nLab A first idea of quantum field theory -- Lagrangians

Lagrangians

In this chapter we discuss the following topics:

• Lagrangian densities

• Euler-Lagrange forms and Presymplectic currents

• Euler-Lagrange equations of motion

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Given any type of fields (def. ), those field histories that are to be regarded as “physically realizable” (if we think of the field theory as a description of the observable universe) should satisfy some differential equation – the equation of motion – meaning that realizability of any field histories may be checked upon restricting the configuration to the infinitesimal neighbourhoods (example ) of each spacetime point. This expresses the physical absence of “action at a distance” and is one aspect of what it means to have a local field theory. By remark this means that equations of motion of a field theory are equations among the coordinates of the jet bundle of the field bundle.

For many field theories of interest, their differential equation of motion is not a random partial differential equations, but is of the special kind that exhibits the “principle of extremal action” (prop. below) determined by a local Lagrangian density (def. below). These are called Lagrangian field theories, and this is what we consider here.

Namely among all the variational differential forms (def. ) two kinds stand out, namley the 0-forms in $\Omega^{0,0}_\Sigma(E)$ – the smooth functions – and the horizontal $p+1$-forms $\Omega^{p+1,0}_\Sigma(E)$ – to be called the Lagrangian densities $\mathbf{L}$ (def. below) – since these occupy the two “corners” of the variational bicomplex (?). There is not much to say about the 0-forms, but the Lagrangian densities $\mathbf{L}$ do inherit special structure from their special position in the variational bicomplex:

Their variational derivative $\delta \mathbf{L}$ uniquely decomposes as

1. the Euler-Lagrange derivative $\delta_{EL} \mathbf{L}$ which is proportional to the variation of the fields (instead of their derivatives)

2. the total spacetime derivative $d \Theta_{BFV}$ of a potential $\Theta_{BFV}$ for a presymplectic current $\Omega_{BFV} \coloneqq \delta \Theta_{BFV}$.

This is prop. below:

$$\delta \mathbf{L} \;=\; \underset{ \text{Euler-Lagrange variation} }{\underbrace{\delta_{EL}\mathbf{L}}} - d \underset{\text{presymplectic current}}{\underbrace{\Theta_{BFV}}} \,.$$

These two terms play a pivotal role in the theory: The condition that the first term vanishes on field histories is a differential equation on field histories, called the Euler-Lagrange equation of motion (def. below). The space of solutions to this differential equation, called the on-shell space of field histories

(1)$$\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \overset{\phantom{AAA}}{\hookrightarrow} \Gamma_\Sigma(E)$$

has the interpretation of the space of “physically realizable field histories”. This is the key object of study in the following chapters. Often this is referred to as the space of classical field histories, indicating that this does not yet reflect the full quantum field theory.

Indeed, there is also the second term in the variational derivative of the Lagrangian density, the presymplectic current $\Theta_{BFV}$, and this implies a presymplectic structure on the on-shell space of field histories (def. below) which encodes deformations of the algebra of smooth functions on $\Gamma_\Sigma(E)$. This deformation is the quantization of the field theory to an actual quantum field theory, which we discuss below.

$$\array{ &&& \delta \mathbf{L} \\ &&& = \\ & & \delta_{EL}\mathbf{L} &- & d \Theta_{BFV} & \\ & \swarrow && && \searrow \\ \array{ \text{classical} \\ \text{field theory} } && && && \array{ \text{deformation to} \\ \text{quantum} \\ \text{field theory} } }$$

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Lagrangian densities

Definition

(local Lagrangian density)

Given a field bundle $E$ over a $(p+1)$-dimensional Minkowski spacetime $\Sigma$ as in example , then a local Lagrangian density $\mathbf{L}$ (for the type of field thus defined) is a horizontal differential form of degree $(p+1)$ (def. ) on the corresponding jet bundle (def. ):

$$\mathbf{L} \;\in \; \Omega^{p+1,0}_{\Sigma}(E) \,.$$

By example in terms of the given volume form on spacetimes, any such Lagrangian density may uniquely be written as

(2)$$\mathbf{L} = L \, dvol_\Sigma$$

where the coefficient function (the Lagrangian function) is a smooth function on the spacetime and field coordinates:

$$L = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots ) \,.$$

where by prop. $L((x^\mu), \cdots)$ depends locally on an arbitrary but finite order of derivatives $\phi^a_{,\mu_1 \cdots \mu_k}$.

We say that a field bundle $E \overset{fb}{\to} \Sigma$ (def. ) equipped with a local Lagrangian density $\mathbf{L}$ is (or defines) a prequantum Lagrangian field theory on the spacetime $\Sigma$.

Remark

(parameterized and physical unit-less Lagrangian densities)

More generally we may consider parameterized collections of Lagrangian densities, i.e. functions

$$\mathbf{L}_{(-)} \;\colon\; U \longrightarrow \Omega^{p+1,0}_\Sigma(E)$$

for $U$ some Cartesian space or generally some super Cartesian space.

For example all Lagrangian densities considered in relativistic field theory are naturally smooth functions of the scale of the metric $\eta$ (def. )

$$\array{ \mathbb{R}_{\gt 0} &\overset{}{\longrightarrow}& \Omega^{p+1,0}_\Sigma(E) \\ r &\mapsto& \mathbf{L}_{r^2\eta} }$$

But by the discussion in remark , in physics a rescaling of the metric is interpreted as reflecting but a change of physical units of length/distance. Hence if a Lagrangian density is supposed to express intrinsic content of a physical theory, it should remain unchanged under such a change of physical units.

This is achieved by having the Lagrangian be parameterized by further parameters, whose corresponding physical units compensate that of the metric such as to make the Lagrangian density “physical unit-less”.

This means to consider parameter spaces $U$ equipped with an action of the multiplicative group $\mathbb{R}_{\gt 0}$ of positive real numbers, and parameterized Lagrangians

$$\mathbf{L}_{(-)} \;\colon\; U \longrightarrow \Omega^{p+1,0}_\Sigma(E)$$

which are invariant under this action.

Remark

(locally variational field theory and Lagrangian p-gerbe connection)

If the field bundle (def. ) is not just a trivial vector bundle over Minkowski spacetime (example ) then a Lagrangian density for a given equation of motion may not exist as a globally defined differential $(p+1)$-form, but only as a p-gerbe connection. This is the case for locally variational field theories such as the charged particle, the WZW model and generally theories involving higher WZW terms. For more on this see the exposition at Higher Structures in Physics.

Example

(local Lagrangian density for free real scalar field on Minkowski spacetime)

Consider the field bundle for the real scalar field from example , i.e. the trivial line bundle over Minkowski spacetime.

According to def. its jet bundle $J^\infty_\Sigma(E)$ has canonical coordinates

$$\left\{ \{x^\mu\}, \phi, \{\phi_{,\mu}\}, \{\phi_{,\mu_1 \mu_2}\}, \cdots \right\} \,.$$

In these coordinates, the local Lagrangian density $L \in \Omega^{p+1,0}(\Sigma)$ (def. ) defining the free real scalar field of mass $m \in \mathbb{R}$ on $\Sigma$ is

$$L \coloneqq \tfrac{1}{2} \left( \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu} - m^2 \phi^2 \right) \mathrm{dvol}_\Sigma \,.$$

This is naturally thought of as a collection of Lagrangians smoothly parameterized by the metric $\eta$ and the mass $m$. For this to be physical unit-free in the sense of remark the physical unit of the parameter $m$ must be that of the inverse metric, hence must be an inverse length according to remark This is the inverse Compton wavelength $\ell_m = \hbar / m c$ (?) and hence the physical unit-free version of the Lagrangian density for the free scalar particle is

$$\mathbf{L}_{\eta,\ell_m} \:\coloneqq\; \tfrac{\ell_m^2}{2} \left( \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu} - \left( \tfrac{m c}{\hbar} \right)^2 \phi^2 \right) \mathrm{dvol}_\Sigma \,.$$

Example

(phi^n theory)

Consider the field bundle for the real scalar field from example , i.e. the trivial line bundle over Minkowski spacetime. More generally we may consider adding to the free field Lagrangian density from example some power of the field coordinate

$$\mathbf{L}_{int} \;\coloneqq\; g \phi^n \, dvol_\Sigma \,,$$

for $g \in \mathbb{R}$ some number, here called the coupling constant.

