We investigate the Wigner rotation for photons, which governs the change in the polarization of the photon as it propagates through an arbitrary gravitational field. We give explicit examples in Schwarzschild spacetime, and compare with the corresponding flat spacetime results, which by the equivalence principle, holds locally at each spacetime point. We discuss the implications of the Wigner rotation for entangled photon states in curved spacetime, and lastly develop a sufficient condition for special (Fermi-Walker) frames in which the observer would detect no Wigner rotation.
In this article, we study the Sagnac effect for spin-$1/2$ particles through local Wigner rotations according to the framework developed by [H. Terashima and M. Ueda, Phys. Rev. A 69, 032113 (2004)]. Since the spin of the particle plays the role of a quantum `clock', as the quanton moves in a superposed path it gets entangled with the momentum (or the path), and this will cause the interferometric visibility to drop, since there is a difference in proper time elapsed along the two trajectories, which is known as the Sagnac time delay.
We investigate in a covariant manner the spin-induced non-geodesic motion of massive spin 1/2 particles in an arbitrary gravitational field for trajectories that are initially geodesic when spin is ignored. Using the WKB approximation for the wave function in curved spacetime, we compute the O(hbar) correction to the Wigner rotation of the spin 1/2 particle, whose O(1) contribution is zero on timelike geodesics. We develop conditions for the motion of observers in which the Wigner rotation is null. For the spherically symmetric Schwarzschild metric, we consider specific examples of particle mo…
The Wigner rotation angle for a particle in a circular motion in the Schwarzschild spacetime is obtained via the Fermi-Walker transport of spinors. Then, by applying the WKB approximation, a possible application of the Fermi-Walker transport of spinors in relativistic EPR correlations is discussed, where it is shown that the spins of the correlated particle undergo a precession in an analogous way to that obtained by Terashima and Ueda [H. Terashima and M. Ueda, Phys. Rev. A {\bf69}, 032113 (2004)] via the application of successive infinitesimal Lorentz transformations. Moreover, from the WKB …
Using a position operator obtained for spin 1 particles by the present author and Wigner, we obtain a quantum relativistic result for the hidden momentum force experienced by particles with structure. In particular, our result applies to the hidden magnetic forces manifest in some problems of electromagnetism. We also discuss spin and orbital angular momentum operators, as well as Wigner rotation.
As Lanzagorta and Crowder have shown in their Comment [Phys. Rev. A 96, 026101 (2017)], the linear application of the Wigner rotations to the quantum state of two massive relativistic particles does not entail the instantaneous transmission of information as we concluded in our paper [Phys. Rev. A 87, 042102 (2013)]. But in the new version of our paper in arXiv [arXiv:1303.4367v2] we show (and solve) another paradox that is generated by the linear application of the Wigner rotations to a system with a single relativistic particle. So the conclusion of our paper that we cannot in general linear…
The deformation quantization formalism is applied to the linearized gravitational field. Standard aspects of this formalism are worked out before the ground state Wigner functional is obtained. Finally, the propagator for the graviton is also discussed within the context of this formalism.
We study relativistic effects on polarised photons that travel in a curved spacetime. As a concrete application, we consider photons in the gravitational field of the Earth, on a closed path that starts at a terrestial laboratory, is reflected at one or more satellites, and finally returns to the laboratory. We find that the photons acquire a non-trivial Wigner phase already when the gravitational field is static, such as the Schwarzschild spacetime, where previous studies have found a trivial Wigner phase for closed photon paths. A gauge-invariant description of this nontrivial Wigner phase r…
Wigner rotations are transformations that affect spinning particles and cause the observable phenomenon of Thomas precession. Here we study these rotations for arbitrary symmetry groups with a semi-direct product structure. In particular we establish a general link between Wigner rotations and Thomas precession by relating the latter to the holonomies of a certain Berry connection on a momentum orbit. Along the way we derive a formula for infinitesimal, Lie-algebraic transformations of one-particle states.
A new uniform asymptotic approximation for the Wigner $6j$ symbol is given in terms of Wigner rotation matrices ($d$-matrices). The approximation is uniform in the sense that it applies for all values of the quantum numbers, even those near caustics. The derivation of the new approximation is not given, but the geometrical ideas supporting it are discussed and numerical tests are presented, including comparisons with the exact $6j$-symbol and with the Ponzano-Regge approximation.
