On the existence of noncontinuous irreducible representations of a locally compact group.
Nondiscrete locally compact group having noncontinuous irreducible representations, proving theorem
Bichteler, K.↗
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Locally compact topologies on a group and the corresponding continuous irreducible representations
Locally compact topologies on group G and corresponding continuous irreducible unitary representations, noting existence of subgroup
Bichteler, K.↗
Wave functions of multiquark hadrons from representations of the symmetry groups S n
Construction of the wave functions of multiquark hadrons by a traditional method based on the tensor products of colors, flavors, spins (and orbital) parts becomes quite complex when quark numbers grow n = 5 , 6 … 12 , as it gets difficult to satisfy the requirements of Fermi statistics. Our novel approach is focused directly on representations of the permutation symmetry generators. After showing how C 3 is manifested in the wave functions of (excited) baryons, we use it to construct the wave functions for a set of pentaquarks and hexaquarks ( n = 5 , 6 ). We also have some partial results for larger systems, with n = 9 and 12, and even beyond that as far as n = 24 . Published by the American Physical Society 2024
Miesch, Nicholas (ORCID:0000000205937535)↗
Photon topology
The topology of photons in vacuum is interesting because there are no photons with k = 0, creating a hole in momentum space. We show that while the set of all photons forms a trivial vector bundle $γ$ over this momentum space, the R and L photons form topologically nontrivial subbundles $γ±$ with first Chern numbers ∓2. In contrast, $γ$ has no linearly polarized subbundles, and there is no Chern number associated with linear polarizations. It is a known difficulty that the standard version of Wigner’s little group method produces singular representations of the Poincaré group for massless particles. By considering representations of the Poincaré group on vector bundles we obtain a version of Wigner’s little group method for massless particles which avoids these singularities. Here we show that any massless bundle representation of the Poincaré group can be canonically decomposed into irreducible bundle representations labeled by helicity, which in turn can be associated to smooth irreducible Hilbert space representations. This proves that the R and L photons are globally well defined as particles and that the photon wave function can be uniquely split into R and L components. This formalism offers a method of quantizing the electromagnetic field without invoking discontinuous polarization vectors as in the traditional scheme. We also demonstrate that the spin-Chern number of photons is not a purely topological quantity. Lastly, there has been an extended debate on whether photon angular momentum can be split into spin and orbital parts. Our work explains the precise issues that prevent this splitting. Photons do not admit a spin operator; instead, the angular momentum associated with photons’ internal degree of freedom is described by a helicity-induced subalgebra corresponding to the translational symmetry of $γ$.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Completing Multiparticle Representations of the Poincaré Group
Not Available
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Towards AI-assisted neutrino flavor theory design
Particle physics theories, such as those which explain neutrino flavor mixing, arise from a vast landscape of model-building possibilities. A model’s construction typically relies on the intuition of theorists. It also requires considerable effort to identify appropriate symmetry groups, assign field representations, and extract predictions for comparison with experimental data. We develop Autonomous Model Builder (AMBer), a framework in which a reinforcement learning agent interacts with a streamlined physics software pipeline to search these spaces efficiently. AMBer selects symmetry groups, particle content, and group representation assignments to construct models while minimizing the number of free parameters introduced. We validate our approach in well-studied regions of theory space and extend the exploration to a previously unexamined symmetry group. While demonstrated in the context of neutrino flavor theories, this approach of reinforcement learning with physics software feedback may be extended to other theoretical model-building problems in the future.
Baretz, Jason Benjamin↗
On the representation matrices of the spin permutation group
A method is presented for the determination of the representation matrices of the spin permutation group (symmetric group), a detailed knowledge of these matrices being required in the study of the electronic structure of atoms and molecules. The method is characterized by the use of two different coupling schemes. Unlike the Yamanouchi spin algebraic scheme, the method is not recursive. The matrices for the fundamental transpositions can be written down directly in one of the two bases. The method results in a computationally significant reduction in the number of matrix elements that have to be stored when compared with, say, the standard Young tableaux group theoretical approach.
Wilson, S.↗
Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
We present a pedagogical review of projective representations of finite groups and their physical applications in quantum many-body systems. Some of our physical results are new. We begin with a self-contained introduction to projective representations, highlighting the role of group cohomology, representation theory, and classification of irreducible projective representations. We then focus on a special subset of cohomology classes, known as the Bogomolov multiplier, which consists of cocycles that are symmetric on commuting pairs but remain nontrivial in group cohomology. Such cocycles have important physical implications: they characterize (1+1)D SPT phases that cannot be detected by string order parameters and give rise, upon gauging, to distinct gapped phases with completely broken non-invertible Rep(G) symmetry. We construct explicit lattice models for these phases and demonstrate how they are distinguished by the fusion rules of local order parameters. We show that a pair of completely broken Rep(G) SSB phases host nontrivial interface modes at their domain walls. As an example, we construct a lattice model where the ground state degeneracy on a ring increases from 32 without interfaces to 56 with interfaces.
