# Lagrangian formalism for fields

Post-publication activity

Curator: Dmitri Sorokin

The Lagrangian formalism is one of the main tools of the description of the dynamics of a vast variety of physical systems including systems with finite (particles) and infinite number of degrees of freedom (strings, membranes, fields). It is based on the action principle, a fundamental theoretical concept which, in particular, for more than a century has been a leading principle for the construction and development of the theory of fundamental (electro-magnetic, weak, strong and gravitational) interactions of elementary particles based on Quantum Field Theory. The action principle states that the classical motion of a given physical system is such that it extremizes a certain functional of dynamical variables called action. The form of the action determines the equations of motion (Euler-Lagrange equations) of the physical system, its symmetries and (via Noether's theorem) the corresponding conserved quantities, the integrals of motion. The action implements its full power in quantum theory where, via the Feynman path integral techniques, it determines the transition amplitude from an initial to final quantum state.

## General properties

The Field Theory action has the following generic properties. Let $$\Phi(x)$$ be a generic field in a D–dimensional space–time parametrized by the coordinates $$x^\mu$$ $$(\mu=0,1,\cdots,D-1)\ .$$ The coordinate $$x^0=ct$$ is the time coordinate times the speed of light whose value is often conventionally put equal to one (as in this article). The other $$D-1$$ coordinates parametrize the space directions. $$\Phi(x)$$ may stand, e.g. for a scalar field $$\phi(x)\ ,$$ a vector (gauge) field $$A_\mu(x)\ ,$$ the gravitational symmetric–tensor field $$g_{\mu\nu}(x)$$ or a fermionic spin 1/2 field $$\psi^\alpha(x)$$ (where $$\alpha$$ is the index of a spinorial representation of the Lorentz group $$O(1,D-1)$$). The action is the space–time integral of the Lagrangian density $$L(\Phi(x),\, \partial_\mu\,\Phi(x))$$ which depends on the field and its derivatives $\tag{1} S=\int\,d^D\,x\,L(\Phi(x),\, \partial_\mu\Phi(x))\,.$

The explicit form of the Lagrangian depends on which field theory is considered, but in all the cases it is usually assumed to satisfy a number of generic requirements:

• The Field Theory describing the fundamental interactions of elementary particles is assumed to be relativistic, i.e. invariant under $$O(1,D-1)$$ Lorentz group transformations in the D–dimensional space–time. Therefore, the action is required to be Lorentz invariant.
• Other symmetries, so called internal (global and local or gauge) symmetries, under which a given field-theoretical system is assumed to be invariant, further specify the structure of the action.
• The action is the real functional of the fields and their derivatives. This insures e.g. that in quantum theory the total probability is a conserved quantity.
• In the case of the bosonic fields, the Lagrangian is usually assumed to contain the derivatives of fields only up to the second order, while in the case of the fermionic fields it is of the first order in derivatives. These restrictions are imposed by the consistency requirement for the Quantum Field Theory to be unitary and causal. However, the Quantum Field Theory can still be consistent even if its Lagrangian contains higher order derivative terms. For instance, in String Theory the effective field theory action describing massless excitations of the string receives higher order (stringy) corrections whose form complies with the quantum consistency of the theory.
• Another restriction on the possible form of the Lagrangian is put by the requirement that the Quantum Field Theory be renormalizable, i.e. that a (finite) number of divergencies that can appear in quantum integrals can be hidden by redefining (renormalizing) basic parameters of the theory (such as mass and charge). For instance, this restricts a possible choice of the terms in the Lagrangian which depend on the fields but not on their derivatives. Such terms are called potential energy terms.

In simple cases the Lagrangian can be written as the difference of the kinetic and potential energy density $L=K(\partial_\mu\Phi\partial^\mu\Phi)-V(\Phi)\,,$ therefore, the action has a natural dimension of energy times time $$(E\,t)$$ which is the dimension of the Plank's constant $$\hbar\ .$$ This defines the dimension of the fields which constitute the action.