The interacting Lagrangian field theory defined by the resulting Lagrangian density

$$\mathbf{L} + \mathbf{L}_{int} \;=\; \tfrac{1}{2} \left( \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu} - m^2 \phi^2 + g \phi^n \right) \mathrm{dvol}_\Sigma$$

is usually called just phi^n theory.

Example

(local Lagrangian density for free electromagnetic field)

Consider the field bundle $T^\ast \Sigma \to \Sigma$ for the electromagnetic field on Minkowski spacetime from example , i.e. the cotangent bundle, which over Minkowski spacetime happens to be a trivial vector bundle of rank $p+1$. With fiber coordinates taken to be $(a_\mu)_{\mu = 0}^p$, the induced fiber coordinates on the corresponding jet bundle $J^\infty_\Sigma(T^\ast \Sigma)$ (def. ) are $( (x^\mu), (a_\mu), (a_{\mu,\nu}), (a_{\mu,\nu_1 \nu_2}), \cdots )$.

Consider then the local Lagrangian density (def. ) given by

(3)$$\mathbf{L} \;\coloneqq\; \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu} dvol_\Sigma \;\in\; \Omega^{p+1,0}_\Sigma(T^\ast \Sigma) \,,$$

where $f_{\mu \nu} \coloneqq \tfrac{1}{2}(a_{\nu,\mu} - a_{\mu,\nu})$ are the components of the universal Faraday tensor on the jet bundle from example .

This is the Lagrangian density that defines the Lagrangian field theory of free electromagnetism.

Here for $A \in \Gamma_\Sigma(T^\ast \Sigma)$ an electromagnetic field history (vector potential), then the pullback of $f_{\mu \nu}$ along its jet prolongation (def. ) is the corresponding component of the Faraday tensor (?):

$$\begin{aligned} \left( j^\infty_\Sigma(A) \right)^\ast(f_{\mu \nu}) & = (d A)_{\mu \nu} \\ & = F_{\mu \nu} \end{aligned}$$

It follows that the pullback of the Lagrangian (3) along the jet prologation of the electromagnetic field is

$$\begin{aligned} \left( j^\infty_\Sigma(A) \right)^\ast \mathbf{L} & = \tfrac{1}{2} F_{\mu \nu} F^{\mu \nu} dvol_\Sigma \\ & = \tfrac{1}{2} F \wedge \star_\eta F \end{aligned}$$

Here $\star_\eta$ denotes the Hodge star operator of Minkowski spacetime.

More generally:

Example

(Lagrangian density for Yang-Mills theory on Minkowski spacetime)

Let $\mathfrak{g}$ be a finite dimensional Lie algebra which is semisimple. This means that the Killing form invariant polynomial

$$k \colon \mathfrak{g} \otimes \mathfrak{g} \longrightarrow \mathbb{R}$$

is a non-degenerate bilinear form. Examples include the special unitary Lie algebras $\mathfrak{so}(n)$.

Then for $E = T^\ast \Sigma \otimes \mathfrak{g}$ the field bundle for Yang-Mills theory as in example , the Lagrangian density (def. ) $\mathfrak{g}$-Yang-Mills theory on Minkowski spacetime is

$$\mathbf{L} \;\coloneqq\; \tfrac{1}{2} k_{\alpha \beta} f^\alpha_{\mu \nu} f^{\beta \mu \nu} \, dvol_\Sigma \;\in\; \Omega^{p+1,0}_\Sigma(T^\ast \Sigma) \,,$$

where

$$f^\alpha_{\mu \nu} \;\coloneqq\; \tfrac{1}{2} \left( a^\alpha_{\nu,\mu} - a^\alpha_{\mu,\nu} + \gamma^{\alpha}{}_{\beta \gamma} a^\beta_{\mu} a^\gamma_{\nu} \right) \;\in\; \Omega^{0,0}_\Sigma(E)$$

is the universal Yang-Mills field strength (?).

For the purposes of perturbative quantum field theory (to be discussed below in chapter 15. Interacting quantum fields) we may allow for a rescaling of the structure constants by (at this point) a real number $g$, to be called the coupling constant, and decompose the Lagrangian into a sum of a free field theory Lagrangian (def. ) and an interaction term:

$$\begin{aligned} \mathbf{L} & = \tfrac{1}{2} k_{\alpha \beta} \tfrac{1}{2} \left( a^\alpha_{\nu,\mu} - a^\alpha_{\mu,\nu} + g \gamma^{\alpha}{}_{\beta' \gamma'} a^{\beta'}_{\mu} a^{\gamma'}_{\nu} \right) \tfrac{1}{2} \left( a^{\beta\nu,\mu} - a^{\beta \mu,\nu} + g \gamma^{\beta}{}_{\beta'' \gamma''} a^{\beta''}_{\mu} a^{\gamma''}_{\nu} \right) \,dvol_\Sigma \\ & = \underset{ \mathbf{L}_{\mathrm{free}} }{ \underbrace{ \tfrac{1}{2} k_{\alpha \beta} \tfrac{1}{2} \left( a^\alpha_{\nu,\mu} - a^\alpha_{\mu,\nu} \right) \tfrac{1}{2} \left( a^{\beta\nu,\mu} - a^{\beta \mu,\nu} \right) \,dvol_\Sigma } } \\ & \phantom{=} + \underset{ \mathbf{L}_{int} }{ \underbrace{ g \, k_{\alpha \beta} \tfrac{1}{2} \left( a^\alpha_{\nu,\mu} - a^\alpha_{\mu,\nu} \right) \tfrac{1}{2} \left( \gamma^{\beta}{}_{\beta'' \gamma''} a^{\beta''}_{\mu} a^{\gamma''}_{\nu} \right) \,dvol_\Sigma \; + \; g^2 \, \tfrac{1}{2} k_{\alpha \beta} \tfrac{1}{2} \left( \gamma^{\alpha}{}_{\beta' \gamma'} a^{\beta'}_{\mu} a^{\gamma'}_{\nu} \right) \tfrac{1}{2} \left( \gamma^{\beta}{}_{\beta'' \gamma''} a^{\beta''}_{\mu} a^{\gamma''}_{\nu} \right) \,dvol_\Sigma } } \\ \end{aligned} \,,$$

Notice that $\mathbf{L}_{free}$ is equivalently a sum of $dim(\mathfrak{g})$-copies of the Lagrangian for the electromagnetic field (example ).

On the other hand, for the purpose of exhibiting “non-perturbative effects due to instantons” in Yang-Mills theory, one consider the rescaled Yang-Mills field coordinate

$$\tilde a^\alpha_\mu \;\coloneqq\; \frac{1}{g} a^\alpha_\mu$$

with corresponding field strength

$$\tilde f^\alpha_{\mu \nu} \;\coloneqq\; \tfrac{1}{2} \left( \tilde a^\alpha_{\nu,\mu} - \tilde a^\alpha_{\mu,\nu} + \gamma^{\alpha}{}_{\beta \gamma} \tilde a^\beta_{\mu} \tilde a^\gamma_{\nu} \right) \;\in\; \Omega^{0,0}_\Sigma(E) \,.$$

In terms of this the expression for the Lagrangian is brought back to the abstract form it had before rescaling the structure constants by the coupling constant, up to a global rescaling of all terms by the inverse square of the coupling constant:

(4)$$\mathbf{L} \;=\; \frac{1}{g^2} \tfrac{1}{2} k_{\alpha \beta} \tilde f^\alpha_{\mu \nu} \tilde f^{\beta \mu \nu} \, dvol_\Sigma \,.$$

Example

(local Lagrangian density for free B-field)

Consider the field bundle $\wedge^2_\Sigma T^\ast \Sigma \to \Sigma$ for the B-field on Minkowski spacetime from example . With fiber coordinates taken to be $(b_{\mu \nu})$ with

$$b_{\mu \nu} = - b_{\nu \mu} \,,$$

the induced fiber coordinates on the corresponding jet bundle $J^\infty_\Sigma(T^\ast \Sigma)$ (def. ) are $( (x^\mu), (b_{\mu \nu}), (b_{\mu \nu, \mu_1}), (b_{\mu \nu, \mu_1 \mu_2}), \cdots )$.

Consider then the local Lagrangian density (def. ) given by

(5)$$\mathbf{L} \;\coloneqq\; \tfrac{1}{2} h_{\mu_1 \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3} \, dvol_\Sigma \;\in\; \Omega^{p+1,0}_\Sigma(\wedge^2_\Sigma T^\ast \Sigma) \,,$$

where $h_{\mu_1 \mu_2 \mu_3}$ are the components of the universal B-field strength on the jet bundle from example .

