Spinors of Spin-one-half Fields

This paper reviews how a two-state, spin-one-half system transforms under rotations. It then uses that knowledge to explain how momentum-zero, spin-one-half annihilation and creation operators transform under rotations. The paper then explains how a spin-one-half field transforms under rotations. The momentum-zero spinors are found from the way spin-one-half systems transform under rotations and from the Dirac equation. Once the momentum-zero spinors are known, the Dirac equation immediately yields the spinors at finite momentum. The paper then shows that with these spinors, a Dirac field transforms appropriately under charge conjugation, parity, and time reversal. The paper also describes how a Dirac field may be decomposed either into two 4-component Majorana fields or into a 2-component left-handed field and a 2-component right-handed field. Wigner rotations and Weinbergs derivation of the properties of spinors are also discussed.

Is the local Lorentz invariance of general relativity implemented by gauge bosons that have their own Yang-Mills-like action?

General relativity with fermions has two independent symmetries: general coordinate invariance and local Lorentz invariance. General coordinate invariance is implemented by the Levi-Civita connection and by Cartans tetrads both of which have as their action the Einstein-Hilbert action. It is suggested here that local Lorentz invariance is implemented not by a combination of the Levi-Civita connection and Cartans tetrads known as the spin connection, but by independent Lorentz bosons that gauge the Lorentz group, that couple to fermions like Yang-Mills fields, and that have their own Yang-Mills-like action. Because the Lorentz bosons couple to fermion number and not to mass, they generate a static potential that violates the weak equivalence principle. If a Higgs mechanism makes them massive, then the static potential also violates the inverse-square law. Experiments put upper bounds on the strength of such a potential for masses less than ~20 eV. These upper limits imply that Lorentz bosons, if they exist, are nearly stable and contribute to dark matter.

Flat Space, Dark Energy, and the Cosmic Microwave Background

This paper reviews some of the results of the Planck collaboration and shows how to compute the distance from the surface of last scattering, the distance from the farthest object that will ever be observed, and the maximum radius of a density fluctuation in the plasma of the CMB. It then explains how these distances together with well-known astronomical facts imply that space is flat or nearly flat and that dark energy is 69% of the energy of the universe.

Zero-point energies, dark matter, and dark energy

A quantum field theory has finite zero-point energy if the sum over all boson modes $b$ of the $n$th power of the boson mass $ m_b^n $ equals the sum over all fermion modes $f$ of the $n$th power of the fermion mass $ m_f^n $ for $n= 0$, 2, and 4. The zero-point energy of a theory that satisfies these three conditions with otherwise random masses is huge compared to the density of dark energy. But if in addition to satisfying these conditions, the sum of $m_b^4 log m_b/mu$ over all boson modes $b$ equals the sum of $ m_f^4 log m_f/mu $ over all fermion modes $f$, then the zero-point energy of the theory is zero. The value of the mass parameter $mu$ is irrelevant in view of the third condition ($n=4$). The particles of the standard model do not remotely obey any of these four conditions. But an inclusive theory that describes the particles of the standard model, the particles of dark matter, and all particles that have not yet been detected might satisfy all four conditions if pseudomasses are associated with the mean values in the vacuum of the divergences of the interactions of the inclusive model. Dark energy then would be the finite potential energy of the inclusive theory.

Path integrals for awkward actions

Time derivatives of scalar fields occur quadratically in textbook actions. A simple Legendre transformation turns the lagrangian into a hamiltonian that is quadratic in the momenta. The path integral over the momenta is gaussian. Mean values of operators are euclidian path integrals of their classical counterparts with positive weight functions. Monte Carlo simulations can estimate such mean values. This familiar framework falls apart when the time derivatives do not occur quadratically. The Legendre transformation becomes difficult or so intractable that one cant find the hamiltonian. Even if one finds the hamiltonian, it usually is so complicated that one cant path-integrate over the momenta and get a euclidian path integral with a positive weight function. Monte Carlo simulations dont work when the weight function assumes negative or complex values. This paper solves both problems. It shows how to make path integrals without knowing the hamiltonian. It also shows how to estimate complex path integrals by combining the Monte Carlo method with parallel numerical integration and a look-up table. This Atlantic City method lets one estimate the energy densities of theories that, unlike those with quadratic time derivatives, may have finite energy densities. It may lead to a theory of dark energy. The approximation of multiple integrals over weight functions that assume negative or complex values is the long-standing sign problem. The Atlantic City method solves it for problems in which numerical integration leads to a positive weight function.

The eras of radiation, matter, and dark energy: new information from the Planck Collaboration

Data released by the Planck Collaboration in 2015 imply new dates for the era of radiation, the era of matter, and the era of dark energy. The era of radiation ended, and the era of matter began, when the density of radiation dropped below that of matter. This happened 50,953 pm 2236 years after the time of infinite redshift when the ratio a(t)/a_0 of scale factors was (2.9332 pm 0.0711) x 10^{-4}. The era of matter ended, and the era of dark energy began, when the density of matter dropped below that of dark energy (assumed constant). This happened 10.1928 pm 0.0375 Gyr after the time of infinite redshift when the scale-factor ratio was 0.7646 pm 0.0168. The era of dark energy started 3.606 billion years ago. In this pedagogical paper, five figures trace the evolution of the densities of radiation and matter, the scale factor, and the redshift through the eras of radiation, matter, and dark energy.

An extension of the standard model in which parity is conserved at high energies

To be compatible with general relativity, every fundamental theory should be invariant under general coordinate transformations including spatial reflection. This paper describes an extension of the standard model in which the action is invariant under spatial reflection, and the vacuum spontaneously breaks parity by giving a mean value to a pseudoscalar field. This field and the scalar Higgs field make the gauge bosons, the known fermions, and a set of mirror fermions suitably massive while avoiding flavor-changing neutral currents. In the model, there is no strong-CP problem, there are no anomalies, fermion number (quark-plus-lepton number) is conserved, and heavy mirror fermions form heavy neutral mirror atoms which are dark-matter candidates. In models with extended gauge groups, nucleons slowly decay into pions, leptons, and neutrinos.

Path integrals for actions that are not quadratic in their time derivatives

The standard way to construct a path integral is to use a Legendre transformation to find the hamiltonian, to repeatedly insert complete sets of states into the time-evolution operator, and then to integrate over the momenta. This procedure is simple when the action is quadratic in its time derivatives, but in most other cases Legendres transformation is intractable, and the hamiltonian is unknown. This paper shows how to construct path integrals when one cant find the hamiltonian because the first time derivatives of the fields occur in ways that make a Legendre transformation intractable; it focuses on scalar fields and does not discuss higher-derivative theories or those in which some fields lack time derivatives.

New dates for the eras of the universe from the Planck data

Data from the Planck satellite imply new dates for the major eras of the universe. The era of radiation ended 50,150 years after inflation and the era of matter 10.31 billion years after inflation or 3.51 billion years ago.

Some nonrenormalizable theories are finite

Some nonrenormalizable theories are less singular than all renormalizable theories, and one can use lattice simulations to extract physical information from them. This paper discusses four nonrenormalizable theories that have finite euclidian and minkowskian Greens functions. Two of them have finite euclidian action densities and describe scalar bosons of finite mass. The space of nonsingular nonrenormalizable theories is vast.