Lagrangian Mechanics for Infinite-Dimensional Systems

The elegance of the Lagrangian formalism (from previous post) extends seamlessly from discrete particle systems to continuous physical systems, such as fields and fluids, which possess an infinite number of degrees of freedom.

Lagrangian Density

For classical field theory, the concept of a single Lagrangian $\displaystyle L$ is replaced by a Lagrangian density, denoted by $\displaystyle \mathcal{L}$ . A field, say $\displaystyle \phi(\mathbf{x},t)$ , is a quantity defined at every point in space and time, effectively representing an infinite number of generalized coordinates. The configuration space for classical field theory is thus an infinite-dimensional vector space of functions or tensor fields on space or spacetime.

The Lagrangian density $\displaystyle \mathcal{L}$ is a function of the field variables $\displaystyle \phi_r$ (where $\displaystyle r$ indexes different fields if multiple are present) and their first derivatives with respect to spacetime coordinates $\displaystyle x^\mu$ (where $\displaystyle x^0=ct$ is the time component and $\displaystyle x^j$ are spatial components). It is typically written as $\displaystyle \mathcal{L}(\phi_r, \partial_\mu \phi_r)$ . The total Lagrangian $\displaystyle L$ for the system is then obtained by integrating this density over the spatial volume:

$\displaystyle L=\int\mathcal{L}(\phi_r, \partial_\mu \phi_r)d^3x$

The action integral $\displaystyle S$ for a field theory is defined as the integral of the Lagrangian density over a four-dimensional spacetime region $\displaystyle \Omega$ (volume element $\displaystyle d^4x=d^3xdt$ ):

$\displaystyle S(\Omega)=\int_{\Omega}\mathcal{L}(\phi_r, \partial_\mu \phi_r)d^4x$

The equations of motion for fields are derived by applying Hamilton’s Principle ( $\displaystyle \delta S=0$ ) to this action integral. This process involves functional derivatives, naturally extending the calculus of variations to infinite-dimensional spaces. The result is the continuum Euler-Lagrange equation, also known as the Euler-Lagrange field equation:

Continuum Euler-Lagrange equation

Consider a small variation $\displaystyle \delta\phi_r(\mathbf{x},t)$ in the field $\displaystyle \phi_r$ . The variation in the action is:

$\displaystyle \delta S = \int \delta\mathcal{L} d^4x$

Using the chain rule for the variation of the Lagrangian density:

$\displaystyle \delta\mathcal{L} = \frac{\partial\mathcal{L}}{\partial\phi_r}\delta\phi_r + \sum_{\mu}\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_r)}\delta(\partial_\mu\phi_r)$

Since variation and differentiation commute, $\displaystyle \delta(\partial_\mu\phi_r) = \partial_\mu(\delta\phi_r)$ . So:

$\displaystyle \delta S = \int \left( \frac{\partial\mathcal{L}}{\partial\phi_r}\delta\phi_r + \sum_{\mu}\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_r)}\partial_\mu(\delta\phi_r) \right) d^4x$

The second term can be transformed using integration by parts (the four-dimensional generalization of $\displaystyle \int u dv = uv - \int v du$ , where boundary terms vanish for variations zero at the boundary):

$\displaystyle \int \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_r)}\partial_\mu(\delta\phi_r) d^4x = \left[ \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_r)}\delta\phi_r \right]_{\text{boundary}} - \int \partial_\mu \left( \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_r)} \right)\delta\phi_r d^4x$

Since variations $\displaystyle \delta\phi_r$ are assumed to be zero at the integration boundaries, the boundary term vanishes. Thus, $\displaystyle \delta S$ becomes:

$\displaystyle \delta S = \int \left( \frac{\partial\mathcal{L}}{\partial\phi_r} - \sum_{\mu}\partial_\mu \left( \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_r)} \right) \right)\delta\phi_r d^4x$

For $\displaystyle \delta S$ to be zero for arbitrary variations $\displaystyle \delta\phi_r$ , the integrand must be identically zero. This yields the continuum Euler-Lagrange equation:

$\displaystyle \frac{\partial\mathcal{L}}{\partial\phi_r} - \sum_{\mu} \partial_\mu \left( \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_r)} \right) = 0$

This partial differential equation describes how the field $\displaystyle \phi_r$ evolves in spacetime. The Lagrangian acts as a concise blueprint for the partial differential equations that govern the system’s dynamics. This implies a powerful, systematic way to construct physical theories: postulate a Lagrangian density based on symmetries and known interactions, and the equations of motion.

Example 1: Electromagnetic Field (Maxwell’s Equations)

This is a cornerstone example, demonstrating how the fundamental laws governing classical electromagnetism can be elegantly derived from a single scalar Lagrangian density. This derivation is a profound validation of the Lagrangian formalism’s power in describing continuous fields and forms the bedrock for quantum electrodynamics.

Field Variables: The fundamental field variable is the four-vector potential $\displaystyle A^\mu(\mathbf{x},t)$ . This relativistic four-vector unifies the scalar electric potential $\displaystyle \Phi$ and the three-dimensional magnetic vector potential $\displaystyle \mathbf{A}$ :

$\displaystyle A^\mu = \left( \frac{\Phi}{c}, \mathbf{A} \right)$

(using conventional units, with $\displaystyle A^0=\Phi/c$ and $\displaystyle A^i$ as spatial components).

Lagrangian Density (L): For the free (source-free) electromagnetic field, the Lorentz-invariant Lagrangian density is:

$\displaystyle \mathcal{L}=-\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu}$

Here, $\displaystyle \mu_0$ is the permeability of free space. The core of this Lagrangian is the electromagnetic field tensor $\displaystyle F_{\mu\nu}$ , curvature defined solely in terms of the derivatives of the four-potential:

$\displaystyle F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$

This anti-symmetric tensor compactly encodes both the electric and magnetic fields. For instance, $\displaystyle F_{0i} = \frac{E_i}{c}$ and $\displaystyle F_{ij} = -\epsilon_{ijk}B_k$ . The term $\displaystyle F^{\mu\nu}F_{\mu\nu}$ is a Lorentz scalar, ensuring the action’s invariance under relativistic transformations.

Euler-Lagrange Field Equation : The equations of motion for the electromagnetic field are derived by applying the field Euler-Lagrange equation, varying with respect to $\displaystyle A_\sigma$ (where $\displaystyle \sigma$ is a specific index, running from $\displaystyle 0$ to $\displaystyle 3$ ):

$\displaystyle \frac{\partial\mathcal{L}}{\partial A_\sigma} - \partial_\mu \left( \frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\sigma)} \right) = 0$

This set of four equations (one for each value of $\displaystyle \sigma = 0, 1, 2, 3$ ) represents two of Maxwell’s four equations in their homogeneous, source-free form.

