called the "Earth's lungs", although it is now known that rainforests contribute little net oxygen addition to the atmosphere through photosynthesis.

Temperate rainforest

General distribution of temperate rainforests

Temperate rainforest in Pacific Rim National Park Reserve in Canada

Main article: Temperate rainforest

Tropical forests cover a large part of the globe, but temperate rainforests only occur in a few regions around the world. Temperate rainforests are rainforests in temperate regions. They occur in North America (in the Pacific Northwest in Alaska, British Columbia, Washington, Oregon and California), in Europe (parts of the British Isles such as the coastal areas of Ireland and Scotland, southern Norway, parts of the western Balkans along the Adriatic coast, as well as in Galicia and coastal areas of the eastern Black Sea, including Georgia and coastal Turkey), in East Asia (in southern China, Highlands of Taiwan, much of Japan and Korea, and on Sakhalin Island and the adjacent Russian Far East coast), in South America (southern Chile) and also in Australia and New Zealand.

Dry rainforest

Dry rainforests have a more open canopy layer than other rainforests, and are found in areas of lower rainfall (630–1,100 mm (25–43 in)). They generally have two layers of trees.

Layers

Main article: Stratification (vegetation)

A tropical rainforest typically has a number of layers, each with different plants and animals adapted for life in that particular area. Examples include the emergent, canopy, understory and forest floor layers.

Emergent layer

The emergent layer contains a small number of very large trees called emergents, which grow above the general canopy, reaching heights of 45–55 m, although on occasion a few species will grow to 70–80 m tall. They need to be able to withstand the hot temperatures and strong winds that occur above the canopy in some areas. Eagles, butterflies, bats and certain monkeys inhabit this layer.

Canopy layer

Main article: Canopy (biology)

The canopy at the Forest Research Institute Malaysia showing crown shyness

The canopy layer contains the majority of the largest trees, typically 30 metres (98 ft) to 45 metres (148 ft) tall. The densest areas of biodiversity are found in the forest canopy, a more or less continuous cover of foliage formed by adjacent treetops. The canopy, by some estimates, is home to 50 percent of all plant species. Epiphytic plants attach to trunks and branches, and obtain water and minerals from rain and debris that collects on the supporting plants. The fauna is similar to that found in the emergent layer but more diverse. A quarter of all insect species are believed to exist in the rainforest canopy. Scientists have long suspected the richness of the canopy as a habitat, but have only recently developed practical methods of exploring it. As long ago as 1917, naturalist William Beebe declared that "another continent of life remains to be discovered, not upon the Earth, but one to two hundred feet above it, extending over thousands of square miles." A true exploration of this habitat only began in the 1980s, when scientists developed methods to reach the canopy, such as firing ropes into the trees using crossbows. Exploration of the canopy is still in its infancy, but other methods include the use of balloons and airships to float above the highest branches and the building of cranes and walkways planted on the forest floor. The science of accessing tropical forest canopy using airships or similar aerial platforms is called dendronautics.

Understory layer

Main article: Understory

The understory or understorey layer lies between the canopy and the forest floor. It is home to a number of birds, snakes and lizards, as well as predators such as jaguars, boa constrictors and leopards. The leaves are much larger at this level and insect life is abundant. Many seedlings that will grow to the canopy level are present in the understory. Only about 5% of the sunlight shining on the rainforest canopy reaches the understory. This layer can be called a shrub layer, although the shrub layer may also be considered a separate layer.

Forest floor

Main article: Forest floor

Rainforest in the Blue Mountains, Australia

The forest floor, the bottom-most layer, receives only 2% of the sunlight. Only plants adapted to low light can grow in this region. Away from riverbanks, swamps and clearings, where dense undergrowth is found, the forest floor is relatively clear of vegetation because of the low sunlight penetration. It also contains decaying plant and animal matter, which disappears quickly, because the warm, humid conditions promote rapid decay. Many forms of fungi growing here help decay the animal and plant waste.

Flora and fauna

More than half of the world's species of plants and animals are found in rainforests. Rainforests support a very broad array of fauna, including mammals, reptiles, amphibians, birds and invertebrates. Mammals may include primates, felids and other families. Reptiles include snakes, turtles, chameleons and other families; while birds include such families as vangidae and Cuculidae. Dozens of families of invertebrates are found in rainforests. Fungi are also very common in rainforest areas as they can feed on the decomposing remains of plants and animals.

The great diversity in rainforest species is in large part the result of diverse and numerous physical refuges, i.e. places in which plants are inaccessible to many herbivores, or in which animals can hide from predators. Having numerous refuges available also results in much higher total biomass than would otherwise be possible.

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  A Kermode bear from the Great Bear Rainforest, Canada

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  A Bengal tiger in Mudumalai National Park, India

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  A jaguar in the Amazon Rainforest, South America

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  Western lowland gorilla in the African rainforest

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  Yellow anacondas reside in the Amazon basin

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  Lion-tailed macaque in Silent Valley National Park, India

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  A Macaw in the Amazon rainforest

Some species of fauna show a trend towards declining populations in rainforests, for example, reptiles that feed on amphibians and reptiles. This trend requires close monitoring. The seasonality of rainforests affects the reproductive patterns of amphibians, and this in turn can directly affect the species of reptiles that feed on these groups, particularly species with specialized feeding, since these are less likely to use alternative resources.

