# Mathematical Physics

## New submissions

[ total of 35 entries: 1-35 ]
[ showing up to 2000 entries per page: fewer | more ]

### New submissions for Tue, 20 Feb 18

[1]
Title: Hamiltonian Zoo for the system with one degree of freedom
Subjects: Mathematical Physics (math-ph)

We present alternative forms of the standard Hamiltonian called Newton-equivalent Hamiltonian Zoo, giving the same equation of motion, for a system with one degree of freedom. These Hamiltonians are solved directly from the Hamilton's equations and come with extra-parameters which are interpreted as time scaling factors.

[2]
Title: New spinor classes on the Graf-Clifford algebra
Authors: R. Lopes, R. da Rocha
Subjects: Mathematical Physics (math-ph)

Pinors and spinors are defined as sections of the subbundles whose fibers are the representation spaces of the Clifford algebra of the forms equipped with the Graf product. In this context, pinors and spinors are here considered and the geometric generalized Fierz identities provide the necessary framework to derive and construct new spinor classes on the space of smooth sections of the exterior bundle, endowed with the Graf product, for prominent specific signatures.

[3]
Title: On superintegrable monopole systems
Journal-ref: Journal of Physics: Conference Series 965 (2018), 012018
Subjects: Mathematical Physics (math-ph)

Superintegrable systems with monopole interactions in flat and curved spaces have attracted much attention. For example, models in spaces with a Taub-NUT metric are well-known to admit the Kepler-type symmetries and provide non-trivial generalizations of the usual Kepler problems. In this paper, we overview new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We present their higher-order, algebraically independent integrals of motion via the direct and constructive approaches which prove the superintegrability of the models. The integrals form symmetry polynomial algebras of the systems with structure constants involving Casimir operators of certain Lie algebras. Such algebraic approaches provide a deeper understanding to the degeneracies of the energy spectra and connection between wave functions and differential equations and geometry.

[4]
Title: Construction of the raising operator for Rosen-Morse eigenstates in terms of the Weyl fractional integral
Authors: Felipe Freitas
Subjects: Mathematical Physics (math-ph)

The raising operator relating adjacent bound states for the general, non-symmetric Rosen-Morse potential is constructed explicitly. It is demonstrated that, in constrast to the symmetric (modified P\"oschl-Teller) potential, the operator is non-local and must be expressed applying techniques from fractional calculus. A recurrence relation between adjacent states is derived applying the Weyl fractional integral, which, in contrast to standard recurrence relations, allows the efficient numerical computation of the coefficients of all Jacobi polynomials necessary for the evaluation of the bound state wave functions, providing an application of fractional calculus to exactly solvable quantum systems.

[5]
Title: Bogoliubov excitation spectrum for Bose-Einstein condensates
Authors: Benjamin Schlein
Comments: 18 pages. Contribution to Proceedings of the International Congress of Mathematicians, Rio de Janeiro, 2018
Subjects: Mathematical Physics (math-ph)

We consider interacting Bose gases trapped in a box $\Lambda = [0;1]^3$ in the Gross-Pitaevskii limit. Assuming the potential to be weak enough, we establish the validity of Bogoliubov's prediction for the ground state energy and the low-energy excitation spectrum. These notes are based on \cite{BBCS3}, a joint work with C. Boccato, C. Brennecke and S. Cenatiempo.

[6]
Title: Adiabatic Limit in QFT and Spectral Geometry
Subjects: Mathematical Physics (math-ph)

In this work we give positive solution to the adiabatic limit problem in causal perturbative QED, as well as give a contribution to the solution of the convergence problem for the perturbative series in QED. The method is general enough to be applicable to more general causal perturbative QFT, such as Standard Model with the Higgs field. As a by-product we provide the spatial-infinity asymtotics of the interacting fields in QED, and realize the proof of charge univerality outlined by Staruszkiewicz. As another byproduct we give a completely knew perspective on the relation between the metric structure of space-time (understood as a spectrum of a certain commutative algebra of operators, with the metric structure determined likewise by operators in the way practiced in spectral formulation of geometry due to Connes) and the energy-momentum tensor undersood as an operator valued distribution. We show that there is a deep bi-unique interrelation between space-time geometry and free quantum fields which persists when passing to interacting fields. We show in particular that passing from free to interacting fields will necessary disturb space-time geometry. Under assumption (which we make precise in this work) that Einstein equations stay valid for coherent states (say in a quasi-classical limit), the gravitational constant can be computed from the relation joining space-time geometry with interacting fields.