Example

(Lagrangian density for free Dirac field on Minkowski spacetime)

For $\Sigma$ Minkowski spacetime of dimension $p + 1 \in \{3,4,6,10\}$ (def. ), consider the field bundle $\Sigma \times S_{odd} \to \Sigma$ for the Dirac field from example . With the two-component spinor field fiber coordinates from remark , the jet bundle has induced fiber coordinates as follows:

$$\left( \left(\psi^\alpha\right) , \left( \psi^\alpha_{,\mu} \right) , \cdots \right) \;=\; \left( \left( (\chi_a), (\chi_{a,\mu}), \cdots \right), \left( ( \xi^{\dagger \dot a}), (\xi^{\dagger \dot a}_{,\mu}), \cdots \right) \right)$$

All of these are odd-graded elements (def. ) in a Grassmann algebra (example ), hence anti-commute with each other, in generalization of (?):

(6)$$\psi^\alpha_{,\mu_1 \cdots \mu_r} \psi^\beta_{,\mu_1 \cdots \mu_s} \;=\; - \psi^\beta_{,\mu_1 \cdots \mu_s} \psi^\alpha_{,\mu_1 \cdots \mu_r} \,.$$

The Lagrangian density (def. ) of the massless free Dirac field on Minkowski spacetime is

(7)$$\mathbf{L} \;\coloneqq\; \overline{\psi} \, \gamma^\mu \psi_{,\mu}\, dvol_\Sigma \,,$$

given by the bilinear pairing $\overline{(-)}\Gamma(-)$ from prop. of the field coordinate with its first spacetime derivative and expressed here in two-component spinor field coordinates as in (?), hence with the Dirac conjugate $\overline{\psi}$ (?) on the left.

Specifically in spacetime dimension $p + 1 = 4$, the Lagrangian function for the massive Dirac field of mass $m \in \mathbb{R}$ is

$$\begin{aligned} L & \coloneqq \underset{ \text{kinetic term} }{ \underbrace{ i \, \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} } } + \underset{ \text{mass term} }{ \underbrace{ m \overline{\psi} \psi }} \end{aligned}$$

This is naturally thought of as a collection of Lagrangians smoothly parameterized by the metric $\eta$ and the mass $m$. For this to be physical unit-free in the sense of remark the physical unit of the parameter $m$ must be that of the inverse metric, hence must be an inverse length according to remark This is the inverse Compton wavelength $\ell_m = \hbar / m c$ (?) and hence the physical unit-free version of the Lagrangian density for the free Dirac field is

$$\mathbf{L}_{\eta,\ell_m} \;\coloneqq\; \ell_m \left( i \overline{\psi} \gamma^\mu \psi_{,\mu} + \left( \tfrac{m c}{\hbar} \right) \overline{\psi} \psi \right) dvol_\Sigma \,.$$

Remark

(reality of the Lagrangian density of the Dirac field)

The kinetic term of the Lagrangian density for the Dirac field form def. is a sum of two contributions, one for each chiral spinor component in the full Dirac spinor (remark ):

$$\begin{aligned} i \overline{\psi} \gamma^\mu \psi_{,\mu} & = i \underset{ -(\partial_\mu \xi^a ) \sigma^\mu_{a \dot c} \xi^{\dagger \dot c} + \partial_\mu(\chi^a \sigma^\mu_{a \dot c} \chi^{\dagger \dot c}) }{ \underbrace{ \xi^a \sigma^\mu_{a \dot c} \partial_\mu \xi^{\dagger \dot c} } } + \xi^\dagger_{\dot a} \tilde \sigma^{\mu \dot a c} \partial_\mu \xi_c \\ & = \xi^\dagger \tilde \sigma^\mu \partial_\mu \xi + \chi^\dagger \tilde \sigma^\mu \partial_\mu \chi + \partial_\mu(\xi \sigma^\mu \xi^\dagger) \end{aligned}$$

Here the computation shown under the brace crucially uses that all these jet coordinates for the Dirac field are anti-commuting, due to their supergeometric nature (6).

Notice that a priori this is a function on the jet bundle with values in $\mathbb{K}$. But in fact for $\mathbb{K} = \mathbb{C}$ it is real up to a total spacetime derivative:, because

$$\begin{aligned} \left( i \chi^\dagger \tilde \sigma^\mu \partial_\mu\chi \right)^\dagger & = -i \left( \partial_\mu \chi\right)^\dagger \sigma^\mu \chi \\ & = i \chi^\dagger \sigma^\mu \partial_\mu \chi + i \partial_\mu\left( \chi^\dagger \sigma^\mu \chi \right) \end{aligned}$$

and similarly for $i \xi^\dagger \tilde \sigma^\mu \partial_\mu\xi$

(e.g. Dermisek I-9)

Example

(Lagrangian density for quantum electrodynamics)

Consider the fiber product of the field bundles for the electromagnetic field (example ) and the Dirac field (example ) over 4-dimensional Minkowski spacetime $\Sigma \coloneqq \mathbb{R}^{3,1}$ (def. ):

$$E \;\coloneqq\; \underset{ \array{ \text{electromagnetic} \\ \text{field} } }{\underbrace{T^\ast \Sigma}} \times \underset{ \array{ \text{Dirac} \\ \text{field} } }{ \underbrace{ S_{odd} } } \,.$$

This means that now a field history is a pair $(A,\Psi)$, with $A$ a field history of the electromagnetic field and $\Psi$ a field history of the Dirac field.

On the resulting jet bundle consider the Lagrangian density

(8)$$L_{int} \;\coloneqq\; i g \, \overline{\psi} \gamma^\mu \psi a_\mu$$

for $g \in \mathbb{R}$ some number, called the coupling constant. This is called the electron-photon interaction.

Then the sum of the Lagrangian densities for

1. the free electromagnetic field (example );

2. the free Dirac field (example )

3. the above electron-photon interaction

$$\mathbf{L}_{EM} + \mathbf{L}_{Dir} + \mathbf{L}_{int} \;=\; \left( \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu} \;+\; i \, \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} + m \overline{\psi} \psi \;+\; i g \, \overline{\psi} \gamma^\mu \psi a_\mu \right) \, dvol_\Sigma$$

defines the interacting field theory Lagrangian field theory whose perturbative quantization is called quantum electrodynamics.

In this context the square of the coupling constant

$$\alpha \coloneqq \frac{g^2}{4 \pi}$$

is called the fine structure constant.

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Euler-Lagrange forms and presymplectic currents

The beauty of Lagrangian field theory (def. ) is that a choice of Lagrangian density determines both the equations of motion of the fields as well as a presymplectic structure on the space of solutions to this equation (the “shell”), making it the “covariant phase space” of the theory. All this we discuss below. But in fact all this key structure of the field theory is nothing but the shadow (under “transgression of variational differential forms”, def. below) of the following simple relation in the variational bicomplex:

Proposition

(Euler-Lagrange form and presymplectic current)

Given a Lagrangian density $\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)$ as in def. , then its de Rham differential $\mathbf{d}\mathbf{L}$, which by degree reasons equals $\delta \mathbf{L}$, has a unique decomposition as a sum of two terms

(9)$$\mathbf{d} \mathbf{L} = \delta_{EL} \mathbf{L} - d \Theta_{BFV}$$

such that $\delta_{EL}\mathbf{L}$ is proportional to the variational derivative of the fields (but not their derivatives, called a “source form”):

$$\delta_{EL} \mathbf{L} \;\in\; \Omega^{p+1,0}_{\Sigma}(E) \wedge \delta C^\infty(E) \;\subset\; \Omega^{p+1,1}_{\Sigma}(E) \,.$$

The map

$$\delta_{EL} \;\colon\; \Omega^{p+1,0}_{\Sigma}(E) \longrightarrow \Omega^{p+1,0}_{\Sigma}(E) \wedge \delta \Omega^{0,0}_{\Sigma}(E)$$

thus defined is called the Euler-Lagrange operator and is explicitly given by the Euler-Lagrange derivative:

(10)$$\begin{aligned} \delta_{EL} L \, dvol_\Sigma & \coloneqq \frac{\delta_{EL} L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma \\ & \coloneqq \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^{\mu_1} d x^{\mu_2}} \frac{\partial L}{\partial \phi^a_{\mu_1, \mu_2}} - \cdots \right) \delta \phi^a \wedge dvol_\Sigma \,. \end{aligned}$$

The smooth subspace of the jet bundle on which the Euler-Lagrange form vanishes

(11)$$\mathcal{E} \;\coloneqq\; \left\{ x \in J^\infty_\Sigma(E) \;\vert\; \delta_{EL}\mathbf{L}(x) = 0 \right\} \;\overset{i_{\mathcal{E}}}{\hookrightarrow}\; J^\infty_\Sigma(E) \,.$$

is called the shell. The smaller subspace on which also all total spacetime derivatives vanish (the “formally integrable prolongation”) is the prolonged shell

(12)$$\mathcal{E}^\infty \;\coloneqq\; \left\{ x \in J^\infty_\Sigma(E) \;\vert\; \left( \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \delta_{EL}\mathbf{L} \right)(x) = 0 \right\} \overset{i_{\mathcal{E}^\infty}}{\hookrightarrow} J^\infty_\Sigma(E) \,.$$

Saying something holds “on-shell” is to mean that it holds after restriction to this subspace. For example a variational differential form $\alpha \in \Omega^{\bullet,\bullet}_\Sigma(E)$ is said to vanish on shell if $\alpha\vert_{\mathcal{E}^\infty} = 0$.

