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Halycon Days: John Smith: Savilian Chair at Oxford 1766-1797 Working with GoLive Web Design
Many of the files below require the 'Ghostscript' or 'Ghostview' viewer program. If you do not have a version of this viewer, you may download it for most major operating systems at Ghostview Homepage. The more recent talks are in .pdf format and when you click on the talk will be opened by acroread. They are opened in presentation format. To get to the more standard adobe acrobat reader screen, type ctrl-L. Or to simply quit, use ctrl-Q.

Contents:

    Solutions and information files not explicitly labelled as 'PDF' require the Ghostscript or Ghostview programs available as described near the top of my home page. However, the download is a bit non-trivial and so I will also provide these course materials in the Adobe Acrobat Reader format. These versions are the 'PDF version' files below. The Adobe Acrobat Reader program is freely available and usually easy to download from the Adobe Acrobat Reader Download Site
Abstract Algebra
Some ideas on Perturbation Theory
Articles by Stephen P. Smith
Experimentalist's Guide to Photon Fluxes Is There Any Physical Significance To Fractional Derivatives? -- a presentation by John R. Smith
Problems with Half-Derivatives - Argument of Chris Freiling
Flocking of Birds
3D BioPrinting Historical Photographs of the Golden Gate Bridge
Useful Mathematical Tricks of the Physics Trade John Bird and John Fortune on the "Subprime Crisis"
Market Meltdown February 14, 2008
Writings and Observations by Christopher F. Freiling
Growing Grapes in Suisun Valley
Recent Research Papers
1) Vertex algorithm: Geometrical Discriminant for Higgs -> 4 Lepton Searches
Abstract: We propose a new variable to be considered for H -> 4 lepton searches based on minimizing the spatial variance between 4 track helices. This spatial variance can be used to discriminate between signal events and typical background sources for the H -> ZZ* ->4 lepton decay mode. The analysis is based on the method of Least Squares and also includes a track-error based analysis which uses the method of Restricted Maximum Likelihood (ReML). In the ReML analysis, for most cases, only two constraints are required to form a consistent set of equations for Newton-Raphson maximization of the likelihood. The paper discusses the relationship between observational bias of the hits in the tracking chamber and the nearly rank-2 characteristic of the spatial error matrix for positions along the extrapolated track helix. The paper also discusses some aspect of robustness related to non-gaussian spatial errors and compares the performance of the Least Square methods, ReML and three-dimensional impact parameter significance (SIP3D) on various signal and background Monte Carlo samples.

This algorithm is useful in the search for Higgs in the 4-lepton channel to discriminate signal from background. Higgs -> 4 Leptons is characterized by the property that the 4 leptons are all produced at the same point (near the beam-beam axis due to the short lifetime of the Higgs). There is NO NEED to introduce auxilliary and irrelevant dependencies on the impact parameter with respect to the primary vertex. In the presence of High-Pileup introducing the Primary Vertex is a distraction -- and good candidate events will be lost due to misidentification of the correct Primary Event Vertex. A good Higgs decay is characterized by all 4 leptons forming their own vertex with small Chi-Square -- that is sufficient for distinguishing them from Zbb and T-Tbar events -- application is described in

Hunting the Higgs Boson using the Cholesky Decomposition of an Indefinite Matrix
Abstract: Linear models have found widespread use in statistical investigations. For every linear model there exists a matrix representation for which the ReML (Restricted Maximum Likelihood) can be constructed from the elements of the corresponding matrix. This method works in the standard manner when the covariance structure is non-singular. It can also be used in the case where the covariance structure is singular, because the method identifies particular non-stochastic linear combinations of the observations which must be constrained to zero. In order to use this method, the Cholesky decomposition has to be generalized to symmetric and indefinite matrices using complex arithmetic methods. This method is applied to the problem of determining the spatial size (vertex) for the Higgs Boson decay in the Higgs -> 4 lepton channel. A comparison based on the Chi-square variable from the vertex fit for Higgs signal and tt background is presented and shows that the background can be greatly suppressed using the Chi-square variable. One of the major advantages of this method over the currently adopted technique of b-tagging is that it is not affected by multiple interactions (pile up).