### Cross-lists for Tue, 20 Feb 18

[7]  arXiv:1712.03840 (cross-list from cond-mat.stat-mech) [pdf, ps, other]
Title: Calculating eigenvalues of many-body systems from partition functions
Subjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

A method for calculating the eigenvalue of a many-body system without solving the eigenfunction is suggested. In many cases, we only need the knowledge of eigenvalues rather than eigenfunctions, so we need a method solving only the eigenvalue, leaving alone the eigenfunction. In this paper, the method is established based on statistical mechanics. In statistical mechanics, calculating thermodynamic quantities needs only the knowledge of eigenvalues and, then, the information of eigenvalues is embodied in thermodynamic quantities. The method suggested in the present paper is indeed a method for extracting the eigenvalue from thermodynamic quantities. As applications, we calculate the eigenvalues for some many-body systems. Especially, the method is used to calculate the quantum exchange energies in quantum many-body systems. Moreover, using the method we calculate the influence of the topological effect on eigenvalues.

[8]  arXiv:1802.05719 (cross-list from quant-ph) [pdf, other]
Title: Generic emergence of objectivity of observables in infinite dimensions
Subjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); History and Philosophy of Physics (physics.hist-ph)

Quantum Darwinism posits that information becomes objective whenever multiple observers indirectly probe a quantum system by each measuring a fraction of the environment. It was recently shown that objectivity of observables emerges generically from the mathematical structure of quantum mechanics, whenever the system of interest has finite dimensions and the number of environment fragments is large [Brand\~ao et. al. (2015), Nat. Commun. 6, 7908]. Despite the importance of this result, it necessarily excludes many practical systems of interest that live in infinite dimensions, including harmonic oscillators. Extending the study of Quantum Darwinism to infinite dimensions is a nontrivial task: we tackle it here by introducing a modified diamond norm, suitable to quantify the distinguishability of channels in infinite dimensions. We prove two theorems that bound the emergence of objectivity, first for finite energy systems, and then for systems that can only be prepared in states with an exponential energy cut-off. We show that the latter class of states includes any bounded-energy subset of single-mode Gaussian states.

[9]  arXiv:1802.06087 (cross-list from cond-mat.str-el) [pdf, other]
Title: Boson-fermion duality in three dimensions
Subjects: Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)

We study the 2+1 dimensional boson-fermion duality in the presence of background curvature and electromagnetic fields. The main players are, on the one hand, free massive complex scalar fields coupled to U(1) Maxwell-Chern-Simons gauge fields at Chern-Simons levels $\pm1$, representing relativistic composite bosons with one unit of attached flux, and on the other hand, free massive Dirac fermions. We prove, in a curved background and at the level of the partition function, that a doublet of relativistic composite bosons, in the infinite coupling limit, is dual to a doublet of Dirac fermions. The spin connection arises from the expectation value of the Wilson loop in the Chern-Simons theory, whereas a non-minimal coupling of bosons to the scalar curvature is necessary in order to obtain agreement between partition functions. Remarkably, we find that the correspondence does not hold in the presence of background electromagnetic fields, a pathology rooted to the coupling of electromagnetism to the spin angular momentum of the Dirac spinor, which can not be reproduced from minimal coupling in the bosonic side. The presence of framing and parity anomalies in the Chern-Simons and fermionic theories, respectively, poses a difficulty in realizing the duality as an exact mapping between partition functions. The existence of non matching anomalies is circumvented by the Dirac fermions coming in pairs, making the fermionic theory parity anomaly free, and by the inclusion of a Maxwell term in the bosonic side, acting as a regulator forcing the CS theory to be quantized in a non-topological way. The Coulomb interaction stemming from the Maxwell term is also of key importance to prevent intersections of worldlines in the path integral. An extension of the duality to the massless case fails if bosons and fermions are in a topological phase, but is possible when the mapping is between trivial theories.