The equations of motion of the field are obtained by requiring that the variation of the action vanish for every classical field configuration, namely $\tag{2} \delta S=\int\,d^Dx\,\left(\frac{\delta L}{\delta\Phi}\,\delta\Phi +\,\frac{\delta L}{\delta(\partial_\mu\Phi)}\,\delta(\partial_\mu\Phi)\right) =\int\,d^Dx\,\left(\frac{\delta L}{\delta\Phi} -\partial_\mu\,\frac{\delta L}{\delta(\partial_\mu\Phi)}\right)\,\delta\Phi+\int\,d^Dx\,\partial_\mu\left(\frac{\delta L}{\delta(\partial_\mu\Phi)}\,\delta\Phi\right)=0\,,$

where the last integral is a total derivative. It vanishes if the field is not varied on the $$(D-1)$$–dimensional integration boundary $$\Sigma_{D-1}\ ,$$ i.e. $$\delta\Phi|_{\Sigma_{D-1}}=0\ ,$$ $\tag{3} \int\,d^Dx\,\partial_\mu\left(\frac{\delta L}{\delta(\partial_\mu\Phi)}\,\delta\Phi\right)=\oint\,d\Sigma^m_{D-1}\,\frac{\delta L}{\delta(\partial_\mu\Phi)}\,\delta \Phi=0\,.$

Since, the field variation $$\delta\Phi$$ is arbitrary everywhere except for the boundary, the first integral in eq. (2) vanishes if $\tag{4} \partial_\mu\,\frac{\delta L}{\delta(\partial_\mu\Phi)} -\frac{\delta L}{\delta\Phi}=0\,.$

These differential Euler-Lagrange equations are the equations of motion of the classical field $$\Phi(x)\ .$$ Since the first variation (2) of the action is zero when the field satisfies the equations of motion, the action itself, computed for the classical (on-shell) field configurations, takes an extremal (possibly, infinite) value.

Consider now several simple examples of field theory actions which are building blocks of the field–theoretical description of the theory of fundamental interactions.

## Scalar field

The action for a complex scalar field $$\phi(x)$$ of mass $$m$$ in a flat D–dimensional space-time with the Minkowski metric $$\eta_{\mu\nu}=\eta^{\mu\nu}={\rm diag}\,(-1,1\cdots,1),\,\,\eta_{\mu\nu}\,\eta^{\nu\rho}=\delta^{\rho}_\mu$$ has the following form $\tag{5} S=\int\,d^{D}x\,\left(-\frac{1}{2}\,\partial_\mu\phi^*\,\partial_\nu\phi\,\eta^{\mu\nu}-V(\phi^*\phi)\right)\,$

where $$\phi^*(x)$$ is the complex conjugate of $$\phi(x)\ ,$$ the first term in (5) is the kinetic term, and the second term is a potential $$V(\phi^*\phi)$$ which includes the mass term $$\frac{1}{2}\,m^2\,\phi^*\phi\ .$$ Notice that the action is the real functional of the complex field and its derivatives.

The form of the potential depends on the properties of a (usually) more general field theory which includes the scalar field. One of the restrictions on the form of $$V$$ is imposed by the quantum renormalizability of the theory. For instance, in (3+1)–dimensional space–time the potential of a renormalizable scalar field theory is a fourth order polynomial in the field $\tag{6} V=\frac{1}{2}\,m^2\,\phi^*\phi+\frac{g}{4}\,(\phi^*\phi)^2\,,$

where $$g$$ is a dimensionless coupling constant. Note that the canonical dimension of the scalar field is the inverse length $$L^{-1}\sim m$$ (in the units in which the speed of light is one).

The Euler-Lagrange equations (4) for the scalar field take the form $\tag{7} \partial_\mu\,\partial^\mu\,\phi-\frac{\delta\,V(\phi^*\phi)}{\delta\phi^*}=0\,$

plus their complex conjugate.

