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Polarisation is described by an SU(2) state due to the coherence of photons emitted from a ubiquitous laser source. We propose a polarisation modulator, called as a Poincaré rotator, which enables dynamic control over a polarisation state by executing an arbitrary rotation on the Poincaré sphere. The Poincaré rotator works as an arbitrary operator in the SU(2) Lie group, combining two U(1) operations to change the phase and the amplitude of the state, respectively. By modulating both the phase and amplitude, we have achieved distinguishable 4×4=16, 8×8=64, and 10×10=100 states on the Poincaré sphere. As a locus of the polarisation states on the sphere, we have successfully drawn the molecular structure of Buckminsterfullerene (C60) and the coastline of the Earth.

Stokes parameters (S) on the Poincaré sphere are very useful values to describe the polarisation state of photons. However, the fundamental principle on the nature of polarisation is not completely understood, yet, because we have no concrete consensus on how to describe spin of photons, quantum-mechanically. Here, we have considered a monochromatic coherent ray of photons, described by a many-body coherent state, and established a fundamental basis to describe the spin state of photons, in connection with a classical description based on Stokes parameters. We show that a spinor description of the coherent state is equivalent to Jones vector for polarisation states, and obtain the spin operators (Ŝ) of all components based on rotators in an SU(2) group theory. Polarisation controllers such as phase-shifters and rotators are also obtained as quantum-mechanical field operators to change the phase of the wavefunction for polarisation states. We show that the Stokes parameters are quantum-mechanical average of the spin operators, S=⟨Ŝ⟩.

The speed of light (c) in a vacuum is independent of the choice of frames to describe the propagation, according to the theory of relativity. We consider how light is characterised in a material, where the speed of light is different from that in a vacuum due to the finite dielectric constant. The phase velocity in a material is smaller than c, such that the speed of a moving frame can be larger than the phase velocity, such that the frame can move faster than the speed of light in a material. Consequently, an unusual Doppler effect is expected, and the wavelength in the moving frame changes from the red-shift to the blue-shift upon increasing the speed of the frame. The corresponding energy of the light also changes sign from positive to negative, while momentum is always positive, leading to the changes of signs for the phase velocity and the helicity. In a graded index fibre, where the exact solution is available, even more complicated phenomena are expected, due to the finite effective mass of photons. Upon the increase of the energy gap, generated by optical confinements and optical orbital angular momentum, the effective mass of photons increases. If the gap is large enough, momentum starts to change the sign upon increasing the frame velocity, while the energy of photons is always positive. In this case, the phase velocity diverges if momentum is in agreement with the fame velocity. Contrary to the unusual behaviours of the phase velocity, the group velocity is always below c. This thought experiment might be useful for considering insight into the polarisation state of light.

Lie algebra is a hidden mathematical structure behind various quantum systems realised in nature. Here, we consider SU(2) wavefunctions for polarisation states of coherent photons emitted from a laser source, and discuss the relationship to spin expectation values with SO(3) symmetry based on isomorphism theorems. In particular, we found rotated half-wave-plates correspond to mirror reflections in the Poincaré sphere, which do not form a subgroup in the projected O(2) plane due to anti-hermitian property. This could be overcome experimentally by preparing another half-wave-plate to realise a pristine rotator in SU(2), which allows arbitrary rotation angles determined by the physical rotation. By combining another 2 quarter-wave-plates, we could also construct a genuine phase-shifter, thus, realising passive control over the full Poincaré sphere.

The algebraic group theory approach to pairing in nuclei is an old subject and yet it continues to be important in nuclear structure, giving new results. It is well known that for identical nucleons in the shell model approach with j − j coupling, pairing algebra is SU(2) with a complementary number-conserving Sp(N) algebra and for nucleons with good isospin, it is SO(5) with a complementary number-conserving Sp(2Ω) algebra. Similarly, with L − S coupling and isospin, the pairing algebra is SO(8). On the other hand, in the interacting boson models of nuclei, with identical bosons (IBM-1) the pairing algebra is SU(1, 1) with a complementary number-conserving SO(N) algebra and for the proton–neutron interacting boson model (IBM-2) with good F-spin, it is SO(3, 2) with a complementary number-conserving SO(ΩB) algebra. Furthermore, in IBM-3 and IBM-4 models several pairing algebras are possible. With more than one j or ℓ orbit in shell model, i.e., in the multi-orbit situation, the pairing algebras are not unique and we have the new paradigm of multiple pairing [SU(2), SO(5) and SO(8)] algebras in shell models and similarly there are multiple pairing algebras [SU(1, 1), SO(3, 2) etc.] in interacting boson models. A review of the results for multiple multi-orbit pairing algebras in shell models and interacting boson models is presented in this article with details given for multiple SU(2), SO(5), SU(1, 1) and SO(3, 2) pairing algebras. Some applications of these multiple pairing algebras are discussed. Finally, multiple SO(8) pairing algebras in shell model and pairing algebras in IBM-3 model are briefly discussed.

We show conceptually that the edge of a two-dimensional topological insulator can be used to construct a solid-state Stern–Gerlach spin splitter. By threading such a Stern–Gerlach apparatus with a magnetic flux, Aharanov–Bohm-like interference effects are introduced. Using ferromagnetic leads, the setup can be used to both measure magnetic flux and as a spintronics switch. With normal metallic leads a switchable spintronics NOT-gate can be implemented. Furthermore, we show that a sequence of such devices can be used to construct a single-qubit SU(2)-gate, one of the two gates required for a universal quantum computer. The field sensitivity, or switching field, b, is related to the characteristic size of the device, r, through b = h/(2πqr2), with q being the unit of electric charge.

The Stern–Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern–Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles θ between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2±μB. The values ±μB thus do not correspond to the continuum of values −μ·B Einstein and Ehrenfest had conjectured. The energy term V=−μ·B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies ±μB, and as such not valid for individual fermions. The two fermion states with energy ±μB are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.

The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) quantum bits or qubits (and more generally qudits), the analogues of the usual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantum mechanics, namely, linearity, which manifests itself through the superposition of qubits and the action of unitary operators on qubits, and entanglement of certain multi-qubit states, a resource that is specific to quantum mechanics. A, second, focus is on some mathematical problems related to the so-called mutually unbiased bases used in quantum computing and quantum information processing. In this direction, the construction of mutually unbiased bases is presented via two distinct approaches: one based on the group SU(2) and the other on Galois fields and Galois rings.

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Abstract: It is shown that the classical laws of thermodynamics require that mechanical systems must exhibit energy that becomes unavailable to do useful work. In thermodynamics, this type of energy is called entropy. It is further shown that these laws require two metrical manifolds, equations of motion, field equations, and Weyl's quantum principles. Weyl's quantum principle requires quantization of the electrostatic potential of a particle and that this potential be non-singular. The interactions of particles through these non-singular electrostatic potentials are analyzed in the low velocity limit and in the relativistic limit. It is shown that writing the two particle interactions for unlike particles allows an examination in two limiting cases: large and small separations. These limits are shown to have the limiting motions of: all motions are ABOUT the center of mass or all motion is OF the center of mass. The first limit leads to the standard Dirac equation. The second limit is shown to have equations of which the electroweak theory is a subset. An extension of the gauge principle into a five-dimensional manifold, then restricting the generality of the five-dimensional manifold by using the conservation principle, shows that the four-dimensional hypersurface that is embedded within the 5-D manifold is required to obey Einstein's field equations. The 5-D gravitational quantum equations of the solar system are presented.