Henryk Arodź

DIRECTOR CONFIGURATION OF PLANAR SOLITONS IN NEMATIC LIQUID CRYSTALS

The director configuration of disclination lines in nematic liquid crystals in the presence of an external magnetic field is evaluated. Our method is a combination of a polynomial expansion for the director and of further analytical approximations which are tested against a numerical shooting method. The results are particularly simple when the elastic constants are equal, but we discuss the general case of elastic anisotropy. The director field is continuous everywhere apart from a straight line segment whose length depends on the value of the magnetic field. This indicates the possibility of an elongated defect core for disclination lines in nematics due to an external magnetic field.

Disclination Lines in an External Magnetic Field

Abstract The director configuration of disclination lines in the presence of an external magnetic field is evaluated. Applying a polynomial expansion, we obtain approximate analytical solutions for the director field which are discussed for the general case of elastic anisotropy. They point out the possibility of an extended defect core. The actual size of the core is estimated by minimizing the total energy, including both the energy of the nematic phase and the core energy.

The Renormalization Group

The relation between the subtracted Green’s functions with different choices of subtraction point in the \(\phi ^4_4\) model. The running coupling constant. Functional equations of the renormalization group. Differential renormalization group equations of the Gell-Mann–Low and the Callan–Symanzik type. The \(\beta \) function. Reliability of the perturbative approximations. The phenomenon of dimensional transmutation in renormalized quantum field theory.

The Euler–Lagrange Equations and Noether’s Theorem

The stationary action principle and the general form of the Euler–Lagrange equations. The notion of symmetry in classical field theory. Noether’s conserved currents.

The Perturbative Expansion for Non-Abelian Gauge Fields

The invariant volume element in the SU(N) group (the Haar measure). The Faddeev–Popov–DeWitt determinant for a given gauge condition. The Faddeev–Popov ghost fields. The correct path integral representation of the Green’s functions of local gauge-invariant operators. Feynman diagrams for the pure non-Abelian gauge field theory. The essential role of the gauge fixing term in the classical effective action. BRST invariance of the effective action and of the measure in the path integral. The Slavnov–Taylor identity for the generating functional of Green’s functions.

The Simplest Supersymmetric Models

The generating elements and their (anti-)commutation relations in the \(N=1\) superalgebra. Multiplets of quantum states generated by elements of the superalgebra. An example of a supersymmetric Lagrangian with free fields. The notions of superspace, superfield, and chiral superfield. The Wess–Zumino model and the Feynman diagrams for it. Examples of the mutual cancellation of ultraviolet divergences. The supersymmetric gauge theory. The \(N=2\) extended supersymmetry. A glossary of formulas used in the analysis of supersymmetric models.

Relativistic Spinor Fields

The Dirac equation. The transformation law of a relativistic bispinor. The SL(2, C) and Spin(4) groups. The free classical Dirac field. The Weyl spinor fields. The \(U(1) \times U(1)\) symmetry of the massless Dirac field. The Majorana field. The Grassmann versions of the classical (bi-)spinor fields.

Perturbative Expansion in the \( \phi ^4_4\) Model

Problems with an exact construction of the quantum \(\phi ^4_4\) model. The interaction picture. The Gell-Mann–Low formula for Greeen’s functions. The generating functional for Green’s functions. The exponential Wick formula. The Feynman free propagator. Regularized Feynman diagrams in four-momentum space. Normal ordered interactions. Cancelation of vacuum bubbles.

The Quantum Theory of Free Fields

Canonical quantization of the free real scalar field. Difficulties with the Schroedinger representation. Inequivalent representations of the canonical commutation relations. The Fock representation. Basic quantum observables: the total energy and momentum of the field. Description of quantum states in terms of particles. The field operator as a generalized function. The classical Dirac field as a system with constraints. The Faddeev-Jackiw method and quantization of the free Dirac field. The Dirac vacuum, the Fock representation and the appearance of a free, relativistic, spin 1/2 particle and its antiparticle. Extraction of the physical of degrees of freedom of the free electromagnetic field. The canonical quantization of the electromagnetic field, the Fock representation and the appearance of a free, massless particle (the photon).

Dynamics of cylindrical domain walls in nematic liquid crystals

Analytical calculations of the dynamics of a curved domain wall in a nematic liquid crystal are performed. The core of the wall is assumed to form a cylinder, whose axis coincides with the direction of an external magnetic field. The equation of motion for the nematic director field is solved in a comoving coordinate frame by applying a polynomial expansion of the tilt angle with respect to the radial distance from the wall core. Starting from a cylindrical domain wall at rest as initial conditions, the shrinking of the cylinder and the change of the wall width is analysed in detail. In particular, we find that the Neel wall decays faster than the Bloch wall, in agreement with energy considerations.