Specifically, for $\displaystyle \sigma = 0$ , it yields $\displaystyle \nabla\cdot\mathbf{E}=0$ . (Gauss Law)

For $\displaystyle \sigma = i$ (spatial indices), it yields $\displaystyle \nabla\times\mathbf{B}=\frac{1}{c^2}\frac{\partial\mathbf{E}}{\partial t}$ . (Ampere’s Law)

To obtain the inhomogeneous Maxwell’s equations (those that include sources like charge density $\displaystyle \rho$ and current density $\displaystyle \mathbf{J}$ ), a matter current term $\displaystyle -\mu_0 J^\mu A_\mu$ must be added to the Lagrangian density. The total Lagrangian density for electromagnetism with sources is the sum of the free field Lagrangian density and this interaction term:

$\displaystyle \mathcal{L}_{\text{total}} = \mathcal{L}_{\text{free EM}} + \mathcal{L}_{\text{int}} = -\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu} - \mu_0 J^\mu A_\mu$

Example 2: Real Scalar Field (Klein-Gordon Equation)

This example is about a real-valued scalar field, $\displaystyle \phi(\mathbf{x},t)$ , which often represents a spinless neutral boson (e.g., the Higgs boson in some contexts) with mass $\displaystyle m$ .

Field Variable: $\displaystyle \phi(x^\mu)$ (where $\displaystyle x^\mu$ denotes spacetime coordinates).
Lagrangian Density (L): The dynamics of this field are governed by the following Lagrangian density: $\displaystyle \mathcal{L}=\frac{1}{2}(\partial_\mu\phi\partial^\mu\phi-m^2\phi^2)$ Here, $\displaystyle \partial_\mu\phi\partial^\mu\phi$ represents a Lorentz-invariant kinetic term. In expanded form (using the metric signature $\displaystyle (+,-,-,-)$ ( setting $\displaystyle c=1$ for simplicity): $\displaystyle \partial_\mu\phi\partial^\mu\phi = (\partial_t\phi)^2-(\nabla\phi)^2$ . The $\displaystyle m^2\phi^2$ term represents the field’s mass.
Euler-Lagrange Field Equation: The equation of motion is derived from the standard field Euler-Lagrange equation:
- First, compute the partial derivatives: $\displaystyle \frac{\partial\mathcal{L}}{\partial\phi}=-m^2\phi$
  $\displaystyle \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}=\partial^\mu\phi$ (arising from $\displaystyle \frac{1}{2}\frac{\partial}{\partial(\partial_\mu\phi)}(\partial_\nu\phi\partial^\nu\phi) = \frac{1}{2}\frac{\partial}{\partial(\partial_\mu\phi)}(\eta^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi)$ )
- Substituting these into the Euler-Lagrange equation: $\displaystyle -m^2\phi-\partial_\mu(\partial^\mu\phi)=0$
  Rearranging terms: $\displaystyle (\partial^\mu\partial_\mu+m^2)\phi=0$ .
- Expanding the D’Alembertian operator $\displaystyle \partial^\mu\partial_\mu=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2$ (reinserting $\displaystyle c$ for clarity):
  
  $\displaystyle \left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2+m^2\right)\phi=0$ . This is the Klein-Gordon equation, a fundamental relativistic wave equation describing the dynamics of massive scalar particles.

Example 3: Complex Scalar Field

A complex scalar field $\displaystyle \psi(\mathbf{x},t)$ is essential for describing charged spinless particles, unlike its real counterpart. It can be thought of as combining two real scalar fields, e.g., $\displaystyle \psi=\frac{1}{\sqrt{2}}(\phi_1+i\phi_2)$ . For variational purposes, $\displaystyle \psi$ and its complex conjugate $\displaystyle \psi^*$ are treated as independent field variables.

Field Variables: $\displaystyle \psi(x^\mu)$ and $\displaystyle \psi^*(x^\mu)$ .
Lagrangian Density (L): The Lagrangian density for a complex scalar field with mass $\displaystyle m$ is:
$\displaystyle \mathcal{L}=(\partial_\mu\psi^*)(\partial^\mu\psi)-m^2\psi^*\psi$
This Lagrangian is notably invariant under a global U(1) phase transformation $\displaystyle \psi\to e^{i\alpha}\psi$ (where $\displaystyle \alpha$ is a constant real number). By Noether’s Theorem, this continuous symmetry directly implies the existence of a conserved quantity, which is interpreted as the electric charge of the field.
Euler-Lagrange Field Equation: To obtain the equation of motion for , we vary with respect to :
- First, compute the partial derivatives: $\displaystyle \frac{\partial\mathcal{L}}{\partial\psi^*}=-m^2\psi$
  $\displaystyle \frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi^*)}=\partial^\mu\psi$
- Substituting these into the Euler-Lagrange equation: $\displaystyle -m^2\psi-\partial_\mu(\partial^\mu\psi)=0$
  Rearranging terms: $\displaystyle (\partial^\mu\partial_\mu+m^2)\psi=0$ This is the Klein-Gordon equation for the complex scalar field. A completely analogous equation, $\displaystyle (\partial^\mu\partial_\mu+m^2)\psi^*=0$ , is obtained by varying the Lagrangian with respect to $\displaystyle \psi$ . This system describes charged, spinless particles consistent with special relativity.

Example 4: Nonlinear Sigma Models

A nonlinear sigma model describes a field $\displaystyle \Sigma$ that takes values not in a flat Euclidean space, but in a curved target manifold T. The field $\displaystyle \Sigma$ is thus a differentiable map from spacetime to this target manifold. This geometric constraint on the field’s values is a defining feature and inherently introduces nonlinearity into the theory.