2) Automatic Methods for Handling Nearly Singular Covariance Structures Using the Cholesky Decomposition of an Indefinite Matrix
Abstract: Linear models have found widespread use in statistical investigations. For every linear model there exists a matrix representation for which the ReML (Restricted Maximum Likelihood) can be constructed from the elements of the corresponding matrix. This method works in the standard manner when the covariance structure is non-singular. It can also be used in the case where the covariance structure is singular, because the method identifies particular non-stochastic linear combinations of the observations which must be constrained to zero. In order to use this method, the Cholesky decomposition has to be generalized to symmetric and indefinite matrices using complex arithmetic methods. This method is ap- plied to the problem of determining the spatial size (vertex) for the Higgs Boson decay in the Higgs -> 4 lepton channel. A comparison based on the Chi-square variable from the vertex fit for Higgs signal and tt background is presented and shows that the background can be greatly suppressed using the Chi-square variable. One of the major advantages of this novel method over the currently adopted technique of b-tagging is that it is not affected by multiple interactions (pile up).

Complex arithmetic is used to construct an automatic method for identifying linear combinations of "data" which must be constrained to zero in the case of a nearly singular Covariance Matrix. In the application used, the "data" are the x,y & z coordinates along an extrapolated curve (helical charged-track trajectory in a magnetic field) and the constraint represents specifying a particular point along the trajectory as the one used in the maximum likelihood problem. The technique of automatic construction of the likelihood and also the constraint conditions (which are enforced by the method of Lagrange multipliers or by including a quadratic penalty term to the likelihood, penalized likelihood method). This technique is employed to construct the vertex algorithm used in discriminating signal from background in the search for the Higgs Boson in the 4-Lepton channel.

3) Published Paper on Speckle Patterns and 2-Dimensional Models
Speckle Patterns and 2-Dimensional Spatial Models.pdf
Abstract:
The result of 2-dimensional Gaussian lattice fit to a speckle intensity pattern based on a linear model that includes nearest-neighbor interactions is presented. We also include a Monte Carlo simulation of the same spatial speckle pattern that takes the nearest-neighbor interactions into account. These nearest-neighbor interactions lead to a spatial variance structure on the lattice. The resulting spatial pattern fluctuates in value from point to point in a manner characteristic of a stationary stochastic process. The value at a lattice point in the simulation is interpreted as an intensity level and the difference in values in neighboring cells produces a fluctuating intensity pattern on the lattice. Changing the size of the mesh changes the relative size of the speckles. Increasing the mesh size tends to average out the intensity in the direction of the mean of the stationary process.
4) Leptoquark Generator for Hadron Colliders
Abstract: A new Monte Carlo Generator for 3rd Generation Vector Leptoquark production is now available for CDF. This program implements the appropriate Feynman Rules which follow from an effective Lagrangian[1] into a matrix element calculation using The Grace System[2]. This effective Lagrangian is parameterized in terms of kappa_G and lambda_V which allow for couplings to anomalous ‘magnetic’ mu_V and ‘electric’ quadrupole moments q_V
mu_V= (g_s / 2M_V)(2 - kappa_G + lambda_G)
q_V =-(g_s /M_v^2) (2 - kappa_G - lambda_G).
The Monte Carlo generator was implemented for proton-antiproton collider via the GR@PPA program[3].

Monte Carlo Methods Applied in the Metropolis Algorithm of the Classical Relativistic Oscillator

5) A Monte Carlo Study of the Relativistic Harmonic Oscillator
Abstract: We present a Monte Carlo calculation of the ground-state wave function for the relativistic harmonic oscillator using the classical action in the path integral representation of quantum mechanics. The ground-state wave function has the usual Gaussian shape in the non-relativistic limit, but is not described by a single Gaussian function in the extreme relativistic limit and appears to remain square-integrable for arbitrary values of mc^2/(hbar*omega). A comparison is made with the ground state of the Klein-Gordon equation in the presence of the harmonic oscillator potential and the differences are presented. Also we present a realization of the non-relativistic ground-state problem in terms of a first-order auto-regressive time series with an extension for periodicity which can be implemented as a 3n computer algorithm.

There appears to be a logarithmic divergence as the step size is reduced in the above Monte Carlo program. The Path Integral is based on a non-Gaussian function and therefore it is not clear whether or not such a Path Integral exists (is finite) or is non-trivial (not identically zero). Such behavior indicates existence problems for arbitrary Path Integrals in the presence of non-Gaussian functions. If you look at a Path Integral as an infinite product, then it is not clear that the product converges to a finite and non-trivial result.