The action and the equations of motion are invariant under the $$U(1)-$$group phase transformations of the scalar field with a constant parameter $$\alpha$$ $\tag{8} \phi'=e^{i\alpha}\,\phi\,.$

An example of the scalar potential which leads to the so called spontaneous breaking of this symmetry is $\tag{9} V=\lambda(\phi^*\phi-v^2\,)^2,$

where $$\lambda$$ and $$v$$ are constants. In this case the solution of the equations of motion (7) with the minimal (zero) energy is $\tag{10} \phi=v\,e^{i\theta}\,,$

where $$\theta$$ is a constant. Since $$\phi$$ satisfying (10) has the zero energy, this value of $$\phi$$ is called the vacuum expectation value. It is obviously not invariant under the phase transformations (8). The symmetry is thus said to be broken spontaneously by the non–zero vacuum expectation value of the scalar field. This leads to profound physical effects governed by the Nambu-Goldstone theorem in the case of global continuous symmetries (e.g. presence of massless scalar fields called Nambu-Goldstone bosons) and by the Brout–Englert–Higgs–Guralnik–Hagen–Kibble mechanism in the case of gauge symmetries. The chase for scalar particles called Higgs bosons which are predicted by this mechanism is part of the experimental program of the Large Hadron Collider (LHC) at CERN (Geneva).

One can introduce the interaction of the complex scalar field with an electromagnetic field $$A_\mu(x)$$ by requiring that the scalar field action be invariant under the local phase (or $$U(1)$$ gauge) transformations whose parameters depend on the coordinates of space–time $\tag{11} \phi'=e^{iq\,\alpha(x)}\,\phi\,,$

where $$q$$ is an electric charge of the field. This is achieved by replacing the partial derivative $$\partial_m$$ with the covariant derivative $\tag{12} D_\mu=\partial_\mu+iq\,A_\mu(x)$

and requiring that the electromagnetic field has the following transformation properties $\tag{13} A'_\mu(x)=A_\mu(x)-\partial_\mu\,\alpha(x)\,.$

The scalar field action takes the form $\tag{14} S=\int\,d^{D}x\,\left(-\frac{1}{2}\,(D_\mu\phi)^*\,D_\nu\phi\,\eta^{\mu\nu}-V(\phi^*\phi)\right)\,.$

To consider a closed system of the scalar and electromagnetic fields one should add to eq. (14) the action which describes the dynamics of the vector field $$A_\mu(x)$$ itself. Note that the vector field $$A_{\mu}$$ describes relativistic particles which have the internal angular momentum or spin 1.

## Vector field

The action for the massless gauge field $$A_\mu(x)$$ (e.g. for the electromagnetic field) must be invariant under the gauge transformations (13). To construct such an action one introduces the gauge invariant anti-symmetric tensor called the field strength $\tag{15} F_{\mu\nu}=\partial_{\mu}\,A_\nu(x)-\partial_\nu\,A_\mu(x)\,.$

By construction, the field strength satisfies the Bianchi identity $\tag{16} \partial_{[\mu}\,F_{\nu\lambda]}= \frac{1}{3}(\partial_{\mu}\,F_{\nu\lambda}+\partial_{\nu}\,F_{\lambda\mu}+\partial_{\lambda}\,F_{\mu\nu})\equiv 0\,.$

The action is quadratic in $$F_{\mu\nu}$$ $\tag{17} S=-\int\, d^Dx\, \frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}\,.$

The Euler-Lagrange equations which follow from the action (17) are $\tag{18} \partial_\mu\,F^{\mu\nu}=0\,.$

Together with the Bianchi identity (16) they form the (Lorentz-covariant) Maxwell equations.

Adding to the action (17) the mass term $\tag{19} S_m=-\frac{1}{2}\,\int\,d^Dx\,m^2\,A_\mu A^\mu$

clearly breaks the gauge invariance.

The Euler-Lagrange equations for the coupled system of the scalar field and the electromagnetic field described by the sum of the actions (14) and (17) are $\tag{20} \partial_\mu\,F^{\mu\nu}=j^{\nu}\,,\qquad D_\mu\,D^\mu\phi-\frac{\delta V(\phi^*\phi)}{\delta\phi^*}=0\,,$

where $$j^\nu=\frac{iq}{2}\,(\phi(D^\nu\phi)^* -\phi^*D^{\nu}\phi)$$ is the scalar field electric current which is conserved ($$\partial_\nu\,j^\nu=0$$) in virtue of the scalar field equations of motion.