[10]  arXiv:1802.06118 (cross-list from math.DS) [pdf, other]
Title: Tracking critical points on evolving curves and surfaces
Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Numerical Analysis (math.NA)

In recent years it became apparent that geophysical abrasion can be well characterized by the time evolution $N(t)$ of the number $N$ of static balance points of the abrading particle. Static balance points correspond to the critical points of the particle's surface represented as a scalar distance function $r$, measured from the center of mass, so their time evolution can be expressed as $N(r(t))$. The mathematical model of the particle can be constructed on two scales: on the macro scale the particle may be viewed as a smooth, convex manifold described by the smooth distance function $r$ with $N=N(r)$ equilibria, while on the micro scale the particle's natural model is a finely discretized, convex polyhedral approximation $r^{\Delta}$ of $r$, with $N^{\Delta}=N(r^{\Delta})$ equilibria. There is strong intuitive evidence suggesting that under some particular evolution models $N(t)$ and $N^{\Delta}(t)$ primarily evolve in the opposite manner. Here we create the mathematical framework necessary to understand these phenomenon more broadly, regardless of the particular evolution equation. We study micro and macro events in one-parameter families of curves and surfaces, corresponding to bifurcations triggering the jumps in $N(t)$ and $N^{\Delta}(t)$. We show that the intuitive picture developed for curvature-driven flows is not only correct, it has universal validity, as long as the evolving surface $r$ is smooth. In this case, bifurcations associated with $r$ and $r^{\Delta}$ are coupled to some extent: resonance-like phenomena in $N^{\Delta}(t)$ can be used to forecast downward jumps in $N(t)$ (but not upward jumps). Beyond proving rigorous results for the $\Delta \to 0$ limit on the nontrivial interplay between singularities in the discrete and continuum approximations we also show that our mathematical model is structurally stable, i.e. it may be verified by computer simulations.

[11]  arXiv:1802.06198 (cross-list from hep-lat) [pdf, other]
Title: Mass gap in the weak coupling limit of $(2+1)$ SU(2) lattice gauge theory
Subjects: High Energy Physics - Lattice (hep-lat); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)

We develop the dual description of $2+1$ SU(2) lattice gauge theory as interacting abelian like' electric loops by using Schwinger bosons. "Point splitting" of the lattice enables us to construct explicit Hilbert space for the gauge invariant theory which in turn makes dynamics more transparent. Using path integral representation in phase space, the interacting closed loop dynamics is analyzed in the weak coupling limit to get the mass gap.

[12]  arXiv:1802.06410 (cross-list from math.AP) [pdf, other]
Title: Emergence of oscillatory behaviors for excitable systems with noise and mean-field interaction, a slow-fast dynamics approach
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Dynamical Systems (math.DS); Probability (math.PR); Adaptation and Self-Organizing Systems (nlin.AO)

We consider the long-time dynamics of a general class of nonlinear Fokker-Planck equations, describing the large population behavior of mean-field interacting units. The main motivation of this work concerns the case where the individual dynamics is excitable, i.e. when each isolated dynamics rests in a stable state, whereas a sufficiently strong perturbation induces a large excursion in the phase space. We address the question of the emergence of oscillatory behaviors induced by noise and interaction in such systems. We tackle this problem by considering this model as a slow-fast system (the mean value of the process giving the slow dynamics) in the regime of small individual dynamics and by proving the existence of a positively stable invariant manifold, whose slow dynamics is at first order the dynamics of a single individual averaged with a Gaussian kernel. We consider applications of this result to Stuart-Landau and FitzHugh-Nagumo oscillators.