The Wigner rotation is a key word in many branches of physics, chemistry and engineering sciences. It is a group theoretical effect resulting from two Lorentz boosts. The net effect is one boost followed or preceded by a rotation. This rotation can therefore be formulated as a product of three boosts. In relativistic kinematics, it is a rotation in the Lorentz frame where the particle is at rest. This rotation does not change its momentum, but it rotates the direction of the spin. The Wigner rotation is not confined to relativistic kinematics. It manifests itself in physical systems where the …
Wigner rotations and Iwasawa decompositions are manifestations of the internal space-time symmetries of massive and massless particles, respectively. It is shown to be possible to produce combinations of optical filters which exhibit transformations corresponding to Wigner rotations and Iwasawa decompositions. This is possible because the combined effects of rotation, phase-shift, and attenuation filters lead to transformation matrices of the six-parameter Lorentz group applicable to Jones vectors and Stokes parameters for polarized light waves. The symmetry transformations in special relativi…
We derive the leading asymptotic approximation, for low angle θ, of the Wigner rotation matrix elements $d^j_{m_1m_2}(θ)$, uniform in $j,m_1$ and $m_2$. The result is in terms of a Bessel function of integer order. We numerically investigate the error for a variety of cases and find that the approximation can be useful over a significant range of angles. This approximation has application in the partial wave analysis of wavepacket scattering.
We compute, for massive particles, the explicit Wigner rotations of one-particle states for arbitrary Lorentz transformations; and the explicit Hermitian generators of the infinite-dimensional unitary representation. For a pair of spin 1/2 particles, Einstein-Podolsky-Rosen-Bell entangled states and their behaviour under the Lorentz group are analysed in the context of quantum field theory. Group theoretical considerations suggest a convenient definition of the Bell states which is slightly different from the conventional assignment. The behaviour of Bell states under arbitrary Lorentz transfo…
It is shown that a general model for particle detection in combination with a linear application of the Wigner rotations, which correspond to momentum-dependent changes of the particle spin under Lorentz transformations, to the state of a massive relativistic particle in a superposition of two counter-propagating momentum states leads to a paradox. The paradoxical behavior is that the probability of finding the particle at different positions would depend on the reference frame. A solution to the paradox is given when the physical construction of the corresponding state is taken into account, …
We present a new treatment for the spin of a massive relativistic particle in the context of quantum information based on a physical interpretation of the Wigner rotations, obtaining different results in relation to the previous works. We are lead to the conclusions that it is not possible to define a reduced density matrix for the particle spin and that the Pauli-Lubanski (or similar) spin operators are not suitable to describe measurements where spin couples to an electromagnetic field in the measuring apparatus. These conclusions contradict the assumptions made by most of the previous paper…
The effective mass that approximately describes the effect of a classical homogeneous gravitational field on an interacting atom-radiation field system is determined within the framework of the Jaynes-Cummings model. By taking into account both the atomic motion and gravitational field, a full quantum treatment of the internal and external dynamics of the atom is presented. By solving exactly the Schrodinger equation in the interaction picture, the evolving state of the system is found. Influence of a classical homogeneous gravitational field on the energy eigenvalues, the effective mass of at…
In this work, we report that the Hawking radiation effect on fermions is fundamentally different from the case of scalar particles. Intrinsic properties of fermions (exclusion principle and spin) affect strongly the interaction of fermions with both Hawking radiation and metric of the spacetime. In particular we have found the following: first, while the fermion vacuum state seen by the Rindler observer is an entangled state in which the right and left Rindler wedge states appear in correlated pairs as in the case of the scalar particles, the entanglement disappears in the excited state due to…
Wigner's Phase Space Current for Variable Beam Splitters -- Phase Space Rotations and Newtonian Trajectories
Institute of Photonics Technologies, National Tsing Hua University（国立清华大学光子技术研究所）；Department of Physics, National Tsing Hua University（国立清华大学物理系）；Center for Theory and Computation, National Tsing Hua University（国立清华大学理论与计算中心）
A recent paper (gr-qc/9909017) criticizes our work on the structure of spacetime foam. Its authors argue that the quantum uncertainty limit for the position of the quantum clock in a gedanken timing experiment, obtained by Wigner and used by us, is based on unrealistic assumptions. Here we point out some flaws in their argument. We also discuss their other comments and some other issues related to our work, including a simple connection to the holographic principle. We see no reason to change our cautious optimism on the detectability of spacetime foam with future refinements of modern gravita…
We obtain the accuracy limit for estimating the expectation value of the position of a relativistic particle for an observer moving along one direction at a constant velocity. We use a specific model of a relativistic spin-1/2 particle described by a gaussian wave function with a spin down in the rest frame. To derive the state vector of the particle for the moving observer, we use the Wigner rotation that entangles the spin and the momentum of the particle. Based on this wave function for the moving frame, we obtain the symmetric logarithmic derivative (SLD) Cramer-Rao bound that sets the est…
A comprehensive theory of the Weyl-Wigner formalism for the canonical pair angle-angular momentum is presented, with special emphasis in the implications of rotational periodicity and angular-momentum discreteness.