The Square-Root Klein-Gordon Operator

My attempts to find the eigenvalues analytically for the Relativistic Coulomb bound states based on the Square-Root Klein-Gordon Operator -- a fractional power of the modified Helmholtz operator.

6) The Square-Root Klein-Gordon Operator With An Application To The Relativistic Coulomb Bound States and Resolution Of The Klein Paradox -- the method fails because the Fourier Transforms contain Cosine Transforms, but Cosine Transforms do not contain all Fourier Transforms -- and therefore the reasoning leading up to Eq. 10 has a gap, indeed the Sine Transform is also required and was overlooked and this spoils the very nice identity which was used to derive Eq. 10.

The spectral properties of the Square-Root Klein-Gordon Equation in the presence of the Coulomb Interaction was written up in Spectral Theory of the Operator (p^2 + m^2)^(1/2) -- Ze^2/r by Ira W. Herbst, Commun. Math. Phys. 53, 285 (1977). The eigenvalues of the Square-Root Klein-Gordon Operator are intermediate between the Dirac Equation and the Klein-Gordon Equation. This can be shown by simple Rayleigh-Ritz techniques in the case of the Coulomb Interaction. However the problem of how to introduce Lorentz-Invariant interactions remains an open one. I will attached an example from Fortran of how to explore the Rayleigh-Ritz method in this case. You can see that there is a problem for the ground-state eigenvalue when the charge exceeds 2*alpha/pi -- the Rayleigh-Ritz "upper bound" on the ground state energy eigenvalue appears to fall to minus infinity. This breakdown point was first discussed in the above cited paper by Herbst.

Further properties of the Square-Root Klein Gordon Equation can be found in Klein-Gordon and square-root operator equations for two-spinors and scalars: perturbation calculations for hydrogen-like systems by Tobias Gleim.

7) Square-Root Klein-Gordon Operator, Microscopic Causality, Propagators, and Interactions -- the crucial step of introducing gauge interactions via the Mandelstam Representation has a gap -- see Section 8 (Interacting Fields in the Mandelstam Representation). This method of introducing Lorentz invariant interactions appears very attractive on the surface because of the manifest Lorentz invariance of Mandelstam's line integral. However, the basic problem is that the Mandelstam Representation uses a path-dependent "function" for which mixed partial derivatives with respect to the end points by Leibnitz's Rule do not commute -- this pathological non-commutivity of the mixed partial derivatives is the cornerstone of this method for introducing interactions and is often pointed out as the main feature of the technique. Therefore caution must be exercised because the line-integral is not a function merely of the end points but also for the variation of the end points according the pre-assigned path. Therefore even though the Mandelstam method is illustrative for introducing interactions in the context of local differential equations (e.g., such as the Klein-Gordon or Dirac Equations), it appears to break down in the presence of non-local operators because one cannot define the integration of such a pathological function over all of space -- integration over all paths might be a way around this problem however the basic problem appears to be dependence of the integral on the order of integration of the spatial coordinates of the endpoints of the line-integral. This implies that Integration-by-Parts and other normal concepts from Calculus cannot be defined! I do not see how path integral methods can remove this problem since the fundamental problem still appears to be the dependence on the order of integration of the spatial variables of the end points of the Mandelstam Representation which exists path-by-path and therefore must still be present when all paths are summed over. Here is a later attempt at second quantization: Second Quantization of the Square-Root Klein-Gordon Operator.

Further publications on the subject of the Second Quantization of the Square-Root Klein-Gordon Operator see: The pseudodifferential operator square root of the Klein-Gordon equation by Claus Lammerzahl, J. Math. Phys. 34, 9, (1993).

The fundamental problem with the Square-Root Klein-Gordon Operator, (as pointed out many years ago in Relativistic Invariance and the Square-Root Klein-Gordon Equation by Joseph Sucher, J. Math. Phys. 4, 17 [1963]), is: How to introduce gauge interactions (or any interactions) with external fields in a Lorentz covariant manner inside the Square-Root Klein-Gordon operator? The Mandelstam technique appeared to be the most promissing because of the manifest Lorentz invariance -- by there is no way to take such a representation seriously for a non-local operator.