In this paper we compare the rotation of a rigid body in the real three-dimensional Euclidean space E3 and its representation in the complex plane (Klein space), on one hand, with the transformation of polarization states of light (SOPs) by the phase-shifters figured in the complex plane and on the Poincaré sphere, on the other hand. Both the Klein space, in classical mechanics, and the Poincaré sphere, in polarization theory, are abstract spaces, whose construction is based on the classical stereographical projection between Riemann sphere and the simple complex plane C1. They are classical abstract spaces, even if they have been largely used for representing quantum spinorial physical realities too. At the interface of classical/quantum physics persist some misaperceptions about what is intrinsically of quantum origin and nature, and what is imported from the classical domain. In this context we examine some misunderstandings that take place in the field of these spaces. I shall focus on the double angle relationship between the rotation of representative points of the SOPs on the Poincaré sphere with respect to the corresponding rotations of the azimuthal and ellipticity angles of the “form of the SOPs”, at a transformation of state given by a phase shifter. This is a classical result, that is transferred on the sphere from the complex plane, on the basis of the stereographic bijective connection between the points on the sphere and those in the complex plane. In any textbook of quantum mechanics “the double angle/half angle problem” is presented as a pure quantum spinorial one, avoiding its classical face and origin. A quantum spinorial approach, obviously, recovers the classical results, together with the specific spinorial ones, but with regards to the double angle/half angle issue it is superfluous. We shall also briefly examine the classical and quantum spinorial content of what we know today under the global name of Pauli spin matrices. Often in papers or textbooks of physics the results are presented in a mélange in which it is difficult to establish from which point on one needs to appeal to spinorial or quantum aspects.
In this paper we compare the rotation of a rigid body in the real three-dimensional Euclidean space E3 and its representation in the complex plane (Klein space), on one hand, with the transformation of polarization states of light (SOPs) by the phase-shifters figured in the complex plane and on the Poincaré sphere, on the other hand. Both the Klein space, in classical mechanics, and the Poincaré sphere, in polarization theory, are abstract spaces, whose construction is based on the classical stereographical projection between Riemann sphere and the simple complex plane C1. They are classical abstract spaces, even if they have been largely used for representing quantum spinorial physical realities too. At the interface of classical/quantum physics persist some misaperceptions about what is intrinsically of quantum origin and nature, and what is imported from the classical domain. In this context we examine some misunderstandings that take place in the field of these spaces. I shall focus on the double angle relationship between the rotation of representative points of the SOPs on the Poincaré sphere with respect to the corresponding rotations of the azimuthal and ellipticity angles of the “form of the SOPs”, at a transformation of state given by a phase shifter. This is a classical result, that is transferred on the sphere from the complex plane, on the basis of the stereographic bijective connection between the points on the sphere and those in the complex plane. In any textbook of quantum mechanics “the double angle/half angle problem” is presented as a pure quantum spinorial one, avoiding its classical face and origin. A quantum spinorial approach, obviously, recovers the classical results, together with the specific spinorial ones, but with regards to the double angle/half angle issue it is superfluous. We shall also briefly examine the classical and quantum spinorial content of what we know today under the global name of Pauli spin matrices. Often in papers or textbooks of physics the results are presented in a mélange in which it is difficult to establish from which point on one needs to appeal to spinorial or quantum aspects.
Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an α∈(0,2π). The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle α (or Fubini–Study distance α) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when H=C2 we do this for every α, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for α≤4π. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability cos2α in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when H=C2 and α=4π where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the α=2π case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.
The foundations of quantum theory are closely tied to a formulation of classical relativistic physics. In Clifford's geometric algebra classical relativistic physics has a spinorial formulation that is closely related to the standard Dirac equation. The algebra of physical space, APS, gives clear insight into the quantum/classical interface. Here, APS is compared to other formulations of relativistic quantum theory, especially the Dirac equation. These formulations are shown to be effectively equivalent to each other and to the standard theory, as demonstrated by establishing several isomorphisms. Dirac spinors are four-component complex entities, and so must be represented by objects containing 8 real degrees of freedom in the standard treatment (or 7 if a normalization constant is added). The relation [special characters omitted]indicates that the 8-dimensional even subalgebra [special characters omitted] of the Space-time algebra, STA is isomorphic to APS Cl 3, which is isomorphic to complex quaternions [special characters omitted]. The complex quaternions should not be confused with the biquaternions, a name sometimes used for them. The biquaternions are more generally elements of the algebra [special characters omitted]. The algebras Cl1,3 and Cl 3,1 are not isomorphic but their even sub-algebras are[1]. The Klein paradox is resolved in APS by considering Feynman's picture of antiparticles as negative energy solutions traveling backward in time. It is also shown that the algebra of physical space can naturally describe an extended version of the De Broglie-Bohm approach to quantum theory. A relativistic causal account of a spin measurement in APS is given. The Stern-Gerlach magnet acts on the eigenspinor Λ field of a charged particle in a way that is analogous to the interaction of a birefringent medium acts on a beam of light. Then we introduce a covariant interpretation of complex algebra of physical space, CAPS, the complex extension of APS. This is done to solve a problem in that the space-time inversion, PT transformation, when P is parity inversion and T time reversal, although it is a proper transformation is not physical, yet, it has the form of a physical rotation in the traditional Dirac theory. The CAPS form of the PT transformation does not have the form of a physical rotation. A further problem is that the explicit form of the time inversion, T and charge conjugation, C transformations depends on the matrix representation chosen. A representation-independent form would be more fundamental. In CAPS, the complex extension of APS these problems are resolved.
