# Chryssomalis Chryssomalakos's research while affiliated with Universidad Nacional Autónoma de México and other places

## Publications (55)

Pure quantum spin-$s$ states can be represented by $2s$ points on the sphere, as shown by Majorana in 1932 --- the description has proven particularly useful in the study of rotational symmetries of the states, and a host of other properties, as the points rotate rigidly on the sphere when the state undergoes an $SU(2)$ transformation in Hilbert sp...

Geometric phases, accumulated when a quantum system traces a cycle in quantum state space, do not depend on the parametrization of the cyclic path, but do depend on the path itself. In the presence of noise that deforms the path, the phase gets affected, compromising the robustness of possible applications, e.g., in quantum computing. We show that...

Any acceptable quantum gravity theory must allow us to recover the classical spacetime in the appropriate limit. Moreover, the spacetime geometrical notions should be intrinsically tied to the behavior of the matter that probes them. We consider some difficulties that would be confronted in attempting such an enterprise. The problems we uncover see...

Spin states of maximal projection along some direction in space are called (spin) coherent, and are, in many aspects, the "most classical" available. For any spin $s$, the spin coherent states form a 2-sphere in the projective Hilbert space $\mathbb{P}$ of the system. We address several questions regarding that sphere, in particular its possible in...

We look for optimal quantum rotosensors, i.e., quantum spin states that are optimal in detecting rotations by a given angle. The exact quantity to be minimized is the probability that the rotated state projects onto the original one, averaged uniformly over all rotation axes. We show analytically that, for small rotation angles, the solution is giv...

We calculate Berry’s phase when the driving field, to which a spin-1 2 is coupled adiabatically, rather than the familiar classical magnetic field, is a quantum vector operator, of noncommuting, in general, components, e.g. the angular momentum of another particle, or another spin. The geometric phase of the entire system, spin plus “quantum drivin...

We comment on a fatal flaw in the analysis contained in the work of Martínez-y-Romero et al., [J. Math. Phys. 54, 053509 (2013)], which concerns the motion of a point particle in an inverse square potential, and show that most conclusions reached there are wrong. In particular, the manifestly senseless claim that, in the attractive potential case,...

We present a quantum description of the mechanism by which a free-falling cat manages to reorient itself and land on its feet, having all along zero angular momentum. Our approach is geometrical, making use of the fiber bundle structure of the cat configuration space. We show how the classical picture can be recovered, but also point out a purely q...

In the separation of rotations from internal motions in the n-body problem, there appear some gauge fields which physically represent Coriolis effects. These fields are also present in the "falling cat" problem: at the kinematical level they map changes in the cat's shape to changes in its orientation whereas at the dynamical level they show up as...

We point out that a proper treatment of quantum gravity ought to take into account the quantum nature of the probes used to unravel spacetime geometry. As a first step in this direction, we use extended classical probes in the study of the geometry of a classical manifold. We comment on a limitation of the standard Dixon-Beiglböck center-of-mass pr...

After reviewing the work of Pryce on Center-of-Mass (CoM) definitions in
special relativity, and that of Jordan and Mukunda on position operators
for relativistic particles with spin, we propose two new criteria for a
CoM candidate: associativity, and compatibility with the Poisson bracket
structure. We find that they are not satisfied by all of Pr...

We consider particles constrained to move in the close vicinity of a space curve through a steep quadratic potential in the plane normal to the curve. As is known, the effective 1D Hamiltonian that governs the motion along the curve involves both its curvature and torsion. Thus, the adiabatic cyclic deformations of the curve might give rise to geom...

After reviewing the work of Pryce on Center-of-Mass (CoM) definitions in
special relativity, and that of Jordan and Mukunda on position operators for
relativistic particles with spin, we propose two new criteria for a CoM
candidate: associativity, and compatibility with the Poisson bracket structure.
We find that they are not satisfied by all of Pr...

Traditional geometry employs idealized concepts like that of a point or a
curve, the operational definition of which relies on the availability of
classical point particles as probes. Real, physical objects are quantum in
nature though, leading us to consider the implications of using realistic
probes in defining an effective spacetime geometry. As...

We show that the algebra of the recently proposed Triply Special Relativity can be brought to a linear (i.e., Lie) form by a correct identification of its generators. The resulting Lie algebra is the stable form proposed by Vilela Mendes a decade ago, itself a reapparition, up to some important sign, of Yang's algebra, dating from 1947. As a coroll...