The remaining term $d \Theta_{BFV}$ in (9) is unique, while the presymplectic potential

(13)$$\Theta_{BFV} \in \Omega^{p,1}_{\Sigma}(E)$$

is not unique.

(For a field bundle which is a trivial vector bundle (example over Minkowski spacetime (def. ), prop. says that $\Theta_{BFV}$ is unique up to addition of total spacetime derivatives $d \kappa$, for $\kappa \in \Omega^{p-1,1}_\Sigma(E)$.)

One possible choice for the presymplectic current $\Theta_{BFV}$ is

(14)$$\begin{aligned} \Theta_{BFV} & \coloneqq \phantom{+} \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \; \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & \phantom{=} + \left( \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu} \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & \phantom{=} + \cdots \,, \end{aligned}$$

where

$$\iota_{\partial_{\mu}} dvol_\Sigma \;\coloneqq\; (-1)^{\mu} d x^0 \wedge \cdots d x^{\mu-1} \wedge d x^{\mu+1} \wedge \cdots \wedge d x^p$$

denotes the contraction (def. ) of the volume form with the vector field $\partial_\mu$.

The vertical derivative of a chosen presymplectic potential $\Theta_{BFV}$ is called a pre-symplectic current for $\mathbf{L}$:

(15)$$\Omega_{BFV} \;\coloneqq\; \delta \Theta_{BFV} \;\;\; \in \Omega^{p,2}_{\Sigma}(E) \,.$$

Given a choice of $\Theta_{BFV}$ then the sum

(16)$$\mathbf{L} + \Theta_{BFV} \;\in\; \Omega^{p+1,0}_\Sigma(E) \oplus \Omega^{p,1}_\Sigma(E)$$

is called the corresponding Lepage form. Its de Rham derivative is the sum of the Euler-Lagrange variation and the presymplectic current:

(17)$$\mathbf{d}( \mathbf{L} + \Theta_{BFV} ) \;=\; \delta_{EL} \mathbf{L} + \Omega_{BFV} \,.$$

(Its conceptual nature will be elucidated after the introduction of the local BV-complex in example below.)

Proof

Using $\mathbf{L} = L dvol_\Sigma$ and that $d \mathbf{L} = 0$ by degree reasons (example ), we find

$$\begin{aligned} \mathbf{d}\mathbf{L} & = \left( \frac{\partial L}{\partial \phi^a} \delta \phi^a + \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} + \frac{\partial L}{\partial \phi^a_{,\mu_1 \mu_2}} \delta \phi^a_{,\mu_1 \mu_2} + \cdots \right) \wedge dvol_{\Sigma} \end{aligned} \,.$$

The idea now is to have $d \Theta_{BFV}$ pick up those terms that would appear as boundary terms under the integral $\int_\Sigma j^\infty_\Sigma(\Phi)^\ast \mathbf{d}L$ if we were to consider integration by parts to remove spacetime derivatives of $\delta \phi^a$.

We compute, using example , the total horizontal derivative of $\Theta_{BFV}$ from (14) as follows:

$$\begin{aligned} d \Theta_{BFV} & = \left( d \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \right) + d \left( \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{\mu \nu}} \delta \phi^a \right) + \cdots \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & = \left( \left( \left( d \frac{\partial L}{\partial \phi^a_{,\mu}} \right) \wedge \delta \phi^a - \frac{\partial L}{\partial \phi^a_{,\mu}} \delta d \phi^a \right) + \left( \left(d \frac{\partial L}{\partial \phi^a_{,\nu \mu}}\right) \wedge \delta \phi^a_{,\nu} - \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta d \phi^a_{,\nu} - \left( d \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \right) \wedge \delta \phi^a + \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta d \phi^a \right) + \cdots \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & = - \left( \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a + \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} \right) + \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} + \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu \mu} - \frac{d^2}{ d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu} \right) + \cdots \right) \wedge dvol_\Sigma \,, \end{aligned}$$

where in the last line we used that

$$d x^{\mu_1} \wedge \iota_{\partial_{\mu_2}} dvol_\Sigma = \left\{ \array{ dvol_\Sigma &\vert& \text{if}\, \mu_1 = \mu_2 \\ 0 &\vert& \text{otherwise} } \right.$$

Here the two terms proportional to $\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu}$ cancel out, and we are left with

$$d \Theta_{BFV} \;=\; - \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} - \frac{d^2}{ d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} + \cdots \right) \delta \phi^a \wedge dvol_\Sigma - \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} + \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu \mu} + \cdots \right) \wedge dvol_\Sigma$$

Hence $-d \Theta_{BFV}$ shares with $\mathbf{d} \mathbf{L}$ the terms that are proportional to $\delta \phi^a_{,\mu_1 \cdots \mu_k}$ for $k \geq 1$, and so the remaining terms are proportional to $\delta \phi^a$, as claimed:

$$\mathbf{d}\mathbf{L} + d \Theta_{BFV} = \underset{ = \delta_{EL}\mathbf{L} }{ \underbrace{ \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu}\frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}} + \cdots \right) \delta \phi^a \wedge dvol_\Sigma }} \,.$$

The following fact is immediate from prop. , but of central importance, we futher amplify this in remark below:

Proposition

(total spacetime derivative of presymplectic current vanishes on-shell)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ). Then the Euler-Lagrange form $\delta_{EL} \mathbf{L}$ and the presymplectic current (prop. ) are related by

$$d \Omega_{BFV} = - \delta(\delta_{EL}\mathbf{L}) \,.$$

In particular this means that restricted to the prolonged shell $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ (12) the total spacetime derivative of the presymplectic current vanishes:

(18)$$d \Omega_{BFV} \vert_{\mathcal{E}^\infty} \;=\; 0 \,.$$

Proof

By prop. we have

$$\delta \mathbf{L} = \delta_{EL} \mathbf{L} - d \Theta_{BFV} \,.$$

The claim follows from applying the variational derivative $\delta$ to both sides, using (?): $\delta^2 = 0$ and $\delta \circ d = - d \circ \delta$.

Many examples of interest fall into the following two special cases of prop. :

Example

(Euler-Lagrange form for spacetime-independent Lagrangian densities)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ) whose field bundle $E$ is a trivial vector bundle $E \simeq \Sigma \times F$ over Minkowski spacetime $\Sigma$ (example ).

In general the Lagrangian density $\mathbf{L}$ is a function of all the spacetime and field coordinates

$$\mathbf{L} = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots) dvol_\Sigma \,.$$

Consider the special case that $\mathbf{L}$ is spacetime-independent in that the Lagrangian function $L$ is independent of the spacetime coordinate $(x^\mu)$. Then the same evidently holds for the Euler-Lagrange form $\delta_{EL}\mathbf{L}$ (prop. ). Therefore in this case the shell (12) is itself a trivial bundle over spacetime.

In this situation every point $\varphi$ in the jet fiber defines a constant section of the shell:

(19)$$\Sigma \times \{\varphi\} \subset \mathcal{E}^\infty \,.$$

Example

(canonical momentum)

Consider a Lagrangian field theory $(E, \mathbf{L})$ (def. ) whose Lagrangian density $\mathbf{L}$

1. does not depend on the spacetime-coordinates (example );

2. depends on spacetime derivatives of field coordinates (hence on jet bundle coordinates) at most to first order.