Soils

Despite the growth of vegetation in a tropical rainforest, soil quality is often quite poor. Rapid bacterial decay prevents the accumulation of humus. The concentration of iron and aluminium oxides by the laterization process gives the oxisols a bright red colour and sometimes produces mineral deposits such as bauxite. Most trees have roots near the surface because there are insufficient nutrients below the surface; most of the trees' minerals come from the top layer of decomposing leaves and animals. On younger substrates, especially of volcanic origin, tropical soils may be quite fertile. If rainforest trees are cleared, rain can accumulate on the exposed soil surfaces, creating run-off, and beginning a process of soil erosion. Eventually, streams and rivers form and flooding becomes possible. There are several reasons for the poor soil quality. First is that the soil is highly acidic. The roots of plants rely on an acidity difference between the roots and the soil in order to absorb nutrients. When the soil is acidic, there is little difference, and therefore little absorption of nutrients from the soil. Second, the type of clay particles present in tropical rainforest soil has a poor ability to trap nutrients and stop them from washing away. Even if humans artificially add nutrients to the soil, the nutrients mostly wash away and are not absorbed by the plants. Finally, these soils are poor due to the high volume of rain in tropical rainforests washes nutrients out of the soil more quickly than in other climates.

Effect on global climate

A natural rainforest emits and absorbs vast quantities of carbon dioxide. On a global scale, long-term fluxes are approximately in balance, so that an undisturbed rainforest would have a small net impact on atmospheric carbon dioxide levels, though they may have other climatic effects (on cloud formation, for example, by recycling water vapour). No rainforest today can be considered to be undisturbed. Human-induced deforestation plays a significant role in causing rainforests to release carbon dioxide, as do other factors, whether human-induced or natural, which result in tree death, such as burning and drought. Some climate models operating with interactive vegetation predict a large loss of Amazonian rainforest around 2050 due to drought, forest dieback and the subsequent release of more carbon dioxide.

Human uses

Aerial view of the Amazon rainforest, taken from a plane.

Further information: Tropical rainforest § Human dimensions

Tropical rainforests provide timber as well as animal products such as meat and hides. Rainforests also have value as tourism destinations and for the ecosystem services provided. Many foods originally came from tropical forests, and are still mostly grown on plantations in regions that were formerly primary forest. Also, plant-derived medicines are commonly used for fever, fungal infections, burns, gastrointestinal problems, pain, respiratory problems, and wound treatment. At the same time, rainforests are usually not used sustainably by non-native peoples but are being exploited or removed for agricultural purposes.

Native people

On January 18, 2007, FUNAI reported also that it had confirmed the presence of 67 different uncontacted tribes in Brazil, up from 40 in 2005. With this addition, Brazil has now overtaken the island of New Guinea as the country having the largest number of uncontacted tribes. The province of Irian Jaya or West Papua in the island of New Guinea is home to an estimated 44 uncontacted tribal groups. The tribes are in danger because of the deforestation, especially in Brazil.

Central African rainforest is home of the Mbuti pygmies, one of the hunter-gatherer peoples living in equatorial rainforests characterised by their short height (below one and a half metres, or 59 inches, on average). They were the subject of a study by Colin Turnbull, The Forest People, in 1962. Pygmies who live in Southeast Asia are, amongst others, referred to as "Negrito". There are many tribes in the rainforests of the Malaysian state of Sarawak. Sarawak is part of Borneo, the third largest island in the world. Some of the other tribes in Sarawak are: the Kayan, Kenyah, Kejaman, Kelabit, Punan Bah, Tanjong, Sekapan, and the Lahanan. Collectively, they are referred to as Dayaks or Orangulu which means "people of the interior".

About half of Sarawak's 1.5 million people are Dayaks. Most Dayaks, it is believed by anthropologists, came originally from the South-East Asian mainland. Their mythologies support this.

Deforestation

Further information: Deforestation in Southeast Asia, Deforestation in Madagascar, and Deforestation of the Amazon Rainforest

Satellite photograph of the haze above Borneo and Sumatra, 24 September 2015

Tropical and temperate rainforests have been subjected to heavy legal and illegal logging for their valuable hardwoods and agricultural clearance (slash-and-burn, clearcutting) throughout the 20th century and the area covered by rainforests around the world is shrinking. Biologists have estimated that large numbers of species are being driven to extinction (possibly more than 50,000 a year; at that rate, says E. O. Wilson of Harvard University, a quarter or more of all species on Earth could be exterminated within 50 years) due to the removal of habitat with destruction of the rainforests.

Another factor causing the loss of rainforest is expanding urban areas. Littoral rainforest growing along coastal areas of eastern Australia is now rare due to ribbon development to accommodate the demand for seachange lifestyles.

Forests are being destroyed at a rapid pace. Almost 90% of West Africa's rainforest has been destroyed. Since the arrival of humans, Madagascar has lost two thirds of its original rainforest. At present rates, tropical rainforests in Indonesia would be logged out in 10 years and Papua New Guinea in 13 to 16 years. According to Rainforest Rescue, an important reason for the increasing deforestation rate, especially in Indonesia, is the expansion of oil palm plantations to meet growing demand for cheap vegetable fats and biofuels. In Indonesia, palm oil is already cultivated on nine million hectares and, together with Malaysia, the island nation produces about 85 percent of the world's palm oil.

Several countries, notably Brazil, have declared their deforestation a national emergency. Amazon deforestation jumped by 69% in 2008 compared to 2007's twelve months, according to official government data.

However, a January 30, 2009 New York Times article stated, "By one estimate, for every acre of rainforest cut down each year, more than 50 acres of new forest are growing in the tropics." The new forest includes secondary forest on former farmland and so-called degraded forest.