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[EQUATION] then [EQUATION] Since the graph is connected, by the definition of [MATH] and ( .2 ), we know that [MATH] where [MATH] is the agent with smallest opinion value larger than [MATH] . Hence by ( 2.1 ) and Lemma 3.3 |
[EQUATION] hence [MATH] This implies that at [MATH] the opinion value of agent [MATH] is more than [MATH] apart from agent [MATH] . If agent [MATH] is still a neighbor of agent [MATH] at [MATH] , by ( .2 ), we have [MATH] . Otherwise, repeating the above procedure no more than [MATH] times. Since the graph is always co... |
# Source: arxiv 1806.03858 # Title: Quantum structure of glasses and the boson peak: a theory of vibrations # Sections: all # Downloaded: 2026-03-03T05:15:25.462585+00:00 |
Quantum structure of glasses and the boson peak: a theory of vibrations Abstract We present a novel analytical model for glasses, starting from the first principle that the disorder in a glass mimics the disorder in a fluid. The origin of the boson peak is attributed to the intrinsically noncommutative geometry of the ... |
Introduction Glasses are a peculiar state of matter whose mechanical rigidity is similar to that of crystalline solids, while the molecular disorder makes them akin to liquids berthier . The significant differences in the thermodynamical and transport properties of glasses as compared to crystalline solids are generall... |
However, none of these advances has allowed the derivation of a complete, widely-accepted theory of amorphous solids. Thus, the physical origin of two most outstanding anomalous behaviors displayed by glasses at intermediate temperatures ( [MATH] K), i.e., the excess of heat capacity and the boson peak, is still under ... |
In this work, we present a novel theory for the vibrations in amorphous solids, relying on the fundamental principle that the disorder in a glass is a reminiscence of the disorder in a fluid. The analytical model for glasses based on this hypohesis naturally contains the boson peak as a manifestation of an extended van... |
II Fluids in Lagrangian description A fluid can be described in the Lagrangian picture (see, e.g., Mechanics ), in which one follows the motion of each individual fluid particle. The Lagrange coordinates are co-moving with the fluid. Assuming that the fluid is a continuum, it has been shown Susskind-Bahcall Susskind Ja... |
[EQUATION] with [MATH] For simplicity, we consider a two-dimensional space. Then the latter condition can be written in terms of a scalar function [MATH] as |
[EQUATION] It has been proven (see also Jackiw2 Polyn ) that in this approach, the reparametrization symmetry can be re-cast in the form of noncommuting coordinates, namely by introducing additional Poisson brackets |
[EQUATION] with an arbitrary set of constants [MATH] . The isotropy of the fluid is ensured through the arbitrariness of the elements of the matrix [MATH] . Rescaling [MATH] by [MATH] , we can re-write [MATH] as |
[EQUATION] The Poisson brackets ( ) are invariant under the transformations ( ), the elements [MATH] remaining invariant under the reparametrization of coordinates |
Jackiw1 . The generalization to three- and higher-dimensional spaces is straightforward. In the quantum treatment of the fluid, the Poisson bracket of coordinates ) becomes the commutator |
[EQUATION] (with an antisymmetric matrix [MATH] ), which has to be added to the usual canonical commutation relations [EQUATION] |
The theory based on the commutators ( ) and ( ) is customarily called noncommutative quantum mechanics. To the matrix [MATH] we can associate a vector [MATH] , whose components are given by [MATH] . We note that there are volume-preserving diffeomorphisms which leave invariant simultaneously [MATH] and the surface dens... |
[MATH] leads to a quantization of the two-dimensional phase space in cells of area [MATH] . In the case of noncommuting coordinates, the commutator ( ) leads to a quantization of the reference configuration space in cells of area [MATH] . We attribute to this basic quantum of area the meaning of surface “occupied” by a... |
[EQUATION] Incidentally, this approach has been applied to the quantum Hall fluid, providing an alternative description to Laughlin’s theory by a noncommutative Chern–Simons quantum field theory Susskind |
Let us dwell for a while on the physical significance of the commutator ( ): its presence in the quantum algebra introduces an additional uncertainty relation , which manifests itself as a “blurriness” of space points. Precise localization, even theoretical, of the particles becomes impossible. This suggests an intrins... |
The dynamics in noncommutative quantum mechanics is described by the Schrödinger equation for [MATH] degrees of freedom, with the Hamiltonian |
[EQUATION] where the canonical coordinates [MATH] satisfy the extended Heisenberg algebra ( )-( ). The mathematical manipulations are considerably simplified by noting (see, e.g., CDP_98 Bigatti_Susskind ) that the shifted coordinates |
