Left-right symmetric gauge model in a noncommutative geometry on M4×Z4M_4\times Z_4

The left-right symmetric gauge model with the symmetry of $SU(3)_c\times SU(2)_L\times SU(2)_R\times U(1)$ is reconstructed in a new scheme of the noncommutative differential geometry (NCG) on the discrete space $M_4\times Z_4$ which is a product space of Minkowski space and four points space. The characteristic point of this new scheme is to take the fermion field to be a vector in a 24-dimensional space which contains all leptons and quarks. Corresponding to this specification, all gauge and Higgs boson fields are represented in $24\times 24$ matrix forms. We incorporate two Higgs doublet bosons $h$ and $SU(2)_R$ adjoint Higgs $\xi_R$ which are as usual transformed as $(2,2^\ast,0)$ and $(1,3,-2)$ under $SU(2)_L\times SU(2)_R\times U(1)$, respectively. Owing to the revise of the algebraic rules in a new NCG, we can obtain the necessary potential and interacting terms between these Higgs bosons which are responsible for giving masses to the particles included. Among the Higgs doublet bosons, one CP-even scalar boson survives in the weak energy scale and other four scalar bosons acquire the mass of the $SU(2)_R\times U(1)$ breaking scale, which is similar to the situation in the standard model. $\xi_R$ is responsible to spontaneously break SU(2)\ma{R} \times U(1) down to U(1)\ma{Y} and so well explains the seesaw mechanism. Up and down quarks have the different masses through the vacuum expectation value of $h$.Comment: 22 pages, LaTex fil

Generalized Chern-Simons Form and Descent Equation

We present the general method to introduce the generalized Chern-Simons form and the descent equation which contain the scalar field in addition to the gauge fields. It is based on the technique in a noncommutative differential geometry (NCG) which extends the $N$-dimensional Minkowski space $M_N$ to the discrete space such as $M_N\times Z_2$ with two point space $Z_2$. However, the resultant equations do not depend on NCG but are justified by the algebraic rules in the ordinary differential geometry.Comment: 7 page

Analytic approaches to relativistic hydrodynamics

I summarize our recent work towards finding and utilizing analytic solutions of relativistic hydrodynamic. In the first part I discuss various exact solutions of the second-order conformal hydrodynamics. In the second part I compute flow harmonics $v_n$ analytically using the anisotropically deformed Gubser flow and discuss its dependence on $n$, $p_T$, viscosity, the chemical potential and the charge.Comment: 8 pages, contribution to Proceedings of Quark Matter 201

On zeros of exponential polynomials and quantum algorithms

We calculate the zeros of an exponential polynomial of some variables by a classical algorithm and quantum algorithms which are based on the method of van Dam and Shparlinski, they treated the case of two variables, and compare with the complexity of those cases. Further we consider the ratio (classical/quantum) of the complexity. Then we can observe the ratio is virtually 2 when the number of the variables is sufficiently large.Comment: 8 pages, LaTe

Real-Time and Imaginary-Time Quantum Hierarchal Fokker-Planck Equations

We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time, but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possess the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state, and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy

Generalized Covariant Derivative on Extra Dimension and Weinberg-Salam Model

The generalized covariant derivative on 5-dimen-sional space including 1-dimensional extra compact space is defined, and, by use of it, the Weinberg-Salam model is reconstructed. The spontaneous breakdown of symmetry takes place owing to the extra dimension under the settings that the Higgs field exists in the extra dimensional space depending on the argument $y$ of this extra space, whereas the gauge and fermion fields do not depend on $y$. Both Yang-Mills-Higgs and fermion Lagrangians in Weinberg-Salam model are correctly reproduced.Comment: 5 page

