curvature quantum curvature and feynman diagrams

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Curvature Quasi Particles

and Feynman Diagrams

2 [CONSIDERATION OF CURVATURE ON THE QUANTUM SCALE]

2

Forward This paper was born from an undetected need that I had been having for a long time. The birth was uneventful but happy. During its gestation time I happened upon many new and exciting ideas about curvature. I consider myself to be a beginner in to-pology an enthusiastic and very interested beginner. In order to learn something I take it and “fool around with it”, draw things, solve problems (not too often on this because I have found that most writer’s problems are totally useless!). Well, as I was “fooling around with” the Gauss Bonnet and Bertrand-Diquet-Puiseuax Theorems I was some-what lost because I had no perspective. So I read how they worked, plugged in something I really knew well from quantum mechanics (the instructions were quite clear) but you had to know quite a bit about things and not get “cold feet”. And I got my first surprising result. Please read on…..

Table of Contents FORWARD ..................................................................................................................................... 2

INTRODUCTION .......................................................................................................................... 3 A HISTORICAL NOTE .................................................................................................................... 3 CURVATURE ................................................................................................................................. 4

Second Derivative as Curvature .............................................................................................. 5 Curvature- Second Derivative (1 D curve) .............................................................................. 5 Principal Curvature ................................................................................................................. 6 Principal Curvature ................................................................................................................. 6 Gaussian Curvature ................................................................................................................. 6 Gaussian Curvature of a Two-Dimensional Surface ............................................................... 8 Gaussian Curvature as a Product of Principle Radii of Curvature ........................................ 8 Formulas for Riemann Curvature ............................................................................................ 9 Riemann Curvature .................................................................................................................. 9 Formulas for Ricci Curvature .................................................................................................. 9 Ricci Curvature ........................................................................................................................ 9

TOPOLOGY .................................................................................................................................. 10 The Euler Characteristic ....................................................................................................... 10 Curvature and the Gauss Bonnet Equation ........................................................................... 11 The Angle Deficit ................................................................................................................... 12

THE PHASE SPACE OF CLASSICAL STATISTICAL MECHANICS AND QUANTUM MECHANICS .... 12 TOPOLOGY AND UNCERTAINTY ................................................................................................. 14 GETTING FROM A TO B ARE YOU CERTAIN? .............................................................................. 14 QUANTUM CURVATURE AND TOPOLOGY I, II AND III .............................................................. 14

Statement I ............................................................................................................................. 14 Bertrand-Diquet-Puiseux Theorem ........................................................................................ 15 Quantum Curvature I from Topology .................................................................................... 15 Statement II ............................................................................................................................ 16 Quantum Curvature - II ......................................................................................................... 16 Statement III ........................................................................................................................... 16 Commutation in Relation to Curvature ................................................................................. 17

PHASE SPACE .............................................................................................................................. 17

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SOME EVIDENCE PLEASE! .......................................................................................................... 19 MOLECULES AND CRYSTALS ............................................................................................... 19

Counting Guidelines for Molecules ....................................................................................... 19 MOLECULES, BONDING AND COORDINATION NUMBER ............................................................ 19

Rules of Identification-Molecules and Crystals ..................................................................... 19 MOLECULES ............................................................................................................................... 20

CH4+2O2→CO2+H2O ........................................................................................................... 22 FEYNMAN DIAGRAMS ............................................................................................................ 26

Summary of Measurements Made on Feynman Diagrams .................................................... 27 Questions and Answers .......................................................................................................... 27 Elements of Feynman Diagrams ............................................................................................ 28 Identification of Measured Features (above) ........................................................................ 29 Histories and Many Body Theory .......................................................................................... 41

Introduction A Historical Note

Over the centuries geometers, mathematicians and scientists have chosen to work on curvature. For quite some time curvature was considered to be a curved line in space. Gauss Riemann and Ricci changed things so that curvature involved the relationship be-tween a curved line in space and a surface. This is how the historical run-down lines up. Mathematician/ Scientist Year of Birth/ Death Appolonius of Perga 262-190 BCE Diadochus Proclus 412 -417 BCE ? N.icole Orseme 1323-1382 Johannes Kepler 1571-1630 Pierre de Fermat 1601-1665 Rene Descartes 1596-1650 Christaan Huygens 1629-1695 Isaac Newton 1642-1727 Godfried Leibniz 1646-1716 Alexis Clairaut 1713-? Leonard Euler 1707-1783 Augustin Cauchy 1789-1757 Johann Gauss 1777-1855 Bernhard Riemann 1826-1866 Cubastro Ricci 1853-1925


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Curvature of a line in space has occupied inquiring human minds since the time of the early Greeks. The Greeks were piqued by their interest in geometry, in particular, they noticed the difference between a line that was straight and one that was a circle. Their analytical skills were limited to approaching things geometrically and so their interest in curvature would be passed on to Proclus (412-417 BCE) where it languished and proceed no further.

