Rotational Covariance Research Articles

Graph neural network coarse-grain force field for the molecular crystal RDX

Condense phase molecular systems organize in wide range of distinct molecular configurations, including amorphous melt and glass as well as crystals often exhibiting polymorphism, that originate from their intricate intra- and intermolecular forces. While accurate coarse-grain (CG) models for these materials are critical to understand phenomena beyond the reach of all-atom simulations, current models cannot capture the diversity of molecular structures. We introduce a generally applicable approach to develop CG force fields for molecular crystals combining graph neural networks (GNN) and data from an all-atom simulations and apply it to the high-energy density material RDX. We address the challenge of expanding the training data with relevant configurations via an iterative procedure that performs CG molecular dynamics of processes of interest and reconstructs the atomistic configurations using a pre-trained neural network decoder. The multi-site CG model uses a GNN architecture constructed to satisfy translational invariance and rotational covariance for forces. The resulting model captures both crystalline and amorphous states for a wide range of temperatures and densities.

Light-front puzzles

Light-front formulations of quantum field theories have many advantages for computing electroweak matrix elements of strongly interacting systems and other quantities that are used to study hadronic structure. The theory can be formulated in Hamiltonian form so non-perturbative calculations of the strongly interacting initial and final states are in principle reduced to linear algebra. These states are needed for calculating parton distribution functions and other types of distribution amplitudes that are used to understand the structure of hadrons. Light-front boosts are kinematic transformations so the strongly interacting states can be computed in any frame. This is useful for computing current matrix elements involving electroweak probes where the initial and final hadronic states are in different frames related by the momentum transferred by the probe. Finally in many calculations the vacuum is trivial so the calculations can be formulated in Fock space. The advantages of light front-field theory would not be interesting if the light-front formulation was not equivalent to the covariant or canonical formulations of quantum field theory. Many of the distinguishing properties of light-front quantum field theory are difficult to reconcile with canonical or covariant formulations of quantum field theory. This paper discusses the resolution of some of the apparent inconsistencies in canonical, covariant and light-front formulations of quantum field theory. The puzzles that will be discussed are (1) the problem of inequivalent representations (2) the problem of the trivial vacuum (3) the problem of ill-posed initial value problems (4) the problem of rotational covariance (5) the problem of zero modes and (6) the problem of spontaneously broken symmetries.

Accurate and Interpretable Dipole Interaction Model-Based Machine Learning for Molecular Polarizability.

Polarizabilities play significant roles in describing dispersive and inductive interactions of the atom and molecular systems. However, an accurate prediction of molecular polarizabilities from first principles is computationally prohibitive. Although physical models or statistical machine learning models have been proposed, either a lack of accurate description of local chemical environments or demanding a large number of samples for training has limited their practical applications. In this study, we combine a physically inspired dipole interaction model and an accurate neural network method for predicting the polarizability tensors of molecules. With the local chemical environment precisely described and the requirement of rotational covariance naturally fulfilled, this hybrid model is proven to give an accurate molecular polarizability prediction, essentially reducing the number of training samples. The atomic polarizabilities are physically interpretable and transferable to larger molecules unseen in the training set. This promising method may find its wide range of applications, such as spectroscopic simulations and the construction of polarizable force fields.

Relation between instant and light-front formulations of quantum field theory

The scattering equivalence of quantum field theories formulated with light-front and instant-form kinematic subgroups is established using non-perturbative methods. The difficulty with field theoretic formulations of Dirac's forms of dynamics is that the free and interacting unitary representations of the Poincar\'e group are defined on inequivalent representations of the Hilbert space, which means that the concept of kinematic transformations must be modified on the Hilbert space of the field theory. This work addresses this problem by assuming the existence of a field theory with the expected properties and constructs equivalent representations with instant and front form kinematic subgroups. In this construction both the light-front and instant-form formulations share the same vacuum and one-particle states. The free field Fock space plays no role. There is no "quantization" of a classical theory. The property that survives from the perturbative approach is the notion of a kinematic subgroup, which means kinematic Poincar\'e transformations can be trivially implemented by acting on suitable basis vectors. This non-perturbative approach avoids dealing with issues that arise in perturbative treatments where is it necessary to have a consistent treatment of renormalization, rotational covariance, and the structure of the light-front vacuum. While addressing these issues in a computational framework is important for applications, this work may provide some insight into the nature of the expected resolution and identifies the origin of some of differences between the perturbative and non-perturbative approaches.