Physical models in which symmetries of the fields form a non-Abelian gauge group, such as the $$SU(3)\times SU(2)\times U(1)$$ group of the Standard Model of elementary particles, are described by the Yang-Mills gauge theory which is the non-Abelian generalization of the $$U(1)$$ gauge invariant Maxwell theory. Note that the quantum theory of massive vector fields is generically not renormalizable unless the mass is generated upon spontaneous breaking the gauge symmetry via the Brout–Englert–Higgs–Guralnik–Hagen-Kibble mechanism.

## Spinor fields

The spinor fields, i.e. the ones which transform under a spinor representation of the Lorentz group, describe relativistic particles of spin 1/2, for example electrons, neutrinos, quarks etc. The types of the spinor fields and the number of their components depend on the number of space-time dimensions. In what follows the spinor fields in four-dimensional space-time will be considered.

The most general spinor field $$\psi^\alpha(x)$$ in D=3+1 has four complex (or eight real) independent components $$(\alpha= 1,2,3,4)$$ it is called Dirac spinor. The complex spinor which has only two independent complex (or four real) components is called Weyl spinor. Two Weyl spinors form a Dirac spinor. The four-component spinor whose components can be made real in a certain (Majorana) basis is called Majorana spinor. In four-dimensional space-time the Weyl and Majorana spinors are related to each other (see below).

In contrast to the scalar and the vector fields, which are bosons i.e. obey the Bose-Einstein statistics, the spinor fields are fermions, i.e. obey the Fermi-Dirac statistics. This is required by the unitarity of the quantum theory. In classical field theory the fermionic fields anticommute with each other $\tag{21} \psi^\alpha(x)\,\psi^\beta(x)+\psi^\beta(x)\,\psi^\alpha(x)=0\,.$

The functions (and numbers) which have above properties are often called Grassmann-odd, while the conventional commuting functions (and numbers) are called Grassmann-even.

To construct the action and equations of motion for the spinor field one needs four-by-four constant Dirac matrices $$(\gamma^\mu)_\alpha{}^\beta$$ which act on the spinor indices of $$\psi^\alpha(x)\ .$$ There are four such matrices labeled by the index $$\mu=0,1,2,3$$ of the vector representation of the Lorentz group $$O(1,3)\ .$$ The simultaneous action of the Lorentz group on the vector and spinor indices leaves the Dirac matrices intact. In other words the constant Dirac matrices are $$O(1,3)$$ invariant. They satisfy the Clifford algebra $\tag{22} (\gamma^\mu)_\alpha{}^\gamma\,(\gamma^\nu)_\gamma{}^\beta +(\gamma^\nu)_\alpha{}^\gamma\,(\gamma^\mu)_\gamma{}^\beta=2\eta^{\mu\nu}\,\delta_\alpha{}^\beta\,,$

where $$\eta^{\mu\nu}$$ is the Minkowski metric and $$\delta_\alpha{}^\beta$$ is the unit matrix.

In the four-dimensional space-time the spinor indices can be lowered and raised by constant antisymmetric (charge conjugation) matrices $\tag{23} C_{\alpha\beta}=-C_{\beta\alpha}=-C^{\alpha\beta}=C^{\beta\alpha}\,,\qquad C_{\alpha\gamma}C^{\gamma\beta}=\delta_\alpha{}^\beta$

$\tag{24} \psi_\alpha=C_{\alpha\beta}\,\psi^{\beta}\,,\qquad \psi^{\alpha}=C^{\alpha\beta}\psi_\beta\,.$

These matrices are Lorentz-invariant.

The Lorentz-invariant action for a free Dirac field of mass $$m$$ has the following form $\tag{25} S=-\int\,d^4x\,\left[\frac{i}{2}(\bar\psi\,\gamma^\mu\partial_\mu\,\psi-\partial_\mu\bar\psi\,\gamma^\mu\,\psi) +i\,m\,\bar\psi\,\psi\right]\,$

where, for simplicity, the contracted spinor indices have been skipped and $$\bar\psi$$ is the so called Dirac adjoint spinor which is constructed by taking the complex conjugate and transpose spinor $$\psi^\dagger$$ of $$\psi$$ and multiplying it by $$\gamma^0$$ $\tag{26} \bar\psi=\psi^\dagger\,\gamma^0\,.$

