Flat Phase Space Research Articles

On Born’s Reciprocal Relativity, Algebraic Extensions of the Yang and Quaplectic Algebra, and Noncommutative Curved Phase Spaces

After a brief introduction of Born’s reciprocal relativity theory is presented, we review the construction of the deformed quaplectic group that is given by the semi-direct product of U(1,3) with the deformed (noncommutative) Weyl–Heisenberg group corresponding to noncommutative fiber coordinates and momenta [Xa,Xb]≠0; [Pa,Pb]≠0. This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks [Xa1a2⋯an,Xb1b2⋯bn]≠0; [Pa1a2⋯an,Pb1b2⋯bn]≠0. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting xμ,pμ operator variables (associated to an 8D curved phase space) to the canonical YA,ΠA operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, the embedding functions YA(x,p),ΠA(x,p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric gμν(x,p), the fiber metric of the vertical space hab(x,p), and the nonlinear connection Naμ(x,p) associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions.

Algebraic Symplectic Reduction and Quantization of Singular Spaces

The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of singular Poisson spaces the deformation quantization is explicitly constructed. In is shown that for the flat phase space with the classical moment map and the orthogonal group action the deformation quantization converges for the entire arguments of exponential type.

Quantum fluctuations of the compact phase space cosmology

In the recent article (2019 Phys. Rev. D 100 043533) a compact phase space generalization of the flat de Sitter cosmology has been proposed. The main advantages of the compactification is that physical quantities are bounded, and the quantum theory is characterized by finite dimensional Hilbert space. Furthermore, by considering the phase space, quantum description is constructed with the use SU(2) representation theory. The purpose of this article is to apply effective methods to analyze quantum dynamics of the model at the semi-classical regime. The analysis is performed both without prior solving of the quantum constraint and by extracting physical Hamiltonian of the model. At the effective level, the results of the two procedures are shown to be equivalent. We find a nontrivial behavior of the fluctuations around the recollapse of the Universe, which is distinct from what is found after quantization with the standard flat phase space. The behavior is reflected at the level of the modified Friedmann equation with quantum back-reaction effects, which is derived. Finally, a relation between quantum fluctuations in the minisuperspace model under consideration and the Bousso bound has been found. It has been shown that the Bousso bound can be violated for certain choices of semiclassical states.

Truncated Wigner approximation as non-positive Kraus map

We show that the Truncated Wigner Approximation developed in the flat phase-space is mapped into a Lindblad-type evolution with an indefinite metric in the space of linear operators. As a result, the classically evolved Wigner function corresponds to a non-positive operator , which does not describe a physical state. The rate of appearance of negative eigenvalues of can be efficiently estimated. The short-time dynamics of the Kerr and second harmonic generation Hamiltonains are discussed.

Is dark matter and black hole cosmology an effect of Born’s reciprocal relativity theory?

Born’s reciprocal relativity theory (BRRT), based on a maximal proper-force, maximal speed of light, and inertial and non-inertial observers, is re-examined in full detail. Relativity of locality and chronology are natural consequences of this theory, even in flat phase space. The advantage of BRRT is that Lorentz invariance is preserved and there is no need to introduce Hopf algebraic deformations of the Poincaré algebra, de Sitter algebra, nor non-commutative space–times. After a detailed study of the notion of generalized force, momentum, and mass in phase space, we explain that what one may interpret as “dark matter” in galaxies, for example, is just an effect of observing ordinary galactic matter in different accelerating frames of reference than ours. Explicit calculations are provided that explain these novel relativistic effects due to the accelerated expansion of the Universe, and which may generate the present-day density parameter value ΩDM ∼ 0.25 of dark matter. The physical origins behind the numerical coincidences in black hole cosmology are also explored. We finalize with a rigorous study of the curved geometry of (co)tangent bundles (phase space) within the formalism of Finsler geometry, and provide a short discussion on Hamilton spaces.

Asymptotic symmetries from finite boxes

It is natural to regulate an infinite-sized system by imposing a boundary condition at finite distance, placing the system in a ‘box.’ This breaks symmetries, though the breaking is small when the box is large. One should thus be able to obtain the asymptotic symmetries of the infinite system by studying regulated systems. We provide concrete examples in the context of Einstein–Hilbert gravity (with negative or zero cosmological constant) by showing in 4 or more dimensions how the anti-de Sitter and Poincaré asymptotic symmetries can be extracted from gravity in a spherical box with Dirichlet boundary conditions. In 2 + 1 dimensions we obtain the full double-Virasoro algebra of asymptotic symmetries for AdS3 and, correspondingly, the full Bondi–Metzner–Sachs (BMS) algebra for asymptotically flat space. In higher dimensions, a related approach may continue to be useful for constructing a good asymptotically flat phase space with BMS asymptotic symmetries.