[EQUATION] satisfy the usual Heisenberg algebra [MATH] [MATH] [MATH] (summation over the repeated index is assumed in ( )). This allows a physical interpretation of the [MATH] -term in ( ) as a quantum shift operator Anca , namely, a translation in space by [MATH] , while [MATH] is the classical geometrical coordinate.... |
III Model for glassy materials: reduced specific heat and the boson peak The glass structure model we propose avails itself of the noncommutative fluid picture. Glasses are not normal liquids due to their rigidity , nor regular solids, due to their disorder The rigidity is introduced by means of a simple cubic lattice.... |
We assume the glass composed of a simple cubic lattice of neutral atoms of mass [MATH] , with the unit cell vector [MATH] . The dynamics of the disordered lattice is described by a noncommutative harmonic oscillator potential function and we consider harmonic interactions only between the first neighbors of the atoms i... |
[EQUATION] where [MATH] is the momentum canonically conjugated to the displacement [MATH] and [MATH] is the usual harmonic oscillator frequency. |
In order to make sure that the disorder effects are not doubly-counted, nor washed out, we make an additional assumption about the dynamics of the atoms on the lattice. Namely, we consider an alternation of ordered and disordered atoms. By ordered atoms , we mean atoms whose quantum coordinates and momenta satisfy the ... |
Let us specify the Hamiltonian ( 23 ) for the two types of atoms. Considering that the atom [MATH] is a disordered one, which suffers itself the effects of the noncommutativity of coordinates, we perform the shift ( ) by replacing in ( 23 |
[EQUATION] while for all the displacements of the nearest neighbors we have [MATH] etc. If the generic atom [MATH] is a ordered one, the shifts ( ) in the Hamiltonian ( 23 ) have to be performed for the coordinates of the neighbors, i.e. |
[EQUATION] while [MATH] , for [MATH] In both cases, [MATH] The equations of motion are obtained by applying the Heisenberg equations |
[MATH] taking into account the canonical commutation relations [EQUATION] The equations of motion (see Appendices) are differential equations which couple displacements in all three directions. We introduce the notation [MATH] and [MATH] . The latter is a dimensionless expansion parameter, proportional to [MATH] . We h... |
Further, we use Born’s approach to the vibrations of a lattice, expanding in Fourier series the operators [EQUATION] and applying the Born–von Kármán boundary conditions (see, e.g., Kantorovich ). In the above expression, [MATH] represent the wave vector projections and [MATH] the mode vibration frequencies. We obtain ... |
So far, the model has two sources of anisotropy: the crystal lattice and the vector [MATH] . As far as the noncommutativity is concerned, the disorder effect is manifest only in the plane orthogonal to [MATH] , and not in the direction of [MATH] . In order to render the model isotropic, we choose as representative axis... |
The isotropised model gives the following dispersion relations: [EQUATION] for disordered atoms, and [EQUATION] for ordered atoms. These expressions are obtained in the small-angle approximation (i.e. [MATH] ), but they are valid with excellent accuracy over the whole Brillouin zone, due to the smallness of the [MATH] ... |
Due to the rotational symmetry, the VDOS is easily derived from ( 18 ) and ( 19 ) using standard methods, with the result: [EQUATION] |
where [EQUATION] is the contribution to VDOS from the ordered atoms. We note the divergence in VDOS, i.e. a van Hove singularity, which occurs for [MATH] . In the limit [MATH] , we recover the VDOS of a usual simple cubic lattice with one atom per cell. |
The reduced specific heat is determined from the following expression (see, e.g., Kantorovich [EQUATION] in which [MATH] is the number of formula units per unit cell (in our model, [MATH] ) and [MATH] is the Boltzmann constant. We take [MATH] as the frequency of the optical branches at [MATH] |
The motif of the disordered lattice can be a single atom, a small molecule, a protein or any combination thereof. Hereafter, three distinct disordered materials, amorphous silicon ( [MATH] -Si) queen , vitreous GeO |
GeO2 and [MATH] Ba Ga 16 Sn 30 clathrate BGS , are considered. The model has two free parameters: the characteristic frequency [MATH] and the noncommutativity parameter [MATH] . They are determined by fitting the theoretical curve ( 22 ) to the experimental data, such that the frequency and reduced specific heat at the... |
Let us recall the physical meaning of the parameter [MATH] : according to ( ), [MATH] is the area “occupied” by a particle in the quantized configuration space, or the area within which the uncertainty in the position of the particle is significant. On the other hand, the mass density of the glass leads to a value of t... |
IV Conclusions and outlook In summary, an amorphous solid can be interpreted as a system with a frozen-in liquid-type of disorder, implemented mathematically as a noncommutative algebra of coordinate operators. Intuitively, this is equivalent to a blurriness or uncertainty in the positions of the particles that form th... |
