Time-dependent Gauge Research Articles

Ultracold gases in presence of time-dependent synthetic gauge field

We analyze the effect of a time-dependent synthetic magnetic field on the Mott-Insulating (MI) to superfluid (SF) phase transition of a system of bosons via a site decoupling mean-field approach. The synthetic field is incorporated through Peierls coupling that induces a phase shift in the hopping matrix of the Bose-Hubbard model (BHM). We observe that the behavior of critical hopping strength, Jc, vary differently for different time-dependent synthetic fields. The Mott insulating lobe either stabilizes or contracts depending on the nature of the tuning of the time-dependent synthetic field. Also the inclusion of time in the synthetic field destroys the lattice periodicity, however with proper tuning of the time variation of the gauge field, the periodicity can be restored back. Hence the periodicity of the one-dimensional superfluid order parameter is very sensitive to the variation of the time-dependent field. Further, the order parameter either becomes site dependent or independent depending on the nature of the tuning of the these fields. The time-dependent synthetic magnetic field also induces an asymmetry in the discrete energy spectrum of the system which is also clear from the vanishing ‘Butterfly’ pattern of the single particle energy spectrum.

Schwinger-Dyson equations in Coulomb gauge consistent with numerical simulation

In the present work we undertake a study of the Schwinger-Dyson equation (SDE) in the Euclidean formulation of local quantum gauge field theory, with Coulomb gauge condition $\partial_i A_i = 0$. We continue a previous study which kept only instantaneous terms in the SDE that are proportional to $\delta(t)$ in order to calculate the instantaneous part of the time component of the gluon propagator $D_{A_0 A_0}(t, R)$. We compare the results of that study with a numerical simulation of lattice gauge theory and find that the infrared critical exponents and related quantities agree to within 1\% to 3\%. This raises the question, "Why is the agreement so good, despite the systematic neglect of non-instantaneous terms?" We discovered the happy circumstance that all the non-instantaneous terms are in fact zero. They are forbidden by the symmetry of the local action in Coulomb gauge under time-dependent gauge transformations $g(t)$. This remnant gauge symmetry is not fixed by the Coulomb gauge condition. The numerical result of the present calculation is the same as in the previous study; the novelty is that we now demonstrate that all the non-instantaneous terms in the SDE vanish. We derive some elementary properties of propagators which are a consequence of the remnant gauge symmetry. In particular the time component of the gluon propagator is found to be purely instantaneous $D_{A_0 A_0}(t, R) = \delta(t) V(R)$, where $V(R)$ is the color-Coulomb potential. Our results support the simple physical scenario in which confinement is the result of a linearly rising color-Coulomb potential, $V(R) \sim \sigma R$ at large $R$.

Constructing a boosted, spinning black hole in the damped harmonic gauge

The damped harmonic gauge is important for numerical relativity computations based on the generalized harmonic formulation of Einstein's equations, and is used to reduce coordinate distortions near binary black hole mergers. However, currently there is no prescription to construct quasiequilibrium binary black hole initial data in this gauge. Instead, initial data are typically constructed using a superposition of two boosted analytic single black hole solutions as free data in the solution of the constraint equations. Then, a smooth time-dependent gauge transformation is done early in the evolution to move into the damped harmonic gauge. Using this strategy to produce initial data in damped harmonic gauge would require the solution of a single black hole in this gauge, which is not known analytically. In this work we construct a single boosted, spinning, equilibrium BH in damped harmonic coordinates as a regular time-independent coordinate transformation from Kerr-Schild coordinates. To do this, we derive and solve a set of 4 coupled, nonlinear, elliptic equations for this transformation, with appropriate boundary conditions. This solution can now be used in the construction of damped harmonic initial data for binary black holes.

Investigation of rotameric conformations of substituted imidazo-[1,2-a]pyrazine: experimental and theoretical approaches.