New Incorporation of the Strong Interaction in NCG and Standard Model

The standard model is reconstructed by new method to incorporate strong interaction into our previous scheme based on the non-commutative geometry. The generation mixing is also taken into account. Our characteristic point is to take the fermion field so as to contain quarks and leptons all together which is almost equal to that of SO(10) grand unified theory(GUT). The space-time $M_4\times Z_2$; Minkowski space multiplied by two point discrete space is prepared to express the left-handed and right-handed fermion fields. The generalized gauge field $A(x,y)$ written in one-differential form extended on $M_4\times Z_2$ is well built to give the correct Dirac Lagrangian for fermion sector. The fermion field is a vector in 24-dimensional space and gauge and Higgs fields are written in $24\times24$ matrices. At the energy of the equal coupling constants for both sheets $y=\pm$ expected to be amount to the energy of GUT scale, we can get $\sin^2\theta_{_{W}}=3/8$ and $m_{_{H}}=\sqrt{2}m_{_{W}}$. In general, the equation m\ma{H}=(4/\sqrt {3})m\ma{W}\sin\theta\ma{W} is followed. Then, it should be noticed that the same result as that of the grand unified theory such as SU(5) or SO(10) GUT is obtained without GUT but with the approach based on the non-commutative geometry and in addition the Higgs mass is related to other physical quantities as stated above.Comment: LaTeX file, 20 page

Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities

For a system strongly coupled to a heat bath, the quantum coherence of the system and the heat bath plays an important role in the system dynamics. This is particularly true in the case of non-Markovian noise. We rigorously investigate the influence of system-bath coherence by deriving the reduced hierarchal equations of motion (HEOM), not only in real time, but also in imaginary time, which represents an inverse temperature. It is shown that the HEOM in real time obtained when we include the system-bath coherence of the initial thermal equilibrium state possess the same form as those obtained from a factorized initial state. We find that the difference in behavior of systems treated in these two manners results from the difference in initial conditions of the HEOM elements, which are defined in path integral form. We also derive HEOM along the imaginary time path to obtain the thermal equilibrium state of a system strongly coupled to a non-Markovian bath. Then, we show that the steady state hierarchy elements calculated from the real-time HEOM can be expressed in terms of the hierarchy elements calculated from the imaginary-time HEOM. Moreover, we find that the imaginary-time HEOM allow us to evaluate a number of thermodynamic variables, including the free energy, entropy, internal energy, heat capacity, and susceptibility. The expectation values of the system energy and system-bath interaction energy in the thermal equilibrium state are also evaluated.Comment: J. Chem. Phys. Accepte

Noncommutative gauge theory with arbitrary U(1) charges

It is well-known that the charge of fermion is 0 or $\pm1$ in the U(1) gauge theory on noncommutative spacetime. Since the deviation from the standard model in particle physics has not yet observed, and so there may be no room to incorporate the noncommutative U(1) gauge theory into the standard model because the quarks have fractional charges. However, it is shown in this article that there is the noncommutative gauge theory with arbitrary charges which symmetry is for example SU(3+1)$\ast$. This enveloping gauge group consists of elements $\exp i {\sum_{a=0}^8 T^a \alpha^a(x,\theta)\ast+ Q \beta(x,\theta)\ast}$ with $Q=\text{diag}(e,e,e,e')$ and the restriction $\lim_{\theta\to0}\alpha^0(x,\theta)=0.$ This type of gauge theory is emergent from the spontaneous breakdown of the noncommutative SU(N)$\ast$ or SO(N)$\ast$ gauge theory in which the gauge field contains the 0 component $A_\mu^0(x,\theta)$. $A_\mu^0(x,\theta)$ can be eliminated by gauge transformation. Thus, the noncommutative gauge theory with arbitrary U(1) charges can not exist alone, but it must coexist with noncommutative nonabelian gauge theory. This suggests that the spacetime noncommutativity requires the grand unified theory which spontaneously breaks down to the noncommutative standard model with fractionally charged quarks.Comment: 7 page

Lorentz covariant field theory on noncommutative spacetime based on DFR algebra

Lorentz covariance is the fundamental principle of every relativistic field theory which insures consistent physical descriptions. Even if the space-time is noncommutative, field theories on it should keep Lorentz covariance. In this letter, it is shown that the field theory on noncommutative spacetime is Lorentz covariant if the noncommutativity emerges from the algebra of spacetime operators described by Doplicher, Fredenhagen and Roberts.Comment: 5 page