Publications

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Interpolating between the gauge and Schrödinger pictures of quantum dynamics

Sayak Guha Roy, Kevin Slagle

SciPost Phys. Core 6, 081 (2023) · published 24 November 2023 | · pdf

Although spatial locality is explicit in the Heisenberg picture of quantum dynamics, spatial locality is not explicit in the Schrödinger picture equations of motion. The gauge picture is a modification of Schrödinger's picture such that locality is explicit in the equations of motion. In order to achieve this explicit locality, the gauge picture utilizes (1) a distinct wavefunction associated with each patch of space, and (2) time-dependent unitary connections to relate the Hilbert spaces associated with nearby patches. In this work, we show that by adding an additional spatially-local term to the gauge picture equations of motion, we can effectively interpolate between the gauge and Schrödinger pictures, such that when this additional term has a large coefficient, all of the gauge picture wavefunctions approach the Schrödginer picture wavefunction (and the connections approach the identity).
Extensions of realisations for low-dimensional Lie algebras

Iryna Yehorchenko

SciPost Phys. Proc. 14, 048 (2023) · published 24 November 2023 | · pdf

We find extensions of realisations of some low-dimensional Lie algebras, in particular, for the Poincaré algebra for one space dimension. Using inequivalent extensions, we performed comprehensive classification of relative differential invariants for these Lie algebras. We show difference between classification of extensions of realisations, and classification of nonlinear realisations of Lie algebras.
On the spin content of the classical massless Rarita-Schwinger system

Mauricio Valenzuela, Jorge Zanelli

SciPost Phys. Proc. 14, 047 (2023) · published 24 November 2023 | · pdf

We analyze the Rarita-Schwinger massless theory in the Lagrangian and Hamiltonian approaches. At the Lagrangian level, the standard gamma-trace gauge fixing constraint leaves a 1/2 and a 3/2 propagating Poincaré group helicities. At the Hamiltonian level, the result depends on whether the Dirac conjecture is assumed or not. In the affirmative case, a secondary first class constraint is added to the total Hamiltonian and a corresponding gauge fixing condition must be imposed, completely removing the 1/2 sector. In the opposite case, the 1/2 field propagates and the Hamilton field equations match the Euler-Lagrange equations.

3 citations
The parastatistics of braided Majorana fermions

Francesco Toppan

SciPost Phys. Proc. 14, 046 (2023) · published 24 November 2023 | · pdf

This paper presents the parastatistics of braided Majorana fermions obtained in the framework of a graded Hopf algebra endowed with a braided tensor product. The braiding property is encoded in a t-dependent 4×4 braiding matrix $B_t$ related to the Alexander-Conway polynomial. The nonvanishing complex parameter t defines the braided parastatistics. At t=1 ordinary fermions are recovered. The values of t at roots of unity are organized into levels which specify the maximal number of braided Majorana fermions in a multiparticle sector. Generic values of t and the t=-1 root of unity mimick the behaviour of ordinary bosons.
Construction of matryoshka nested indecomposable N-replications of Kac-modules of quasi-reductive Lie superalgebras, including the sl(m/n) and osp(2/2n) series

Jean Thierry-Mieg, Peter D. Jarvis, Jerome Germoni, Maria Gorelik

SciPost Phys. Proc. 14, 045 (2023) · published 24 November 2023 | · pdf

We construct a new class of finite dimensional indecomposable representations of simple superalgebras which may explain, in a natural way, the existence of the heavier elementary particles. In type I Lie superalgebras sl(m/n) and osp(2/2n), one of the Dynkin weights labeling the finite dimensional irreducible representations is continuous. Taking the derivative, we show how to construct indecomposable representations recursively embedding N copies of the original irreducible representation, coupled by generalized Cabibbo angles, as observed among the three generations of leptons and quarks of the standard model. The construction is then generalized in the appendix to quasi-reductive Lie superalgebras.
Towards higher super-$\sigma$-model categories

