Wigner function for SU(1,1)

In spite of their potential usefulness, Wigner functions for systems with SU(1,1) symmetry have not been explored thus far. We address this problem from a physically-motivated perspective, with an eye towards applications in modern metrology. Starting from two independent modes, and after getting rid of the irrelevant degrees of freedom, we derive in a consistent way a Wigner distribution for SU(1,1). This distribution appears as the expectation value of the displaced parity operator, which suggests a direct way to experimentally sample it. We show how this formalism works in some relevant examples.Dedication: While this manuscript was under review, we learnt with great sadness of the untimely passing of our colleague and friend Jonathan Dowling. Through his outstanding scientific work, his kind attitude, and his inimitable humor, he leaves behind a rich legacy for all of us. Our work on SU(1,1) came as a result of long conversations during his frequent visits to Erlangen. We dedicate this paper to his memory.

Superconductivity in the Hubbard model: a hidden-order diagnostics from the Luther-Emery phase on ladders

Short-range antiferromagnetic correlations are known to open a spin gap in the repulsive Hubbard model on ladders with \boldsymbol{M}𝐌 legs, when \boldsymbol{M}𝐌 is even. We show that the spin gap originates from the formation of correlated pairs of electrons with opposite spin, captured by the hidden ordering of a spin-parity operator. Since both spin gap and parity vanish in the two-dimensional limit, we introduce the fractional generalization of spin parity and prove that it remains finite in the thermodynamic limit. Our results are based upon variational wave functions and Monte Carlo calculations: performing a finite size-scaling analysis with growing \boldsymbol{M}𝐌, we show that the doping region where the parity is finite coincides with the range in which superconductivity is observed in two spatial dimensions. Our observations support the idea that superconductivity emerges out of spin gapped phases on ladders, driven by a spin-pairing mechanism, in which the ordering is conveniently captured by the finiteness of the fractional spin-parity operator.

Superposition and entanglement of two-mode squeezed vacuum states

In this paper, by using the parity operator as well as the two-mode squeezing operator, we define new operators which by the action of them on the vacuum state of the two-mode radiation field, superposition of two two-mode squeezed vacuum states and entangled two-mode squeezed vacuum states are generated.

Polynomial algebra with reflection symmetry and thermostatistics

In this paper, we use the reflection (or parity) operator to construct the new algebra whose maximum occupation number is finite. For the Hamiltonian proportional to the number operator, we discuss the thermostatistics and compute the thermodynamical quantities such as distribution function, multiparticle distribution function, mean energy and specific heat. We also calculate the intercept for this algebra to show that a particle obeying this algebra is an exotic particle which is neither a boson nor a fermion.

An identification of the Dirac operator with the parity operator

In this paper, we provide a new derivation of the Dirac equation which promptly generalizes to higher spins. We apply this idea to spin-half Elko dark matter.

PARITY ANOMALY IN DIRAC EQUATION IN (1+2) DIMENSIONS, QUANTUM ELECTRODYNAMICS AND PAIR PRODUCTION

We show that parity operator plays an interesting role in Dirac equation in (1+2) dimensions and can be used for defining chiral charges. Further the "anomalous" current induced by an external gauge field can be related to the anomalous divergence of an axial vector current which arises due to quantum radiative corrections provided by triangular loop Feynman diagrams in analogy with the corresponding axial anomaly in (1+3) dimensions. It is shown that the nonconservation of "chiral charge" due to anomaly is related with the topological Chern–Simons charge. As an application pair creation of massless fermions in electric field is discussed.

SUPERCONFORMAL QUANTUM MECHANICS VIA WIGNER–HEISENBERG ALGEBRA

We show the natural relation between the Wigner Hamiltonian and the conformal Hamiltonian. A model in (super)conformal quantum mechanics with (super)conformal symmetry in the Wigner–Heisenberg algebra picture [x, px] = i(1+cP) (P being the parity operator) is presented. In this context, the energy spectrum, the Casimir operator, raising and lowering operators are defined.