arXiv subjects

Richard Shurtleff

Publications and source records attributed to Richard Shurtleff.

31 records · Page 2Linked to original sources

arXiv2006

First-Order Field Equations in Spin 1/2 Form

From one point of view in the quantum theory of fields, free quantum fields are uniquely determined, not by field equations, but by the transformations of the field and the annihilation and creation operators from which the field is constructed. One says that a free field equation merely records the fact that some field components are superfluous. Here, free field equations that are first order and covariant are derived so that the already determined field is one solution. The unknowns are the vector matrices that combine with the known gradient of the field to make an invariant equation: the scalar product of the vector matrices and the gradient are proportional to the field. Thus these free field equations are direct consequences of the transformation properties of the annihilation and creation operators and the transformation properties of the field.

hep-th↗

arXiv2005

The Lorentz Force Law and Spacetime Symmetries

Assume that arc length is measured with the flat spacetime metric. Then, for the most general Poincare group representation for translating 4-vectors, curves with parallel translated tangent vectors must have accelerations that are the scalar product of the tangent vector with an antisymmetric tensor. Such curves are the paths of charged particles in an electromagnetic field.

math-ph↗

arXiv2005

Poincare Connections in Flat Spacetime

In flat spacetime, as a simple 4-vector, a particle's 4-velocity cannot be changed by translation. Parallel translation then produces constant velocity, motion without force. Here we consider a richer, but less well-known, representation of the Poincare group of symmetries of flat spacetime in which translation adds a non-linear connection term to the 4-vector. Then 4-vectors can be meaningfully parallel translated, curved geodesics can be developed, and the curvature for the connection can be found. By adding an assumption, that the arc length is calculated with a position dependent metric, it is shown that a geodesic has a 4-acceleration which is the sum of a Christoffel connection term and a term that is the scalar product of 4-velocity and an antisymmetric tensor. This is just like the force law obeyed by a charged, massive particle in general relativity. Thus the dynamical laws of electrodynamics and general relativity can be deduced as geodesic equations from the way particle proper time depends on position and the way 4-vectors can change upon translation in flat spacetime.

gr-qc↗

arXiv2004

Spin 1/2 and Invariant Coefficients II. Massless

A `covariant' field that transforms like a relativistic field operator is required to be a linear combination of `canonical' fields that transform like annihilation and creation operators and with invariant coefficients. The Invariant Coefficient Hypothesis contends that this familiar construction by itself yields useful results. Thus, just the transformation properties are considered here, not the specific properties of annihilation or creation operators. The results include Weyl wave equations for some massless fields and, for other fields, Weyl-like noncovariant wave equations that are allowed here because no assumptions are made to exclude them. The hypothesis produces wave equations for translation-matrix-invariant fields while translation-matrix-dependent coefficient functions have currents that are the vector potentials of the coefficient functions of those translation-matrix-invariant fields. The statement is proven by showing that Maxwell equations are satisfied, though in keeping with the hypothesis they are not assumed to hold. The underlying mechanism is the same for the massless class here as it is for the massive class in a previous paper, suggesting that spin 1/2 particles may have a universal electromagnetic-type charge whether they are massive or massless. Keywords: Relativistic quantum fields; neutrino; Poincare transformations

hep-th↗

arXiv2004

Massless Particle Fields, with Momentum Matrices

Nontrivial translation matrices occur for spin (A,B)+(C,D) with |A-C| = |B-D| = 1/2, necessarily associating a (C,D) field with a spin (A,B) field. Including translation matrices in covariant non-unitary Poincare representations also introduces new gauge terms in the construction of massless particle fields from canonical unitary fields. In the usual procedure without spacetime translation matrices, gauge terms arise from `translations' of the massless little group; the little group combines spacetime rotations and boosts making a group isomorphic with the Euclidean group E2, including E2 translations. The usual remedy is to invoke gauge invariance. But here, the spacetime translation gauge terms can cancel the little group gauge terms, trading the need for gauge invariance with the need to specify displacements and to freeze two little group degrees of freedom that are not wanted anyway. The cancelation process restricts the helicity to A-B-1 for A-C = -(B-D) = 1/2 and A-B+1 for A-C = -(B-D) = -1/2. However, the cancelation only works for the little group standard momentum and specific transformations and, in general, gauge invariance is still needed to obtain massless particle fields. Expressions for massless particle fields for each spin type are found.

hep-th↗

arXiv2000

Rotations and e, $ν$ Propagators, Part III

In Parts I and II we showed that e, $ν$ propagators can be derived from rotation invariant projection operators, thereby providing examples of how quantities with spacetime symmetry can be obtained by constraining rotationally symmetric objects. One constraint is the restriction of the basis; only two kinds of bases were considered, one for the electron and one for the neutrino. In this part, we find that, of a wide range of bases each consistent with the constraint process, only the two kinds of bases considered in Parts I and II give spacetime symmetric propagators. We interpret the result geometrically. The spinor representation is unfaithful in four dimensional Euclidean space which explains why spin 1/2 wave functions have four, not two, components. Then we show how a basis relates to two planes in four dimensional Euclidean space. A pair of planes spanning two or three dimensions does not allow spacetime symmetry. Spacetime symmetry requires two planes that span four dimensions. PACS: 11.30.-j, 11.30.Cp, and 03.65.Fd

hep-th↗

arXiv2000

Rotations and e, $ν$ Propagators, Part II

We continue to derive spacetime quantities and spin 1/2 propagators from rotations. Rotation-invariant projection operators are found for each element of a four element basis, i.e. a basis for four component quantities with specific transformation rules under rotations. With these four projection operators, we make two spacetime invariant projection operators, i.e. once space, time, energy, and momentum are identified. The spacetime invariant operators are propagators for free neutrinos. Except for the substitute basis, the process is the same as the one that gave electron propagators in Part I. PACS number(s): 11.30.-j, 11.30.Cp, and 03.65.Fd

hep-th↗

arXiv2000

Rotations and e, $ν$ Propagators, Part I

Rotation symmetry is less constraining than space-time symmetry. The free electron propagator is a projection operator that we show can be constructed from rotation symmetric projection operators. Rotation-based identifications of time, space, energy, momentum, polarization matrices, and the positron hypothesis are determined by the constraints that turn rotation symmetric projection operators into the electron propagator. PACS: 11.30.-j, 11.30.Cp, and 03.65.Fd

hep-th↗

arXiv2000

Neutrino Proper Time?