From the period of 1323-1650 there were many contributions to developing and un-derstanding curvature but not until the time of Isaac Newton did the real concept of cur-vature take form. With Newton’s newly developed calculus and its principle of exhaus-tion it was possible to analyze the tangency between a line and a curve for the first time.

Leibniz, Cliaraut and Euler were all concerned with curvature being associated with a 1D curvy line embedded in 2D space. It wasn’t until Gauss analyzed the relationship be-tween a line and a surface ( for the first time!) that everything changed dramatically. Fur-thermore he established the difference between the extrinsic and intrinsic properties of a surface. He found an equation for calculating curvature that is now known as the Gaussi-an curvature.

B. Riemann studied under Gauss and founded the field of Riemann geometry. He de-veloped some new groundwork an established equations relevant to curvature. He created equations that would open the door for calculating the Riemannian curvature.

Riemann’s work on Riemannian geometry and specifically, curvature, was subse-quently used by Einstein and others when they studied the large scale structure of the universe.

Curvature For the most part, down through the centuries, the fundamentals of curvature have

been developed by geometers and mathematicians. The radius of curvature of a line could be associated with a circle that touched a line in a number of points. The circle is known as the osculating circle.

Figure 1 An osculating circle used for determining the radius of curvature Indeed, we’ve benefitted from their endeavors in many ways. However most of the

developments were based on mathematical reasoning but could have benefitted if some basis of physical reality were incorporated into their ideas. For example, in considering the radius of curvature in the limit, as it approaches zero and the curvature approaches zero. Mathematically this is

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limr→0

1r→∞ (1

a singularity and even though the ratio isn’t convergent , there are certain ways to go about resolving things. In the physical world things don’t necessarily go to zero.

Types of Curvature that are in Use Today Several different curvatures will be used in this work that are discussed below:

• Second Derivative • Principal Curvature • Gaussian Curvature • Ricci Curvature • Riemann Curvature

Curvature is concerned with how much a surface deviates from a Euclidean plane.

There are two surfaces of importance: the Euclidean plane which is essentially a test sur-face and the surface to be tested whose curvature is unknown. The test surface is generat-ed by moving simultaneously outward from a point P thus increasing r and asking “Is the plane flat?” if it’s not, then it has curvature.

The most natural notion for describing any curvature is that of an angle in a plane with a vector in the same plane pointing perpendicular to the angle. The collection of all possible directions that are swept out by the vectors is the curvature.

Second Derivative as Curvature L. Euler came up with a theorem that maintained that if a curve was parameterized

then the curvature is equal to the second derivative of the parameterization. The second derivative is a curvature that can be made from an infinitely differentiable

function: α : a,b[ ]→ R3 α ' s( ) ≠ 0 ∀ α ∈ a.b[ ]

Curvature- Second Derivative (1 D curve)

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Principal Curvature This curve and its normal describe curvature in 1D but require 2D for illustration

Figure 2 Principal curvature

Principal Curvature The Principal Curvature lies perpendicular to the curve in the plane of the curve

κ p =x! 1 × x! 2

x3 x! 1 × x! 2( )

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Gaussian Curvature

The Gaussian curvature is related to the Riemann curvature by the following equa-tion. Let there be two non-parallel vectors S and T that meet in space at a point P on the surface S

then this is the Gaussian curvature of the subspace defined by S and T times the uncurved area squared of the S T parallelogram.

The Gaussian curvature is a measure of how much the circumference of a small circle deviates from its expected value when it is placed on a flat surface. The devi-

RµνρσSµT νSρT σ

κGauss

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ation is quadratic, as would be expected since the Riemann tensor depends upon the se-cond derivatives of the metric.

Gauss defined curvature in an intrinsic way so that it could be measured without em-bedding it in a higher dimensional space. There are several ways to derive Gaussian cur-vature.

Consider a starting point P in space and move a geodesic distance ε in all directions forming a “circle” in space. If the space is flat, then the circumference C of this circle is C=2πε but in curved space the circumference will be slightly greater or smaller.

Figure 3 The Gaussian curvature applies to surfaces having positive, negative and zero curvatures.


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Gaussian Curvature of a Two-Dimensional Surface Circumference of constructed circle in 3 space

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(6 Alternatively: Let there be a point P on a surface S embedded in R3, then the Gaussian curvature is: where : Δθ= an arc length defined by the intersection of of a curve and the surface S.