Field Variable: $\displaystyle \Sigma^a(x^\mu)$ , where $\displaystyle a$ indexes the coordinates on the target manifold $\displaystyle T$ .
Lagrangian Density (L): The kinetic term in the Lagrangian density is directly related to the geometry of the target manifold. $\displaystyle \mathcal{L}=\frac{1}{2}g_{ab}(\Sigma)(\partial_\mu\Sigma^a)(\partial^\mu\Sigma^b)-V(\Sigma)$ Here, $\displaystyle g_{ab}(\Sigma)$ is the Riemannian metric on the target manifold $\displaystyle T$ . Its dependence on the field $\displaystyle \Sigma$ itself is precisely what makes the Lagrangian intrinsically nonlinear. $\displaystyle V(\Sigma)$ is a potential term, which also depends on the field’s position on the target manifold.
Euler-Lagrange Field Equations: Applying the Euler-Lagrange field equations to such a Lagrangian generally yields complex, highly non-linear partial differential equations. These equations describe the dynamics of the field $\displaystyle \Sigma$ as it evolves on the curved target manifold.
A prominent example is the O(3) nonlinear sigma model in (1+1) dimensions (one spatial, one time dimension). In this case, the field $\displaystyle \mathbf{n}(x^\mu)$ is a unit vector, mapping spacetime to a 2-sphere ( $\displaystyle S^2$ ), i.e., $\displaystyle \mathbf{n}\cdot\mathbf{n}=1$ . The metric $\displaystyle g_{ab}(\Sigma)$ for the $\displaystyle S^2$ manifold ensures this constraint is maintained. Its Lagrangian density (often without a potential term, $\displaystyle V=0$ ) is: $\displaystyle \mathcal{L}=\frac{1}{2}(\partial_\mu\mathbf{n})\cdot(\partial^\mu\mathbf{n})$ with the constraint $\displaystyle \mathbf{n}\cdot\mathbf{n}=1$ . This constraint, fundamental to the manifold’s geometry, is usually either explicitly enforced via Lagrange multipliers or implicitly handled by parameterizing $\displaystyle \mathbf{n}$ using two independent variables (e.g., spherical angles).

Example 5: Proca Field (Massive Vector)

The Proca equation is a relativistic wave equation that describes massive vector bosons (spin-1 particles), in contrast to the massless photon governed by Maxwell’s equations. Such particles include the W and Z bosons that mediate the weak nuclear force.

Field Variable: The fundamental field is a four-vector field $\displaystyle A^\mu(x^\mu)$ , similar to the electromagnetic potential.
Lagrangian Density (L): The Lagrangian density for a free Proca field includes the standard kinetic term for a vector field (identical to the free electromagnetic field Lagrangian) and a new mass term: $\displaystyle \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}m^2A^\mu A_\mu$ (Here, we’ve set $\displaystyle c=1$ for simplicity, and $\displaystyle m$ is the mass of the vector boson). The kinetic term $\displaystyle -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ involves the electromagnetic field tensor $\displaystyle F^{\mu\nu}=\partial^\mu A^\nu - \partial^\nu A^\mu$ . The mass term $\displaystyle \frac{1}{2}m^2A^\mu A_\mu$ is a Lorentz scalar, ensuring the overall Lagrangian’s relativistic invariance. This term is what gives the vector boson its mass, unlike the photon.
Euler-Lagrange Field Equation : We apply the Euler-Lagrange field equation, varying with respect to
- Partial derivatives:
  - Derivative with respect to $\displaystyle A_\nu$ : Only the mass term depends explicitly on $\displaystyle A_\nu$ .
    
    $\displaystyle \frac{\partial\mathcal{L}}{\partial A_\nu} = \frac{\partial}{\partial A_\nu}\left(\frac{1}{2}m^2A^\mu A_\mu\right) = \frac{1}{2}m^2\frac{\partial}{\partial A_\nu}(A^\mu\eta_{\mu\rho}A^\rho)$
    
    $\displaystyle = \frac{1}{2}m^2( \delta^\nu_\mu \eta_{\mu\rho}A^\rho + A^\mu \eta_{\mu\rho}\delta^\nu_\rho ) = \frac{1}{2}m^2(A_\nu + A_\nu) = m^2A_\nu$
  - Derivative with respect to $\displaystyle \partial_\mu A_\nu$ : Only the kinetic term depends on derivatives. As shown in the Maxwell’s equations derivation, this yields:
    
    $\displaystyle \frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)} = -\frac{1}{4}\frac{\partial}{\partial(\partial_\mu A_\nu)}(F^{\alpha\beta}F_{\alpha\beta}) = -F^{\mu\nu}$
- Substituting into Euler-Lagrange Equation:
  $\displaystyle m^2A_\nu - \partial_\mu (-F^{\mu\nu}) = 0$
  
  $\displaystyle \partial_\mu F^{\mu\nu} + m^2A^\nu = 0$ .

This is the Proca equation.

Implication for Massive Vectors ( $\displaystyle m\neq 0$ ): A crucial consequence of the Proca equation for a massive vector field is that it automatically implies the Lorenz gauge condition ( $\displaystyle \partial_\mu A^\mu = 0$ ), which is usually a choice (gauge fixing) for massless photons. To see this, take the four-divergence of the Proca equation: $\displaystyle \partial_\nu (\partial_\mu F^{\mu\nu} + m^2A^\nu) = 0$ $\displaystyle \partial_\nu \partial_\mu F^{\mu\nu} + m^2\partial_\nu A^\nu = 0$ The first term vanishes identically due to the antisymmetry of $\displaystyle F^{\mu\nu}$ ( $\displaystyle F^{\mu\nu}=-F^{\nu\mu}$ ) and the commutativity of partial derivatives ( $\displaystyle \partial_\nu\partial_\mu=\partial_\mu\partial_\nu$ ). Specifically, $\displaystyle \partial_\nu\partial_\mu F^{\mu\nu} = \partial_\nu\partial_\mu (\partial^\mu A^\nu - \partial^\nu A^\mu)$ . By swapping dummy indices $\displaystyle \mu \leftrightarrow \nu$ in the first part, one can show it is identically zero. Thus, we are left with: $\displaystyle m^2\partial_\nu A^\nu = 0$ . Since $\displaystyle m \neq 0$ for a massive particle, this forces $\displaystyle \partial_\nu A^\nu = 0$ . This intrinsic condition means that massive vector fields have three physical degrees of freedom (polarizations)– one constraint coming from the dynamics. The mass term breaks the gauge invariance, so you cannot reduce the degrees of freedom. For photons or case of massless fields, you dont have this dynamic constraint, but the gauge symmetries reduces the degrees of freedom to two. This is a vital distinction in particle physics.

Example 6: Yang-Mills Fields

Yang-Mills theory is a profound generalization of electromagnetism, extending the concept of local gauge invariance from the Abelian group $\displaystyle U(1)$ (for photons) to non-Abelian Lie groups (e.g., $\displaystyle SU(2)$ for electroweak theory, $\displaystyle SU(3)$ for quantum chromodynamics). These theories describe fundamental forces mediated by multiple types of gauge bosons.