An electron neutrino can have the quantum phase of an electron, i.e. share its internal clock, if the neutrino takes a path in space-time that is not in the direction of its energy-momentum. Each flavor neutrino would then have a different internal clock; a muon neutrino would have a muon clock and a tau neutrino would have a tau clock. Perhaps surprisingly, there is some evidence suggesting neutrinos have such clocks. If muon neutrinos travel on space-like paths then some atmospheric muon neutrinos would take such paths backwards into outer space and not be observed. These are lost at the source and have nothing to do with oscillations or flavor-changing in flight. The expected depletion of source muon neutrinos is shown here to be 9%, which accounts for half of the missing muon neutrino source flux reported by Super-Kamiokande. Since there is no depletion in the electron neutrino flux source reported at SK and SN1987A electron neutrinos seem to have traveled at the speed of light, the electron neutrino travels on a light-like path. Accelerator-based experiments could be arranged to confirm the reverse motion of muon neutrinos.

hep-ph↗

arXiv1999

Rotation Representations and e, $pi$, p Masses

Mass is proportional to phase gain per unit time; for e, $π$, and p the quantum frequencies are 0.124, 32.6, and 227 Zhz, respectively. By explaining how these particles acquire phase at different rates, we explain why these particles have different masses. Any free particle spin 1/2 wave function is a sum of plane waves with spin parallel to velocity. Each plane wave, a pair of 2-component rotation eigenvectors, can be associated with a 2x2 matrix representation of rotations in a Euclidean space without disturbing the plane wave's space-time properties. In a space with more than four dimensions, only rotations in a 4d subspace can be represented. So far all is well known. Now consider that unrepresented rotations do not have eigenvectors, do not make plane waves, and do not contribute phase. The particles e, $π,$ and p are assigned rotations in a 4d subspace of 16d, rotations in an 8d subspace of 12d, and rotations in a 12d subspace of 12d, respectively. The electron 4d subspace, assumed to be as likely to align with any one 4d subspace as with any other, produces phase when aligned with the represented 4d subspace in 16d. Similarly, we calculate the likelihood that a 4d subspace of the pion's 8d space aligns with the represented 4d subspace in 12d. The represented 4d subspace is contained in the proton's 12d space, so the proton always acquires phase. By the relationship between mass and phase, the resulting particle phase ratios are the particle mass ratios and these are coincident with the measured mass ratios, within about one percent. 1999 PACS number(s): 03.65.Fd Keywords:Algebraic methods; particle masses; rotation group

quant-ph↗

arXiv1999

Rotation Eigenvectors and Spin 1/2

It is an easily deduced fact that any four-component spin 1/2 state for a massive particle is a linear combination of pairs of two-component simultaneous rotation eigenstates, where `simultaneous' means the eigenspinors of a given pair share the same eigenvalue. The new work here constructs the reverse: Given pairs of simultaneous rotation eigenvectors, the properties of these pairs contains relationships that are equivalent to spin 1/2 single particle equations. Thus the needed aspects of space-time symmetry can be produced as special cases of more general properties already present in the rotation group. The exercise exploits the flexibility of the rotation group in three dimensions to deduce relativistic quantities in four dimensions.

quant-ph↗

arXiv1999

Collapse and the Tritium Endpoint Pileup

The beta-decay of a tritium nucleus produces an entangled quantum system, a beta electron, a helium nucleus, and an antineutrino. For finite collapse times, the post-collapse beta electron energy can originate from a range of pre-collapse energies due to the uncertainty principle. Long collapse times give negligible uncertainty, so the pre-collapse spectrum must approach that of isolated nuclei. We calculate the post-collapse electron spectrum which shows a collapse-dependent pileup near the endpoint. Comparison with observation shows that a collapse time of 1 x 10^-17 s explains the observed pileup. The collapse of the entangled quantum system must be triggered by the environment: most likely an atomic (molecular) electron initially bound to the atomic (molecular) tritium source or perhaps ambient gas molecules. Coincidentally, the 40 eV tritium atom-helium ion energy level shift is unobservably small for times shorter than the system collapse time. We conclude that an atomic (molecular) electron triggers the collapse once the 40 eV shift becomes detectible and the electron detects the helium nucleus. Thus collapse may explain the tritium endpoint pileup.

nucl-th↗

arXiv1999

Four-Spinor Reference Sheets

Some facts about 4-spinors listed and discussed. None, well perhaps some, of the work is original. However, locating formulas in other places has proved a time-consuming process in which one must always worry that the formulas found in any given source assume the other metric (I use {-1,-1,-1,+1}) or assume some other unexpected preconditions. Here I list some formulas valid in general representations first, then formulas using a chiral representation are displayed, and finally formulas in a special reference frame (the rest frame of the `current' j) in the chiral representation are listed. Some numerical and algebraic exercises are provided.

physics.ed-ph↗