The different rotameric conformations of imidazo-[1,2-a]pyrazine have been synthesized and characterized by means of different experimental techniques, such as NMR, FTIR, and absorption spectroscopy and quantum chemical calculations. The different conformations were stabilized by hydrogen bonds, such as OH⋯N, ArH⋯N and ArH⋯ArH. The ground state optimizations and potential energy surface (PES) scanning profiles produced using density functional theory (DFT) show two stable rotameric forms for each molecule. The relative population of the conformations is affected by the strength of the hydrogen bonds. The calculated absorption spectra and isotopic shielding constants were acquired by time-dependent density functional theory (TD-DFT) and gauge invariant atomic orbitals (GIAO)-DFT, respectively. The strength of the hydrogen bonding interactions that resulted in the different conformations was studied by quantum theory of atoms in molecules (QTAIM).

Emergence of electromotive force by time dependent gauge fields in monolayer graphene

We propose a valley-dependent electromotive force in a graphene device based on a deformed graphene in the presence of magnetic field. The coexistence of the strain and magnetic field in the presented mechanism leads to a valley dependent electromotive force in zero Fermi energy. In the non-zero Fermi energies strain alone could induce a valley dependent electromotive force while the magnetic field alone could not induce any electromotive force in zero and any level of Fermi energy. The calculations are based on the Berry curvature approaches.

Equations of motion as constraints: superselection rules, Ward identities

The meaning of local observables is poorly understood in gauge theories, not to speak of quantum gravity. As a step towards a better understanding we study asymptotic (infrared) transformations in local quantum physics. Our observables are smeared by test functions, at first vanishing at infinity. In this context we show that the equations of motion can be seen as constraints, which generate a group, the group of space and time dependent gauge transformations. This is one of the main points of the paper. Infrared nontrivial effects are captured allowing test functions which do not vanish at infinity. These extended operators generate a larger group. The quotient of the two groups generate superselection sectors, which differentiate different infrared sectors. The BMS group changes the super-selection sector, a result long known for its Lorentz subgroup. It is hence spontaneously broken. Ward identities implied by the gauge invariance of the S-matrix generalize the standard results and lead to charge conservation and low energy theorems. Their validity does not require Lorentz invariance.

Non-Abelian Aharonov–Bohm effect with the time-dependent gauge fields

We investigate the non-Abelian Aharonov-Bohm (AB) effect for time-dependent gauge fields. We prove that the non-Abelian AB phase shift related to time-dependent gauge fields, in which the electric and magnetic fields are written in the adjoint representation of $SU(N)$ generators, vanishes up to the first order expansion of the phase factor. Therefore, the flux quantization in a superconductor ring does not appear in the time-dependent Abelian or non-Abelian AB effect.

Three-Dimensional Dynamic Localization of Light from a Time-Dependent Effective Gauge Field for Photons.

We introduce a method to achieve the three-dimensional dynamic localization of light. We consider a dynamically modulated resonator lattice that has been previously shown to exhibit an effective gauge potential for photons. When such an effective gauge potential varies sinusoidally in time, dynamic localization of light can be achieved. Moreover, while previous works on such an effective gauge potential for photons were carried out in the regime where the rotating wave approximation is valid, the effect of dynamic localization persists even when the counterrotating term is taken into count.

Correspondences between quantum and classical orbits Berry phases and Hannay angles for harmonic oscillator system

On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.

Some Remarks on Quantum Mechanics in a Curved Spacetime, Especially for a Dirac Particle

Some precisions are given about the definition of the Hamiltonian operator H and its transformation properties, for a linear wave equation in a general spacetime. In the presence of time-dependent unitary gauge transformations, H as an operator depends on the gauge choice. The other observables of QM and their rates also become gauge-dependent unless a proper account for the gauge choice is done in their definition. We show the explicit effect of these non-uniqueness issues in the case of the Dirac equation in a general spacetime with the Schwinger gauge. We show also in detail why, the meaning of the energy in QM being inherited from classical Hamiltonian mechanics, the energy operator and its mean values ought to be well defined in a general spacetime.