Using the method of canonical group quantization, we construct the angular momentum operators associated to configuration spaces with the topology of (i) a sphere and (ii) a projective plane. In the first case, the angular momentum operators derived this way are the quantum version of Poincaré's vector, i.e., the physically correct angular momentum operators for an electron coupled to the field of a magnetic monopole. In the second case, the operators one obtains represent the angular momentum operators of a system of two indistinguishable spin zero quantum particles in three spatial dimensions. The relevance of the proposed formalism for the progress in our understanding of the spin–statistics connection in nonrelativistic quantum mechanics is discussed. Introduction The connection between the spin of quantum particles and the statistics they obey is a remarkable example of a simply stated physical fact without the recognition of which many physical phenomena (ranging from the stability of matter and the electronic configuration of atoms to Bose–Einstein condensation and superconductivity) would not have an explanation. Nevertheless, the simplicity of the assertion “integer spin particles obey Bose statistics, and half-integer spin particles obey Fermi statistics” stands in bold contrast to its intricate physical origin. Indeed, Pauli's proof of the spin–statistics theorem [Pau40] (improving on earlier work by Fierz [Fie39]), showed that the spin–statistics connection was deeply rooted in relativistic quantum field theory. The path to a rigorous proof of this theorem (from the mathematical point of view) was a long one and involved the efforts of many people (see, e.g., the book by Duck and Sudarshan [DS98b]).
We comprehensively review the quantum theory of the polarization properties of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincaré sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincaré space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored, and their potential applications in quantum technologies are discussed.
Abstract Research on optical modes, such as orbital, temporal, or parity, bring much attention, for these new degrees of freedom allow larger quantum communication alphabets. Each lab usually adapts the Bloch or Poincare sphere to their experiment or light mode. This takes an extra effort and time and produces a plethora of spheres and notations. Yet, we miss a common framework or convention valid among diverse physical-modes. We aim to unite in one representation the best points from many different spheres. Such common-sphere could also help to compare distant experiments, for an intuitive understanding of quantum optical states. We built a common representation by mathematically aligning the Hilbert space and a three dimensional color space. We define a unique color for each one of the three Poincare axes and positive Pauli vectors. Beyond three primary colors and states, our equations associate each Hilbert state to a specific tonality, among the infinite combinations in Color space. These maths achieve a new ability to unequivocally represent any quantum state by its precise combination of colors. Thus, with these equations, quantum states ‘yellow’ or ‘magenta’ are not mere names, rather each one denotes an exact superposition in Hilbert space. To handle disparities between SO3 vs. SU2 space operations, we propose a darkness bit and a Hermite-inspired shape. A simulation of HG modes let us align distinct shapes to quantum optical states. Three examples of applications show our color sphere in practice. First, we apply the Hilbert-Color mapping in Polarization. Then, the same color-space is shown in Orbital Angular Momentum. We also represent location paths in this color-mapping. The simulations and practical comparisons let us refine the proposed color sphere convention. For higher-order and path-to-industry, any sphere section serves as color constellation diagram. One color-space sphere served as common ground to represent coexisting concepts among diverse physical areas. The introduced change diagrams are visual tools to communicate setups and operators. The examples showed a unique notation matches many physical processes. The resulting diagram of superposition of spatially separated optical paths is coherent with a Plate on Polarization or cylinder lens on OAM Hermite Gaussian modes. A unique change diagram describes the three examples. The meaning persist despite the physical implementation. Found also how this color space let us grasp visually some meaning. Thus, the amount of blue in a state representation indicated the degree of its phase shift. Overall, we presented math and visual tools to display and compare experiments. We showed examples in different physical modes, all linked to a unique color sphere.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.