Field Variable: The fundamental field is the gauge field $\displaystyle A^a_\mu(x^\mu)$ , a four-vector field where the index $\displaystyle a$ runs over the generators of the Lie algebra (infinitesimal elements near the identity) associated with the specific gauge group.
Lagrangian Density (L): The Lagrangian density for a free Yang-Mills field is: $\displaystyle \mathcal{L}=-\frac{1}{4}F^{a\mu\nu}F_{a\mu\nu}$ where $\displaystyle F^a_{\mu\nu}$ is the non-Abelian field strength tensor:

$\displaystyle F^a_{\mu\nu}=\partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g f^{abc} A^b_\mu A^c_\nu$

This tensor differs crucially from its Abelian (electromagnetic) counterpart by the presence of the non-linear term $\displaystyle g f^{abc} A^b_\mu A^c_\nu$ . Here, $\displaystyle g$ is the coupling constant, and $\displaystyle f^{abc}$ are the structure constants of the Lie algebra. This non-linear term directly implies that the gauge bosons of Yang-Mills theory (e.g., gluons in QCD) interact with each other, unlike photons.
Euler-Lagrange Field Equations: Applying the Euler-Lagrange equations to this Lagrangian yields the non-linear Yang-Mills equations, which describe the dynamics of these self-interacting gauge fields. This inherent non-linearity, stemming from the structure of $\displaystyle F^a_{\mu\nu}$ , is responsible for key phenomena like asymptotic freedom and confinement in Quantum Chromodynamics.

Example 7: Dirac Field

The Dirac equation is a cornerstone of relativistic quantum mechanics, describing the dynamics of spin-½ particles such as electrons and quarks. These particles are characterized by their intrinsic angular momentum (spin) and fermionic statistics. The field representing these particles is not a simple scalar or vector, but a spinor $\displaystyle \psi$ . A spinor is a mathematical object with components that transforms in a specific way under rotations and Lorentz transformations, uniquely suited to describe particles with half-integer spin. The field is a four-component spinor, which is essential to incorporate both spin and antiparticles.

Field Variables: The primary field variable is the Dirac spinor $\displaystyle \psi(x^\mu)$ . For variational purposes, $\displaystyle \psi$ and its Dirac adjoint $\displaystyle \bar{\psi}(x^\mu)=\psi^\dagger(x^\mu)\gamma^0$ are treated as independent field variables ( $\displaystyle \psi^\dagger$ is the hermitian conjugate, and $\displaystyle \gamma^0$ is a specific Dirac gamma matrix).
Lagrangian Density (L): The Lorentz-invariant Lagrangian density for a free Dirac field with mass $\displaystyle m$ is: $\displaystyle \mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi$ Here, $\displaystyle \gamma^\mu$ are the constant Dirac gamma matrices, which encode the relativistic spacetime algebra and allow the equation to be first-order in derivatives. The term $\displaystyle i\gamma^\mu\partial_\mu$ is the kinetic part, and $\displaystyle m$ is the mass term.
Euler-Lagrange Field Equation: To derive the Dirac equation for , we vary the Lagrangian with respect to :
- First, compute the partial derivatives: $\displaystyle \frac{\partial\mathcal{L}}{\partial\bar{\psi}}=(i\gamma^\mu\partial_\mu-m)\psi$
  
  (treating $\displaystyle \bar{\psi}$ as independent of $\displaystyle \partial_\mu\bar{\psi}$ and applying the rule for matrix products: $\displaystyle \frac{\partial}{\partial \bar{\psi}}(\bar{\psi} M \psi) = M \psi$ )
  
  $\displaystyle \frac{\partial\mathcal{L}}{\partial(\partial_\mu\bar{\psi})}=0$ (as the Lagrangian density does not contain derivatives of $\displaystyle \bar{\psi}$ ).
- Substituting these into the Euler-Lagrange equation:
  
  $\displaystyle (i\gamma^\mu\partial_\mu-m)\psi-0=0$ $\displaystyle (i\gamma^\mu\partial_\mu-m)\psi=0$
  
  This is the Dirac equation, a relativistic quantum mechanical equation for spin-½ particles. It automatically predicts the existence of antiparticles and describes intrinsic spin. A similar variational process with respect to $\displaystyle \psi$ (and $\displaystyle \partial_\mu\psi$ ) yields the adjoint Dirac equation, which governs the dynamics of $\displaystyle \bar{\psi}$ . If we assume $\displaystyle m=0$ , the Massless Dirac fermions admit a second, inequivalent U(1) symmetry.

Example 8: Weyl and Majorana Fermions

Weyl and Majorana fermions are special types of spin-½ particles that represent distinct fundamental symmetries compared to Dirac fermions. They are crucial in theories beyond the Standard Model.

Weyl Fermions: These are massless spin-½ particles with a definite helicity (a projection of spin onto momentum direction). They are described by two-component complex spinors.
- Field Variable: Weyl spinor ( $\displaystyle \chi$ or $\displaystyle \xi$ , two-component).
- Lagrangian Density (L):
  - For a right-handed Weyl fermion ( $\displaystyle \chi$ ): $\displaystyle \mathcal{L}=i\chi^\dagger\bar{\sigma}^\mu\partial_\mu\chi$
  - For a left-handed Weyl fermion ( $\displaystyle \xi$ ): $\displaystyle \mathcal{L}=i\xi^\dagger\sigma^\mu\partial_\mu\xi$ Here, $\displaystyle \sigma^\mu=(I,\sigma^i)$ and $\displaystyle \bar{\sigma}^\mu=(I,-\sigma^i)$ , where $\displaystyle I$ is the $\displaystyle 2\times2$ identity matrix and $\displaystyle \sigma^i$ are the Pauli matrices. The absence of a mass term is intrinsic to Weyl fermions; any explicit mass term would mix left- and right-handed components.
- Euler-Lagrange Field Equations: Applying the Euler-Lagrange equations yields the Weyl equation:
  
  $\displaystyle i\bar{\sigma}^\mu\partial_\mu\chi=0$ for right-handed, and $\displaystyle i\sigma^\mu\partial_\mu\xi=0$ for left-handed.
Majorana Fermions: These are electrically neutral spin-½ particles that are their own antiparticles. They are four-component spinors, like Dirac fermions, but subject to an additional “reality” condition.
- Field Variable: Majorana spinor ( $\displaystyle \psi_M$ , four-component).
- Lagrangian Density (L):
  
  $\displaystyle \mathcal{L}=\frac{1}{2}\bar{\psi}_M(i\gamma^\mu\partial_\mu-m)\psi_M$
  
  This Lagrangian is similar to the Dirac Lagrangian but includes a factor of $\displaystyle 1/2$ and an explicit Majorana condition:
  
  $\displaystyle \psi_M = \psi_M^c$
  
  (where $\displaystyle \psi_M^c$ is the charge conjugate of $\displaystyle \psi_M$ ). This condition forces the particle to be identical to its antiparticle, leading to electrical neutrality. Unlike Weyl fermions, Majorana fermions can be massive.
- Euler-Lagrange Field Equations: Applying the Euler-Lagrange equations to this Lagrangian, while respecting the Majorana condition, yields the Majorana equation:
  
  $\displaystyle (i\gamma^\mu\partial_\mu-m)\psi_M=0$ .