Kinematical superspaces

We classify N =1 d = 4 kinematical and aristotelian Lie superalgebras with spa- tial isotropy, but not necessarily parity nor time-reversal invariance. Employing a quater- nionic formalism which makes rotational covariance manifest and simplifies many of the calculations, we find a list of 43 isomorphism classes of Lie superalgebras, some with pa- rameters, whose (nontrivial) central extensions are also determined. We then classify their corresponding simply-connected homogeneous (4|4)-dimensional superspaces, resulting in a list of 27 homogeneous superspaces, some with parameters, all of which are reductive. We determine the invariants of low rank and explore how these superspaces are related via geometric limits.

The Garrison-Wong phase operator and the rotational covariance of phase states

The cosine and sine phase operators and the Garrison-Wong (GW) operator of the phase have nonorthogonal diagonal representations in terms of the usual phase states. Also their orthogonal diagonal representations are studied theoretically. The phase distributions of the normalized truncated cosine and sine phase states reveal that the usual phase of the cosine and sine phase states is two-valued in general. The phase distributions of 110 normalized truncated GW phase states suggest that the usual phase of the GW phase states is single-valued. The preferred phases and phase dispersions of the normalized truncated GW phase states have been determined.

Minimum error discrimination for an ensemble of linearly independent pure states

Inspired by the work done by Belavkin (1975 Stochastics 1 315) and independently by Mochon, (2006 Phys. Rev. A 73 032328), we formulate the problem of minimum error discrimination (MED) of any ensemble of n linearly independent pure states by stripping the problem of its rotational covariance and retaining only the rotationally invariant aspect of the problem. This is done by embedding the optimal conditions in a matrix equality as well as matrix inequality. Employing the implicit function theorem in these conditions we get a set of first-order coupled ordinary nonlinear differential equations which can be used to drag the solution from an initial point (where solution is known) to another point (whose solution is sought). This way of obtaining the solution can be done through a simple Taylor series expansion and analytic continuation when required. Thus, we complete the work done by Belavkin and Mochon by ultimately leading their theory to a solution for the MED problem of linearly independent pure state ensembles. We also compare the computational complexity of our technique with the barrier-type interior point method of SDP and show that our technique is computationally as efficient as (actually, a bit more than) the SDP algorithm, with the added advantage of being much simpler to implement.

Closed form least-squares solution to 3D symmetric Helmert transformation with rotational invariant covariance structure

Least-squares (LS) solutions to the 3D Helmert and symmetric Helmert transformations with rotational invariant covariance structure are studied in a unified framework. This is an extension of the 3D Helmert transformation with naive identity covariance and the counterpart of the 2D symmetric Helmert transformation with rotational invariant covariance. It is found that the closed form LS solution still exists and that the rotation parameters are still the same between the Helmert and symmetric Helmert transformations.

Rotational covariance and Greenberger-Horne-Zeilinger theorems for three or more particles of any dimension

Greenberger-Horne-Zeilinger (GHZ) states are characterized by their transformation properties under a continuous symmetry group, and $N$-body operators that transform covariantly exhibit a wealth of GHZ contradictions. We show that local or noncontextual hidden variables cannot duplicate this covariance for any state-changing transformations, and we extract specific GHZ contradictions from discrete subgroups, with no restrictions on particle number $N$ or dimension $d$ except for the fundamental requirement that $N \geq 3$ for nonprobabilistic contradictions. However, the specific contradictions fall into three regimes distinguished by increasing demands on the number of measurement operators required for the proofs. We introduce new methods of proof that define these regimes. The first recovers theorems equivalent to those found recently by Ryu et. al. \cite{RLZL}, the first operator-based theorems for all odd dimensions, $d$, covering many (but not all) particle numbers $N$ for each $d$. The second and third produce new theorems that fill all remaining gaps down to $N=3$, for every $d$. The common origin of all such GHZ contradictions is that the GHZ states and measurement operators transform according to different representations of the symmetry group, which has an intuitive physical interpretation.