Hence if the field bundle $E \overset{fb}{\to} \Sigma$ is a trivial vector bundle over Minkowski spacetime (example ) this means to consider the case that

$$\mathbf{L} \;=\; L\left( (\phi^a), (\phi^a_{,\mu}) \right) \wedge dvol_\Sigma \,.$$

Then the presymplectic current (def. ) is (up to possibly a horizontally exact part) of the form

(20)$$\Omega_{BFV} \;=\; \delta p_a^\mu \wedge \delta \phi^a \wedge \iota_{\partial_\mu} dvol_\Sigma$$

where

(21)$$p_a^\mu \;\coloneqq\; \frac{\partial L}{ \partial \phi^a_{,\mu}}$$

denotes the partial derivative of the Lagrangian function with respect to the spacetime-derivatives of the field coordinates.

Here

$$\begin{aligned} p_a & \coloneqq p_a^0 \\ & = \frac{\partial L}{\partial \phi^a_{,0}} \end{aligned}$$

is called the canonical momentum corresponding to the “canonical field coordinate” $\phi^a$.

In the language of multisymplectic geometry the full expression

$$p_a^\mu \wedge \iota_{\partial_\mu} dvol_\Sigma \;\in\; \Omega^{p,1}_\Sigma(E)$$

is also called the “canonical multi-momentum”, or similar.

Proof

We compute:

$$\begin{aligned} \mathbf{d} \mathbf{L} & = \left( \frac{\partial L}{\partial \phi^a} \delta \phi^a + \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} \right) \delta \phi^a \wedge dvol_\Sigma \\ & = \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} \right) \wedge dvol_\Sigma - d \underset{ \Theta_{BFV} }{ \underbrace{ \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \right) \wedge \iota_{\partial_\mu} dvol_\Sigma } } \end{aligned} \,.$$

Hence

$$\begin{aligned} \Omega_{BFV} & \coloneqq \delta \Theta_{BFV} \\ & = \delta \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} \wedge \iota_{\partial_\mu} dvol_\Sigma \right) \\ & = \delta \frac{\partial L}{\partial \phi^a_{,\mu}} \wedge \delta \phi^a_{,\mu} \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & = \delta p_a^\mu \wedge \delta \phi^a \wedge \iota_{\partial_\mu} dvol_\Sigma \end{aligned}$$

Remark

(presymplectic current is local version of (pre-)symplectic form of Hamiltonian mechanics)

In the simple but very common situation of example the presymplectic current (def. ) takes the form (21)

$$\Omega_{BFV} \;=\; \delta p_a^\mu \wedge \delta \phi^a \wedge \iota_{\partial_\mu} dvol_\Sigma$$

with $\phi^a$ the field coordinates (“canonical coordinates”) and $p_a^\mu$ the “canonical momentum” (21).

Notice that this is of the schematic form “$(\delta p_a \wedge \delta q^a) \wedge dvol_{\Sigma_p}$”, which is reminiscent of the wedge product of a symplectic form expressed in Darboux coordinates with a volume form for a $p$-dimensional manifold. Indeed, below in Phase space we discuss that this presymplectic current “transgresses” (def. below) to a presymplectic form of the schematic form “$d P_a \wedge d Q^a$” on the on-shell space of field histories (def. ) by integrating it over a Cauchy surface of dimension $p$. In good situations this presymplectic form is in fact a symplectic form on the on-shell space of field histories (theorem below).

This shows that the presymplectic current $\Omega_{BFV}$ is the local (i.e. jet level) avatar of the symplectic form that governs the formulation of Hamiltonian mechanics in terms of symplectic geometry.

In fact prop. may be read as saying that the presymplectic current is a conserved current (def. below), only that it takes values not in smooth functions of the field coordinates and jets, but in variational 2-forms on fields. There is a conserved charge associated with every conserved current (prop. below) and the conserved charge associated with the presymplectic current is the (pre-)symplectic form on the phase space of the field theory (def. below).

Example

(Euler-Lagrange form and presymplectic current for free real scalar field)

Consider the Lagrangian field theory of the free real scalar field from example .

Then the Euler-Lagrange form and presymplectic current (prop. ) are

(22)$$\delta_{EL}\mathbf{L} \;=\; \left(\eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \right) \delta \phi \wedge dvol_\sigma \;\in\; \Omega^{p+1,1}_{\Sigma}(E) \,.$$

and

$$\Omega_{BFV} \;=\; \left(\eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma} \;\in\; \Omega^{p,2}_{\Sigma}(E) \,,$$

respectively.

Proof

This is a special case of example , but we spell it out in detail again:

We need to show that Euler-Lagrange operator $\delta_{EL} \colon \Omega^{p+1,0}(\Sigma) \to \Omega^{p+1,1}_S(\Sigma)$ takes the local Lagrangian density for the free scalar field to

$$\delta_EL L \;=\; \left( \eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \phi \right) \delta \phi \wedge \mathrm{dvol}_\Sigma \,.$$

First of all, using just the variational derivative (vertical derivative) $\delta$ is a graded derivation, the result of applying it to the local Lagrangian density is

$$\delta L \;=\; \left( \eta^{\mu \nu} \phi_{,\mu} \delta \phi_{,\nu} - m^2 \phi \delta \phi \right) \wedge \mathrm{dvol}_\Sigma \,.$$

By definition of the Euler-Lagrange operator, in order to find $\delta_{EL}\mathbf{L}$ and $\Theta_{BFV}$, we need to exhibit this as the sum of the form $(-) \wedge \delta \phi - d \Theta_{BFV}$.

The key to find $\Theta_{BFV}$ is to realize $\delta \phi_{,\nu}\wedge \mathrm{dvol}_\Sigma$ as a total spacetime derivative (horizontal derivative). Since $d \phi = \phi_{,\mu} d x^\mu$ this is accomplished by

$$\delta \phi_{,\nu} \wedge \mathrm{dvol}_\Sigma = \delta d \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \,,$$

where on the right we have the contraction (def. ) of the tangent vector field along $x^\nu$ into the volume form.

Hence we may take the presymplectic potential (13) of the free scalar field to be

(23)$$\Theta_{BFV} \coloneqq \eta^{\mu \nu} \phi_{,\mu} \delta \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \,,$$

because with this we have

$$d \Theta_{BFV} = \eta^{\mu \nu} \left( \phi_{,\mu \nu} \delta \phi - \eta^{\mu \nu} \phi_{,\mu} \delta \phi_{,\nu} \right) \wedge \mathrm{dvol}_\Sigma \,.$$

In conclusion this yields the decomposition of the vertical differential of the Lagrangian density

$$\delta L = \underset{ = \delta_{EL} \mathcal{L} }{ \underbrace{ \left( \eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \phi \right) \delta \phi \wedge \mathrm{dvol}_\Sigma } } - d \Theta_{BFV} \,,$$

which shows that $\delta_{EL} L$ is as claimed, and that $\Theta_{BFV}$ is a presymplectic potential current (13). Hence the presymplectic current itself is

$$\begin{aligned} \Omega_{BFV} &\coloneqq \delta \Theta_{BFV} \\ & = \delta \left( \eta^{\mu \nu} \phi_{,\mu} \delta \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \right) \\ & = \left(\eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma} \end{aligned} \,.$$

Example

(Euler-Lagrange form for free electromagnetic field)

Consider the Lagrangian field theory of free electromagnetism from example .

The Euler-Lagrange variational derivative is

(24)$$\delta_{EL} \mathbf{L} \;=\; - \frac{d}{d x^\mu} f^{\mu \nu} \delta a_\nu \,.$$

Hence the shell (11) in this case is

$$\mathcal{E} = \Sigma \times \left\{ \left( (a_\mu) , (a_{\mu,\mu_1}), (a_{\mu,\mu_1 \mu_2}), \cdots \right) \;\vert\; f^{\mu \nu}{}_{,\mu} = 0 \right\} \;\subset\; J^\infty_\Sigma(T^\ast \Sigma) \,.$$

Proof

By (10) we have

$$\begin{aligned} \frac{\delta_{EL} L}{\delta a_\mu} \delta a_\mu & = \left( \underset{ = 0 }{ \underbrace{ \frac{\partial}{\partial a_\mu} \tfrac{1}{2} a_{[\mu,\nu]} a^{[\mu,\nu]} } } - \frac{d}{d x^\rho} \frac{\partial}{\partial a_{\alpha,\rho}} \tfrac{1}{2} a_{[\mu,\nu]} a^{[\mu,\nu]} \right) \delta a_\alpha \\ & = - \tfrac{1}{2} \left( \frac{d}{d x^\rho} \frac{\partial}{\partial a_{\alpha,\rho}} a_{\mu,\nu} a^{[\mu,\nu]} \right) \delta a_\alpha \\ & = - \left( \frac{d}{d x^\rho} a^{[\alpha,\rho]} \right) \delta a_{\alpha} \\ & = - f^{\mu \nu}{}_{,\mu} \delta a_{\nu} \,. \end{aligned}$$

More generally:

Example

(Euler-Lagrange form for Yang-Mills theory on Minkowski spacetime)

Let $\mathfrak{g}$ be a semisimple Lie algebra and consider the Lagrangian field theory $(E,\mathbf{L})$ of $\mathfrak{g}$-Yang-Mills theory from example .