Glasses are complex systems, and their quantum behaviour at different temperatures is dominated by different aspects of their structure and dynamics. We remark that the departure of the experimental curves from the analytic curve is natural at temperatures further away from the peak, since formula for VDOS ( 62 ) inclu... |
Acknowledgements The authors are particularly grateful to R. Bufalo and M. Chaichian for valuable discussions and advice. We are also grateful to F. Hellman for kindly sharing the experimental data for amorphous silicon. T. R. Cardoso thanks B. M. Pimentel for discussions on noncommutative field theories and H. S. Mart... |
Appendix A Noncommutative normal modes of vibration Assuming harmonic interactions among the nearest neighbors of a neutral monatomic simple cubic lattice, the dynamics in noncommutative quantum mechanics for normal modes of vibration of a simple cubic lattice is described by the Schrödinger equation for [MATH] degrees... |
[EQUATION] where [MATH] is the momentum canonically conjugated to the displacement [MATH] and [MATH] is the usual harmonic oscillator frequency. The quantum algebra satisfied by the canonical variables is: |
[EQUATION] As explained earlier, the model consists of an alternation of ordered and disordered atoms in a simple cubic lattice. Ordered atoms mean atoms whose quantum coordinates and momenta satisfy the usual Heisenberg algebra, i.e., whose coordinates commute, while the disordered atoms |
have coordinates and momenta satisfying the noncommutative space algebra. (There are “two species” of atoms in this case, like, for example, in a NaCl lattice, just that the atoms are identical, but half are ordered and half are not.) |
We should emphasize that the distinction ordered/disordered atoms is a matter of semantics, because all atoms suffer the effects of the noncommutativity of coordinates either directly or through the dynamical couplings. The lattice, as a whole, will be disordered. |
Appendix B Equations of motion and dispersion relations for disordered atoms Considering that the atom [MATH] is a disordered one, which suffers itself the effects of the noncommutativity of coordinates, we perform in ( 23 the shift |
[EQUATION] while for all the displacements of the nearest (ordered) neighbors we have [EQUATION] For the momenta, [MATH] , with [MATH] . The shifted displacements obey the canonical commutation relations |
[EQUATION] The Hamiltonian ( 23 ) becomes: [EQUATION] where we have defined [MATH] and [MATH] The equations of motion are obtained via Heisenberg’s equations |
[EQUATION] where the generic operator stands for either the displacements or momenta of the atom [MATH] , and [MATH] is [MATH] given by ( ). |
The equations of motion for disordered atoms are [EQUATION] where we kept the terms up to the second order in [MATH] Any function in a space formed by a periodic arrangement of atoms must satisfy periodic boundary conditions, the Born–von Kármán boundary conditions. It is important to highlight that upon the shift of c... |
Since the reciprocal lattice of a simple cubic lattice is another simple cubic lattice, one can expand in Fourier series the operators |
[EQUATION] in the above expression, [MATH] represent the wave vectors and [MATH] the vibrational frequencies. Replacing the Ansatz ( 39 ) into the equations of motion ( 36 ), we find the saecular equation for the disordered atoms: |
[EQUATION] As explained Sect. III, the only physically relevant directions for our model are the directions of coordinate axes. The isotropization of the model is performed by taking the dispersion relations in the direction [MATH] and replicating it by rotational symmetry to all the directions of the Cartesian system,... |
Upon isotropization, the solutions of equation ( 40 (for [MATH] and [MATH] ), are: [EQUATION] where the minus (plus) sign is related to the degenerated acoustic (optical) branch. In the small angle approximation (i.e. [MATH] ), this expressions becomes |
[EQUATION] preserving the relation of the minus (plus) sign for the acoustic (optical) branch. Both functions ( ) and ( 42 ) are monotonically increasing within the Brillouin zone, with the maximum at [MATH] . In Fig. are depicted the dispersion relations for [MATH] We note the characteristic plateau in [MATH] , which ... |
[EQUATION] We may put equation ( 43 ) in the form [EQUATION] and identify the isofrequency surfaces [MATH] in the [MATH] -space as spheres of radius [MATH] . The density of states is obtained by taking the derivative of the volume of the sphere with respect to [MATH] and subsequently dividing by the volume of one cell ... |
[EQUATION] where the prefactor 3/2 accounts for the three acoustic branches and the fact that only half of the atoms are disordered. |
Appendix C Equations of motion and dispersion relations for ordered atoms In a similar manner as described in the preceding section, we proceed with the ordered atoms. If the atom [MATH] is a ordered one, we have |