[13]  arXiv:1802.06419 (cross-list from math.CO) [pdf, other]
Title: Maximizing the number of edges in three-dimensional colored triangulations whose building blocks are balls
Authors: Valentin Bonzom
Comments: 38 pages, generous quantity of figures
Subjects: Combinatorics (math.CO); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)

Colored triangulations offer a generalization of combinatorial maps to higher dimensions. Just like maps are gluings of polygons, colored triangulations are built as gluings of special, higher-dimensional building blocks, such as octahedra, which we call colored building blocks and known in the dual as bubbles. A colored building block is determined by its boundary triangulation, which in the case of polygons is simply characterized by its length. In three dimensions, colored building blocks are labeled by some 2D triangulations and those homeomorphic to the 3-ball are labeled by the subset of planar ones. Similarly to Euler's formula in 2D which provides an upper bound to the number of vertices at fixed number of polygons with given lengths, we look in three dimensions for an upper bound on the number of edges at fixed number of given colored building blocks. In this article we solve this problem when all colored building blocks, except possibly one, are homeomorphic to the 3-ball. To do this, we find a characterization of the way a colored building block homeomorphic to the ball has to be glued to other blocks of arbitrary topology in a colored triangulation which maximizes the number of edges. This local characterization can be extended to the whole triangulation as long as there is at most one colored building block which is not a 3-ball. The triangulations obtained this way are in bijection with trees. The number of edges is given as an independent sum over the building blocks of such a triangulation. In the case of all colored building blocks being homeomorphic to the 3-ball, we show that these triangulations are homeomorphic to the 3-sphere. Those results were only known for the octahedron and for melonic building blocks before. This article is self-contained and can be used as an introduction to colored triangulations and their bubbles from a purely combinatorial point of view.

[14]  arXiv:1802.06435 (cross-list from math.SG) [pdf, other]
Title: Topological Methods in the Quest for Periodic Orbits
Authors: Joa Weber
Subjects: Symplectic Geometry (math.SG); Mathematical Physics (math-ph); Differential Geometry (math.DG); Dynamical Systems (math.DS); Geometric Topology (math.GT)

These are lecture notes on Floer and Rabinowitz-Floer homology written for a graduate course at UNICAMP August-December 2016 and a mini-course held at IMPA in August 2017.

[15]  arXiv:1802.06436 (cross-list from cond-mat.stat-mech) [pdf, other]
Title: Multicritical edge statistics for the momenta of fermions in non-harmonic traps
Comments: 6 pages + 11 pages (Supplementary material), 2 figures
Subjects: Statistical Mechanics (cond-mat.stat-mech); Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph); Probability (math.PR)

We compute the joint statistics of the momenta $p_i$ of $N$ non-interacting fermions in a trap, near the Fermi edge, with a particular focus on the largest one $p_{\max}$. For a $1d$ harmonic trap, momenta and positions play a symmetric role and hence, the joint statistics of momenta is identical to that of the positions. In particular, $p_{\max}$, as $x_{\max}$, is distributed according to the Tracy-Widom distribution. Here we show that novel "momentum edge statistics" emerge when the curvature of the potential vanishes, i.e. for "flat traps" near their minimum, with $V(x) \sim x^{2n}$ and $n>1$. These are based on generalisations of the Airy kernel that we obtain explicitly. The fluctuations of $p_{\max}$ are governed by new universal distributions determined from the $n$-th member of the second Painlev\'e hierarchy of non-linear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.

