The ABC of Physics

Why quantum? Why spacetime? We find that the key idea underlying both is uncertainty. In a world lacking probes of unlimited delicacy, our knowledge of quantities is necessarily accompanied by uncertainty. Consequently, physics requires a calculus of number pairs and not only scalars for quantity alone. Basic symmetries of shuffling and sequencing dictate that pairs obey ordinary component-wise addition, but they can have three different multiplication rules. We call those rules A, B and C. “A” shows that pairs behave as complex numbers, which is why quantum theory is complex. However, consistency with the ordinary scalar rules of probability shows that the fundamental object is not a particle on its Hilbert sphere but a stream represented by a Gaussian distribution. “B” is then applied to pairs of complex numbers (qubits) and produces the Pauli matrices for which its operation defines the space of four vectors. “C” then allows integration of what can then be recognised as energy-momentum into time and space. The picture is entirely consistent. Spacetime is a construct of quantum and not a container for it.

Novel method for one-way local distinguishability of generalized Bell states in arbitrary dimension

In this paper the local distinguishability of generalized Bell states in arbitrary dimension is investigated. We firstly study the decomposition of a basis which consists of $d^{2}$ number of generalized Pauli matrices. We discover that this basis is equal to the union of $D$ number of different sets, where $D=\frac{2}{\phi(d)}\sum_{t\in \mathbb{Z}_{d} \atop gcd(t,d)=1}\sum_{i=2}^{\lfloor\frac{d}{t}\rfloor}\phi(i)+1$ and $\phi$ is Euler $\phi$-function. Then we define the generator of the matrices in this decomposition, and exhibit an algorithm to calculate generators of a given set of matrices. This algorithm shows that generators of a given set can be calculated simply and efficiently. Secondly, we show that a set $\mathcal {L}$ of GBSs can be distinguished by one-way LOCC if the cardinality of $\mathcal {G}_{\mathcal {L}}$ is less than $D\phi(d)$, where $\mathcal {G}_{\mathcal {L}}$ is a set of generators of all the elements in difference set of a set $\mathcal {L}$ of GBSs. The previous results in [2004 Phys. Rev. Lett. \textbf{92} 177905; 2019 Phys. Rev. A \textbf{99} 022307; 2021 Quant. Info. Proc. \textbf{20} 52] can be covered by our result. Finally, for the uncovered cases in [2021 Quant. Info. Proc. \textbf{20} 52], we give a new result to partly solve that problem.

Strong 1-boundedness of unimodular orthogonal free quantum groups

Recently, Brannan and Vergnioux showed that the orthogonal free quantum group factors [Formula: see text] have Jung’s strong [Formula: see text]-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in [Formula: see text] dimensions [Formula: see text]. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in [Formula: see text]-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a [Formula: see text]-bounded set without losing [Formula: see text]-boundedness. In particular, this allows us to include the character of the fundamental representation, proving strong [Formula: see text]-boundedness.

Two-State Systems: The Atomic Operators

In the digital world, the concepts of on and off or high and low or 0 and 1 are common classical two-state systems. Quantum systems can be similarly configured, as we saw in Chapter 9 with the demonstration of Rabi oscillations. Two-state or few-state systems are so important that a powerful algebra has been developed to study and explore these systems. A similar algebra emerged from the algebra developed for spin ½ particles. While Chapter 10 discussed the spinors and spin matrices and the corresponding Pauli matrices, in this chapter the corresponding commutators are determined for the various atomic operators first introduced in Chapter 15. We then move to the Heisenberg picture including the operators for the vacuum field. The Heisenberg equations of motion are derived following the rules in Chapter 8 when a classical electromagnetic field is present and then in the presence of the quantum vacuum to include the effects of decay. This provides the first means of handling the return of an excited population back to the ground state which is very challenging to deal with in the amplitude picture. This chapter is enormously important because it sets the stage for much more advanced studies in advanced texts that determine the impact of fluctuations of the field and correlations measured from single photon emitters.

Electron Spin and General Angular Momenta

The historical background of the discovery of the electron spin is provided. The Stern-Gerlach and Einstein-de Haas experiments are discussed. The operators for a single electron spin are defined, along with the formulation in terms of the 2x2 Pauli matrices. The discussion then moves on to the definition of the spin for many-electron systems and explains how the famous Hund rule (or Hund’s first rule) arises from considering the energy of an open-shell spin singlet vs. triplet state. Next, the generalized angular momentum, ladder operators, and spherical vector operators are defined, and the rules for the addition of angular momenta are derived. The chapter concludes with a discussion of the total spin, orbital, and total angular momentum for open-shell atoms, term symbols, and Hund’s second and third rule.

FeynOnium: using FeynCalc for automatic calculations in Nonrelativistic Effective Field Theories

We present new results on FeynOnium, an ongoing project to develop a general purpose software toolkit for semi-automatic symbolic calculations in nonrelativistic Effective Field Theories (EFTs). Building upon FeynCalc, an existing Mathematica package for symbolic evaluation of Feynman diagrams, we have created a powerful framework for automatizing calculations in nonrelativistic EFTs (NREFTs) at tree- and 1-loop level. This is achieved by exploiting the novel features of FeynCalc that support manipulations of Cartesian tensors, Pauli matrices and nonstandard loop integrals. Additional operations that are common in nonrelativistic EFT calculations are implemented in a dedicated add-on called FeynOnium. While our current focus is on EFTs for strong interactions of heavy quarks, extensions to other systems that admit a nonrelativistic EFT description are planned for the future. All our codes are open-source and publicly available. Furthermore, we provide several example calculations that demonstrate how FeynOnium can be employed to reproduce known results from the literature.

Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System

Here, we derive the Pauli Matrix Equivalent for Spin-1 particles (mainly Z-Boson and W-Boson). Pauli Matrices are generally associated with Spin- 1/2 particles and it is used for determining the properties of many Spin- 1/2 particles. But in our case, we try to expand its domain and attempt to implement it for calculating the Unitary Operators of the Harmonic oscillator involving the Spin-1 system and study it.