Its Euler-Lagrange form (prop. ) is

$$\begin{aligned} \delta_{EL}\mathbf{L} & = - \left( f^{\mu \nu \alpha}_{,\mu} + \gamma^\alpha{}_{\beta' \gamma} a_\mu^{\beta'} f^{\mu \nu \gamma} \right) k_{\alpha \beta} \,\delta a_\mu^\beta \, dvol_\Sigma \,, \end{aligned}$$

where

$$f^\alpha_{\mu \nu} \;\in\; \Omega^{0,0}_\Sigma(E)$$

is the universal Yang-Mills field strength (?).

Proof

With the explicit form (10) for the Euler-Lagrange derivative we compute as follows:

$$\begin{aligned} \delta_{EL} \left( \tfrac{1}{2} k_{\alpha \beta} f^\alpha_{\mu\nu} f^{\beta \mu \nu} \right) & = \left( \left( \frac{\partial}{\partial a_{\mu'}^{\alpha'}} \left( a_{\nu,\mu}^\alpha + \tfrac{1}{2} \gamma^{\alpha}{}_{\alpha_2 \alpha_3} a_{\mu}^{\alpha_2} a_\nu^{\alpha_3} \right) \right) k_{\alpha \beta} f^{\beta \mu \nu} - \left( \frac{d}{d x^{\nu'}} \frac{\partial}{\partial a_{\mu',\nu'}^{\alpha'}} \left( a_{\nu,\mu}^\alpha + \tfrac{1}{2} \gamma^{\alpha}{}_{\alpha_2 \alpha_3} a_{\mu}^{\alpha_2} a_\nu^{\alpha_3} \right) \right) k_{\alpha \beta} f^{\beta \mu \nu} \right) \delta a_{\mu'}^{\alpha'} \\ & = \gamma^{\alpha}{}_{\alpha' \alpha_3} a_\nu^{\alpha_3} f^{\beta \mu \nu} k_{\alpha \beta} \delta a_{\mu}^{\alpha'} - \left( \frac{d}{d x^{\mu}} f^{\beta \mu \nu} \right) k_{\alpha \beta} \delta a_{\nu}^{\alpha} \\ &= - \left( f^{\alpha \mu \nu}_{,\mu} + \gamma^\alpha{}_{\beta \gamma} a_\mu^\beta f^{\gamma \mu \nu} \right) k_{\alpha \beta} \delta a_\nu^\beta \end{aligned}$$

In the last step we used that for a semisimple Lie algebra $\gamma_{\alpha \beta \gamma} \coloneqq k_{\alpha \alpha'} \gamma^{\alpha'}{}_{\beta \gamma}$ is totally skew-symmetric in its indices (this being the coefficients of the Lie algebra cocycle) which is in transgression with the Killing form invariant polynomial $k$.

Example

(Euler-Lagrange form of free B-field)

Consider the Lagrangian field theory of the free B-field from example .

The Euler-Lagrange variational derivative is

$$\delta_{EL} \mathbf{L} \;=\; h^{\mu \nu \rho}{}_{,\rho} \delta b_{\mu \nu} \,,$$

where $h_{\mu_1 \mu_2 \mu_3}$ is the universal B-field strength from example .

Proof

By (10) we have

$$\begin{aligned} \frac{\delta_{EL} L}{\delta b_{\mu \nu}} \delta b_{\mu \nu} & = \left( \underset{ = 0 }{ \underbrace{ \frac{\partial}{\partial b_{\mu \nu}} \tfrac{1}{2} b_{[\mu_1 \mu_2, \mu_3]} b^{[\mu_1 \mu_2, \mu_3]} } } - \frac{d}{d x^\rho} \frac{\partial}{\partial b_{\mu \nu, \rho}} \tfrac{1}{2} b_{[\mu_1 \mu_2, \mu_3]} b^{[\mu_1 \mu_2, \mu_3]} \right) \delta b_{\mu \nu} \\ & = - \left( \frac{d}{d x^\rho} \frac{\partial}{\partial b_{\mu \nu, \rho}} \tfrac{1}{2} b_{\mu_1 \mu_2, \mu_3} b^{[\mu_1 \mu_2, \mu_3]} \right) \delta b_{\mu \nu} \\ & = - \left( \frac{d}{d x^\rho} b^{[\mu \nu, \rho]} \right) \delta b_{\mu \nu} \\ & = - h^{\mu \nu \rho}{}_{,\rho} \delta b_{\mu \nu} \,. \end{aligned}$$

Example

(Euler-Lagrange form and presymplectic current of Dirac field)

Consider the Lagrangian field theory of the Dirac field on Minkowski spacetime of dimension $p + 1 \in \{3,4,6,10\}$ (example ).

Then

• the Euler-Lagrange variational derivative (def. ) in the case of vanishing mass $m$ is

  $$\delta_{EL} \mathbf{L} \;=\; 2 i\, \overline{\delta \psi} \,\gamma^\mu\, \psi_{,\mu} \, \wedge dvol_\Sigma$$

  and in the case that spacetime dimension is $p +1 = 4$ and arbitrary mass $m\in \mathbb{R}$, it is

  $$\delta_{EL} \mathbf{L} \;=\; \left( \overline{\delta \psi} \left( i \gamma^\mu \psi_{,\mu} + m \psi \right) + \left( - i \gamma^\mu\overline{\psi_{,\mu}} + m \overline{\psi} \right) (\delta \psi) \right) \, dvol_\Sigma$$

• its presymplectic current (def. ) is

  $$\Omega_{BFV} \;=\; \overline{\delta \psi}\,\gamma^\mu \,\delta \psi \, \iota_{\partial_\mu} dvol_\Sigma$$

Proof

In any case the canonical momentum of the Dirac field according to example is

$$\begin{aligned} p^\alpha_\mu & \coloneqq \frac{\partial }{\partial \psi^\alpha_{,\mu}} \left( i \overline {\psi} \, \gamma^\nu \, \psi_{,\nu} + m \overline{\psi} \psi \right) \\ & = \overline{\psi}^\beta (\gamma^\mu)_\beta{}^\alpha \end{aligned}$$

This yields the presymplectic current as claimed, by example .

Now regarding the Euler-Lagrange form, first consider the massless case in spacetime dimension $p+1 \in \{3,4,6,10\}$, where

$$L \;=\; i \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} \,.$$

Then we compute as follows:

$$\begin{aligned} \delta_{EL} L & = i \,\overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu} \underset{ = + i \,\overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu} }{ \underbrace{ - i \overline{\psi_{,\mu}} \, \gamma^\mu \, \delta \psi } } \\ & = 2 i \, \overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu} \end{aligned}$$

Here the first equation is the general formula (10) for the Euler-Lagrange variation, while the identity under the braces combines two facts (as in remark above):

1. the symmetry (?) of the spinor pairing $\overline{(-)}\gamma^\mu(-)$ (prop. );

2. the anti-commutativity (6) of the Dirac field and jet coordinates, due to their supergeometric nature (remark ).

Finally in the special case of the massive Dirac field in spacetime dimension $p+1 = 4$ the Lagrangian function is

$$L \;=\; i \, \overline{\psi} \gamma^\mu \psi_{,\mu} + m \overline{\psi}\psi$$

where now $\psi_\alpha$ takes values in the complex numbers $\mathbb{C}$ (as opposed to in $\mathbb{R}$, $\mathbb{H}$ or $\mathbb{O}$). Therefore we may now form the derivative equivalently by treeating $\psi$ and $\overline{\psi}$ as independent components of the field. This immediately yields the claim.