We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincaré-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a thre...

The motion of a quantum particle, constrained to move along a space curve via a confining potential, is governed by a hamiltonian that depends on the geometry of the curve. Adiabatic changes in the shape of the curve are shown to give rise to geometric phases. The effect of the latter on the quantization of the motion of the curve itself can be nat...

We show that quantum particles constrained to move along curves undergoing cyclic deformations acquire, in general, geometric phases. We treat explicitly an example, involving particular deformations of a circle, and ponder on potential applications.

We study Sorkin's proposal of a generalization of quantum mechanics and find that the theories proposed derive their probabilities from kth order polynomials in additive measures, in the same way that quantum mechanics uses a probability bilinear in the quantum amplitude and its complex conjugate. Two complementary approaches are presented, a C* an...

The motion of a quantum particle, constrained to move along a space curve via a confining potential, is governed by a hamiltonian that depends on the geometry of the curve. Adiabatic changes in the shape of the curve are shown to give rise to geometric phases. The effect of the latter on the quantization of the motion of the curve itself can be nat...

In this contribution, we suggest the approach that geometric concepts ought to be defined in terms of physical operations involving quantum matter. In this way it is expected that some (presumably nocive) idealizations lying deep within the roots of the notion of spacetime might be excluded. In particular, we consider that spacetime can be probed o...

We show that quantum particles may acquire geometrical phases when the curves they are constrained to live on undergo cyclic deformations. We also investigate the role of the velocity-dependent term in the hamiltonian, associated with the inertial forces felt in the frame adapted to the curve.

We present a star product for noncommutative spaces of Lie type, including the so called ``canonical'' case by introducing a central generator, which is compatible with translations and admits a simple, manageable definition of an invariant integral. A quasi-cyclicity property for the latter is shown to hold, which reduces to exact cyclicity when t...

Our main thesis in this note is that if spacetime noncommutativity is at all relevant in the quantum gravitational regime, there might be a canonical approach to pinning down its form. We start by emphasizing the distinction between an intrinsically noncommuting "manifold", i.e., one with noncommuting coordinate functions, on the one hand, and part...

We present a covariant form for the dynamics of a canonical GA of arbitrary cardinality, showing how each genetic operator can be uniquely represented by a mathematical object - a tensor - that transforms simply under a general linear coordinate transformation. For mutation and recombination these tensors can be written as tensor products of the an...

We apply Lie algebra deformation theory to the problem of identifying
the stable form of the quantum relativistic kinematical algebra. Three
possible deformations are found, which introduce dimensionful constants.
We also argue that, instead of positions, moments should serve as Lie
algebra generators, leading to a radically different interpretatio...

There are two main points that concern us in this short contribution. The first one is the conceptual distinction between a intrinsically noncommuting spacetime, i.e., one where the coordinate functions fail to commute among themselves, on the one hand, and the proposal of noncommuting position operators, on the other. The second point concerns a p...

We consider anew some puzzling aspects of the equivalence of the quantum field theoretical description of bremsstrahlung from the inertial and accelerated observer’s perspectives. More concretely, we focus on the seemingly paradoxical situation that arises when noting that the radiating source is in thermal equilibrium with the thermal state of the...

A number of problems in theoretical physics share a common nucleus of combinatoric nature. It is argued here that Hopf algebraic concepts and techiques can be particularly efficient in dealing with such problems. As a first example, a brief review is given of the recent work of Connes, Kreimer and collaborators on the algebraic structure of the pro...

We show how the idea of coarse graining can be applied fruitfully to the area of genetic dynamics, both in the context of ¿effective¿ theories - leading to more appropriate effective degrees of freedom with which to describe the dynamics - as well as in terms of integrating out degrees of freedom, using the Renormalization Group as a systematic cal...

We study the motion of a quantum charged particle, constrained on the surface of a cylinder, in the presence of a radial magnetic field. When the spin of the particle is neglected, the system essentially reduces to an infinite family of simple harmonic oscillators, equally spaced along the axis of the cylinder. Interestingly enough, it can be used...

We present a covariant form for genetic dynamics and show how different formulations are simply related by linear coordinate trans- formations. In particular, in the context of the simple genetic algorithm, we show how the Vose model, in either the string or Walsh bases, is rela- ted to recent coarse-grained formulations that are naturally interpre...

