(a) Since the generators Ta are Hermitian, the transformation laws are δψ = iǫaTaψ (2) δψ̄ = −i ̄ Ta ψǫa (3) The Lagrangian is quite trivially seen to be invariant δL = −iδ ψ̄(∂/ −m)ψ − iψ̄(∂/ −m)δψ = −iψ̄(−iǫaTa)(∂/ −m)ψ − iψ̄(∂/ −m)(iǫaTa)ψ = 0 , (4) where we used the trivial relation [Ta, γ ] = 0. (b) We use the so called Noether method (described in e.g. Di Francesco et al.: CFT Section 2.4.2) to...

1 Kinetic terms A quantum field theory is defined by a Lagrangian. The first step is to define the fields. such that they represent particles. That means that there should be no terms with only one field present and all vacuum expectation values should have been subtracted The kinetic terms, those containing exactly two fields, are assumed to be diagonalized. This means that all terms with two ...

Relativistic elasticity on an arbitrary spacetime is formulated as a Lagrangian field theory which is covariant under spacetime diffeomorphisms. This theory is the relativistic version of classical elasticity in the hyperelastic, materially frame-indifferent case and, on Minkowski space, reduces to the latter in the limit c → ∞. The field equations are cast into a first – order symmetric hyperb...

I’d like to start the story on relatively stable ground: the land of classical physics. More specifically, I’ll introduce the idea of a classical Lagrangian field theory. The idea is that many physical systems take a similar form; there is a space of possible configurations of the system (which we usually refer to as fields), but those that arise in reality are those which minimise a certain ac...

Relativistic elasticity on an arbitrary spacetime is formulated as a Lagrangian field theory which is covariant under spacetime diffeomorphisms. This theory is the relativistic version of classical elasticity in the hyperelastic, materially frame-indifferent case and, on Minkowski space, reduces to the latter in the limit c → ∞. The field equations are cast into a first – order symmetric hyperb...

We consider the theories obtained by dimensional reduction to D = 1, 2, 3 of 4D supersymmetric Yang–Mills theories and calculate there the effective low-energy lagrangia describing moduli space dynamics — the low-dimensional analogs of the Seiberg–Witten effective lagrangian. The effective theories thus obtained are rather beautiful and interesting from mathematical viewpoint. In addition, thei...

The main result of this paper is the construction of a conformally covariant operator in two dimensions acting on scalar fields and containing fourth order derivatives. In this way it is possible to derive a class of Lagrangians invariant under conformal transformations. They define conformal field theories satisfying equations of the biharmonic type. Two kinds of these biharmonic field theorie...

This paper presents some analytic methods for studying uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues λ and λmin of 2-graphs. First, these two parameters are extended to weighted uniform hypergraphs; second, the eigenvalues-numbers λ and λmin are extended to eigenvalues-functions λ(p) and λ (p) min, which also ...

For large values of the Higgs boson mass the low energy structure of the gauged linear sigma model in the spontaneously broken phase can adequately be described by an effective field theory. In this work we present a manifestly gauge invariant technique to explicitly evaluate the corresponding effective Lagrangian from the underlying theory. In order to demonstrate the application of this funct...

We discuss some of the topological features of a non-interacting two (1 + 1)-dimensional Abelian gauge theory in the framework of superfield formulation. This theory is described by a BRST invariant Lagrangian density in the Feynman gauge. We express the local and continuous symmetries, Lagrangian density, topological invariants and symmetric energy momentum tensor of the theory in the language...