Bogomolov multiplier↗
Pitfalls in the n -mode representation of vibrational potentials
Simulations of anharmonic vibrational motion rely on computationally expedient representations of the governing potential energy surface. The n-mode representation (n-MR)—effectively a many-body expansion in the space of molecular vibrations—is a general and efficient approach that is often used for this purpose in vibrational self-consistent field (VSCF) calculations and correlated analogues thereof. In the present analysis, a lack of convergence in many VSCF calculations is shown to originate from negative and unbound potentials at truncated orders of the n-MR expansion. For cases of strong anharmonic coupling between modes, the n-MR can both dip below the true global minimum of the potential surface and lead to effective single-mode potentials in VSCF that do not correspond to bound vibrational problems, even for bound total potentials. The present analysis serves mainly as a pathology report of this issue. Furthermore, this insight into the origin of VSCF non-convergence provides a simple, albeit ad hoc, route to correct the problem by “painting in” the full representation of groups of modes that exhibit these negative potentials at little additional computational cost. Somewhat surprisingly, this approach also reasonably approximates the results of the next-higher n-MR order and identifies groups of modes with particularly strong coupling. The method is shown to identify and correct problematic triples of modes—and restore SCF convergence—in two-mode representations of challenging test systems, including the water dimer and trimer, as well as protonated tropine.
Chemistry↗
NASA's Metrology and Calibration program
The success of the National Aeronautics and Space Administration's scientific and engineering projects is often based on the capability and quality of the metrology and calibration programs of its field centers and facilities. A Metrology and Calibration Working Group, having representation from each of the NASA field centers and facilities, was established to develop and implement a program to provide a centralized direction and agency wide focus. A brief history of the Group is presented. The development of the Group's operating philosophy, the long term objectives, the measurement research and development program, and review of accomplishments is discussed.
Kern, Frederick A.↗
Remote Instrumentation AMPATH Astronomy Working Group
This viewgraph representation presents an overview of the Telescience Resource Kit (TReK) and the Internet Voice Distribution System (IVoDS) applications which are being used to provide support for science experiments aboard the International Space Station (ISS). Operational logistics, structural design, technical capabilities for both systems are discussed. Additional topics covered include: general discussion of remote instrumentation and tools, and hypothetical examples where these tools are used.
Bradford, Robert N.↗
Hilbert series for covariants and their applications to minimal flavor violation
We elaborate how to apply the Hilbert series method to enumerating group covariants, which transform under any given representation, including but going beyond group invariants. Mathematically, group covariants form a module over the ring of the invariants. The number of independent covariants is given by the rank of the module, which can be computed by taking a ratio of two Hilbert series. In many cases, the rank equals the dimension of the group covariant representation. When this happens, we say that there is a rank saturation. We apply this technology to revisit the hypothesis of Minimal Flavor Violation in constructing Effective Field Theories beyond the Standard Model. We find that rank saturation is guaranteed in this case, leading to the important consequence that the MFV symmetry principle does not impose any restriction on the EFT, i.e. MFV SMEFT = SMEFT, in the absence of additional assumptions.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Unification of finite symmetries in the simulation of many-body systems on quantum computers
Symmetry is fundamental in the description and simulation of quantum systems. Leveraging symmetries in classical simulations of many-body quantum systems can result in significant overhead due to the exponentially growing size of some symmetry groups as the number of particles increases. Quantum computers hold the promise of achieving exponential speedup in simulating quantum many-body systems; however, a general method for utilizing symmetries in quantum simulations has not yet been established. In this work, we present a unified framework for incorporating symmetry group transforms on quantum computers to simulate many-body systems. The core of our approach lies in the development of efficient quantum circuits for symmetry-adapted projection onto irreducible representations of a group or pairs of commuting groups. We provide resource estimations for common groups, including the cyclic and permutation groups. Our algorithms demonstrate the capability to prepare coherent superpositions of symmetry-adapted states and to perform quantum evolution across a wide range of models in condensed-matter physics and ab initio electronic structure in quantum chemistry. Specifically, we execute a symmetry-adapted quantum subroutine for small molecules in first-quantization on noisy hardware and demonstrate the emulation of symmetry-adapted quantum phase estimation for preparing coherent superpositions of quantum states in various irreducible representations of a symmetry group. In addition, we present a discussion of open problems regarding treating symmetries in digital quantum simulations of many-body systems, paving the way for future systematic investigations into leveraging symmetries quantumly for practical quantum advantage. The broad applicability and rigorous resource estimation for symmetry transformations make our framework appealing for achieving provable quantum advantage on fault-tolerant quantum computers, especially for symmetry-related properties.