The issue of whether some manifestations of gravitation in the quantum domain, are indicative or not of a non-geometrical aspect in gravitation is discussed. We focus on gedanken experiments, involving generalizations of the flavor-oscillation clocks of Ahluwalia and Burgard, and provide a critical analysis of previous interpretations. A detailed q...

We examine the equilibria of a rigid loop in the plane, characterized by an energy functional quadratic in the curvature,
subject to the constraints of fixed length and fixed enclosed area. Whereas the only non self-intersecting equilibrium corresponding
to the fixed length constraint is the circle, the area constraint gives rise to distinct equili...

We examine the equilibrium conditions of a curve in space when a local energy penalty is associated with its extrinsic geometrical state characterized by its curvature and torsion. To do this we tailor the theory of deformations to the Frenet–Serret frame of the curve. The Euler–Lagrange equations describing equilibrium are obtained; Noether's theo...

We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area constraint gives rise to equilibria with remarkable geometrical properties; not only can the Euler-Lagrange equa...

We introduce normal coordinates on the infinite dimensional group $G$ introduced by Connes and Kreimer in their analysis of the Hopf algebra of rooted trees. We study the primitive elements of the algebra and show that they are generated by a simple application of the inverse Poincar\'e lemma, given a closed left invariant 1-form on $G$. For the sp...

We give a pedagogical introduction to integration techniques appropriate for non-commutative spaces while presenting some new results as well. A rather detailed discussion outlines the motivation for adopting the Hopf algebra language. We then present some trace formulas for the integral on Hopf algebras and show how to treat the $\int 1=0$ case. W...

We study maps preserving the Heisenberg commutation relation $ab - ba=1$. We find a one-parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dual derivative. It involves a non-local ``coordinate'' operator while the dual ``derivative'' is just the Jackson finite-difference operator. Substitution of...

We argue that a description of supersymmetric extended objects from a unified geometric point of view requires an enlargement of superspace. To this aim we study in a systematic way how superspace groups and algebras arise from Grassmann spinors when these are assumed to be the only primary entities. In the process, we recover generalized space-tim...

After defining cohomologically higher order BRST and anti-BRST operators for
a compact simple algebra {\cal G}, the associated higher order Laplacians are
introduced and the corresponding supersymmetry algebra $\Sigma$ is analysed.
These operators act on the states generated by a set of fermionic ghost fields
transforming under the adjoint represen...

We give a general integration prescription for finite dimensional braided Hopf algebras, deriving the N-dimensional quantum superplane integral as an example. The transformation properties of the integral on the quantum plane are found. We also discuss integration on quantum group modules that lack a Hopf structure. Comment: 25 pages, many pictures...

We examine the two parameter deformed superalgebra $U_{qs}(sl(1|2))$ and use the results in the construction of quantum chain Hamiltonians. This study is done both in the framework of the Serre presentation and in the $R$-matrix scheme of Faddeev, Reshetikhin and Takhtajan (FRT). We show that there exists an infinite number of Casimir operators, in...

We introduce a *-structure on the quantum double and its dual in order to make contact with various approaches to the enveloping algebras of complex quantum groups. Furthermore, we introduce a canonical basis in the quantum double, its universalR-matrices and give its relation to subgroups in the dual Hopf algebra.

The non-commutative differential calculus on quantum groups can be extended by introducing, in analogy with the classical case, inner product operators and Lie derivatives. For the case of $\GL$ we show how this extended calculus induces by coaction a similar extended calculus, covariant under $\GL$, on the quantum plane. In this way, inner product...

The main theme of this thesis is the search for applications of Quantum
Group and Hopf algebraic concepts and techniques in Physics. We
investigate in particular the possibilities that exist in deforming, in
a self consistent way, the symmetry structure of physical theories with
the hope that the resulting scheme will be of relevance in the
descrip...

We review the construction of the cross product algebra $\A\rtimes\U$ from two dually paired Hopf algebras $\U$ and $\A$. The canonical element in $\U \otimes \A$ is then introduced, and its properties examined. We find that it is useful for giving coactions on $\A\rtimes\U$, and it allows the construction of objects with specific invariance proper...

We present a formulation of covariant translations in the quan- tum plane. We are led to an extension of the algebra of the coordi- nate functions and their dual derivatives by the quantum analogue of their eigenvalues. Jackson exponentials emerge as the corresponding eigenfunctions. An integral invariant under quantum translations is introduced an...