The Wigner rotation matrix ($d$-function), which appears as a part of the angular-momentum-projection operator, plays a crucial role in modern nuclear-structure models. However, it is a long-standing problem that its numerical evaluation suffers from serious errors and instability, which hinders precise calculations for nuclear high-spin states. Recently, Tajima [Phys. Rev. C 91, 014320 (2015)] has made a significant step toward solving the problem by suggesting the high-precision Fourier method, which however relies on formula-manipulation softwares. In this paper we propose an effective and …
The ``little group'' for massless particles (namely, the Lorentz transformations $Λ$ that leave a null vector invariant) is isomorphic to the Euclidean group E2: translations and rotations in a plane. We show how to obtain explicitly the rotation angle of E2 as a function of $Λ$ and we relate that angle to Berry's topological phase. Some particles admit both signs of helicity, and it is then possible to define a reduced density matrix for their polarization. However, that density matrix is physically meaningless, because it has no transformation law under the Lorentz group, even under ordinary…
We analyze the Wigner function constructed on the basis of the discrete rotation and displacement operators labeled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyze the algebraic origin of the non uniqueness of the representation of the Wigner function. Explicit expressions for the Wigner kernel are given in both cases.
We discuss a recent provocative suggestion by Amelino-Camelia and others that classical spacetime may break down into ``quantum foam'' on distance scales many orders of magnitude larger than the Planck length, leading to effects which could be detected using large gravitational wave interferometers. This suggestion is based on a quantum uncertainty limit obtained by Wigner using a quantum clock in a gedanken timing experiment. Wigner's limit, however, is based on two unrealistic and unneccessary assumptions: that the clock is free to move, and that it does not interact with the environment. Re…
We study the effects of frequency-dependent squeeze amplitude attenuation and squeeze angle rotation by electromagnetically induced transparency (EIT) on gravitational wave (GW) interferometers. We propose the use of low-pass, band-pass, and high-pass EIT filters, an S-shaped EIT filter, and an intra-cavity EIT filter to generate frequency-dependent squeezing for injection into the antisymmetric port of GW interferometers. We find that the EIT filters have several advantages over the previous filter designs with regard to optical losses, compactness, and the tunability of the filter linewidth.
Quantization of a random-walk model is performed by giving a qudit (a multi-component wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine…
The connection between spin and symmetry was established by Wigner in his 1939 paper on the Poincaré group. For a massive particle at rest, the little group is O(3) from which the concept of spin emerges. The little group for a massless particle is isomorphic to the two-dimensional Euclidean group with one rotational and two translational degrees of freedom. The rotational degree corresponds to the helicity, and the translational degrees to the gauge degree of freedom. The question then is whether these two different symmetries can be united. Another hard-pressing problem is Feynman's parton p…
We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $\lnπ+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural min…
We investigate the rotation of the polarization of a light ray propagating in the gravitational field of a circularly polarized laser beam. The rotation consists of a reciprocal part due to gravitational optical activity, and a non-reciprocal part due to the gravitational Faraday effect. We discuss how to distinguish the two effects: Letting light propagate back and forth between two mirrors, the rotation due to gravitational optical activity cancels while the rotation due to the gravitational Faraday effect accumulates. In contrast, the rotation due to both effects accumulates in a ring cavit…
The analysis of a physical problem simplifies considerably when one uses a suitable coordinate system. We apply this approach to the discrete-time quantum walks with coins given by $2j+1$-dimensional Wigner rotation matrices (Wigner walks), a model which was introduced in T. Miyazaki et al., Phys. Rev. A 76, 012332 (2007). First, we show that from the three parameters of the coin operator only one is physically relevant for the limit density of the Wigner walk. Next, we construct a suitable basis of the coin space in which the limit density of the Wigner walk acquires a much simpler form. This…