Lagrangian Formulation of Fluid Mechanics (Clebsch variables)

Fluid dynamics can be described from two perspectives: the Eulerian (fixed point in space) and the Lagrangian (following individual fluid parcels). While the Eulerian description is more common for practical applications, formulating fluid dynamics from a Lagrangian action principle is highly insightful, though challenging due to the non-canonical nature of velocity and density as independent generalized coordinates. This challenge is elegantly overcome using Clebsch variables.

Lagrangian Density:

$\displaystyle \mathcal{L}_{\text{fluid}}=\frac{1}{2}\rho v^2-\rho U(\rho,S)+\rho(\partial_t\phi+\mathbf{v}\cdot\nabla\phi+\alpha\dot{\beta}-\beta\dot{\alpha})$

Here, $\displaystyle \rho$ is the fluid density, $\displaystyle \mathbf{v}$ is the fluid velocity, and $\displaystyle U(\rho,S)$ is the specific internal energy as a function of density and entropy $\displaystyle S$ . The scalar fields $\displaystyle \phi, \alpha, \beta$ are the Clebsch potentials.
The first term, $\displaystyle \frac{1}{2}\rho v^2$ , represents the kinetic energy density.
The second term, $\displaystyle -\rho U(\rho,S)$ , represents the potential energy density associated with the fluid’s internal energy.
The crucial third term, $\displaystyle \rho(\partial_t\phi+\mathbf{v}\cdot\nabla\phi+\alpha\dot{\beta}-\beta\dot{\alpha})$ , is an interaction term that enforces the specific parameterization of the velocity field. The fluid velocity $\displaystyle \mathbf{v}$ is implicitly defined in terms of the Clebsch potentials and their derivatives as $\displaystyle \mathbf{v} = \nabla\phi + \alpha\nabla\beta$ (for barotropic fluids, entropy is often incorporated differently). This term effectively regularizes the variational problem, allowing the velocity and density to be treated as independent field variables for the purpose of variation, despite their implicit dynamical relations.
Applying the Euler-Lagrange field equations to this Lagrangian density, with $\displaystyle \phi, \alpha, \beta, \rho, S$ treated as independent fields, one can systematically derive Euler’s equations for inviscid fluid flow along with the continuity equation and entropy transport equation. This provides an elegant variational foundation for fluid dynamics, akin to how other field theories are constructed.

Example: Interaction Lagrangians (e.g., QED)

Interactions between different fields are typically introduced into a theory’s Lagrangian density by adding specific interaction terms to the sum of the individual free field Lagrangians. This approach, often guided by symmetry principles like gauge invariance, provides a systematic way to describe how particles (quanta of fields) interact.

Quantum Electrodynamics (QED) is the quintessential example, describing the interaction between electrons (represented by the Dirac field $\displaystyle \psi$ ) and photons (represented by the electromagnetic field $\displaystyle A^\mu$ ). It’s one of the most precisely tested theories in physics.

Lagrangian Density (LQED):

$\displaystyle \mathcal{L}_{\text{QED}}=\bar{\psi}(i\gamma^\mu D_\mu-m)\psi-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$

This Lagrangian beautifully unifies the free Dirac Lagrangian for electrons and the free electromagnetic Lagrangian for photons. The interaction is encoded within the covariant derivative $\displaystyle D_\mu=\partial_\mu+ieA_\mu$ . This specific modification of the ordinary derivative ensures the entire Lagrangian remains invariant under local gauge transformations, which is a fundamental symmetry of electromagnetism. The parameter $\displaystyle e$ represents the electric charge, acting as the coupling constant.
Euler-Lagrange Field Equations:
- Varying $\displaystyle \mathcal{L}_{\text{QED}}$ with respect to $\displaystyle \bar{\psi}$ yields the Dirac equation coupled to the electromagnetic field. This equation describes how electrons move and are influenced by photons.
- Varying $\displaystyle \mathcal{L}_{\text{QED}}$ with respect to $\displaystyle A^\mu$ yields Maxwell’s equations, where the electromagnetic current (generated by the Dirac field) acts as a source. This unified Lagrangian automatically produces all the dynamics of light and matter interaction from a single coherent principle.

Scalar Electrodynamics

This theory describes a U(1) gauge field (like the electromagnetic field) coupled to a charged spin-0 scalar field (like the complex scalar field we discussed earlier). It’s a simpler model that illustrates gauge coupling for scalar particles.

Field Variables: Complex scalar field $\displaystyle \phi$ (representing the charged spin-0 particle) and the gauge field $\displaystyle A^\mu$ (representing the photon).

Lagrangian Density (L):

$\displaystyle \mathcal{L}=(D^\mu\phi)^*(D_\mu\phi)-m^2\phi^*\phi-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$

Here, $\displaystyle D_\mu=\partial_\mu-ieA_\mu$ is the covariant derivative for a charged scalar field. It ensures gauge invariance of the scalar field’s kinetic term. The first term is the kinetic term for the scalar field, the second is its mass term, and the third is the kinetic term for the electromagnetic field.

Euler-Lagrange Field Equations: Applying the Euler-Lagrange equations to this Lagrangian yields coupled equations for both the scalar field and the electromagnetic field. These equations describe their mutual interaction, analogous to QED but for scalar particles rather than spin-1/2 fermions.

Quantum Chromodynamics (QCD)

Theory of the strong nuclear force, binding quarks into protons and neutrons. It is a non-Abelian gauge theory based on the $\displaystyle SU(3)$ color symmetry. Describes the self-interacting gluons and quarks, leading to phenomena like confinement and asymptotic freedom.

Lagrangian Density (L): $\displaystyle \mathcal{L}_{\text{QCD}}=\bar{\psi}_f(i\gamma^\mu D_\mu-m_f)\psi_f-\frac{1}{4}G^{a\mu\nu}G_{a\mu\nu}$

Here, $\displaystyle \psi_f$ are quark fields of different “flavors” ( $\displaystyle f$ ), $\displaystyle m_f$ are their masses, $\displaystyle D_\mu$ is the QCD covariant derivative, and $\displaystyle G^{a\mu\nu}$ is the gluon field strength tensor.