Students' notions regarding ‘covariance’ of a physical theory

A physical theory is said to be covariant with respect to a certain class of transformations when its basic equations retain their ‘form’ under those transformations. It is one of the basic notions encountered in physics, particularly in the domain of relativity. In this paper we study in some detail how students deal with this notion in different contexts (rotational, Galilean, Lorentz and general covariance). The tool of the study is a diagnostic questionnaire developed through a pilot study and then administered on a sample of 27 senior undergraduates undergoing a course on relativity. The analysis helps us identify a number of conceptual vulnerabilities of students in the domain. The most important of these concerns the non-covariant manifestations of a covariant theory, and the connection between Galilean and Lorentz invariants. The study should be useful in the teaching of relativity at the senior undergraduate and graduate levels.

Electromagnetic form factors in the light-front formalism and the Feynman triangle diagram: Spin-0 and spin-1 two-fermion systems

The connection between the Feynman triangle diagram and the light-front formalism for spin-0 and spin-1 two-fermion systems is analyzed. It is shown that in the limit ${q}^{+}=0$ the form factors for both spin-0 and spin-1 systems can be uniquely determined using only the good amplitudes, which are not affected by spurious effects related to the loss of rotational covariance present in the light-front formalism. At the same time, the unique feature of the suppression of the pair creation process is maintained. Therefore, a physically meaningful one-body approximation, in which all the constituents are on their mass shells, can be consistently formulated in the limit ${q}^{+}=0.$ Moreover, it is shown that the effects of the contact term arising from the instantaneous propagation of the active constituent can be canceled out from the triangle diagram by means of an appropriate choice of the off-shell behavior of the bound state vertices; this implies that in the case of good amplitudes the Feynman triangle diagram and the one-body light-front result match exactly. For illustrative purposes our covariant light-front approach is applied to the evaluation of the $\ensuremath{\rho}$-meson elastic form factors.

SU(6) breaking effects in the nucleon elastic electromagnetic form factors

The effects of SU(6) breaking on the nucleon elastic form factors are investigated within the constituent quark model formulated on the light-front. It is shown that the kinematical SU(6) breaking caused by the Melosh rotations of the quark spins as well as the dynamical SU(6) breaking due to the mixed-symmetry component generated in the nucleon wave function by the spin-dependent terms of the quark-quark interaction, can affect both GEn(Q**2) and GMp(Q**2)/GMn(Q**2). The calculated GEn(Q**2) is found to be qualitatively consistent with existing data. while the calculations of GM(Q**2) based on the plus component of the current are found to be plagued by spurious effects related to the loss of the rotational covariance in the light-front formalism. These unwanted effects can be avoided by considering the transverse y-component of the current. In this way our light-front predictions are found to be consistent with the data on both GEn(Q**2) and GMp(Q**2)/GMn(Q**2). Finally, it is shown that a suppression of the ratio GEp(Q**2)/GMp(Q**2) with respect to the dipole-fit prediction can be expected in the constituent quark model provided the relativistic effects generated by the Melosh rotations of the constituent spins are taken into account.

Electromagnetic form factors of the ϱ meson in a light-front constituent quark model

The electromagnetic form factors of the ϱ meson are evaluated adopting a relativistic constituent quark model based on the light-front formalism, and using a meson wave function with the high-momentum tail generated by the one-gluon-exchange interaction. The breakdown of the rotational covariance for the one-body component of the current operator is investigated and the sensitivity of the ratio of the ϱ-meson form factors to the pion (charge) form factor to the spin-dependent component of the effective q q interaction is illustrated.

Baryon current matrix elements in a light-front framework.

Current matrix elements and observables for electro- and photo-excitation of baryons from the nucleon are studied in a light-front framework. Relativistic effects are estimated by comparison to a nonrelativistic model, where we use simple basis states to represent the baryon wave functions. Sizeable relativistic effects are found for certain transitions, for example, to radial excitations such as that conventionally used to describe the Roper resonance. A systematic study shows that the violation of rotational covariance of the baryon transition matrix elements stemming from the use of one-body currents is generally small.