Equations of motion for natural orbitals of strongly driven two-electron systems

Natural orbital theory is a computationally useful approach to the few and many-body quantum problem. While natural orbitals are known and applied since many years in electronic structure applications, their potential for time-dependent problems is being investigated only since recently. Correlated two-particle systems are of particular importance because the structure of the two-body reduced density matrix expanded in natural orbitals is known exactly in this case. However, in the time-dependent case the natural orbitals carry time-dependent phases that allow for certain time-dependent gauge transformations of the first kind. Different phase conventions will, in general, lead to different equations of motion for the natural orbitals. A particular phase choice allows us to derive the exact equations of motion for the natural orbitals of any (laser-) driven two-electron system explicitly, i.e., without any dependence on quantities that, in practice, require further approximations. For illustration, we solve the equations of motion for a model helium system. Besides calculating the spin-singlet and spin-triplet ground states, we show that the linear response spectra and the results for resonant Rabi flopping are in excellent agreement with the benchmark results obtained from the exact solution of the time-dependent Schr\"odinger equation.

Bianchi type VI1 cosmological model with wet dark fluid in scale invariant theory of gravitation

In this paper, we have investigated Bianchi type VIh, II and III cosmological model with wet dark fluid in scale invariant theory of gravity, where the matter field is in the form of perfect fluid and with a time dependent gauge function (Dirac gauge). A non-singular model for the universe filled with disorder radiation is constructed and some physical behaviors of the model are studied for the feasible VIh (h=1) space-time.

LRS Bianchi Type II Massive String Cosmological Models with Magnetic Field in Lyra's Geometry

