We examine the local conformal invariance (Weyl invariance) in tensor-scalar theories used in recently proposed conformal cosmological models. We show that the Noether currents associated with Weyl invariance in these theories vanish. We assert that
the corresponding Weyl symmetry does not have any dynamical role.
Dedicated to Ludwig Faddeev on his 80th birthday. Ludwig exemplifies perfectly a mathematical physicist: significant contribution to mathematics (algebraic properties of integrable systems) and physics (quantum field theory). In this note I present a
n exercise which bridges mathematics (restricted Clifford algebra) to physics (Majorana fermions).
To mark the 111th birthday of Eugene Wigner, we review topological excitations in diverse dimensions.
Ken Wilson is remembered.
Extending previous work on 2 -- and 3 -- point functions, we study the 4 -- point function and its conformal block structure in conformal quantum mechanics CFT$_1$, which realizes the SO(2,1) symmetry group. Conformal covariance is preserved even tho
ugh the operators with which we work need not be primary and the states are not conformally invariant. We find that only one conformal block contributes to the four-point function. We describe some further properties of the states that we use and we construct dynamical evolution generated by the compact generator of SO(2.1).
Zero modes arising from a planar Majorana equation in the presence of $N$ vortices require an $mathcal{N}$-dimensional state-space, where $mathcal{N} = 2^{N/2}$ for $N$ even and $mathcal{N} = 2^{(N + 1)/2}$ for $N$ odd. The mode operators form a restricted $mathcal{N}$-dimensional Clifford algebra.
A 0+1-dimensional candidate theory for the CFT$_1$ dual to AdS$_2$ is discussed. The quantum mechanical system does not have a ground state that is invariant under the three generators of the conformal group. Nevertheless, we show that there are oper
ators in the theory that are not primary, but whose non-primary character conspires with the non-invariance of the vacuum to give precisely the correlation functions in a conformally invariant theory.
We describe the occurrence and physical role of zero-energy modes in the Dirac equation with a topologically non-trivial background.
We review the relation between scale and conformal symmetries in various models and dimensions. We present a dimensional reduction from relativistic to non-relativistic conformal dynamics.
A Dirac-type matrix equation governs surface excitations in a topological insulator in contact with an s-wave superconductor. The order parameter can be homogenous or vortex valued. In the homogenous case a winding number can be defined whose non-van
ishing value signals topological effects. A vortex leads to a static, isolated, zero energy solution. Its mode function is real, and has been called Majorana. Here we demonstrate that the reality/Majorana feature is not confined to the zero energy mode, but characterizes the full quantum field. In a four-component description a change of basis for the relevant matrices renders the Hamiltonian imaginary and the full, space-time dependent field is real, as is the case for the relativistic Majorana equation in the Majorana matrix representation. More broadly, we show that the Majorana quantization procedure is generic to superconductors, with or without the Dirac structure, and follows from the constraints of fermionic statistics on the symmetries of Bogoliubov-de Gennes Hamiltonians. The Hamiltonian can always be brought to an imaginary form, leading to equations of motion that are real with quantized real field solutions. Also we examine the Fock space realization of the zero mode algebra for the Dirac-type systems. We show that a two-dimensional representation is natural, in which fermion parity is preserved.