Higgs Field

A scalar field responsible for giving mass to elementary particles through spontaneous electroweak symmetry breaking.

Lagrangian Density (L): $\displaystyle \mathcal{L}_{\text{Higgs}}=|D^\mu\phi|^2-\mu^2|\phi|^2-\lambda|\phi|^4$

Here, $\displaystyle \phi$ is the complex Higgs doublet field, $\displaystyle D_\mu$ is its covariant derivative, and $\displaystyle \mu^2,\lambda$ are parameters.

The negative $\displaystyle \mu^2 &fg=000000$ term leads to a non-zero vacuum expectation value, which generates mass for fundamental particles and force carriers.

Standard Model Lagrangian:

The standard model unifies the electromagnetic, weak, and strong forces, describing all known elementary particles and their interactions.

Lagrangian Density (L): This is a complex sum of many terms, schematically:
$\displaystyle \mathcal{L}_{\text{SM}}=\mathcal{L}_{\text{gauge}}+\mathcal{L}_{\text{fermion}}+\mathcal{L}_{\text{Higgs}}+\mathcal{L}_{\text{Yukawa}}+\mathcal{L}_{\text{gauge-fixing}}+\mathcal{L}_{\text{ghost}}$

This single, monumental formula encapsulates almost all of known particle physics.

Supersymmetric Field Theories (Wess-Zumino model)

The Wess–Zumino model is the simplest nontrivial example of a 4D supersymmetric quantum field theory, introducing a symmetry (SUSY) that relates bosons and fermions. It serves as a foundational framework to explore the structure and implications of supersymmetry.

$\displaystyle \mathcal{L} = \frac{1}{2}(\partial^\mu A)^2 + \frac{1}{2}(\partial^\mu B)^2 + \frac{i}{2} \bar{\psi} \gamma^\mu \partial_\mu \psi + \frac{1}{2}F^2 + \frac{1}{2}G^2 + \ldots$

This Lagrangian describes two bosonic scalar fields ( $\displaystyle A, B$ ) and one fermionic Dirac field ( $\displaystyle \psi$ ) which form a supermultiplet. The terms $\displaystyle F$ and $\displaystyle G$ represent auxiliary fields necessary for closing the supersymmetry algebra. The Lagrangian is invariant under SUSY transformations, providing a framework to explore particle unification and solutions to the hierarchy problem.

BF Theory (Topological QFT)

BF theory is a prime example of a Topological Quantum Field Theory (TQFT), meaning its properties depend solely on the topology of the spacetime manifold, not its local geometry (like the metric).

$\displaystyle S = \int_M K(B, F)$

$\displaystyle B$ is a 2-form field and $\displaystyle F$ is a curvature 2-form (derived from a connection field $\displaystyle A$ ). $\displaystyle K$ is an invariant bilinear form. The action is explicitly metric-independent and yields topological invariants upon quantization, making it relevant for quantum gravity and condensed matter topological phases.

Einstein-Hilbert Action (General Relativity)

This is the fundamental action principle from which Einstein’s field equations of General Relativity are derived, describing gravity as the curvature of spacetime.

$\displaystyle S = \frac{1}{16\pi G} \int \sqrt{-g} \, R \, d^4x$

$\displaystyle R$ is the Ricci scalar, representing the spacetime curvature, and $\displaystyle g$ is the determinant of the metric tensor $\displaystyle g_{\mu\nu}$ , which is the dynamical field representing gravity. $\displaystyle G$ is Newton’s gravitational constant. Variation of this action with respect to $\displaystyle g_{\mu\nu}$ yields Einstein’s vacuum field equations.

$\displaystyle f(R)$ Gravity

These theories are direct generalizations of General Relativity, proposed to address cosmological phenomena like cosmic acceleration without requiring dark energy.

$\displaystyle S = \int \sqrt{-g} \, f(R) \, d^4x$

Here, $\displaystyle f(R)$ is an arbitrary function of the Ricci scalar $\displaystyle R$ . This modification to the Einstein-Hilbert action leads to extended gravitational theories whose field equations differ from Einstein’s, potentially providing alternative explanations for cosmological observations.

Brans-Dicke Theory

Brans-Dicke theory is a prominent example of a scalar-tensor theory of gravity, where the gravitational interaction is mediated not only by the metric tensor but also by a dynamical scalar field.

$\displaystyle S = \int \sqrt{-g} \left( \phi R - \frac{\omega}{\phi} \partial_\mu \phi \partial^\mu \phi \right) \, d^4x$

The scalar field $\displaystyle \phi$ is dynamically coupled to gravity ( $\displaystyle \phi R$ term) and also has its own kinetic term ( $\displaystyle \frac{\omega}{\phi} \partial_\mu \phi \partial^\mu \phi$ , where $\displaystyle \omega$ is the Brans-Dicke coupling constant). This field effectively determines the strength of gravity, making Newton’s constant $\displaystyle G$ a dynamical quantity.

Sine-Gordon Model

The Sine-Gordon model is a classic example of a (1+1) dimensional scalar field theory renowned for its stable, localized, particle-like solutions known as solitons.

$\displaystyle \mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^2 - \frac{m^2}{\lambda^2} (1 - \cos \lambda \phi)$

$\displaystyle \phi$ is a scalar field, and $\displaystyle \lambda$ is a coupling constant. The unique periodic potential term $\displaystyle (1 - \cos \lambda \phi)$ creates multiple vacuum states, leading to stable topological solutions (kinks and anti-kinks) that interpolate between these vacua.

$\displaystyle \phi^4$ Theory (Domain walls)

This is a fundamental real scalar field theory frequently used to model spontaneous symmetry breaking and the formation of stable topological defects.

$\displaystyle \mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^2 - \frac{\lambda}{4}(\phi^2 - v^2)^2$

$\displaystyle \phi$ is a real scalar field, and $\displaystyle V(\phi) = \frac{\lambda}{4}(\phi^2 - v^2)^2$ is a “double-well” potential ( $\displaystyle \lambda > 0, v^2 > 0$ ). The potential has multiple minima, leading to spontaneous symmetry breaking. The field equations support stable solutions known as “kinks” or “domain walls,” which are localized structures separating regions of different vacuum states.

Skyrme Model

The Skyrme model is a significant nonlinear field theory that proposes a non-perturbative description of baryons (protons and neutrons) as topological solitons.

