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«STUDENT RESEARCH PAPERS SUMMER 2011 VOLUME 22 REU DIRECTOR UMESH GARG, PH.D. REU Student Research Papers – Summer 2011 University of Notre Dame – ...»

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The shuffling permits for the galaxies to move out of the box at one end and with the application of periodic boundary conditions to return at the other end, staying within the limits of the space as shown in figure 2.

Figure 2 An illustration of the galaxies within a given plane, represented by black dots, the program is able to move planes of galaxies and to rotate the axes from which the box is seen. In addition the program applies periodic boundary conditions to the shuffling to keep all the galaxies within the box limits.

The distance of each galaxy is needed in order to calculate the redshift. The

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In order to apply the periodic boundary conditions the shuffling within the program is written so that the shuffling will not put a galaxy outside of the boundary limit, in the case we used the galaxies had to stay within 25 h−1 Mpc in the x, y and z direction. Therefore was defined as follows = [ + − . ] × (2) Where is the coordinate value from the file containing all galaxies coordinates, is due to the coordinates of the galaxies being in terms of a fraction of the box size, is a term associated with the shuffling motion, which was a random number generator, and . is a term associated with the translation of the galaxies.

Similar to the x value, the y value and z value are as follows.

= [ + − . ] × (3) = [ + − . ] × (4) The position in the sky of the galaxies was used to limit the galaxy distribution to match the SDSS conditions and was calculated as listed below.

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Two methods of interpolation were used with the galaxy distribution to find the redshift from a distance to redshift conversion table that reflects the “concordance model”. The first method used was a simple linear interpolation created in a spread sheet using the distances generated from the galaxy distribution

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distance. This calculation was not applied across all 2097 data points within the galaxy distribution since it was used only as a method to check the other interpolation scheme used.

The other method of interpolation was a linear interpolation as well, but a c program was written to use the redshift table and interpolate the values from the galaxy distribution with the appropriate points, the nearest two points in the redshift table. This task proved to be a challenge as two interpolation schemes were attempted to be implemented, but could not handle the input of the large arrays.

Therefore an interpolation scheme was written based off the algorithm described in Numerical Analysis by the Neville’s iterated interpolation. [Burden & Faires, 2010:3.1] The program uses equation 7, which was adapted from Neville’s iterated


� − − �,−1 − ( − )−1,−1 , = � − − �

And the interpolation scheme used in the program:

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Where ′ is the distance calculated from the galaxy distribution (equation 1) and ′ is the redshift corresponding to the distance. The and are the distances

such that:

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the challenge with other methods attempted. The program then created a commaseparated-values (csv) file that contained the distance, θ, φ and the redshift. This csv file was then cross referenced with the spread sheet interpolation to ensure the program ran properly and correctly.

In addition to the redshift found from the linear interpolation scheme, there is a redshift associated with the particular velocity of the galaxy. The redshift associated with the velocity produces either a redshift, the galaxy is moving away from the observer, or a blueshift in which the galaxy is moving towards the observer.

[Carroll & Ostlie, 2007:98] This was found from the velocity along the line of sight of the particular galaxy as follows.

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Where is the total redshift, is the redshift value from the interpolation, and the last term is the velocity along the line of sight of the galaxy divided by the speed of light.

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In order for the redshift distribution of the simulated model to be similar to the SDSS there were various limits that were applied. These limits, found from the SDSS Constants table, must be applied since the computer produces all data points while observable data would have limits due to the technology used and by the fact that we can only visibly see parts of the electromagnetic spectrum whereas the computer would be able to produce data of all parts of the electromagnetic spectrum. In addition, the SDSS covers a limited area in the sky while the simulation has no such limits on the area of the sky it does calculations on. The area mapped

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longitude limits and mapped in terms of stripes. [Strauss] The program will need to convert the position, so that the area mapped by the stripes in the SDSS can be applied to the simulation.

The next step for the program will be to implement a calculation of the magnitude using the luminosities available for the galaxy distribution through the following equations.

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Where is the radiant flux measured at a distance (already obtained previously in the program) and is the luminosity of the object.

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Where is the apparent magnitude of the object and the subscripts are used so that the flux and magnitude have some object compared to them.

These calculated magnitudes from the galaxy distribution would then be limited just as θ and φ were based off the SDSS Constants table, part of which is listed below.

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The program successfully calculates the redshift from the interpolation of the redshift table along with the redshift due to the velocity of each galaxy within the distribution. This creates a redshift distribution similar to the SDSS and the program applies the conditions of the SDSS to the simulated redshift distribution.

The next step towards verifying the simulated galaxy distribution will be to implement the limiting magnitude conditions within the program.

Acknowledgements I would like to thank my advisor, Dr. Lara Arielle Phillips, for allowing me the opportunity to work on this project. I would also like to thank Ali Snedden for all the help he provided. I would finally like to thank Dr. Umesh Garg and the University of Notre Dame Department of Physics for the opportunity to research with such exceptional researchers, students and staff.

References Burden, Richard, & Faires, J. (2010). Numerical Analysis. Boston, MA: BrooksCole Pub Co.

Carroll, Bradley W., & Ostlie, Dale A. (1996). An Introduction to Modern Astrophysics.

