Session Y16: Advances in Particle Theory

Generalized Pure Density Matrices and the Standard Model

We consider generalizations of pure density matrices that have $\rho\rho=\rho$, but give up the trace=1 requirement. Given a representation of a quantum algebra in $N\times N$ complex matrices, the elements that satisfy $\rho\rho=\rho$ can be taken to be pure density matrix states. In the Standard Model, particles from different ``superselection sectors'' cannot form linear superpositions. For example, it is impossible to form a linear superposition between an electron and a neutrino. We report that some quantum algebras give symmetry, particle and generation content, gauge freedom, and superselection sectors that are similar to those of the Standard Model. Our lecture will consider as an example the $4\times 4$ complex matrices. There are 16 that are diagonal with $\rho\rho=\rho$. The 4 with trace=1 give the usual pure density matrices. We will show that the 6 with trace=2 form an $SU(3)$ triplet of three superselection sectors, with each sector consisting of an $SU(2)$ doublet. Considering one of these sectors, the mapping to $SU(2)$ is not unique; there is an $SU(2)$ gauge freedom. This gauge freedom is an analogy to the $U(1)$ gauge freedom that arises when converting a pure density matrix to a state vector.   

Symmetric blocking and renormalization in lattice N=4 super Yang-Mills

The form of the long distance effective action of the twisted lattice ${\cal N}=4$ super Yang-Mills theory depends on having a real space renormalization group transformation that preserves the original lattice properties, both the symmetries and the geometric interpretation of the fields. We have found such a transformation and have exhibited its behavior through a preliminary Monte Carlo renormalization group calculation. Other results regarding the number of counterterms are also obtained by considering rescalings of the lattice fields.   

Super-Adiabatic Particle Number in Schwinger and de Sitter Particle Production

We consider the time evolution of the adiabatic particle number in both time-dependent electric fields and in de Sitter spaces, and define a super-adiabatic particle number in which the (divergent) adiabatic expansion is truncated at optimal order. In this super-adiabatic basis, the particle number evolves smoothly in time, according to Berry's universal adiabatic smoothing of the Stokes phenomenon. This super-adiabatic basis also illustrates clearly the quantum interference effects associated with particle production, in particular for sequences of time-dependent electric field pulses, and in eternal de Sitter space where there is constructive interference in even dimensions, and destructive interference in odd dimensions.   

The Electron is a Charged Photon

The Dirac equation's relativistic electron is modeled as a helically-circulating charged photon whose helical radius at low electron speeds is the Dirac equation's electron amplitude $hbar/2mc$. The helically-circulating charged photon's longitudinal or $z$-component of velocity equals the velocity of the electron. The electron's relativistic energy-momentum equation $E^{2}=p^{2}c^{2}+m^{2}c^{4}$ corresponds the helically-circulating charged photon's energy $E=\gamma mc^{2}=h\nu $ with the charged photon's total momentum $p_{total} =\gamma mc$, its longitudinal momentum component $p=\gamma mv$ (the electron's linear momentum) and its transverse momentum component $p_{trans} =mc$. The charged photon's circulating transverse momentum component $p_{trans} =mc$, acting at the charged photon's helical radius $hbar/2mc$, generates the spin-up and spin-down $z$-components $\pm hbar/2$ of a slowly-moving electron's spin. The relativistic de Broglie wavelength $h/\gamma mv$ of the electron is easily calculated from the longitudinal component of the circulating charged-photon's wave vector $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over {k}} $.   

Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry

Connes' non-commutative geometry (NCG) is a generalization of Riemannian geometry that is particularly apt for expressing the standard model of particle physics coupled to Einstein gravity. Recently, we suggested a reformulation of this framework that is: (i) simpler and more unified in its axioms, and (ii) allows the Lagrangian for the standard model of particle physics (coupled to Einstein gravity) to be specified in a way that is tighter and more explanatory than the traditional algorithm based on effective field theory. Here we explain how this same reformulation yields a new perspective on the symmetries of a given NCG. Applying this perspective to the NCG traditionally used to describe the standard model we find, instead, an extension of the standard model by an extra $U(1)_{B-L}$ gauge symmetry, and a single extra complex scalar field $\sigma$, which is a singlet under $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$, but has $B-L=2$. This field has cosmological implications, and offers a new solution to the discrepancy between the observed Higgs mass and the NCG prediction.   

The Vacuum in Light Front Field Theory

In the light-front formulation of quantum field theory, one finds that the interacting vacuum and the free-field vacuum are both the same trivial Fock vacuum. This stands in contrast to the more usual equal time formulation, where the interacting vacuum and the free vacuum have a complicated relationship. To examine this apparent inconsistency, we first focus on free-fields with two distinct masses. The characterization of the vacuum by annihilation operators is incomplete, and leads to an apparent contradiction concerning the creation and annihilation operators of the two theories. Alternatively, the vacuum can be considered as a positive linear functional on an operator algebra generated by the field. In this characterization, the definition of the vacuum depends on the choice of algebra. The physically relevant algebra should be Poincare invariant and contain local observables. Extending the light-front algebra to this local algebra provides a resolution to the apparent inconsistency, but allows one to still use the Fock vacuum. These results can then be applied to interacting theories.   

Is a generalized NJL model the effective action of massless QCD?

A local and gauge invariant alternative version of QCD for massive fermions, which was proposed in a previous work, will be presented. It will be underlined that its action includes new vertices which eventually could had been overlooked before, because at first sight, they seem as breaking power counting renormalizability. However, the fact that these terms also modify the quark propagators, to become more convergent at large momenta, strongly suggests that theory is renormalizable. Accepting this view, surprisingly, it follows that all the four fermions terms constituting the Nambu-Jona-Lasinio models, can be included as counterterms in a slightly generalized renormalization procedure for massless QCD.   

Simulation of Relativistic Open Quantum Systems: Antiparticle production in the presence of an environment

We present general strategies for simulating spin 1/2 relativistic open quantum systems interacting with an environment in addition to an external electromagnetic field. We illustrate our approach by simulating the production of antiparticles in the presence of quantum decoherence and energy dissipation. Important observations are made concerning the zitterbewegung effect and the total production of antiparticles.   

Three Higgs-related predictions, including $Z^0 \rightarrow $ new spin 1/2 particles

A fundamental statistical picture that was proposed earlier is shown to lead to three predictions for scalar bosons $\phi _{b}$ that may be testable in the foreseeable future, perhaps at a 13 TeV LHC. The first is a modification of the propagators, and consequently of cross sections involving virtual processes. The second is an unrenormalized value near zero for the self-coupling coefficient $\lambda_b $. The third is an extra term in the Lagrangian with the form \begin{eqnarray} {\mathcal L}_{\chi }= - \chi _{b}^{\dag } \, \phi _{b}^{\dag }\left( x\right) \sigma ^{k} B_{k} \, \phi _{b}\left( x\right) \chi _{b} \quad , \quad B_{k}=-\frac{1}{2}F_{k^{\prime }k^{\prime \prime }}\epsilon ^{k^{\prime }k^{\prime \prime }}\,_{k} \quad , \quad k=1,2,3 \; . \nonumber \end{eqnarray} With $\chi _{b}$ taken to transform as a spinor, this term is invariant under a rotation, but not under a Lorentz boost. There is then a violation of Lorentz invariance, associated with the new ``spinon'' field $\chi _{b}$ and isolated in the term involving this field. This term also predicts new spin 1/2 particles which can be produced in pairs. For example, a sufficiently energetic Z$^0$ can decay to 2 Higgs-related spinons in the presence of a neutral Higgs condensate.