$\displaystyle \mathcal{L} = \frac{1}{4} \text{Tr}(\partial_\mu U \partial^\mu U^\dagger) + \frac{1}{32e^2} \text{Tr}([U^\dagger \partial_\mu U, U^\dagger \partial_\nu U]^2)$

$\displaystyle U$ is an $\displaystyle SU(2)$ -valued field representing the meson fields. The first term is a standard kinetic term. The second term, known as the Skyrme term, is quartic in derivatives. This higher-derivative term is crucial for providing a stable configuration (a skyrmion) that prevents the topological soliton from shrinking to zero size, thus modeling extended particles.

‘t Hooft–Polyakov Monopole

This model provides a theoretical framework for the existence of magnetic monopoles as topological soliton solutions within a non-Abelian gauge theory with a Higgs field.

$\displaystyle \mathcal{L} = -\frac{1}{4} F_{\mu\nu}^a F^{a\mu\nu} + \frac{1}{2} (D_\mu \phi^a)(D^\mu \phi^a) - V(\phi^a)$

$\displaystyle A^a_\mu$ is a non-Abelian gauge field, and $\displaystyle \phi^a$ is a Higgs field in the adjoint representation. The Higgs potential $\displaystyle V(\phi^a)$ induces spontaneous symmetry breaking. The equations of motion derived from this Lagrangian possess smooth, finite-energy solutions that are topologically stable and carry a quantized magnetic charge.

Baby Skyrmions (2+1D Solitons)
Baby Skyrmions are $\displaystyle (2+1)$ -dimensional topological solitons that serve as a simplified, lower-dimensional testbed for properties observed in the $\displaystyle (3+1)$ dimensional Skyrme model.

$\displaystyle \mathcal{L} = \frac{1}{2} (\partial_\mu \mathbf{n}) \cdot (\partial^\mu \mathbf{n}) + \frac{1}{4} (\partial_\mu \mathbf{n} \times \partial_\nu \mathbf{n})^2 - V(\mathbf{n})$

$\displaystyle \mathbf{n}$ is a unit vector field ( $\displaystyle \mathbf{n}\cdot\mathbf{n}=1$ ) mapping spacetime to a sphere. This Lagrangian includes a standard kinetic term, a higher-order Skyrme term (here $\displaystyle (\ldots)^2$ implicitly means squared magnitude of the cross product), and a potential term. It exhibits topological properties and stable solutions analogous to the full Skyrme model.

Nambu-Goto String Action

The Nambu-Goto action is the simplest and most fundamental action for a relativistic string, describing its dynamics as it sweeps out a two-dimensional surface (the world-sheet) in spacetime.

$\displaystyle S = -T_0 \int d\tau d\sigma \sqrt{-\det h_{ab}}$

$\displaystyle T_0$ is the string tension (energy per unit length). $\displaystyle d\tau d\sigma$ are the coordinates on the world-sheet. $\displaystyle h_{ab}$ is the induced metric on the world-sheet, which depends on how the string is embedded in the higher-dimensional spacetime ( $\displaystyle X^\mu(\tau,\sigma)$ ). The action is directly proportional to the invariant area of this world-sheet, making it Lorentz invariant.

Polyakov Action (String Theory)

The Polyakov action is an alternative formulation for a relativistic string that is classically equivalent to the Nambu-Goto action but is significantly more tractable and commonly used for quantization.

$\displaystyle S = -\frac{T_0}{2} \int d\tau d\sigma \sqrt{-h} h^{ab} \partial_a X^\mu \partial_b X_\mu$

Here, $\displaystyle h_{ab}$ is an auxiliary world-sheet metric, which is treated as an independent field variable in the action. Its presence makes the action quadratic in the derivatives of the spacetime coordinates $\displaystyle X^\mu$ , simplifying the quantization procedure. Varying with respect to $\displaystyle h_{ab}$ recovers the Nambu-Goto equations of motion.

Green-Schwarz Superstring

The Green-Schwarz action is a covariant action principle for superstrings, which are fundamental objects in string theory that incorporate supersymmetry (relating bosonic and fermionic degrees of freedom).

$\displaystyle S = \int d^2\sigma \, \mathcal{L}_{GS}(X^\mu, \theta)$

The Lagrangian $\displaystyle \mathcal{L}_{GS}$ depends on both bosonic spacetime coordinates $\displaystyle X^\mu$ and fermionic spinor fields $\displaystyle \theta$ on the world-sheet. This action is designed to be invariant under local world-sheet supersymmetry and global spacetime supersymmetry, which are crucial for the consistency and anomaly cancellation of superstring theories.

DBI Action (D-brane)

The Dirac-Born-Infeld (DBI) action describes the low-energy effective dynamics of D-branes, which are extended objects in string theory where open strings can end.

$\displaystyle S_{DBI} = -T_p \int d^{p+1}\sigma \, e^{-\phi} \sqrt{-\det(g_{ab} + B_{ab} + 2\pi \alpha' F_{ab})}$

$\displaystyle T_p$ is the D-brane tension, $\displaystyle \phi$ is the dilaton field, $\displaystyle g_{ab}$ is the induced metric on the D-brane, $\displaystyle B_{ab}$ is the pullback of the NS-NS B-field, and $\displaystyle F_{ab}$ is the field strength of a $\displaystyle U(1)$ gauge field living on the D-brane. This highly nonlinear action naturally arises from string theory and generalizes the Born-Infeld action.

Landau-Ginzburg Model

The Landau-Ginzburg model is a phenomenological field theory used to describe superconductivity and other continuous phase transitions, where a complex scalar field acts as an order parameter.

$\displaystyle \mathcal{L} = \frac{1}{2m^*} |(\nabla - ie^* \mathbf{A})\psi|^2 - \alpha |\psi|^2 - \frac{\beta}{2} |\psi|^4 + \frac{1}{2\mu_0} \mathbf{B}^2$

$\displaystyle \psi$ is a complex scalar order parameter representing the macroscopic quantum wave function of the superconductor. $\displaystyle m^*, e^*$ are effective mass/charge, $\displaystyle \mathbf{A}$ is the magnetic vector potential. The potential terms ( $\displaystyle -\alpha |\psi|^2 - \frac{\beta}{2} |\psi|^4$ ) describe spontaneous symmetry breaking (for $\displaystyle \alpha>0$ ) as the system transitions into the superconducting state.

Chern-Simons Theory (2+1D)

Chern-Simons theory is a unique example of a topological field theory in $\displaystyle (2+1)$ dimensions, where its properties depend only on the topology of the underlying manifold.