Example

(trivial Lagrangian densities and the Euler-Lagrange complex)

If a Lagrangian density $\mathbf{L}$ (def. ) is in the image of the total spacetime derivative, hence horizontally exact (def. )

$$\mathbf{L} \;=\; d \mathbf{\ell}$$

for any $\mathbf{\ell} \in \Omega^{p,0}_\Sigma(E)$, then both its Euler-Lagrange form as well as its presymplectic current (def. ) vanish:

$$\delta_{EL}\mathbf{L} = 0 \phantom{AA} \,, \phantom{AA} \Omega_{BFV} = 0 \,.$$

This is because with $\delta \circ d = - d \circ \delta$ (?) the defining unique decomposition (9) of $\delta \mathbf{L}$ is given by

$$\begin{aligned} \delta \mathbf{L} & = \delta d \mathbf{\ell} \\ & = \underset{= \delta_{EL}\mathbf{L}}{\underbrace{0}} - d \underset{\Theta_{BFV}}{\underbrace{\delta \mathbf{l}}} \end{aligned}$$

which then implies with (15) that

$$\begin{aligned} \Omega_{BFV} & \coloneqq \delta \Theta_{BFV} \\ & = \delta \delta \mathbf{\ell} \\ & = 0 \end{aligned}$$

Therefore the Lagrangian densities which are total spacetime derivatives are also called trivial Lagrangian densities.

If the field bundle $E \overset{fb}{\to} \Sigma$ is a trivial vector bundle (example ) over Minkowski spacetime (def. ) then also the converse is true: Every Lagrangian density whose Euler-Lagrange form vanishes is a total spacetime derivative.

Stated more abstractly, this means that the exact sequence of the total spacetime from prop. extends to the right via the Euler-Lagrange variational derivative $\delta_{EL}$ to an exact sequence of the form

$$\mathbb{R} \overset{}{\hookrightarrow} \Omega^{0,0}_\Sigma(E) \overset{d}{\longrightarrow} \Omega^{1,0}_\Sigma(E) \overset{d}{\longrightarrow} \Omega^{2,0}_\Sigma(E) \overset{d}{\longrightarrow} \cdots \overset{d}{\longrightarrow} \Omega^{p,0}_\Sigma(E) \overset{d}{\longrightarrow} \Omega^{p+1,0}_\Sigma(E) \overset{\delta_{EL}}{\longrightarrow} \Omega^{p+1,0}_\Sigma(E) \wedge \delta(C^\infty(E)) \overset{\delta_{H}}{\longrightarrow} \cdots \,.$$

In fact, as shown, this exact sequence keeps going to the right; this is also called the Euler-Lagrange complex.

(Anderson 89, theorem 5.1)

The next differential $\delta_{H}$ after the Euler-Lagrange variational derivative $\delta_{EL}$ is known as the Helmholtz operator. By definition of exact sequence, the Helmholtz operator detects whether a partial differential equation on field histories, induced by a variational differential form $P \in \Omega^{p+1,0}_\Sigma(E) \wedge \delta(C^\infty(E))$ as in (25) comes from varying a Lagrangian density, hence whether it is the equation of motion of a Lagrangian field theory via def. .

This way homological algebra is brought to bear on core questions of field theory. For more on this see the exposition at Higher Structures in Physics.

Remark

(supergeometric nature of Lagrangian density of the Dirac field)

Observe that the Lagrangian density for the Dirac field (def. ) makes sense (only) due to the supergeometric nature of the Dirac field (remark ): If the field jet coordinates $\psi_{,\mu_1 \cdots \mu_k}$ were not anti-commuting (6) then the Dirac’s field Lagrangian density (def. ) would be a total spacetime derivative and hence be trivial according to example .

This is because

$$d \left( \tfrac{1}{2} \overline{\psi} \,\gamma^\mu\, \psi \, \iota_{\partial_\mu} dvol_\Sigma \right) = \tfrac{1}{2} \overline{\psi_{,\mu}} \,\gamma^\mu\, \psi \, dvol_\Sigma + \underset{ = (-1) \tfrac{1}{2} \overline{\psi_{,\mu}} \,\gamma^\mu\, \psi \, dvol_\Sigma }{ \underbrace{ \tfrac{1}{2}\overline{\psi} \,\gamma^\mu\, \psi_{,\mu} \, dvol_\Sigma }} \,.$$

Here the identification under the brace uses two facts:

1. the symmetry (?) of the spinor bilinear pairing $\overline{(-)}\Gamma (-)$;

2. the anti-commutativity (6) of the Dirac field and jet coordinates, due to their supergeometric nature (remark ).

The second fact gives the minus sign under the brace, which makes the total expression vanish, if the Dirac field and jet coordinates indeed are anti-commuting (which, incidentally, means that we found an “off-shell conserved current” for the Dirac field, see example below).

If however the Dirac field and jet coordinates did commute with each other, we would instead have a plus sign under the brace, in which case the total horizontal derivative expression above would equal the massless Dirac field Lagrangian (7), thus rendering it trivial in the sense of example .

The same supergeometric nature of the Dirac field will be necessary for its intended equation of motion, the Dirac equation (example ) to derive from a Lagrangian density; see the proof of example below, and see remark below.

$\,$

Euler-Lagrange equations of motion

The key implication of the Euler-Lagrange form on the jet bundle is that it induces the equation of motion on the space of field histories:

Definition

(Euler-Lagrange equation of motion)

Given a Lagrangian field theory $(E,\mathbf{L})$ (def. then the corresponding Euler-Lagrange equations of motion is the condition on field histories (def. )

$$\Phi_{(-)} \;\colon\; U \longrightarrow \Gamma_\Sigma(E)$$

to have a jet prolongation (def. )

$$j^\infty_\Sigma(\Phi_{(-)}(-) ) \;\colon\; U \times \Sigma \longrightarrow J^\infty_\Sigma(E)$$

that factors through the shell inclusion $\mathcal{E} \overset{i_{\mathcal{E}}}{\hookrightarrow} J^\infty_\Sigma(E)$ (11) defined by vanishing of the Euler-Lagrange form (prop. )

(25)$$j^\infty_\Sigma(\Phi_{(-)}(-)) \;\colon\; U \times \Sigma \longrightarrow \mathcal{E} \overset{i_{\mathcal{E}}}{\hookrightarrow} J^\infty_\Sigma(E) \,.$$

(This implies that $j^\infty_\Sigma(\Phi_{(-)})$ factors even through the prolonged shell $\mathcal{E}^\infty \overset{i_{\mathcal{E}^\infty}}{\hookrightarrow} J^\infty_\Sigma(E)$ (12).)

In the case that the field bundle is a trivial vector bundle over Minkowski spacetime as in example this is the condition that $\Phi_{(-)}$ satisfies the following differential equation (again using prop. ):

$$\frac{\delta_{EL} L}{\delta \phi^a} \;\coloneqq\; \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}} - \cdots \right) \left( (x^\mu), (\Phi^a), \left( \frac{\partial \Phi^a_{(-)}}{\partial x^\mu}\right), \left( \frac{\partial^2 \Phi^a_{(-)}}{\partial x^\mu \partial x^\nu} \right), \cdots \right) \;=\; 0 \,,$$

where the differential operator (def. )

(26)$$j^\infty_\Sigma(-)^\ast \left( \frac{\delta_{EL}L}{\delta \phi^{(-)}} \right) \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(T^\ast_\Sigma E)$$

from the field bundle (def. ) to its vertical cotangent bundle (def. ) is given by the Euler-Lagrange derivative (10).

The on-shell space of field histories is the space of solutions to this condition, namely the the sub-super smooth set (def. ) of the full space of field histories (?) (def. )

(27)$$\Gamma_\Sigma(E)_{\delta_{EL} L = 0} \overset{\phantom{AAA}}{\hookrightarrow} \Gamma_\Sigma(E)$$

whose plots are those $\Phi_{(-)} \colon U \to \Gamma_\Sigma(E)$ that factor through the shell (25).

More generally for $\Sigma_r \hookrightarrow \Sigma$ a submanifold of spacetime, we write

(28)$$\Gamma_{\Sigma_r}(E)_{\delta_{EL} L = 0} \overset{\phantom{AAA}}{\hookrightarrow} \Gamma_{\Sigma_r}(E)$$

for the sub-super smooth ste of on-shell field histories restricted to the infinitesimal neighbourhood of $\Sigma_r$ in $\Sigma$ (?).