[EQUATION] while its nearest neighbors are disordered and the shift of noncommutative coordinates has to be applied to them, as follows: |
[EQUATION] As for the momenta, [MATH] The Hamiltonian ( 23 ) becomes: [EQUATION] Proceeding as in the case of disordered atoms, we find the equations of motion up to the second order in [MATH] |
[EQUATION] The saecular equation for the ordered atoms is given by [EQUATION] where the elements of the matrix [MATH] are [EQUATION] |
[EQUATION] [EQUATION] Upon isotropization and in the small angle approximation, the saecular equation for ordered atoms becomes [EQUATION] |
or [EQUATION] The solutions for the saecular equation coming from ( 56 ) are: [EQUATION] Accordingly, in the small angle approximation, this expression is rewritten as: |
[EQUATION] Similarly to the case of the disordered atoms, the ordered atoms contribution to the density of states is found to be: |
[EQUATION] Appendix D Specific heat of the glass The complete glass DOS in the small angle approximation is equal to the sum of the disordered and ordered atoms expressions for DOS, namely |
[EQUATION] where [MATH] is the contribution to VDOS from the ordered atoms given by 61 The reduced specific heat is determined from the following expression |
[EQUATION] in which [MATH] is the number of formula units per unit cell (in our model, [MATH] ) and [MATH] is the Boltzmann constant. We take [MATH] as the frequency of the optical branches at [MATH] . We adopt this value for the maximum frequency because our small angle approximation makes the acoustic branch frequenc... |
In order to establish an order of magnitude for the noncommutativity parameter [MATH] , implicitly a measure of nonlocality, one has to resort to the available experimental data: from the match between the theoretical prediction with the experimental data for the reduced specific heat one can assign that the unique pai... |
# Source: arxiv 1806.03867 # Title: Ladder system uniformization on trees I & II # Sections: all # Downloaded: 2026-03-03T02:35:06.373364+00:00 |
Ladder system uniformization on trees I & II Abstract. Suppose that [MATH] is a tree of height [MATH] . We say that a ladder system colouring [MATH] has a [MATH] -uniformization if there is a function [MATH] defined on a subtree [MATH] of [MATH] so that for any [MATH] of limit height and almost all [MATH] [MATH] . In s... |
Furthermore, it is consistent that for any Aronszajn tree [MATH] and ladder system [MATH] there is a colouring of [MATH] without a [MATH] -uniformization; however, and quite surprisingly, [MATH] implies that for any ladder system [MATH] there is an Aronszajn tree [MATH] so that any monochromatic colouring of [MATH] has... |
Key words and phrases: ladder system, uniformization, Suslin tree, special tree, Aronszajn tree, diamond, colouring 2010 Mathematics Subject Classification: 03E05,03E35,03E50 |
1. Introduction A sequence [MATH] indexed by the set of countable limit ordinals is a ladder system on [MATH] if [MATH] is a cofinal subset of [MATH] of type [MATH] . A colouring of the ladder system is a sequence of maps [MATH] so that [MATH] . Now, given such a ladder system colouring, one might ask if there is a sin... |
Apriori, nothing prevents the existence of [MATH] since two local maps [MATH] and [MATH] are only defined on finitely many common points and finite errors are allowed (see Figure ). |
The existence of [MATH] -uniformizations for arbitrary colourings is not decided by the usual ZFC axioms. This topic has been extensively studied due to various connections to algebra, in particular to the Whitehead problem and its relatives |
, to topology , and to fundamental questions in set theory , in particular, to the study of forcing axioms that are compatible with the Continuum Hypothesis |
Our current interest lies in understanding a relatively new version of the uniformization property introduced by Justin Moore, a notion that played a key role in understanding uncountable minimal linear orders |
. If [MATH] is a tree of height [MATH] , we say that [MATH] is a subtree if [MATH] is downward closed and pruned in [MATH] i.e., if any [MATH] has extensions with arbitrary large height below [MATH] . Given some [MATH] and [MATH] , we let [MATH] be the unique predecessor of [MATH] in [MATH] of height [MATH] |
Now, the main definition is the following. Definition 1.1 Suppose that [MATH] is a tree of height [MATH] is a ladder system and is a colouring of . A [MATH] -uniformization of is a map [MATH] on a subtree [MATH] of [MATH] so that for all [MATH] of limit height [MATH] and almost all [MATH] |
[EQUATION] In the above situation, we say that [MATH] uniformizes [MATH] (on [MATH] . In Figure , we aimed to emphasize that [MATH] is only required to agree with the local colouring [MATH] along those branches that are bounded in [MATH] . One can imagine this map [MATH] as a coherent collection of countable maps that ... |
