### Conveners

#### Mathematical Problems of Relativistic Physics: Classical and Quantum: Block 1

- Michael Kiessling (Rutgers University, Dept. of Mathematics)
- A. Shadi Tahvildar-Zadeh (Rutgers University)

#### Mathematical Problems of Relativistic Physics: Classical and Quantum: Block 2

- Michael Kiessling (Rutgers University, Dept. of Mathematics)
- A. Shadi Tahvildar-Zadeh (Rutgers University)

### Description

We will survey recent advances in mathematical analysis of relativistic and semi-relativistic phenomena, including:

1. Joint classical and quantum evolution of charged point particles and fields in special and general relativity;

2. Dirac's equation on electromagnetic background spacetimes;

3. Schroedinger-Newton equation and bosonic stars;

4. Interacting photon-electron systems in Dirac's multi-time formalism;

5. The ground state of Positronium as an ultralight spin-zero boson and its application to the dark matter puzzle;

6. Divison-algebraic underpinnings of the Standard Model of Elementary Particles.

In relativistic quantum mechanics, the point spectrum of the Dirac Hamiltonian with Coulomb potential famously agrees with Sommerfeld's fine structure formula for Hydrogen. In the Coulomb approximation, the proton is assumed to only have an electric charge. However, the physical proton also appears to have a magnetic moment. The resulting hyperfine structure of Hydrogen is computed...

The second Bianchi identity is a well-known and fundamental differential identity which holds on any smooth (semi-)Riemannian manifold. In general relativity, due to the relation of the curvature tesnor and the energy-momentum tensor via the Einstein equations, this identity then naturally implies energy and momentum conservation for matter fields. What happens in situations where curvature...

Arrival-time operators (or observables) describing time-of-flight experiments are naturally constrained by gauge invariance requirements. Surveying the literature on time operators, including POVMs, I will show that a natural generalization of Aharonov-Bohm-Kijowski's arrival-time distribution (referred to as the ``standard arrival-time distribution'' by some authors) fails to be gauge...

The theory of causal fermion systems is an approach to fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. The dynamics of causal fermion systems is described by a variational principle called the causal action principle (for more details see...

Can the 32C-dimensional algebra R(x)C(x)H(x)O offer anything new for particle physics? Indeed it can. Here we identify a sequence of complex structures within R(x)C(x)H(x)O which induces a cascade of breaking symmetries: Spin(10) -> Pati-Salam -> Left-Right symmetric -> Standard model + B-L (both pre- and post-Higgs-mechanism). These complex structures derive from the octonions, then from the...

Physical reasoning give expressions for the Hamiltonian of a system. These Hamiltonians are differential operators that are mostly symmetric in a densely defined domain.

However, to study the dynamics of the unitary group corresponding to a Hamiltonian, it is

required that the Hamiltonian be self-adjoint or essentially self-adjoint. I will present our study

on how the static non-linear...

We give a lower bound for the ADM mass of 3-dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth `spacetime harmonic functions' in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary is a new proof...

My goal in this talk is to address some of the fundamental mathematical questions in the field of relativistic dissipative fluid dynamics. This is an area that has witnessed progress within the physics community but for which many foundational mathematical questions remain open. Some of these problems, such as the study of causality, local well-posedness and breakdown of solutions, are...

In this talk, we discuss the existence of a static, spherically symmetric spacetime that is the solution of the Einstein field equations coupled with an electric field obeying the equations of electromagnetism of Maxwell-Bopp-Lande-Thomas-Podolsky for a static point charge. Contrary to what happens with the Reissner-Nordstrom spacetime, it is shown that the electric field energy is finite,...

Élie Cartan’s invariant integral formalism is extended to gauge field theory, including general relativity. This constitutes an alternative procedure that is equivalent to the Rosenfeld, Bergmann, Dirac algorithm. In addition, a Hamilton-Jacobi formalism is developed for constructing explicit phase space functions in general relativity that are invariant under the full four-dimensional...

Bondi's celebrated mass loss formula measures the rate of change of energy carried away from an isolated system (in asymptotically flat space-time) by gravitational radiation. In this talk, we generalize this idea to the de Sitter setting. We derive a formula for the total canonical energy, and its flux, of weak gravitational waves on a de Sitter background. Based on arXiv:2003.09548 [gr-qc],...

An optical medium can be represented by a Riemannian manifold $(\mathcal{B}, g)$ where $\mathcal{B}$ is consider to be the physical space and $g$ the optical spatial metric. A geodesic flow in the unitary tangent bundle can be represented by a contact transformation in the space of contact elements. This fact, allows us to describe the wavefront evolution in an optical medium solely in terms...

Our talk will start with a (hopefully) pedagogical introduction to the

topic of asymptotically de Sitter spacetimes. Afterward, we will

construct 'conserved' charges a'la Wald and Zoupas at a conformal

infinity and prove their uniqueness under a natural set of

assumptions. We will finish with a small comment on how to distinguish

the de Sitter group within a group of all asymptotic...

We use Weinberg’s trick for adiabatic modes in a Manton approximation for general relativity on manifolds with spatial boundary. This results in a description of the time dependent solutions as null geodesics on the space of boundary diffeomorphisms, with respect to **a metric we prove to be composed solely of the boundary data**. We show how the solutions in the bulk space is determined with...

In this talk I will discuss a systematic and rigorous classification of all the possible choices for averaging observables in cosmology. In this regard, the use of the so-called Geodesic Light-cone gauge provides simple expressions as I will show here. These new results will be compared with the recent literature. Moreover, I will discuss their impact on the bias that they can induce in the...

We show that spinors propagating in curved gravitational background

acquire an interaction with spacetime curvature, which leads to a

quantum mechanical geometric effect. This is similar to what happens

in the case of magnetic fields, known as the Pancharatnam-Berry phase.

As the magnetic and gravitational fields have certain similar

properties, e.g. both contribute to curvature, this...

Using Newman-Penrose formalism in tetrad and spinor notation, we perform separation of variables in the wave equations for massless fields of various spins s=1/2, 1, 3/2, 2 on the background of exact plane-fronted gravitational wave metrics. Then, applying Wald's method of adjoint operators, we derive equations for Debye potentials generating these fields and find inverse projection operators...

Free massless fields of any spin in flat D-dimensional spacetime propagate at the speed of light. But the retarded fields produced by the corresponding point-like moving sources share this property only for even D. Since the Green’s functions of the d’Alembert equation are localized on the light cone in even-dimensional spacetime, but not in odd dimensions, extraction of the emitted part of...

Orientability is an important topological property of spacetime manifolds.

It is generally assumed that a test for spatial orientability requires a journey

across the whole 3-space to check for orientation-reversing paths.

Since such a global expedition is not feasible, theoretical arguments that combine

universality of physical experiments with local arrow of time, CP violation and...

Gravitational shockwaves are simple exact solutions of Einstein equations representing the fields of ultrarelativistic sources and idealized gravitational waves (shocks). Historically, much work has focused on shockwaves in the context of possible black hole formation in high energy particle collisions, yet they remain at the forefront of research even today. Representing hard modes in the...

In a brief review, we draw attention to the “hypercomplex medium” with the accent on quaternion (Q) algebra and contiguous areas: biquaternion (BQ) numbers, Q-spinors, and related groups. Due to Heaviside-&-Gibbs’ vector algebra, this “Q-set” was nearly abandoned in XX century rarely emerging as a math tool [1]. However, it turns out to contain many geometric images and equations related to...