ΔA= a smooth area surrounding the poin

Gaussian Curvature as a Product of Principle Radii of Curvature

Gaussian Curvature (int rinsic) κGauss =κ p1κ p2

where κ p1κ p2 is the product of the two principal radii of curvature

C = 2π sin ερ

⎛⎝⎜

⎞⎠⎟≈ 2πε 1− ε 2

6ρ 2

⎛⎝⎜

⎞⎠⎟

κG = limε→06ε 2 1− C

2πε⎛⎝⎜

⎞⎠⎟

κGauss = limΔA→0ΔθΔA

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Formulas for Riemann Curvature

Riemann Curvature

Formulas for Ricci Curvature

Ricci Curvature

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[CONSIDERATION OF CURVATURE ON THE QUANTUM SCALE]

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Topology

The Euler Characteristic Topology is an essential part of the Euler characteristic. Euler found that for al-

most every 3D polygon there was an exceptionally simple relationship existing be-tween the vertices, the edges and the faces of the polygon, in fact, it didn’t matter how many sides or edges or vertices because as long as it was a convex polygon then:

= 2 (7 χ(N) is known as the Euler characteristic. As a result of this relationship counting su-

persedes everything else and topological structures become ones of mathematics. Shown below are some convex polygons and their atributes.

Figure 4 Topologically equivalent polygons (2D)

Figure 5 Attributes counted for assessing the Euler characteristic

χ(N ) =Vertices − Edges + Faces

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Figure 5 15 different figures with their topological attributes . The convex polygons have χ=2 the others do not

Curvature and the Gauss Bonnet Equation Carl Fedrick Gauss (1777-1855).called his curvature theorem “Theorema Egregium”

…”The Remarkable Little Theorem”. With it Gauss was able to prove that it was possible to determine the metric properties of a surface (without recourse to 3D embedding) by measuring distances and angles on the surface. This meant that phys-ical measurements could be made that determined the intrinsic property of a surface that were not dependent on an external coordinate system. His work laid the foundations of differential geometry.

Gaussian curvature determines the curvature of a surface and is incorporated into the Gauss-Bonnet equation.

(8 This equation is important for the following reasons:

• Since it uses Gaussian curvature the curvature is intrinsic • The equation a great deal of disparate information • The equation describes a relationship between local geometry and

global topology .

κG dA = 2πχ N( )∫

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Considering a surface that is piece-wise smooth over a cell structure but also having a boundary creates additional terms:

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1. On 2-cells of M -κ dA, Gauss curvature times an area element 2. On 1-cells of M, κdA means κg ds, geodesic curvature times the length element 3. On 0-cells of M, κdA means the angle defect

The Angle Deficit The terms vertices, edges and faces refer to the geometry of a polygon. The Gauss-

Bonnet equation is a topological statement. The angle deficit can be understood in the following manner: position a cone on top

of a sphere and draw a circle on the cone where the cone and sphere are in contact; call this the cone-sphere circumference. Now remove the cone and slit it along one side up to its apex and note that the circumference is no longer joined to itself, there’s a deficiency in its length. Before slitting the cone it could not be flattened on a surface but after slit-ting the cone can be flattened and placed on the flat surface (Figure 5 (a) The opened cone has a gap, called the angle deficit. The figure below shows just such a relationship for an angle deficicit (left) and an angle excess (right)

Figure 8(a Angle deficiency and (b angle excess

The Phase Space of Classical Statistical Mechanics and Quantum Mechanics

Phase space is an abstract space composed of the x and p coordinates of N parti-

cles. The particle density, ρ is conserved and the trajectory of a system can be fol-lowed as a function of time. Below is presented expositions on phase space for both Classical and Quantum Mechanics. This is required for developing the fundamental physics for understanding how to insert topology into phase space.

κG dA =σ∫ κ o dA

N 0∫ + κ g ds + κ h dα

N 3∫

N1∫ = 2πχ N( )

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Classical Statistical Mechanics A brief but relevant presentation of statistical Mechanics is as follows. For th

most part I am using a one dimensional model for the development.

Hh

=dx dp∫∫h

This is the Gauss Bonnet equation that stems from empirical observations made

by Euler. He found that were well defined rules associated with the topological prop-erties of a great variety of solids

The object of these realizations is to use phase space for the concise definition of volume. The Gauss Bonnet equation puts this form so that it can be compared with quantum curvature.

dAr 2∫ = 2πχ N( )

Both equations are associated with integrals and curvature but they have been

used in very separate ways. One equation deals with particles, the other with topolo-gy.

If I were to consider them as being related somehow what would I have to consider. The Gauss-Bonnet has nothing to do with interactions it is a representation of something that just exists, it’s a rule of Nature. The place that’s one of nonexist-ence where there are virtually no interactions ( interactions might create verticies and edges etc and I’m interested in an environment where that’s impossible}. There are no interactions but the Gauss Bonnet says that there is a potential for topologies to be in existence without interactions! If they had a propensity to form, there’d be a shift in the ground state energy.