Bianchi type II massive string cosmological models with magnetic field and time dependent gauge function (<svg style="vertical-align:-4.15506pt;width:13.6px;" id="M1" height="17.049999" version="1.1" viewBox="0 0 13.6 17.049999" width="13.6" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.813)"><path id="x1D719" d="M542 242q0 -106 -80 -176.5t-192 -72.5l-55 -251l-6 -3q-18 7 -28 38t1 91l23 124q-85 16 -133.5 72.5t-48.5 137.5q0 95 62.5 159.5t166.5 97.5l14 -31q-77 -32 -115 -82.5t-38 -128.5q0 -63 31.5 -107t74.5 -59l125 597l56 43l16 -10l-40 -168q-9 -40 3 -45&#xA;q77 -33 120 -85.5t43 -140.5zM469 232q0 67 -37 115t-77 62l-76 -370q89 -4 139.5 52.5t50.5 140.5z" /></g> <g transform="matrix(.012,-0,0,-.012,9.663,16.838)"><path id="x1D456" d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30&#xA;q16 0 76 48z" /></g> </svg>) in the frame work of Lyra's geometry are investigated. The magnetic field is in <svg style="vertical-align:-0.0pt;width:23.924999px;" id="M2" height="11.6" version="1.1" viewBox="0 0 23.924999 11.6" width="23.924999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.537)"><path id="x1D44C" d="M667 650l-9 -28q-53 -5 -76 -17t-64 -59q-51 -61 -175 -225q-21 -29 -27 -55l-27 -136q-13 -65 -0.5 -80t83.5 -22l-7 -28h-280l8 28q64 4 81 19t30 83l25 128q6 35 -7 65l-98 231q-17 41 -32.5 52t-68.5 16l8 28h252l-6 -28l-40 -4q-27 -3 -33 -12.5t2 -31.5&#xA;q8 -26 43 -107.5t61 -134.5q114 145 174 240q14 24 8 33t-37 13l-34 4l8 28h238z" /></g><g transform="matrix(.017,-0,0,-.017,11.605,11.537)"><path id="x1D44D" d="M698 636l-541 -596q60 -5 176 -5q91 0 138.5 5t69.5 24q44 36 85 124l29 -15q-38 -125 -64 -173h-559l-9 16l545 598h-182q-81 0 -109 -8t-48 -31q-26 -29 -55 -103l-29 3q23 86 42 200h22q11 -16 21 -20.5t34 -4.5h428z" /></g> </svg>-plane. To get the deterministic solution, we have assumed that the shear (<svg style="vertical-align:-0.1638pt;width:9.7250004px;" id="M3" height="8.3000002" version="1.1" viewBox="0 0 9.7250004 8.3000002" width="9.7250004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,8.038)"><path id="x1D70E" d="M548 455q-32 -77 -74 -77q-37 0 -103 10l-3 -2q36 -30 48.5 -62t12.5 -80q0 -103 -75 -179.5t-175 -76.5q-70 0 -113 45t-43 128q0 115 83.5 200.5t209.5 85.5q31 0 77.5 -4t67.5 -4q36 0 62 30zM350 274q0 54 -17 86q-19 35 -57 35q-50 0 -90 -37.5t-59 -91.5t-19 -109&#xA;q0 -61 24.5 -96.5t65.5 -35.5q51 0 87.5 46.5t50.5 100.5t14 102z" /></g> </svg>) is proportional to the expansion (<svg style="vertical-align:-0.1638pt;width:8.5874996px;" id="M4" height="12.4375" version="1.1" viewBox="0 0 8.5874996 12.4375" width="8.5874996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D703" d="M475 507q0 -83 -20 -172t-56 -167.5t-93.5 -129t-125.5 -50.5q-157 0 -157 227q0 78 21.5 164t59 161t96.5 123.5t126 48.5q79 0 114 -58t35 -147zM391 522q0 155 -81 155q-62 0 -111 -82.5t-73 -200.5h253q12 81 12 128zM373 346h-255q-12 -91 -12 -150q0 -72 20 -123&#xA;t63 -51q34 0 64 28.5t52.5 77t39 103t28.5 115.5z" /></g> </svg>). This leads to <svg style="vertical-align:-0.23206pt;width:45.25px;" id="M5" height="13.9" version="1.1" viewBox="0 0 45.25 13.9" width="45.25" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,13.55)"><path id="x1D445" d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117&#xA;t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" /></g><g transform="matrix(.017,-0,0,-.017,15.718,13.55)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,30.422,13.55)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68&#xA;l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g> <g transform="matrix(.012,-0,0,-.012,38.588,5.388)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g> </svg>, where <svg style="vertical-align:-0.1092pt;width:11.075px;" id="M6" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D445"/></g> </svg> and <svg style="vertical-align:-0.23206pt;width:8.2875004px;" id="M7" height="11.75" version="1.1" viewBox="0 0 8.2875004 11.75" width="8.2875004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D446"/></g> </svg> are metric potentials and <svg style="vertical-align:-0.1638pt;width:8.6625004px;" id="M8" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"/></g> </svg> is a constant. We find that the models start with a big bang at initial singularity and expansion decreases due to lapse of time. The anisotropy is maintained throughout but the model isotropizes when <svg style="vertical-align:-0.1638pt;width:36.237499px;" id="M9" height="11.125" version="1.1" viewBox="0 0 36.237499 11.125" width="36.237499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,13.305,10.862)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,28.008,10.862)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> </svg>. The physical and geometrical aspects of the model in the presence and absence of magnetic field are also discussed.

A new loss mechanism in graphene nanoresonators due to the synthetic electric fields caused by inherent out-of-plane membrane corrugations

For the first time the influence of out-of-plane deformations, which always exist in graphene, on the non-stationary processes is considered for the case of a monolayer graphene nanoresonator. A new loss mechanism for this device caused by dissipative intra-valley currents stipulated by synthetic electric fields is studied. These fields are generated by time-dependent gauge fields arising in a graphene membrane due to its intrinsic out-of-plane distortions and the influence of the external periodic electromotive force. The corresponding formula for the quality factor has a quantum mechanical origin and includes quantum mechanical parameters. This loss mechanism accounts for an essential part (about 40%) of losses in a graphene nanoresonator and it is specific just for graphene. The ways of minimization of this kind of dissipation (an increase in the quality factor of the electromechanical system) are discussed. It is explained why one can enhance the quality factor by correctly choosing a combination of strains (by strain engineering). In addition, it is shown that the quality factor can be increased by switching on a magnetic field perpendicular to the graphene membrane.