The introduction of the Dirac adjoint spinor is required for the action to be Lorentz-invariant. Note also that the action is real with the complex conjugation of the product of two spinors being defined as follows $(\bar\psi_1\,\psi_2)^*=-\bar\psi_2\,\psi_1\,\qquad (\bar\psi_1\,\gamma^\mu\psi_2)^*=\bar\psi_2\,\gamma^\mu\psi_1.$ Varying the action (25) independently with respect to $$\bar\psi$$ and $$\psi$$ one gets the Dirac equations $\tag{27} \gamma^\mu\partial_\mu\psi+m\,\psi=0\,,\qquad \partial_\mu\bar\psi\gamma^\mu-m\,\bar\psi=0\,.$

The spinor is of the Majorana type (i.e. has four independent real components) if $\tag{28} \psi=C\bar\psi^T\,,$

where $$T$$ stands for transposition. On the other hand the Dirac spinor can be split into a left- and right-handed (Weyl) spinors which have two independent complex components each $\tag{29} \psi=\psi_L+\psi_R\,,$

where $\tag{30} \psi_L= \frac{1}{2}(1 + \gamma^5)\,\psi\,,\qquad \psi_R= \frac{1}{2}(1 - \gamma^5)\,\psi \qquad {\rm and}\qquad \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3\,.$

The matrices $$\frac{1}{2}(1 \pm \gamma^5)$$ are projectors having two non–zero unit eigenvalues each. Thus the spinors $$\psi_L$$ and $$\psi_R$$ have two independent complex components. In four-dimensional space-time a Majorana spinor is related to e.g. the left-handed Weyl spinor as follows $\tag{31} \psi_M=\psi_L+i\gamma^2\psi^*_L\,\qquad {\rm note~~that} \qquad \psi_M=i\gamma^2\psi^*_M\,.$

In terms of the left- and right-handed spinors the Dirac action (25) is (up to a total derivative) $\tag{32} S=-\int\,d^4x\,\left[i\bar\psi_L\,\gamma^\mu\partial_\mu\,\psi_L+i\bar\psi_R\,\gamma^\mu\partial_\mu\,\psi_R +m\,(i\bar\psi_L\,\psi_R+i\bar\psi_R\,\psi_L)\right]\,.$

When the mass is zero the action splits into two independent actions for the left- and right-handed spinors. This manifests the fact that massless fermions in four-dimensional space-time are described by the Weyl spinors and carry a definite left- or right-handed helicity. Nevertheless, a Majorana spinor field can have a non-zero mass, in which case in (32) $$\psi_R=i\gamma^2\psi^*_L\ .$$ E.g. according to present experiments, neutrinos (which previously were believed to be massless) possess tiny Majorana masses.

The actions (25) and (32) are invariant under the rigid $$U(1)$$ transformations of the Dirac field $\psi'=e^{i\alpha}\,\psi\,.$ Promoting this symmetry to the local Abelian gauge symmetry and replacing the partial derivative $$\partial_\mu$$ with the covariant derivative (12), as in the case of the scalar fields, one couples charged fermions to an electromagnetic field.

## Supersymmetric field theories

Bosonic and fermionic fields may form supersymmetric systems. Supersymmetry is a radically new type of (still hypothetical) symmetry of space-time and fundamental fields. It establishes a relationship between elementary particles of different quantum nature, bosons and fermions, unifies in a non-trivial way space-time and internal symmetries of the microscopic World and drastically improves properties of the Quantum Field Theory. Supersymmetry was first revealed in two-dimensional dual models of spinning strings by P. Ramond, A. Neveu and J.H. Schwarz, and by J.-L. Gervais and B. Sakita in 1971. Supersymmetry in four-dimensional field theory was put forward independently by Yu. A. Golfand and E.P. Likhtman (in 1971), Volkov and V.P. Akulov (in 1972) and by J. Wess and B. Zumino (in 1973).