The search for a quantum theory of gravity must include the recovery of the classical space-time. We consider some of the difficulties that must be confronted in any such enterprise. These problems seem to go beyond the technical level, to the point of questioning the overall feasibility of the project. The main issue is related to the fact that, i...

## Citations

... Proof. This result has been proven in previous work [9, 13, 14]. We keep here the proof for completeness and because the notation is slightly different. ...

... In order to guarantee background-independence without explicit diffeomorphism-invariance, it should be possible to formulate the theory in a way that does not directly refer to spacetime geometry, but only to the algebraic and statistical relations between quantum operators, while the effective spacetime geometry is extracted a posteriori. (See, e.g., [15,16,17,18,19,20,21,22] for a small subset of works in this direction.) With this goal in mind, in [23] we formulated a spacetime-free framework for quantum theory. ...

... The problem of derivation of such charges for noncommutative models is open. The integration over noncommutative spaces namely braided linear space (including q-Minkowski) is well developed [3,4,8,14] but subintegrals and commutation rules for subintegrals and derivatives need further study. In particular the generalized version of the property (64) ∂ t sub = sub ∂ t must be derived. ...

... After the pioneering work of Wess and Zumino, the differential calculi with nilpotencies 2 and 3 on quantum (super)spaces have been extensively studied by many authors in the past years (see, for example, Refs. [3][4][5][6][7][8][9][11][12][13][14][15][16]). ...

... In these and the following expressions T may either be interpreted simply as a Matrix T ∈ M n (F ) or, much more general, as the canonical element of U ⊗ F : Let {e i } and {f i } be dual linear bases of U and F respectively. It is very convenient to work with the canonical element in U ⊗ F (also called the universal T -matrix [9], for an elementary overview see [5]), ...

Reference: Twisted Quantum Lax Equations

... Rather, we compute the time evolution with the adapted frame Hamiltonian, project to the initial state to find the total phase, and identify, in the end, its geometric, ω-independent part. Finally, we would like to draw the reader's attention to the fact that explicit calculations, performed in the example of harmonic deformations of a circle, give identical results by either method (see [5] and [6]). Although far from conclusive, this is further evidence that our expectations about the ultimate agreement between (62) and (75) might be well founded. ...

... To investigate this point it is advantageous to use the Majorana stellar representation [343], which allows us to uniquely depict a spin state state living in H by 2 points on the unit sphere [344]. Several decades after its conception, this representation has recently attracted a great deal of attention in several fields [345][346][347][348][349][350][351][352][353][354][355][356][357][358]. ...

Reference: Quantum concepts in optical polarization

... An involution of the algebra of the latter, made possible by the invertibility of the Killing form, gives rise to a dual object, the anti-BRST operator, and a grade-preserving Laplacian. Further generalizations, involving higher order invariant tensors of the algebra, have been explored in [7]. We have developed similar techniques to deal with the non-compact case, reinstating the connection term, and used them in one of our programming approaches — we defer further details to a future publication. ...

... This gives, ... φ + 1 2φ 2 − H φ + P ud = 0. (A2) Equation (A2) describes balance of normal forces on an infinitesimal element of the sheet. The investigation of this equation has been the subject of many recent studies that are related to cylindrical configurations of fluid membranes [47,57,58], folding of pressurized rings [59,60], deformation of a carbon nanotube [61], and the mathematical investigation of area-constrained elastica [46,62,63]. Not only that Eq. (A2) acquires an exact solution in terms of the Jacobi Elliptic functions, its corresponding embeddings, i.e., the configuration on the xy plane, are given as a function of the tangent angle φ(s) and its derivatives alone [57]. ...

... This gives, ... φ + 1 2φ 2 − H φ + P ud = 0. (A2) Equation (A2) describes balance of normal forces on an infinitesimal element of the sheet. The investigation of this equation has been the subject of many recent studies that are related to cylindrical configurations of fluid membranes [47,57,58], folding of pressurized rings [59,60], deformation of a carbon nanotube [61], and the mathematical investigation of area-constrained elastica [46,62,63]. Not only that Eq. (A2) acquires an exact solution in terms of the Jacobi Elliptic functions, its corresponding embeddings, i.e., the configuration on the xy plane, are given as a function of the tangent angle φ(s) and its derivatives alone [57]. ...