In-out formalism for the QED effective action in a constant electromagnetic field

In the in-out formalism, the effective action is given by the vacuum persistence amplitude and is expressed by using Bogoliubov coefficients between the in-vacuum and the out-vacuum. In the second quantized field theory, the Jost functions with proper boundary conditions determine the Bogoliubov coefficients for spin-1/2 fermions or spinless, charged bosons in electric fields. We apply the in-out formalism to compute the complex effective action in a constant electric field parallel to a constant magnetic field in the space-dependent gauge and in the time-dependent gauge. The vacuum polarization has the proper-time integral representation through the gamma-function regularization, and the vacuum persistence is twice the sum of the residues at the poles.

Spin-depairing transition of attractive Fermi gases on a ring driven by synthetic gauge fields

Motivated by the recent experimental realization of synthetic gauge fields in ultracold atoms, we investigate one-dimensional attractive Fermi gases with a time-dependent gauge flux on the spin sector. By combining the methods of the Bethe ansatz with complex twists and Landau-Dykhne, it is shown that a spin-depairing transition occurs, which may represent a nonequilibrium transition from fermionic superfluids to normal states with spin currents caused by a many-body quantum tunneling. For the case of the Hubbard ring at half filling, our finding forms a dual concept with the dielectric breakdown of the Mott insulator discussed in [Phys. Rev. B 81, 033103 (2010)]. We analyze cases of arbitrary filling and continuum model, and show how filling affects the transition probability.

SCHRÖDINGER EQUATIONS WITH TIME-DEPENDENT UNBOUNDED SINGULAR POTENTIALS

We consider time-dependent perturbations by unbounded potentials of Schrödinger operators with scalar and magnetic potentials which are almost critical for the selfadjointness. We show that the corresponding time-dependent Schrödinger equations generate a unique unitary propagator if perturbations of scalar and magnetic potentials are differentiable with respect to the time variable and they increase at the spatial infinity at most quadratically and at most linearly, respectively, where both have mild local singularities. We use time-dependent gauge transforms and apply Kato's abstract theorem on evolution equations.

Tunneling of anyonic Majorana excitations in topological superconductors

We consider topological superconductors and topological insulator/superconductor structures in the presence of multiple static vortices that host Majorana modes and focus on the Majorana tunneling processes between vortices. It is shown that these tunnelings generally lift the degeneracy of the many-body ground state in a non-universal way, sensitive to microscopic details at the smallest length-scales determined by the underlying physical problem. We also discuss an explicit realization of the Jackiw-Rossi zero-mode in a topological insulator/superconductor structure with zero chemical potential. In this case, the exact degeneracy of the many-anyon ground state is protected by an additional chiral symmetry and can be linked to the rigorous index theorem. However, the existence of a non-zero chemical potential, as expected in realistic solid state structures, breaks chiral symmetry and removes protection, which leads to the degeneracy being lifted. Finally, we discuss the implications of our results for the collective states of many-anyon systems. We argue that quantum dynamics of vortices in realistic systems is generally important and may give rise to effective time-dependent gauge factors that enter interaction terms between Majorana modes in many-anyon systems.

BIANCHI TYPE-I STRING DUST COSMOLOGICAL MODELS IN LYRA GEOMETRY

Bianchi type-I string cosmological models with time dependent gauge function (β) within the framework of Lyra geometry is presented. To get the deterministic model of the universe, we assume that the eigenvalue [Formula: see text] of shear tensor [Formula: see text] is proportional to the expansion (θ). This leads to A = (BC)nwhere A, B, C are metric potential and n is a constant. The physical and geometrical aspects of the model are discussed in detail. In a special case, the behavior of the model in terms of cosmic time t for [Formula: see text] is also discussed.

Gauge transformation through an accelerated frame of reference

The Schrödinger equation of a charged particle in a uniform electric field can be specified in either a time-independent or a time-dependent gauge. The wave functions in these two gauges are related by a phase factor reflecting the gauge symmetry of the problem. We show that the effect of such a gauge transformation connecting the two wave functions can be mimicked by the effect of two successive extended Galilean transformations. An extended Galilean transformation connects two reference frames such that one is accelerating with respect to the other.