$\displaystyle S = \frac{k}{4\pi} \int d^3x \, \epsilon^{\mu\nu\rho} A_\mu \partial_\nu A_\rho$

$\displaystyle k$ is an integer coefficient, and $\displaystyle \epsilon^{\mu\nu\rho}$ is the Levi-Civita symbol. This action is distinct because it is metric-independent and only involves first derivatives of the gauge field $\displaystyle A_\mu$ . It yields topological invariants rather than propagating dynamics and is crucial in describing the fractional quantum Hall effect and other topological phases in condensed matter physics.

Gross–Neveu Model

The Gross-Neveu model is a $\displaystyle (1+1)$ -dimensional quantum field theory famous for describing interacting fermions and exhibiting dynamical symmetry breaking.

$\displaystyle \mathcal{L} = \bar{\psi}(i \gamma^\mu \partial_\mu - m) \psi - \frac{g^2}{2N} (\bar{\psi}\psi)^2$

$\displaystyle \psi$ is an $\displaystyle N$ -component fermion field, $\displaystyle g$ is a coupling constant. The quartic interaction term $\displaystyle (\bar{\psi}\psi)^2$ is a four-fermion interaction. This model provides an exactly solvable example where fermion masses can be dynamically generated through spontaneous chiral symmetry breaking.

Thirring Model

The Thirring model is another exactly solvable $\displaystyle (1+1)$ -dimensional quantum field theory, notable for describing self-interacting Dirac fermions.

$\displaystyle \mathcal{L} = \bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi - \frac{g}{2} (\bar{\psi}\gamma^\mu \psi)(\bar{\psi}\gamma_\mu \psi)$

$\displaystyle \psi$ is a Dirac fermion field, $\displaystyle g$ is the coupling constant. The interaction term represents a current-current coupling. This model exhibits intriguing properties like bosonization, where fermionic degrees of freedom can be shown to be equivalent to bosonic ones at low energies.

Chiral Perturbation Theory (ChPT)

Chiral Perturbation Theory is an effective field theory used to describe the low-energy dynamics of strongly interacting particles, particularly the nearly massless Goldstone bosons that emerge from spontaneously broken symmetries.

$\displaystyle \mathcal{L}_{\chi PT} = \frac{F_\pi^2}{4} \text{Tr}(\partial_\mu U \partial^\mu U^\dagger) + \ldots$

$\displaystyle U$ is an $\displaystyle SU(2)$ matrix field (containing the pion fields), and $\displaystyle F_\pi$ is the pion decay constant. This Lagrangian is constructed based on the symmetries of Quantum Chromodynamics ( $\displaystyle QCD$ ) at low energies, where quarks and gluons are confined, and pions behave as Goldstone bosons.

Inflaton Field Lagrangian (Cosmology)

This Lagrangian describes a hypothetical scalar field, the inflaton, which is central to the theory of cosmic inflation, explaining the very early, rapid expansion of the universe.

$\displaystyle \mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^2 - V(\phi)$

$\displaystyle \phi$ is the inflaton scalar field. The kinetic term $\displaystyle \frac{1}{2} (\partial_\mu \phi)^2$ describes its dynamics, and $\displaystyle V(\phi)$ is its potential energy. The specific shape of $\displaystyle V(\phi)$ is designed to drive a period of exponential expansion, which solves cosmological puzzles like the horizon problem, flatness problem, and the origin of large-scale structure.

Horndeski Theory

Horndeski’s theory represents the most general class of scalar-tensor theories of gravity in four dimensions that yields second-order equations of motion, avoiding problematic higher-derivative ghost instabilities.

$\displaystyle S = \int d^4x \sqrt{-g} \sum_{i=2}^5 \mathcal{L}_i[g_{\mu\nu}, \phi]$

The action is a sum of specific Lagrangian densities $\displaystyle \mathcal{L}_i$ (known as G-terms), each constructed from the metric $\displaystyle g_{\mu\nu}$ and a scalar field $\displaystyle \phi$ and their derivatives. This comprehensive framework encompasses many other gravitational theories, such as General Relativity, Brans-Dicke theory, and k-essence, as special cases.

Born–Infeld Electrodynamics

This is a non-linear theory of electromagnetism that classically resolves the issue of infinite self-energy for a point charge by introducing a maximum electric field strength.

$\displaystyle \mathcal{L}_{BI} = b^2 \left(1 - \sqrt{-\det(\eta_{\mu\nu} + F_{\mu\nu}/b)}\right)$

$\displaystyle b$ is a fundamental constant representing the limiting electric field strength. $\displaystyle \eta_{\mu\nu}$ is the Minkowski metric, and $\displaystyle F_{\mu\nu}$ is the electromagnetic field tensor. For small field strengths ( $\displaystyle F_{\mu\nu} \ll b$ ), this Lagrangian reduces to the familiar Maxwell’s theory. It naturally arises as the low-energy effective action for D-branes in string theory.

Axion Lagrangian

The Axion Lagrangian describes a hypothetical elementary particle, the axion, proposed to solve the strong CP problem in Quantum Chromodynamics (QCD) and also considered a candidate for dark matter.

$\displaystyle \mathcal{L} = \frac{1}{2}(\partial_\mu a)^2 - V(a) - \frac{g^2}{32\pi^2 f_a} a \, F_{\mu\nu}^a \tilde{F}^{a\mu\nu}$

$\displaystyle a$ is the axion scalar field, $\displaystyle V(a)$ is its potential, and $\displaystyle f_a$ is the axion decay constant. The final term couples the axion field to the gluon field strength tensor $\displaystyle F^a_{\mu\nu}$ and its dual $\displaystyle \tilde{F}^{a\mu\nu}$ . This specific coupling is what allows the axion to dynamically relax the problematic strong CP phase to zero.

Noncommutative Scalar Field Theory

Noncommutative field theories are speculative theories that modify the fundamental structure of spacetime by replacing the usual commuting spacetime coordinates with non-commuting operators, leading to a “fuzzy” or “granular” spacetime.

$\displaystyle \mathcal{L} = \frac{1}{2} \partial_\mu \phi \star \partial^\mu \phi - V(\phi \star \phi)$

Here, $\displaystyle \phi$ is a scalar field, and the standard pointwise product of fields is replaced by a Moyal product ( $\displaystyle \star$ ). This modification fundamentally alters the field interactions and propagation, and can lead to intriguing consequences such as Lorentz violation, non-unitarity, and a minimum measurable length in spacetime.