[16]  arXiv:1802.06452 (cross-list from nlin.SI) [pdf, ps, other]
Title: Direct linearisation of the discrete-time two-dimensional Toda lattices
Authors: Wei Fu
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)

The discrete-time two-dimensional Toda lattice of $A_\infty$-type is studied within the direct linearisation framework, which allows us to deal with several nonlinear equations in this class simultaneously and to construct more general solutions of these equations. The periodic reductions of this model are also considered, giving rise to the discrete-time two-dimensional Toda lattices of $A_{r-1}^{(1)}$-type for $r\geq 2$ together with their integrable structures. Particularly, the $A_{r-1}^{(1)}$ classes cover some of the discrete integrable systems very recently found by Fordy and Xenitidis [J. Phys. A: Math. Theor. 50 (2017) 165205], which amount to the negative flows of members in the discrete Gel'fand-Dikii hierarchy.

[17]  arXiv:1802.06499 (cross-list from math.QA) [pdf, ps, other]
Title: Higher order Hamiltonians for the trigonometric Gaudin model
Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph)

We consider the trigonometric classical $r$-matrix for $\mathfrak{gl}_N$ and the associated quantum Gaudin model. We produce higher Hamiltonians in an explicit form by applying the limit $q\to 1$ to elements of the Bethe subalgebra for the $XXZ$ model.

[18]  arXiv:1802.06672 (cross-list from math.PR) [pdf, ps, other]
Title: Martingale representation for degenerate diffusions
Subjects: Probability (math.PR); Mathematical Physics (math-ph)

Let $(W,H,\mu)$ be the classical Wiener space on $\R^d$. Assume that $X=(X_t)$ is a diffusion process satisfying the stochastic differential equation $dX_t=\sigma(t,X)dB_t+b(t,X)dt$, where $\sigma:[0,1]\times C([0,1],\R^n)\to \R^n\otimes \R^d$, $b:[0,1]\times C([0,1],\R^n)\to \R^n$, $B$ is an $\R^d$-valued Brownian motion. We suppose that the weak uniqueness of this equation holds for any initial condition. We prove that any square integrable martingale $M$ w.r.t. to the filtration $(\calF_t(X),t\in [0,1])$ can be represented as $$M_t=E[M_0]+\int_0^t P_s(X)\alpha_s(X).dB_s$$ where $\alpha(X)$ is an $\R^d$-valued process adapted to $(\calF_t(X),t\in [0,1])$, satisfying $E\int_0^t(a(X_s)\alpha_s(X),\alpha_s(X))ds<\infty$, $a=\sigma^\star\sigma$ and $P_s(X)$ denotes a measurable version of the orthogonal projection from $\R^d$ to $\sigma(X_s)^\star(\R^n)$. In particular, for any $h\in H$, we have $$\label{wick} E[\rho(\delta h)|\calF_1(X)]=\exp\left(\int_0^1(P_s(X)\dot{h}_s,dB_s)-\half\int_0^1|P_s(X)\dot{h}_s|^2ds\right)\,,$$ where $\rho(\delta h)=\exp(\int_0^1(\dot{h}_s,dB_s)-\half |H|_H^2)$. This result gives a new development as an infinite series of the $L^2$-functionals of the degenerate diffusions. We also give an adequate notion of "innovation process" associated to a degenerate diffusion which corresponds to the strong solution when the Brownian motion is replaced by an adapted perturbation of identity. This latter result gives the solution of the causal Monge-Amp\ere equation.}

[19]  arXiv:1802.06717 (cross-list from nlin.CD) [pdf, other]
Title: Emission of autoresonant trajectories and thresholds of resonant pumping
Authors: O.M. Kiselev
Subjects: Chaotic Dynamics (nlin.CD); Mathematical Physics (math-ph); Dynamical Systems (math.DS)

We study an autoresonant asymptotic behaviour for nonlinear oscillators under slowly changing frequency and amplitude of external driver. As a result we obtain formulas for threshold values of amplitude and frequency of the driver when autoresonant behaviour for the nonlinear oscillator is observed. Also we study a capture into resonance and emission out of the resonance for trajectories of the oscillator. A measure of autoresonant asymptotic behaviours for nonlinear oscillator is obtained.

