Elementary Retracements (spin 14)

In this work we present a matrix generalization of the Euler identity about exponential representation of a complex number. The concept of matrix exponential is used in a fundamental way. We define a notion of matrix imaginary unit which generalizes the usual complex imaginary unit. The Euler-like identity so obtained is compatible with the classical one. Also, we derive some exponential representation for matrix real and imaginary unit, and for the first Pauli matrix

The Lorentz group consists, unsurprisingly, of the Lorentz transformations, which are the linear transformations preserving the Minkowski dot product. Equivalently, they are the linear transformations fixing that hyperboloid of two sheets. If we discard one of the sheets, we obtain the orthochronous (time-preserving) subgroup.

• From the perspective of the centre of the cone, the hyperboloid looks like an open disc. The orthochronous Lorentz transformations precisely correspond to distance-preserving transformations of the hyperbolic plane. These are themselves determined uniquely by a conformal (or anticonformal) transformation of the ‘circle at infinity’.
• Adding an extra dimension, the orthochronous Lorentz group O^{+}(3,1) is isomorphic to the group of distance-preserving transformations of hyperbolic 3-space, which is again isomorphic to the group of (anti-)conformal transformations of the ‘sphere at infinity’, namely our index-2 supergroup of the Möbius group.

Moreover, this nicely generalises: the group generated by geometric inversions on the n-sphere is abstractly isomorphic to the orthochronous Lorentz group O^{+}(n+1,1). And when n = 24, we get a very beautiful discrete subgroup, namely the automorphism group of the II(25,1) lattice intimately related to the Leech lattice. (Complex Projective 4-Space)

The Lie group structure of the Lorentz group is explored. Its generators and its Lie algebra are exhibited, via the study of infinitesimal Lorentz transformations.

• The exponential map is introduced and it is shown that the study of the Lorentz group can be reduced to that of its Lie algebra.
• Finally, the link between the restricted Lorentz group and the special linear group is established via the spinor map.

The Lie algebras of these two groups are shown to be identical (up to some isomorphism).

The four pairwise disjoint and non-compact connected components of the Lorentzgroup L = O(1, 3) and corresponding subgroups:

• the proper Lorentz group L+ = SO(1, 3),
• the orthochronous Lorentz group L↑,
• the orthochronous Lorentz group Lo = L↑ + ∪ TL↑+ (see below) and
• the proper orthochronous Lorentz group L↑+ = SO+(1, 3), which contains the identity element.

Of course, the sets L↓−, L↑− and L↓+ do not represent groups due to the missing identity element. ([The-four-pairwise-disjoint)

SU(5) fermions of standard model in 5+10 representations. The sterile neutrino singlet’s 1 representation is omitted. Neutral bosons are omitted, but would occupy diagonal entries in complex superpositions. X and Y bosons as shown are the opposite of the conventional definition

The color strong force is the strong interaction between the three quarks that a proton or neutron is made of.

• It is called the color strong force because, like the electromagnetic force, the strong force has charges.
• The electromagnetic force has only one type of charge, which can be either positive or negative (magnetic charges are just slow-moving electric charges), but the strong force has three types.
• These three types of charges are named after colors: red, green, and blue. They also have anti-colors: anti-red, anti-green and anti-blue. Like the electromagnetic force’s positive and negative charges, different colors attract, and the same colors repel. Some particles that have color charge are quarks and antiquarks.
• The type of quark is not related to that quark’s color charge at all. Quarks are one of the smallest particles currently known. They take up no space because they are points, and they are the only particles that we have not been able to break apart from other particles yet. This is because the nature of the strong force between particles is that it becomes stronger the further away the particles are.

The force carrier of the strong force is called the gluon. Gluons also have color charge. Both quarks and gluons have properties that make them unique from other particles, as described in the Standard Model. (Wikipedia).

Shortly after the existence of quarks was proposed by Murray Gell-Mann and George Zweig in 1964, Moo-Young Han and Yoichiro Nambu introduced a hidden internal degree of freedom in which quark wave functions were antisymmetric, thus solving the spin-statistics problem of the Gell Mann-Zweig quark model.