Fréchet-algebraic deformation quantizations

In this review I present some recent results on the convergence properties of formal star products. Based on a general construction of a Fréchet topology for an algebra with countable vector space basis I discuss several examples from deformation quantization: the Wick star product on the flat phase space m2n gives a first example of a Fréchet algebraic framework for the canonical commutation relations. More interesting, the star product on the Poincare disk can be treated along the same lines, leading to a non-trivial example of a convergent star product on a curved Kahler manifold.

(Anti)de Sitter/Poincaré symmetries and representations from Poincaré/Galilei through a classical deformation approach

A classical deformation procedure, based on universal enveloping algebras, Casimirs and curvatures of symmetrical homogeneous spaces, is applied to several cases of physical relevance. Starting from the (3 + 1)D Galilei algebra, we describe at the level of representations the process leading to its two physically meaningful deformed neighbours. The Poincaré algebra is obtained by introducing a negative curvature in the flat Galilean phase space (or space of worldlines), while keeping a flat spacetime. To be precise, starting from a representation of the Galilei algebra with both Casimirs different from zero, we obtain a representation of the Poincaré algebra with both Casimirs necessarily equal to zero. The Poincaré angular momentum, Pauli–Lubanski components, position and velocity operators, etc are expressed in terms of ‘Galilean’ operators through some expressions deforming the proper Galilean ones. Similarly, the Newton–Hooke algebras appear by endowing spacetime with a nonzero curvature, while keeping a flat phase space. The same approach, starting from the (3 + 1)D Poincaré algebra provides representations of the (anti)de Sitter as Poincaré deformations.

Symplectic quaternion scheme for biophysical molecular dynamics

Massively parallel biophysical molecular dynamics simulations, coupled with efficient methods, promise to open biologically significant time scales for study. In order to promote efficient fine-grained parallel algorithms with low communication overhead, the fast degrees of freedom in these complex systems can be divided into sets of rigid bodies. Here, a novel Hamiltonian form of a minimal, nonsingular representation of rigid body rotations, the unit quaternion, is derived, and a corresponding reversible, symplectic integrator is presented. The novel technique performs very well on both model and biophysical problems in accord with a formal theoretical analysis given within, which gives an explicit condition for an integrator to possess a conserved quantity, an explicit expression for the conserved quantity of a symplectic integrator, the latter following and in accord with Calvo and Sanz-Sarna, Numerical Hamiltonian Problems (1994), and extension of the explicit expression to general systems with a flat phase space.

FOCK BUNDLES AND QUANTIZATION OF SYSTEMS WITH FIRST CLASS CONSTRAINTS

The flat phase space (R2n, ω) with a symmetry group G generated by quadratic first class constraints is considered. We analyze the quantization of this constrained system in terms of the Fock bundle [Formula: see text], where S=Sp(2n, R)/U(n) is the space of positive translationally invariant complex structures J on (R2n, ω) and fibers HJ are the Hilbert spaces of quantization associated with the space (R2n, ω, J).

Operator quantization of dynamical systems with curved phase space

The operator quantization method is extended to dynamical systems with curved phase space by treating them effectively as those with a flat phase space of the double dimension and effective second-class constraints.

Hopf algebras for physics at the Planck scale

Applies ideas of non-commutative geometry to reformulate the classical and quantum mechanics of a particle moving on a homogeneous spacetime. The reformulation maintains an interesting symmetry between observables and states in the form of a Hopf algebra structure. In the simplest example both the dynamics and quantum mechanics are completely determined by the Hopf algebra consideration. The simplest example is a two-parameter algebra-co-algebra KAB which is the unique Hopf algebra extension, such as is possible, of the self-dual Hopf algebra of functions on flat phase space C(R*R). In the limit (A=0,B) the author recovers functions on a curved phase space with curvature proportional to B2 and in another limit (A= infinity , B= infinity ), A/B=h(cross), the author recovers quantum mechanics on R>or=0 with an absorbing wall at the origin. The algebra in this way corresponds to a toy model of quantum mechanics of a particle in one space dimension combined with gravity-like forces. It has an interesting Z2 symmetry interchanging A to or from B, and thereby, in some sense, the quantum element with the geometric element. The compatibility conditions that are solved are a generalisation of the classical Yang-Baxter equations.

Geometry of the dynamical systems with time-dependent constraints and time-dependent Hamiltonian: An approach towards quantization

The quantization of a dynamical system with time-dependent constraints or time-dependent Hamiltonians presents particular difficulties not only from a technical, but also from a theoretical point of view. We give here an approach to the evolution of such a quantum system in terms of star products on symplectic or canonical manifolds. These star products are particular deformations of the usual product of functions, deformations that start with the Poisson bracket. The introduction of the star products corresponds to a generalization of the Weyl–Wigner quantization on a flat phase space. This approach leads to the introduction of ‘‘two times’’ denoted t and τ, playing different roles; the time t plays the role of a geometrical time corresponding to the control, while the other, τ, which parametrizes the evolution, plays the role of a dynamical time. The classical as well as the quantum results point out the role played by the variable (t+τ). This reflects the coherence of the formalism.