Reciprocal relativity of noninertial frames and the quaplectic group

Newtonian mechanics has the concept of an absolute inertial rest frame. Special relativity eliminates the absolute rest frame but continues to require the absolute inertial frame. General relativity solves this for gravity by requiring particles to have locally inertial frames on a curved position-time manifold. The problem of the absolute inertial frame for other forces remains. We look again at the transformations of frames on an extended phase space with position, time, energy and momentum degrees of freedom. Under nonrelativistic assumptions, there is an invariant symplectic metric and a line element dt^2. Under special relativistic assumptions the symplectic metric continues to be invariant but the line elements are now -dt^2+dq^2/c^2 and dp^2-de^2/c^2. Max Born conjectured that the line element should be generalized to the pseudo- orthogonal metric -dt^2+dq^2/c^2+ (1/b^2)(dp^2-de^2/c^2). The group leaving these two metrics invariant is the pseudo-unitary group of transformations between noninertial frames. We show that these transformations eliminate the need for an absolute inertial frame by making forces relative and bounded by b and so embodies a relativity that is 'reciprocal' in the sense of Born. The inhomogeneous version of this group is naturally the semidirect product of the pseudo-unitary group with the nonabelian Heisenberg group. This is the quaplectic group. The Heisenberg group itself is the semidirect product of two translation groups. This provides the noncommutative properties of position and momentum and also time and energy that are required for the quantum mechanics that results from considering the unitary representations of the quaplectic group.Comment: Substantial revision, Publicon LaTe

Unitarily localizable entanglement of Gaussian states

We consider generic $m\times n$-mode bipartitions of continuous variable systems, and study the associated bisymmetric multimode Gaussian states. They are defined as $(m+n)$-mode Gaussian states invariant under local mode permutations on the $m$-mode and $n$-mode subsystems. We prove that such states are equivalent, under local unitary transformations, to the tensor product of a two-mode state and of $m+n-2$ uncorrelated single-mode states. The entanglement between the $m$-mode and the $n$-mode blocks can then be completely concentrated on a single pair of modes by means of local unitary operations alone. This result allows to prove that the PPT (positivity of the partial transpose) condition is necessary and sufficient for the separability of $(m + n)$-mode bisymmetric Gaussian states. We determine exactly their negativity and identify a subset of bisymmetric states whose multimode entanglement of formation can be computed analytically. We consider explicit examples of pure and mixed bisymmetric states and study their entanglement scaling with the number of modes.Comment: 10 pages, 2 figure

Chiral surfaces self-assembling in one-component systems with isotropic interactions

We show that chiral symmetry can be broken spontaneously in one-component systems with isotropic interactions, i.e. many-particle systems having maximal a priori symmetry. This is achieved by designing isotropic potentials that lead to self-assembly of chiral surfaces. We demonstrate the principle on a simple chiral lattice and on a more complex lattice with chiral super-cells. In addition we show that the complex lattice has interesting melting behavior with multiple morphologically distinct phases that we argue can be qualitatively predicted from the design of the interaction.Comment: 4 pages, 4 figure

Positive mass theorems for asymptotically AdS spacetimes with arbitrary cosmological constant

We formulate and prove the Lorentzian version of the positive mass theorems with arbitrary negative cosmological constant for asymptotically AdS spacetimes. This work is the continuation of the second author's recent work on the positive mass theorem on asymptotically hyperbolic 3-manifolds.Comment: 17 pages, final version, to appear in International Journal of Mathematic

Localization of quantum wave packets

We study the semiclassical propagation of squeezed Gau{\ss}ian states. We do so by considering the propagation theorem introduced by Combescure and Robert \cite{CR97} approximating the evolution generated by the Weyl-quantization of symbols $H$. We examine the particular case when the Hessian $H^{\prime\prime}(X_{t})$ evaluated at the corresponding solution $X_{t}$ of Hamilton's equations of motion is periodic in time. Under this assumption, we show that the width of the wave packet can remain small up to the Ehrenfest time. We also determine conditions for ``classical revivals'' in that case. More generally, we may define recurrences of the initial width. Some of these results include the case of unbounded classical motion. In the classically unstable case we recover an exponential spreading of the wave packet as in \cite{CR97}

Strong and weak semiclassical limits for some rough Hamiltonians

We present several results concerning the semiclassical limit of the time dependent Schr\"odinger equation with potentials whose regularity doesn't guarantee the uniqueness of the underlying classical flow. Different topologies for the limit are considered and the situation where two bicharateristics can be obtained out of the same initial point is emphasized

The orientation-preserving diffeomorphism group of S^2 deforms to SO(3) smoothly

Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.Comment: 16 page

The contribution of muscle hypertrophy to strength changes following resistance training

Purpose Whilst skeletal muscle hypertrophy is considered an important adaptation to resistance training (RT), it has not previously been found to explain the inter-individual changes in strength after RT. This study investigated the contribution of hypertrophy to individual gains in isometric, isoinertial and explosive strength after 12 weeks of elbow flexor RT. Methods Thirty-three previously untrained, healthy men (18–30 years) completed an initial 3-week period of elbow flexor RT (to facilitate neurological responses) followed by 6-week no training, and then 12-week elbow flexor RT. Unilateral elbow flexor muscle strength [isometric maximum voluntary force (iMVF), single repetition maximum (1-RM) and explosive force], muscle volume (V m), muscle fascicle pennation angle (θ p) and normalized agonist, antagonist and stabilizer sEMG were assessed pre and post 12-week RT. Results Percentage gains in V m correlated with percentage changes in iMVF (r = 0.527; P = 0.002) and 1-RM (r = 0.482; P = 0.005) but not in explosive force (r ≤ 0.243; P ≥ 0.175). Percentage changes in iMVF, 1-RM, and explosive force did not correlate with percentage changes in agonist, antagonist or stabilizer sEMG (all P > 0.05). Percentage gains in θ p inversely correlated with percentage changes in normalized explosive force at 150 ms after force onset (r = 0.362; P = 0.038). Conclusions We have shown for the first time that muscle hypertrophy explains a significant proportion of the inter-individual variability in isometric and isoinertial strength gains following 12-week elbow flexor RT in healthy young men

Tsirelson's problem and Kirchberg's conjecture

Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables located at different sites is assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products of C*-algebras would imply a positive answer to this question for all bipartite scenarios. This remains true also if one considers not only spatial correlations, but also spatiotemporal correlations, where each party is allowed to apply their measurements in temporal succession; we provide an example of a state together with observables such that ordinary spatial correlations are local, while the spatiotemporal correlations reveal nonlocality. Moreover, we find an extended version of Tsirelson's problem which, for each nontrivial Bell scenario, is equivalent to the QWEP conjecture. This extended version can be conveniently formulated in terms of steering the system of a third party. Finally, a comprehensive mathematical appendix offers background material on complete positivity, tensor products of C*-algebras, group C*-algebras, and some simple reformulations of the QWEP conjecture.Comment: 57 pages, to appear in Rev. Math. Phy