[20]  arXiv:1802.06751 (cross-list from hep-th) [pdf, ps, other]
Title: Seiberg-Witten differential via primitive forms
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)

Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional $\mathcal{N}=2$ SCFT. The Seiberg-Witten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding Seiberg-Witten differential is given by the Gelfand-Leray form of K. Saito's primitive form. Our result also extends the Seiberg-Witten solution to include irrelevant deformations.

### Replacements for Tue, 20 Feb 18

[21]  arXiv:1606.04613 (replaced) [pdf, ps, other]
Title: A Nekrasov-Okounkov formula for Macdonald polynomials
Comments: 24 pages; Revised version includes an alternative proof (suggested by Jim Bryan) of an elliptic Nekrasov-Okounkov formula based on the equivariant DMVV formula. In v3 some final corrections have been carried out
Subjects: Combinatorics (math.CO); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
[22]  arXiv:1612.08809 (replaced) [pdf, ps, other]
Title: Mean-field bound on the 1-arm exponent for Ising ferromagnets in high dimensions
Subjects: Mathematical Physics (math-ph); Probability (math.PR)
[23]  arXiv:1705.10103 (replaced) [pdf, ps, other]
Title: Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras
Comments: 57 pages. Minor editing and corrections following the referee suggestions
Subjects: Mathematical Physics (math-ph); Rings and Algebras (math.RA); Representation Theory (math.RT); Exactly Solvable and Integrable Systems (nlin.SI)
[24]  arXiv:1708.04292 (replaced) [pdf, ps, other]
Title: Droplet breakup in the liquid drop model with background potential
Comments: This version will appear in Commun. Contemp. Math
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
[25]  arXiv:1708.07817 (replaced) [pdf, ps, other]
Title: Positive Functionals Induced by Minimizers of Causal Variational Principles
Authors: Felix Finster
Comments: 15 pages, LaTeX, 1 figure, minor improvements (published version)
Subjects: Mathematical Physics (math-ph); Functional Analysis (math.FA)
[26]  arXiv:1709.02943 (replaced) [pdf, ps, other]
Title: Representations of the Quantum Holonomy-Diffeomorphism Algebra
Subjects: Mathematical Physics (math-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Quantum Algebra (math.QA)
[27]  arXiv:1710.03819 (replaced) [pdf, ps, other]
Title: Soliton Resolution for the Derivative Nonlinear Schrödinger Equation
Comments: 44 pages, 4 figures. This article is a revision of sections 5-7 and appendices A and C of arXiv:1706.06252. The larger paper has been split into two articles. This version is the final version to appear in Comm. Math. Phys. and incorporates a number of suggestions by the referee
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Spectral Theory (math.SP)
[28]  arXiv:1711.04028 (replaced) [pdf, other]
Title: The vector field of a rolling rigid body
Comments: Added semi-symplectic reduction. Will be published as e-print only. 8 pages
Subjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS); Classical Physics (physics.class-ph)
[29]  arXiv:1711.04054 (replaced) [pdf, ps, other]
Title: Vector bundles for "Matrix algebras converge to the sphere"
Authors: Marc A. Rieffel
Subjects: Operator Algebras (math.OA); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[30]  arXiv:1712.10209 (replaced) [pdf, other]
Title: Spectral properties of the 2+1 fermionic trimer with contact interactions
Subjects: Mathematical Physics (math-ph); Functional Analysis (math.FA); Spectral Theory (math.SP)
[31]  arXiv:1801.08698 (replaced) [pdf, ps, other]
Title: Average values of functionals and concentration without measure
Authors: Cheng-shi Liu
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Functional Analysis (math.FA); Statistics Theory (math.ST); Data Analysis, Statistics and Probability (physics.data-an)
[32]  arXiv:1801.08885 (replaced) [pdf, other]
Title: Fractional powers and singular perturbations of differential operators
Subjects: Functional Analysis (math.FA); Mathematical Physics (math-ph)
[33]  arXiv:1802.00980 (replaced) [pdf, ps, other]
Title: The first-order flexibility of a crystal framework
Authors: E. Kastis, S.C. Power