quantum algorithms↗
Loop-level double-copy for massive fermions in the fundamental
We find that unitarity cuts and the duality between color and kinematics are sufficient constraints to bootstrap D-dimensional QCD scattering amplitudes starting from three-particle tree-level. Specifically, we calculate tree level amplitudes through six-points, as well as the four-point one-loop correction for massive fermions in the fundamental representation of the gauge group — constructing a color-dual representation of the latter for the first time. To do so we clarify a prescription for functional kinematic ansatze involving fermionic matter. The advantages of color-dual calculation, familiar from particles in the adjoint, also apply here: only a small number of basis topologies must be constrained via physical information of the theory, and algebraic relations propagate this to a full solution. As all the QCD amplitudes we construct here are color-dual, they trivially generate D-dimensional amplitudes in gravitational theories via double-copy construction.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks
There has been much recent interest in designing symmetry-aware neural networks (NNs) exhibiting relaxed equivariance. Such NNs aim to interpolate between being exactly equivariant and being fully flexible, affording consistent performance benefits. In a separate line of work, certain structured parameter matrices -- those with displacement structure, characterized by low displacement rank (LDR) -- have been used to design small-footprint NNs. Displacement structure enables fast function and gradient evaluation, but permits accurate approximations via compression primarily to classical convolutional neural networks (CNNs). In this work, we propose a general framework -- based on a novel construction of symmetry-based structured matrices -- to build approximately equivariant NNs with significantly reduced parameter counts. Our framework integrates the two aforementioned lines of work via the use of so-called Group Matrices (GMs), a forgotten precursor to the modern notion of regular representations of finite groups. GMs allow the design of structured matrices -- resembling LDR matrices -- which generalize the linear operations of a classical CNN from cyclic groups to general finite groups and their homogeneous spaces. We show that GMs can be employed to extend all the elementary operations of CNNs to general discrete groups. Further, the theory of structured matrices based on GMs provides a generalization of LDR theory focussed on matrices with cyclic structure, providing a tool for implementing approximate equivariance for discrete groups. We test GM-based architectures on a variety of tasks in the presence of relaxed symmetry. We report that our framework consistently performs competitively compared to approximately equivariant NNs, and other structured matrix-based compression frameworks, sometimes with a one or two orders of magnitude lower parameter count.
Samudre, Ashwin↗
Lorentz group equivariant autoencoders
Abstract There has been significant work recently in developing machine learning (ML) models in high energy physics (HEP) for tasks such as classification, simulation, and anomaly detection. Often these models are adapted from those designed for datasets in computer vision or natural language processing, which lack inductive biases suited to HEP data, such as equivariance to its inherent symmetries. Such biases have been shown to make models more performant and interpretable, and reduce the amount of training data needed. To that end, we develop the Lorentz group autoencoder (LGAE), an autoencoder model equivariant with respect to the proper, orthochronous Lorentz group $$\textrm{SO}^+(3,1)$$ SO + ( 3 , 1 ) , with a latent space living in the representations of the group. We present our architecture and several experimental results on jets at the LHC and find it outperforms graph and convolutional neural network baseline models on several compression, reconstruction, and anomaly detection metrics. We also demonstrate the advantage of such an equivariant model in analyzing the latent space of the autoencoder, which can improve the explainability of potential anomalies discovered by such ML models.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Quantum Algorithms for Representation-Theoretic Multiplicities
Kostka, Littlewood-Richardson, Plethysm, and Kronecker coefficients are the multiplicities of irreducible representations in the decomposition of representations of the symmetric group that play an important role in representation theory, geometric complexity, and algebraic combinatorics. We give quantum algorithms for computing these coefficients whenever the ratio of dimensions of the representations is polynomial. We show that there is an efficient classical algorithm for computing the Kostka numbers under this restriction and conjecture the existence of an analogous algorithm for the Littlewood-Richardson coefficients. We argue why such classical algorithm does not straightforwardly work for the Plethysm and Kronecker coefficients and conjecture that our quantum algorithms lead to superpolynomial speedups. The conjecture about Kronecker coefficients was disproved by Panova [Polynomial time classical versus quantum algorithms for representation theoretic multiplicities, arXiv:2502.20253] with a classical algorithm which, if optimal, points to a 𝒪(𝑛 4+2𝑘 ) vs $\tilde{Ω}$(𝑛 4𝑘 2 +1 ) polynomial gap in quantum vs classical computational complexity for an integer parameter 𝑘.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
On a class of selection rules without group actions in field theory and string theory
We discuss a class of selection rules which i) do not come from group actions on fields, ii) are exact at tree level in perturbation theory, iii) are increasingly violated as the loop order is raised, and iv) eventually reduce to selection rules associated with an ordinary group symmetry. We start from basic field-theoretical examples in which fields are labeled by conjugacy classes rather than representations of a group, and discuss generalizations using fusion algebras or hypergroups. We also discuss how such selection rules arise naturally in string theory, such as for non-Abelian orbifolds or other cases with non-invertible worldsheet symmetries.
Kaidi, Justin (ORCID:0000000161440729)↗