Definition

(free field theory)

A Lagrangian field theory $(E, \mathbf{L})$ (def. ) with field bundle $E \overset{fb}{\to} \Sigma$ a vector bundle (e.g. a trivial vector bundle as in example ) is called a free field theory if its Euler-Lagrange equations of motion (def. ) is a differential equation that is linear differential equation, in that with

$$\Phi_1, \Phi_2 \;\in\; \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$$

any two on-shell field histories (27) and $c_1, c_2 \in \mathbb{R}$ any two real numbers, also the linear combination

$$c_1 \Phi_1 + c_2 \Phi_2 \;\in\; \Gamma_\Sigma(E) \,,$$

which a priori exists only as an element in the off-shell space of field histories, is again a solution to the equations of motion and hence an element of $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$.

A Lagrangian field theory which is not a free field theory is called an interacting field theory.

Remark

(relevance of free field theory)

In perturbative quantum field theory one considers interacting field theories in the infinitesimal neighbourhood (example ) of free field theories (def. ) inside some super smooth set of general Lagrangian field theories. While free field theories are typically of limited interest in themselves, this perturbation theory around them exhausts much of what is known about quantum field theory in general, and therefore free field theories are of paramount importance for the general theory.

We discuss the covariant phase space of free field theories below in Propagators and their quantization below in Free quantum fields.

Example

(equation of motion of free real scalar field is Klein-Gordon equation)

Consider the Lagrangian field theory of the free real scalar field from example .

By example its Euler-Lagrange form is

$$\delta_{EL}\mathbf{L} \;=\; \left(\eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \right) \delta \phi \wedge dvol_\sigma$$

Hence for $\Phi \in \Gamma_\Sigma(E) = C^\infty(X)$ a field history, its Euler-Lagrange equation of motion according to def. is

$$\eta^{\mu \nu} \frac{\partial^2 }{\partial x^\mu \partial x^\nu} \Phi - m^2 \Phi \;=\; 0$$

often abbreviated as

(29)$$(\Box - m^2) \Phi \;=\; 0 \,.$$

This PDE is called the Klein-Gordon equation on Minowski spacetime. If the mass $m$ vanishes, $m = 0$, then this is the relativistic wave equation.

Hence this is indeed a free field theory according to def. .

The corresponding linear differential operator (def. )

(30)$$(\Box - m^2) \;\colon\; \Gamma_\Sigma(\Sigma \times \mathbb{R}) \longrightarrow \Gamma_\Sigma(\Sigma \times \mathbb{R})$$

is called the Klein-Gordon operator.

For later use we record the following basic fact about the Klein-Gordon equation:

Example

(Klein-Gordon operator is formally self-adjoint )

The Klein-Gordon operator (30) is its own formal adjoint (def. ) witnessed by the bilinear differential operator (?) given by

(31)$$K(\Phi_1, \Phi_2) \;\coloneqq\; \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \,.$$

Proof

$$\begin{aligned} d K(\Phi_1, \Phi_2) & = d \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \\ &= \left( \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 + \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu} \right) - \left( \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu} + \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) \right) dvol_\Sigma \\ & = \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 - \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) dvol_\Sigma \\ & = \Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2) \end{aligned}$$

Example

(equations of motion of vacuum electromagnetism are vacuum Maxwell's equations)

Consider the Lagrangian field theory of free electromagnetism on Minkowski spacetime from example .

By example its Euler-Lagrange form is

$$\delta_{EL}\mathbf{L} \;=\; \frac{d}{d x^\mu}f^{\mu \nu} \delta a_\nu \,.$$

Hence for $A \in \Gamma_{\Sigma}(T^\ast \Sigma) = \Omega^1(\Sigma)$ a field history (“vector potential”), its Euler-Lagrange equation of motion according to def. is

$$\begin{aligned} & \frac{\partial}{\partial x^\mu} F^{\mu \nu} = 0 \\ \Leftrightarrow\;\; & d \star_\eta F = 0 \end{aligned} \,,$$

where $F = d A$ is the Faraday tensor (?). (In the coordinate-free formulation in the second line “$\star_\eta$” denotes the Hodge star operator induced by the pseudo-Riemannian metric $\eta$ on Minkowski spacetime.)

These PDEs are called the vacuum Maxwell's equations.

This, too, is a free field theory according to def. .

Example

(equation of motion of Dirac field is Dirac equation)

Consider the Lagrangian field theory of the Dirac field on Minkowski spacetime from example , with field fiber the spin representation $S$ regarded as a superpoint $S_{odd}$ and Lagrangian density given by the spinor bilinear pairing

$$L \;=\; i \overline{\psi} \gamma^\mu \partial_\mu \psi + m \overline{\psi}\psi$$

(in spacetime dimension $p+1 \in \{3,4,6,10\}$ with $m = 0$ unless $p+1 = 4$).

By example the Euler-Lagrange differential operator (26) for the Dirac field is of the form

(32)$$\array{ \Gamma_\Sigma(\Sigma \times S) &\overset{ }{\longrightarrow}& \Gamma_\Sigma(\Sigma \times S^\ast) \\ \Psi &\mapsto& \overline{(-)} D \psi }$$

so that the corresponding Euler-Lagrange equation of motion (def. ) is equivalently

(33)$$\underset{D}{ \underbrace{ \left(-i \gamma^\mu \partial_\mu + m\right) }} \psi \;=\; 0 \,.$$

This is the Dirac equation and $D$ is called a Dirac operator. In terms of the Feynman slash notation from (?) the corresponding differential operator, the Dirac operator reads

$$D \;=\; \left( - i \partial\!\!\!/\, + m \right) \,.$$

Hence this is a free field theory according to def. .

Observe that the “square” of the Dirac operator is the Klein-Gordon operator $\Box - m^2$ (29)

$$\begin{aligned} \left( +i \gamma^\mu \partial_\mu + m \right) \left(-i \gamma^\mu \partial_\mu + m\right)\psi & = \left(\partial_\mu \partial^\mu - m^2\right) \psi \\ & = \left(\Box - m^2\right) \psi \end{aligned} \,.$$

This means that a Dirac field which solves the Dirac equations is in particular (on Minkowski spacetime) componentwise a solution to the Klein-Gordon equation.

Remark

(supergeometric nature of the Dirac equation as an Euler-Lagrange equation)

While the Dirac equation (33) of example would make sense in itself also if the field coordinates $\psi$ and jet coordinates $\psi_{,\mu}$ of the Dirac field were not anti-commuting (6), due to their supergeometric nature (remark ), it would, by remark , then no longer be the Euler-Lagrange equation of a Lagrangian density, hence then Dirac field theory would not be a Lagrangian field theory.

Example

(Dirac operator on Dirac spinors is formally self-adjoint differential operator)

The Dirac operator, hence the differential operator corresponding to the Dirac equation of example via def. is a formally anti-self adjoint (def. ):

$$D^\ast = - D \,.$$

Proof

By (32) we are to regard the Dirac operator as taking values in the dual spin bundle by using the Dirac conjugate $\overline{(-)}$ (?):

$$\array{ \Gamma_\Sigma(\Sigma \times S) &\overset{}{\longrightarrow}& \Gamma_\Sigma(\Sigma \times S^\ast) \\ \Psi &\mapsto& \overline{(-)} D \Psi }$$

Then we need to show that there is $K(-,-)$ such that for all pairs of spinor sections $\Psi_1, \Psi_2$ we have

$$\overline{\Psi_2}\gamma^\mu (\partial_\mu \Psi_1) - \overline{\Psi_1}\gamma^\mu (-\partial_\mu \Psi_2) \;=\; d K(\psi_1, \psi_2) \,.$$

But the spinor-to-vector pairing is symmetric (?), hence this is equivalent to

$$\overline{\partial_\mu \Psi_1}\gamma^\mu \Psi_2 + \overline{\Psi_1}\gamma^\mu (\partial_\mu \Psi_2) \;=\; d K(\psi_1, \psi_2) \,.$$

By the product law of differentiation, this is solved, for all $\Psi_1, \Psi_2$, by

$$K(\Psi_1, \Psi_2) \;\coloneqq\; \left( \overline{\Psi_1} \gamma^\mu \Psi_2\right) \, \iota_{\partial_\mu} dvol \,.$$

$\,$

This concludes our discussion of Lagrangian densities and their variational calculus. In the next chapter we consider the infinitesimal symmetries of Lagrangians and the conserved currents that these induce via Noether's theorem.