### Wess-Zumino model

The simplest supersymmetric action in four-dimensional space-time is that of a massless complex scalar and Majorana spinor field $\tag{33} S=-\frac{1}{2}\,\int\,d^4x\left(\partial_\mu\phi^*\,\partial_\nu\phi\,\eta^{\mu\nu} +i\bar\psi_M\,\gamma^\mu\partial_\mu\,\psi_M\right).$

It is invariant under supersymmetry with a Grassmann-odd constant Majorana spinor parameter $$\epsilon^\alpha$$ which transforms the bosonic field into the fermionic one and vice versa $\tag{34} \delta\phi=\frac{i}{2}\,\bar\epsilon(1+\gamma^5)\,\psi_M\,,\qquad \delta\phi^*= \frac{i}{2}\,\bar\psi_M(1-\gamma^5)\,\epsilon\,,$

$\delta\psi_M=\frac{1}{4}\,\left[\partial_\mu(\phi+\phi^*)\gamma^\mu\epsilon -\partial_\mu(\phi-\phi^*)\gamma^\mu\gamma^5\epsilon\right].$ Adding to the free action (33) the interacting terms with a dimensionless coupling constant $$g$$ $S_{int}=-\frac{1}{2}\int\,d^4x\,\left[i g\bar\psi_M(1-\gamma_5)\psi_M\,\phi+ig\bar\psi_M(1+\gamma_5)\psi_M\,\phi^*+g^2\,(\phi^*\,\phi)^2\right]$ one gets the supersymmetric model by Wess and Zumino (with the supersymmetry variations (34) acquiring corresponding additional terms).

### Volkov-Akulov model

As has been first shown by Volkov and Akulov in 1972, it is possible to realize supersymmetry on a single fermionic Majorana field $$\theta(x)$$ and the space–time coordinate $$x^\mu$$ $\tag{35} x^{\prime\mu}=x^\mu-i\bar\epsilon\gamma^\mu\theta(x)\,,\qquad \theta'(x')=\theta(x)+\epsilon$

Eqs. (35) imply that the spinor field undergoes infinitesimal supersymmetry variation $\tag{36} \delta\theta(x)=\epsilon+i(\bar\epsilon\gamma^\mu\theta)\,\partial_\mu\theta(x)$

which is non-linear in $$\theta(x)\ .$$

The supersymmetry transformations (35) leave invariant the differential one-form $\tag{37} e^\mu(x)=dx^\mu+i\bar\theta\gamma^\mu\,d\theta(x) =dx^\nu(\delta_\nu^\mu+i\bar\theta\gamma^\mu\,\partial_\nu\theta)=dx^\nu\,e_\nu{}^\mu(x)\,.$

This one-form can be used to construct the supersymmetric Volkov-Akulov action as the wedge product of $$e^\mu(x)$$ $\tag{38} S=-\frac{1}{4!}\,\int\,\varepsilon_{\mu\nu\rho\lambda}\,e^\mu \wedge e^\nu\wedge e^\rho\wedge e^\lambda= -\int\,d^4x\, \det (\delta_\mu^\nu+i\bar\theta\gamma^\nu\,\partial_\mu\,\theta) \,,$

where $$\varepsilon_{\mu\nu\rho\lambda}$$ is the totally anti-symmetric unit tensor $$\varepsilon_{0123}=1\ .$$

The Euler-Lagrange equations obtained by varying (38) are $\tag{39} e_\mu{}^\nu\,\gamma^\mu\,\partial_\nu\theta\, +\frac{1}{{\det}e}\, \partial_\nu\left(e_\mu{}^\nu\gamma^\mu\,\theta\,\det\,e\right)=0\,, \quad {\rm where} \quad \det\,e =\det(\delta_\mu^\nu+i\bar\theta\gamma^\nu\,\partial_\mu\,\theta).$

A vacuum solution of eq. (39) is a constant spinor $\theta^\alpha=c^\alpha\,,$ which in view of the form of the supersymmetry transformations (35) implies that $$\theta(x)$$ are Goldstone fermions that spontaneously break supersymmetry (compare with symmetry breaking by the vacuum expectation value of the scalar field (9) and (10)). The Volkov-Akulov model was the first example of field theory with spontaneously broken supersymmetry.

Search for manifestations of supersymmetry in Nature and, in particular, superpartners of known elementary particles is in the agenda of the LHC experiments at CERN and at other research centers.

Simple examples of Field Theory actions and Euler-Lagrange equations considered above are building blocks of more complicated field-theoretical models of the fundamental constituents of Nature.