Rotational covariance and light-front current matrix elements.

Light-front current matrix elements for elastic scattering from hadrons with spin 1 or greater must satisfy a nontrivial constraint associated with the requirement of rotational covariance for the current operator. Using a model $\ensuremath{\rho}$ meson as a prototype for hadronic quark models, this constraint and its implications are studied at both low and high momentum transfers. In the kinematic region appropriate for asymptotic QCD, helicity rules, together with the rotational covariance condition, yield an additional relation between the light-front current matrix elements.

The non-linear 2D symmetric helmert transformation: An exact non-linear least-squares solution

In this paper a particular class of non-linear least-squares problems for which it is possible to take advantage of the special structure of the non-linear model, is discussed. The non-linear models are of the ruled-type (Teunisson, 1985a). The proposed solution strategy is applied to the2D non-linear Symmetric Helmert transformation which is defined in the paper. An exact non-linear least-squares solution, using a rotational invariant covariance structure is given.

NONLINEARITY AND LEAST SQUARES

Since almost all functional relations in our geodetic models are nonlinear, it is important, especially from a statistical inference point of view, to know how nonlinearity manifests itself at the various stages of an adjustment. In this paper particular attention is given to the effect of nonlinearity on the first two moments of least squares estimators. Expressions for the moments of least squares estimators of parameters, residuals and functions derived from parameters, are given. The measures of nonlinearity are discussed both from a statistical and differential geometric point of view. Finally, our results are applied to the 2D symmetric Helmert transformation with a rotational invariant covariance structure.

Dilepton production from collisions of polarized spin-½ hadrons. I. General kinematic analysis

The kinematics for the production of lepton pairs from collisions of polarized spin-\textonehalf{} particles is discussed on the basis of rotational covariance and the Jacob-Wick helicity formalism. It is assumed that the leptons come from the decay of a single vector particle.

Euclidean Fermi fields

This paper is intended to clarify the relationships among different approaches to Euclidean Fermi fields. The previous work being considered can be found in: Osterwalder and Schrader [7], Gross [5], Schrader and Uhlenbrock [IS] and Osterwalder [6]. The construction of the Osterwalder-Schrader fields is carried out in the first part of this paper. The smeared fields are defined directly in a standard FockCook representation. Thus no estimates are required to establish the existence of the smeared fields, and the consistency and rotational covariance of the construction are immediate. The free Dirac field is then defined. In the standard constructions of the free Dirac field [7] a certain amount of dedication is required to check consistency, Lorentz covariance, and locality. In the construction presented here some effort has been made to make the labor involved in checking the details as insignificant as possible. Once the Schwinger functions for the free Dirac field are defined their representation in terms of the Osterwalder-Schrader fields is demonstrated. This result is proved in much the same manner that Gross [5] used to prove a more complicated result for the Yukawa model. The proof is included partly because it is naturally simpler than that in [5] and partly because the different definitions adopted here make translations from [S] and (71 difficult. Incident]? the complications in this proof are absent from [7] because they use “Wick’s theorem.” the proof of which would necessarily involve similar considerations. ‘I’he next part of the paper is devoted to a comparison of the unitary dilation analysis in [ 151 and the map W constructed in [7]. It is shown that W =:. T(w) where ZL’ is a complex linear map from the ‘one particle’ Euclidean Hilbert space to the ‘one particle’ Dirac space which compresses the unitary Euclidean time translations into the contraction semi-group associated with the Dirac equation. The term ‘one particle’ is qualified since the Hilbert spaces involved are each double the size of the usual one particle spaces. The connection between this doubling and Euclidean rotational covariance is especially transparent.

Rotational covariance and the Yang-Mills monopole

Rotational covariance in the monopole sector of the SU(2) Yang-Mills model with a triplet of Higgs scalars and the possible addition of fermions is examined. Ansatze about the leading-order behavior of monopole-monopole matrix elements are shown to be consistent with rotational symmetry. Spin identification of monopole states is done for all cases. (AIP)