Rafał R. Suszek

SciPost Phys. Proc. 14, 044 (2023) · published 24 November 2023 | · pdf

A simplicial framework for the gerbe-theoretic modelling of supercharged-loop dynamics in the presence of worldsheet defects is discussed whose equivariantisation with respect to global supersymmetries of the bulk theory and subsequent orbit decomposition lead to a natural stratification of and cohomological superselection rules for target-space supergeometry, expected to encode essential information on the quantised 2d field theory. A physically relevant example is analysed in considerable detail.
Segal's contractions, AdS and conformal groups

Daniel Sternheimer

SciPost Phys. Proc. 14, 043 (2023) · published 24 November 2023 | · pdf

Symmetries and their applications always played an important role in I.E. Segal's work. I shall exemplify this, starting with his correct proof (at the Lie group level) of what physicists call the "O'Raifeartaigh theorem", continuing with his incidental introduction in 1951 of the (1953) Inönü-Wigner contractions, of which the passage from AdS (SO(2,3)) to Poincaré is an important example, and with his interest in conformal groups in the latter part of last century. Since the 60s Flato and I had many fruitful interactions with him around these topics. In a last section I succintly relate these interests in symmetries with several of ours, especially elementary particles symmetries and deformation quantization, and with an ongoing program combining both.
Clocking mechanism from a minimal spinning particle model

Tobiasz Pietrzak, Łukasz Bratek

SciPost Phys. Proc. 14, 042 (2023) · published 24 November 2023 | · pdf

The clock hypothesis plays an important role in the theory of relativity. To test this hypothesis, a mechanical model of an ideal clock is needed. Such a model should have the phase of its intrinsic periodic motion increasing linearly with the affine parameter of the clock's center of mass worldline. A class of relativistic rotators introduced by Staruszkiewicz in the context of an ideal clock is studied. A singularity in the inverse Legendre transform leading from the Hamiltonian to the Lagrangian leads to new possible Lagrangians characterized by fixed values of mass and spin. In free motion the rotators exhibit intrinsic motion with the speed of light and fixed frequency.
On Möbius gyrogroup and Möbius gyrovector space

Kurosh Mavaddat Nezhaad, Ali Reza Ashrafi

SciPost Phys. Proc. 14, 041 (2023) · published 24 November 2023 | · pdf

Gyrogroups are new algebraic structures that appeared in 1988 in the study of Einstein's velocity addition in the special relativity theory. These new algebraic structures were studied intensively by Abraham Ungar. The first gyrogroup that was considered into account is the unit ball of Euclidean space $\mathbb{R}^3$ endowed with Einstein's velocity addition. The second geometric example of a gyrogroup is the complex unit disk $\mathbb{D}$={z ∈ $\mathbb{C}: |z|<1$}. To construct a gyrogroup structure on $\mathbb{D}$, we choose two elements $z_1, z_2 ∈ \mathbb{D}$ and define the Möbius addition by $z_1\oplus z_2 = \frac{z_1+z_2}{1+\bar{z_1}z_2}$. Then $(\mathbb{D},\oplus)$ is a gyrocommutative gyrogroup. If we define $r \odot x$ $=$ $\frac{(1+|x|)^r - (1-|x|)^r}{(1+|x|)^r + (1-|x|)^r}\frac{x}{|x|}$, where $x ∈ \mathbb{D}$ and $r ∈ \mathbb{R}$, then $(\mathbb{D},\oplus,\odot)$ will be a real gyrovector space. This paper aims to survey the main properties of these Möbius gyrogroup and Möbius gyrovector space.
Group theoretical derivation of consistent particle theories

Giuseppe Nisticò

SciPost Phys. Proc. 14, 040 (2023) · published 24 November 2023 | · pdf

Current quantum theories of an elementary free particle assume unitary space inversion and anti-unitary time reversal operators. In so doing robust classes of possible theories are discarded. The present work shows that consistent theories can be derived through a strictly deductive development from the principle of relativistic invariance and position covariance, also with anti-unitary space inversion and unitary time reversal operators. In doing so the class of possible consistent theories is extended for positive but also zero mass particles. In particular, consistent theories for a Klein-Gordon particle are derived and the non-localizability theorem for a non zero helicity massless particle is extended.

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Publications

3 citations