We show that the cylindrical symmetry of the eigenvectors of the photon position operator with commuting components, x, reflects the E(2) symmetry of the photon little group. The eigenvectors of x form a basis of localized states that have definite angular momentum, J, parallel to their common axis of symmetry. This basis is well suited to the description of "twisted light" that has been the subject of many recent experiments and calculations. Rotation of the axis of symmetry of this basis results in the observed Berry phase displacement. We prove that {x1,x2,J3} is a realization of the two di…
In this work we analyze the amount of entanglement associated with the spin and momentum degrees of freedom of a single massive spin-$\frac{1}{2}$ particle from a relativistic perspective. The effect of a Lorentz boost introduces a Wigner rotation that correlates the spin and momentum degrees of freedom. We show that the natural basis to discuss the geometrical effects of the boost are the helicity eigenstates in the rest frame. In the mid-relativistic regime (where the Wigner rotation angle is limited by $δ< \fracπ{2}$) we prove for states with equal helicity that the entanglement with respec…
The effect of a classical gravitational field on the spin entanglement of a system of two spin-1/2 particles moving in the curved spacetime is discussed. The system is described by a two-particle Gaussian wave packet represented in the momentum space and both the acceleration of the system and the curvature of the spacetime cause to produce a Wigner rotation acting on the wave packet as it moves along a path in the curved spacetime. By calculating the reduced density operator at a final point, we focus on the spin entanglement of the system. In a spherically symmetric and static gravitational …
We study the loss of quantumness caused by time dilation [1] for a Schrödinger cat state. We give a holistic view of the quantum to classical transition by comparing the dynamics of several nonclassicality indicators, such as the Wigner function interference fringe, the negativity of the Wigner function, the nonclassical depth, the Vogel criterion and the Klyshko criterion. Our results show that only two of these indicators depend critically on the size of the cat, namely on how macroscopic the superposition is. Finally we compare the gravitation-induced decoherence times to the typical decohe…
A gauge-invariant Wigner quasi-distribution function for charged particles in classical electromagnetic fields is derived in a rigorous way. Its relation to the axial gauge is discussed, as well as the relation between the kinetic and canonical momenta in the Wigner representation. Gauge-invariant quantum analogs of Hamilton-Jacobi and Boltzmann kinetic equations are formulated for arbitrary classical electromagnetic fields in terms of the "slashed" derivatives and momenta, introduced for this purpose. The kinetic meaning of these slashed quantities is discussed. We introduce gauge-invariant c…
We present new sum rules for $3jm$ coefficients, which involve, in addition to the usual weighting factor $(2j + 1)$ where $j$ is an angular momentum, the quantity $[j(j + 1)]^k$ with $k \ge 1$. The sum rules appear for instance in the statistical modeling of rotational spectra within the theory of moments, and enable one to deduce the expectation values of $r^k$ (used in the theory of Stark effect for hydrogenic ions) in parabolic coordinates from the expectation values of $r^k$ in spherical coordinates.
The time evolution of a quantum wave packet in the linear gravity potential is known as Quantum Bouncing Ball. The qBounce collaboration recently observed such a system by dropping wave packets of ultracold neutrons by a height of roughly 30 microns. In this article, space and momentum spectra as well as Wigner functions of the neutron wave functions in the gravitational field of the Earth are analyzed. We investigate the quantum states in the "preparation region", into which they transition after exiting a narrow double-mirror system and where we would expect to observe free fall and bounces …
Rotation-induced splitting of the otherwise degenerate photonic bands is predicted for a two-dimensional photonic crystal made of evanescently coupled microcavities. The symmetry-broken energy splitting is similar to the Zeeman splitting of atomic levels or electron's (hole's) magnetic moment sublevels in an external magnetic field. The orbital motion of photons in periodic photonic lattice of microcavities is shown to enhance significantly such Coriolis-Zeeman splitting as compared to a solitary microcavity [D.L. Boiko, Optics Express 2, 397 (1998)]. The equation of motion suggests that nonst…
This thesis deals with the systematic treatment of quantum-mechanical systems in post-Newtonian gravitational fields. Starting from clearly spelled-out assumptions, employing a framework of geometric background structures defining the notion of a post-Newtonian expansion, our systematic approach allows to properly derive the post-Newtonian coupling of quantum-mechanical systems to gravity based on first principles. This sets it apart from more heuristic approaches that are commonly employed, for example, in the description of quantum-optical experiments under gravity. Regarding single particle…