• Han and Nambu initially designated this degree of freedom by the group SU(3)’, but it was referred to in later papers as “the three triplet model.” One feature of the model (which was originally preferred by Han and Nambu) was that it permitted integrally charged quarks, as well as the fractionally charged quarks initially proposed by Zweig and Gell-Mann.
• Somewhat later, in the early 1970s, Gell-Mann, in several conference talks, coined the name “Color” to describe the internal degree of freedom of the three triplet model, and advocated a new field theory, designated as “Quantum Chromodynamics” (QCD) to describe the interaction of quarks and gluons within hadrons. In Gell-Mann’s QCD, each quark and gluon had fractional electric charge, and carried what came to be called “Color Charge” in the space of the Color degree of freedom.In quantum chromodynamics (QCD), a quark’s color can take one of three values or charges: red, green, and blue. An antiquark can take one of three anticolors: called antired, antigreen, and antiblue (represented as cyan, magenta, and yellow, respectively). Gluons are mixtures of two colors, such as red and antigreen, which constitutes their color charge. QCD considers eight gluons of the possible nine color–anticolor combinations to be unique; see eight gluon colors for an explanation.
• All three colors mixed together, or any one of these colors and its complement (or negative), is “colorless” or “white” and has a net color charge of zero. Due to a property of the strong interaction called color confinement, free particles must have a color charge of zero.
• A baryon is composed of three quarks, which must be one each of red, green, and blue colors; likewise an antibaryon is composed of three antiquarks, one each of antired, antigreen and antiblue. A meson is made from one quark and one antiquark; the quark can be any color, and the antiquark has the matching anticolor.

The following illustrates the coupling constants for color-charged particles. In a quantum field theory, a coupling constant and a charge are different but related notions. The coupling constant sets the magnitude of the force of interaction; for example, in quantum electrodynamics, the fine-structure constant is a coupling constant. (Wikipedia)

In physics, and specifically in quantum field theory, a bispinor is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons.

• It is a specific embodiment of a spinor, specifically constructed so that it is consistent with the requirements of special relativity.
• Bispinors transform in a certain “spinorial” fashion under the action of the Lorentz group, which describes the symmetries of Minkowski spacetime.
• They occur in the relativistic spin-1/2 wave function solutions to the Dirac equation.
• Bispinors are so called because they are constructed out of two simpler component spinors, the Weyl spinors. Each of the two component spinors transform differently under the two distinct complex-conjugate spin-1/2 representations of the Lorentz group.
• This pairing is of fundamental importance, as it allows the represented particle to have a mass, carry a charge, and represent the flow of charge as a current, and perhaps most importantly, to carry angular momentum.
• More precisely, the mass is a Casimir invariant of the Lorentz group (an eigenstate of the energy), while the vector combination carries momentum and current, being covariant under the action of the Lorentz group.
• The angular momentum is carried by the Poynting vector, suitably constructed for the spin field.[1]
• A bispinor is more or less “the same thing” as a Dirac spinor. The convention used here is that the article on the Dirac spinor presents plane-wave solutions to the Dirac equation using the Dirac convention for the gamma matrices. That is, the Dirac spinor is a bispinor in the Dirac convention.
• Bispinors are elements of a 4-dimensional complex vector space (1/2, 0) ⊕ (0, 1/2) representation of the Lorentz group.

Dirac bispinor 6D shows eight (8) quantum spin eigenstates in six (6) dimensions of complex spacetime: 0 (the Higgs field), ±½ (fermions), ±1 (bosons), ±⅔ (anti-fermions), 2 (graviton). Top-left Minkowski diagram displays 6D spacetime curvature. Bottom-right projection displays the 2 orthogonal sinusoids of the Dirac harmonic oscillator, and their phase offsets.

They are the imaginary time versions of statistical mechanics partition functions, giving rise to a close connection between these two areas of physics. Partition functions can rarely be solved for exactly, although free theories do admit such solutions. Instead, a perturbative approach is usually implemented, this being equivalent to summing over Feynman diagrams. (Wikiwand)

In this work, we propose a new route to realizing flat band physics in monolayer graphene under a periodic modulation from substrates.

• We take gaphene/SiC heterostructure as a role model and demonstrate experimentally the substrate modulation leads to Dirac fermion cloning and consequently, the proximity of the two Dirac cones of monolayer graphene in momentum space.
• Our theoretical modeling captures the cloning mechanism of Dirac states and indicates that flat bands can emerge at certain magic lattice constants of substrate when the period of modulation becomes nearly commensurate with the (√3 ×√3)R30◦ supercell of graphene.

The results show that the epitaxial monolayer graphene is a promising platform for exploring exotic many-body quantum phases arising from interactions between Dirac electrons. (Dirac Fermion Cloning - pdf)

The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the “mass gap”: the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. (Clay Institute)

And we believe this discovery will have important implications in the knowledge and lives of human beings. For example, we live in a universe full of matter now, but the Big Bang created both matter and antimatter. (Quantized signature of majorana)

Yang–Mills Existence and Mass Gap: Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R^4 and has a mass gap Δ > 0. (In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.) (Wikipedia)