Prior to Moore’s and our work, a similar theme of ’uniformization on trees’ was investigated by Zoran Spasojevic , in connection to tree topologies, but mostly for a restricted class of colourings that correspond to topological separation axioms. |
The goal of our project was to contrast the theory of [MATH] -uniformizations and [MATH] -uniformizations. First, note that any subtree of [MATH] (in our convention of a subtree) must be [MATH] itself, and so [MATH] for any [MATH] -uniformization [MATH] . We will say that a [MATH] -uniformization [MATH] of some ladder ... |
However, there is great flexibility in picking very different domains for [MATH] -uniformizations of different colourings . Furthermore, given a [MATH] -uniformization [MATH] and some level [MATH] of the tree, the number of [MATH] so that [MATH] generally depends on the choice of [MATH] . Thus, it seems (and actually i... |
Given a ladder system on [MATH] , an [MATH] -colouring of is a sequence of maps [MATH] We say that is monochromatic if each [MATH] is constant. Now, we will write |
[MATH] if any [MATH] -colouring of has a [MATH] -uniformization, and [MATH] if all monochromatic [MATH] -colouring of has a [MATH] -uniformization. |
Let us briefly review some fundamental results about ladder system uniformizations: the study was initiated by Saharon Shelah in the 1970s, as he isolated the uniformization property as the combinatorial essence of the Whitehead problem (after solving the problem in |
). First, the two extreme cases are summarized below. (I) [MATH] implies [MATH] for any ladder system , and (II) [MATH] implies [MATH] for any ladder system |
We should mention that for any ladder system colouring, there is a (proper) poset which introduces an [MATH] -uniformization but which adds no new reals. However, the latter result implies that these posets cannot be iterated without adding reals, and hence the uniformization property is one of the fundamental barriers... |
At this point, the following theorem of Moore on [MATH] -uniformizations might come as a surprise: (III) it is consistent with CH that [MATH] holds for any Aronszajn tree |
[MATH] and ladder system 20 , Theorem 1.9] As earlier said, Moore’s main motivation was to study uncountable linear orders, and he proved that in any model as above, the only minimal uncountable linear orders are [MATH] and its reverse [MATH] |
20 , Lemma 3.3] Answering a question of James Baumgartner, this result was later extended by the present author to show that it is consistent with CH that there is a Suslin tree |
[MATH] and [MATH] holds for all and any Aronszajn tree [MATH] that embeds no derived subtree of [MATH] 31 , Theorem 2.1 and Corollary 2.3] . The latter was applied to show that the existence of a Suslin tree does not imply that there are minimal uncountable linear orders beside [MATH] The author have used uniformizatio... |
and ladder systems on trees to study graph chromatic number problems Initially, our motivation was to understand if Moore’s model can contain any Suslin trees, or if our latter theorem is optimal and CH does not allow uniformization on Suslin trees. Upon answering this question, we extended various classical results to... |
Colourings from (weak) diamonds First, in Section , we show that [MATH] implies [MATH] for all Suslin trees [MATH] . In turn, Moore’s model from point ( III ) contains no Suslin trees, and our 31 , Theorem 2.1 and Corollary 2.3] are optimal. |
In Section , we show how variations of the diamond principle imply that there are ladder system colourings without [MATH] -uniformizations for arbitrary Aronszajn trees. As expected, stronger diamonds imply that we can take care of more trees and find even monochromatic 2-colourings without [MATH] -uniformizations. In ... |
Returning to Suslin trees, we show that a natural ccc poset which introduces a uniformization for a given colouring preserves all Suslin trees. Hence, for any Suslin tree [MATH] , the restricted forcing axiom [MATH] implies that any ladder system colouring has an [MATH] -uniformization. This is done in Section |
Constructing trees and uniformizations In Section , we prove one of our main results: if [MATH] is the tree of all well ordered subsets of [MATH] with a maximum then any ladder system colouring has a [MATH] -uniformization; moreover, this fact will be witnessed by a single master colouring |
[MATH] regardless of the choice of ladder system and its colouring. One might say that this is not so surprising as the tree [MATH] is far from Aronszajn: although it has no uncountable chains, the levels are of size continuum and furthermore, [MATH] satisfies rather strong closure properties. We present another variat... |
Next, we show that [MATH] implies that for any ladder system , there is an Aronszajn tree [MATH] so that [MATH] holds. The tree [MATH] can either be made special or alternatively, we can ensure that any colouring of has a [MATH] -uniformization defined on a Suslin subtree of [MATH] . We present these constructions in S... |
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