Home Page for John R. Smith


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Halycon Days: John Smith: Savilian Chair at Oxford 17661797  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 ctrlL. Or to simply quit, use ctrlQ.


Contents:




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 HalfDerivatives  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 trackerror 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 NewtonRaphson maximization of the likelihood. The paper discusses the relationship between observational bias of the hits in the tracking chamber and the nearly rank2 characteristic of the spatial error matrix for positions along the extrapolated track helix. The paper also discusses some aspect of robustness related to nongaussian spatial errors and compares the performance of the Least Square methods, ReML and threedimensional impact parameter significance (SIP3D) on various signal and background Monte Carlo samples. This algorithm is useful in the search for Higgs in the 4lepton 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 beambeam 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 HighPileup 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 ChiSquare  that is sufficient for distinguishing them from Zbb and TTbar events  application is described in Hunting the Higgs Boson using the Cholesky Decomposition of an Indefinite Matrix 

2) Automatic Methods for Handling Nearly Singular Covariance Structures Using the Cholesky Decomposition of an Indefinite Matrix 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 chargedtrack 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 4Lepton channel. 

3) Published Paper on Speckle Patterns and 2Dimensional Models Speckle Patterns and 2Dimensional Spatial Models.pdf Abstract: The result of 2dimensional Gaussian lattice fit to a speckle intensity pattern based on a linear model that includes nearestneighbor interactions is presented. We also include a Monte Carlo simulation of the same spatial speckle pattern that takes the nearestneighbor interactions into account. These nearestneighbor 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 protonantiproton 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 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 nonGaussian function and therefore it is not clear whether or not such a Path Integral exists (is finite) or is nontrivial (not identically zero). Such behavior indicates existence problems for arbitrary Path Integrals in the presence of nonGaussian 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 nontrivial result. 

The SquareRoot KleinGordon Operator
My attempts to find the eigenvalues analytically for the Relativistic Coulomb bound states based on the SquareRoot KleinGordon Operator  a fractional power of the modified Helmholtz operator. 6) The SquareRoot KleinGordon 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 SquareRoot KleinGordon 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 SquareRoot KleinGordon Operator are intermediate between the Dirac Equation and the KleinGordon Equation. This can be shown by simple RayleighRitz techniques in the case of the Coulomb Interaction. However the problem of how to introduce LorentzInvariant interactions remains an open one. I will attached an example from Fortran of how to explore the RayleighRitz method in this case. You can see that there is a problem for the groundstate eigenvalue when the charge exceeds 2*alpha/pi  the RayleighRitz "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 SquareRoot Klein Gordon Equation can be found in KleinGordon and squareroot operator equations for twospinors and scalars: perturbation calculations for hydrogenlike systems by Tobias Gleim. 7) SquareRoot KleinGordon 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 pathdependent "function" for which mixed partial derivatives with respect to the end points by Leibnitz's Rule do not commute  this pathological noncommutivity 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 lineintegral is not a function merely of the end points but also for the variation of the end points according the preassigned path. Therefore even though the Mandelstam method is illustrative for introducing interactions in the context of local differential equations (e.g., such as the KleinGordon or Dirac Equations), it appears to break down in the presence of nonlocal 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 lineintegral. This implies that IntegrationbyParts 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 pathbypath and therefore must still be present when all paths are summed over. Here is a later attempt at second quantization: Second Quantization of the SquareRoot KleinGordon Operator. Further publications on the subject of the Second Quantization of the SquareRoot KleinGordon Operator see: The pseudodifferential operator square root of the KleinGordon equation by Claus Lammerzahl, J. Math. Phys. 34, 9, (1993). The fundamental problem with the SquareRoot KleinGordon Operator, (as pointed out many years ago in Relativistic Invariance and the SquareRoot KleinGordon 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 SquareRoot KleinGordon 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 nonlocal operator. 