Benjamin-Cummings Pub Co.

Carroll, Bradley W., & Ostlie, Dale A. (2007). An Introduction to Modern Astrophysics.

San Francisco, CA: Benjamin-Cummings Pub Co.

Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1 Nagamine, K., Cen, R., & Ostriker, J. P. 2000, ApJ, 541, 25 Sloan Digital Sky Survey. (2011, May). SDSSConstants. from SDSS.

Strauss, Michael. et al., 2003, Sloan Digital Sky Survey. Great Circle Drift Scanning.

Wang, L., Caldwell, R. R., Ostriker, J. P., & Steinhardt, P. J. 2000, ApJ, 530, 17

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1 Abstract In this paper, preliminary results from investigation into the formalism of Quantum Field Theory as well as various numerical techniques used to approximate theoretical models will be presented. First, the two equations of primary interest, the Klein Gordon and Dirac equations, will be introduced and examined for underlying mathematical and physical structure. In the next section, a nave approach to transforming the differential equations into difference equations will be used and their limitations will be discussed. We will then proceed to investigate methods of preserving discrete forms of the continuous symmetries these equations hold.

Finally we will discuss how these methods can be combined together in ways that can balance the need for numerical methods to both well approximate the system as well as preserve the symmetry of the equations.

2 Introduction

The formulation of Quantum Field Theory succeeded in unifying Quantum Mechanics with Special Relativity in a very satisfying way. While the subject is nowhere near complete, the interactions described with QFT explain observed behavior so naturally it is impossible to deny its computational ability. However, while the theory allows us to write down very elegant descriptions of how particles with internal degrees of freedom (spin) interact with each other through the Strong, Weak and Electromagnetic Forces, the process of actually calculating values that are physically observable becomes complicated very quickly because the corresponding integrals have no closed form solution. The physicist must truncate an infinite series at some point to begin making a calculation. These approximations can be carried out to arbitrary precision, but this is not the only issue. Ignoring the common divergences that physicists encounter in these approximations, there is the

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issue of practicality. It does not make sense to carry out these calculations by hand when a computer can do them exponentially faster. However, a very large sacrifice is made when computation is left to a computer because it only has the ability to add and multiply. Binary cannot represent without truncating an infinite sequence, so the reduction of a very complex integral equation to a series of arithmetic operations is not a trivial process.

3 Klein-Gordon and Dirac Equations We begin by stating the two primary field equations of interest1.

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This will be our first relativistic wave equation to consider. As a scalar field it describes the motion of a neutral spin 0 particle who is it’s own antiparticle. If we were to modify it to be complex valued the real and imaginary parts would correspond to a particle-antiparticle pair with opposite Throughout the text the metric signature (+,-,-,-) will be used. Also =c=1

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Before defining the next field of interest, we define a mathematical object vital to working with the field.

Definition 2 (Lorentz Algebra). Let γ µ be 4 n×n matrices satisfying the anti-commutation relations

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The elements of the Lorentz algebra satisfy the following commutation relations [S µν, S ρσ ] = i(g νρ S µσ − g µρ S νσ − g νσ S µρ + g µσ S νρ ) In the case of n = 4, the γ µ matrices are the Dirac matrices. Their particular representation will be specified later. With these in mind, we proceed to the next field under consideration.

Definition 3 (Dirac Field). Let ψ be a 4 component field that transforms under boosts and rotations according to the Lorentz algebra (called a Dirac

spinor). Let ψ satisfy:

(iγ µ ∂µ − m)ψ(x) = 0

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This field describes a spin 1/2 particle and its corresponding antiparticle.

A solution to the Dirac Field equation is automatically a solution to the Klein-Gordon equation, but not vice versa.

These two fields give us the foundational tools to describe the interactions of relativistic quantum systems. It should be noted that both equations have solutions with negative energies, which was initially seen as a weakness of the theories. Reinterpretation of those solutions as a particles corresponding antiparticle resolved those issues. Of course, it is inaccurate to say these two fields describe particles at all, because we have not quantized them yet.

When the quantization procedure is carried out, care must be taken to apply appropriate commutation relations. Specifically, for the Dirac Field we must impose anti-commutation relations to preserve causality and to explain the negative energy solutions in terms of antiparticles.

One final thing to make note of with these equations is the conserved energy norm. Because solutions to the Dirac equation are automatically solutions to the Klein-Gordon equation, it is sufficient to just show the conserved norm

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And for our solution φ(x,t), the energy should be conserved (i.e. E(t) = E(0) ∀ t). This quantity is important in its own right, because it allows us to define a topology on the space of solutions to the Klein-Gordon equation.

Any approximation should then have a discrete conserved energy norm in order to recover the structure of the continuous solution.

4 2nd Order Finite Difference Method - A Careless Attempt

To begin, we consider the Klein-Gordon equation in (1+1) dimensions:

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There are several conditions that we must impose to effectively translate this differential equation to a difference equation. The first is that we must restrict the equation to some finite region. For simplicity, we will consider a box of side length L and partition it into n equal sized intervals. In other words, ∆t = ∆x = L/n = h. Additionally, boundary conditions must be specified so that the value of the field is known along the x = -h and x = L + h lines.

Again, we strive for simplicity and make the field equal to zero everywhere

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