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Topology and Uncertainty

Getting from A to B are you Certain?

Quantum Curvature and Topology I, II and III

Statement I

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Bertrand-Diquet-Puiseux Theorem Let p be a point on a smooth surface M. The geodesic circle of radius r centered at p is the set of all points whose geodesic distance from p is equal to r. Let A® de-note the area of the disc contained within the circle, then the Gaussian curvature is:

κ N( ) = limr→0+12πr2 − A r( )

πr4 (11

Therefore, using the BDT theorem and the Heisenberg Commutator it’s possible to show:

κQ = limh→0

π! ± X,P[ ]π!2

Quantum Curvature I from Topology

κQ = 1h1+ i

π⎛⎝⎜

⎞⎠⎟

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Statement II

This is supporting evidence based on quantum operators

Quantum Curvature - II

the momentum operator

the coordinate operator

the Uncertainty Principle putting everything together

Curvature (15

Statement III This is more supporting evidence based on quantum mechanics.

Δp = -i! ∂

∂x

Δx = i! ∂

∂p

ΔpΔx ≥ !

ΔpΔx ≥ ! = −i! ∂

∂xi! ∂∂p

⎛⎝⎜

⎞⎠⎟= !2 ∂2

∂x∂p

κQ = ∂2

∂x∂p= 1!

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Commutation in Relation to Curvature

Phase Space Phase space is a 6N dimensional construct that plays a vital role in our under-standing of physical and chemical phenomena. The geometric element that forms the space is that of Planck’s constant, h. Phase space isn’t smooth but it’s granular on the atomic scale. Planck’s constant is the tile that covers the area.

The phase integral is used in many ways but one of the most important ones is to develop a partition function and then to evaluate entropy which is a volume element in phase space.

u,v[ ] = ∂u∂qr

∂v∂pr

− ∂u∂pr

∂v∂qr

⎧⎨⎩

⎫⎬⎭r

∑definition for the Quantum Poisson Brackets u,v[ ]of any two var iables u,v uv - vu = ih u,v[ ]for any two particles r and sxr , xs[ ] = 0 pr , ps[ ] = 0

xr , ps[ ] = δ rs

The Commutator is XP − PX[ ] = i!u sing X and P as operators in the Commutator

i! ∂∂p

−i! ∂∂x

⎛⎝⎜

⎞⎠⎟ − −i! ∂

∂p⎛⎝⎜

⎞⎠⎟i! ∂∂x

⎡

⎣⎢

⎤

⎦⎥

i!2 ∂2

∂p∂x− ∂2

∂x∂p⎛⎝⎜

⎞⎠⎟= i!

thus

∂2

∂p∂x− ∂2

∂x∂p⎛⎝⎜

⎞⎠⎟= 1!

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This is quite straightforward. The analysis seeks to evaluate energy as determined by products of differential length dx times differential momentum dp. The integration is done over the coordinates of phase space and then these probabilities are multiplies to-gether for all of the particles. The “volume element for doing all of this is h6N. To do this for one particle:

Classical Statistical Mechanics A brief but relevant presentation of statistical Mechanics is as follows. For the

most part I am using a one dimensional model for the development.

Hh

=dx dp∫∫h

This is the Gauss Bonnet equation that stems from empirical observations made

by Euler. He found that were well defined rules associated with the topological prop-erties of a great variety of solids

The object of these realizations is to use phase space for the concise definition of volume. The Gauss Bonnet equation puts this form so that it can be compared with quantum curvature.

dAr 2∫ = 2πχ N( )

Both equations are associated with integrals and curvature but they have been

used in very separate ways. One equation deals with particles, the other with topolo-gy.

If I were to consider them as being related somehow what would I have to consider? The Gauss-Bonnet has nothing to do with interactions it is a representation of something that just exists, it’s a rule of Nature. I think that a place that I should consider is one of nonexistence where there are virtually no interactions ( interactions might create vertices and edges etc and I’m interested in an environment where that’s impossible}. There are no interactions but the Gauss Bonnet says that there is a poten-tial for topologies to form and to be in existence without interactions! If they had that

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propensity to form, there’d be a shift in the ground state energy , however, they’ve probably already formed many times so that their activities would not be observable.

ΔEnergy⇒ ΔTopology

E ⇒ h ω 2πχ N( ) Just saying that there’s a relationship between energy and topology, then making a gener-ic commitment E = hω2π(N1 +N2...)

Some Evidence Please!

I’ve looked and found some candidates molecules and crystals and Fennman dia-grams.

Molecules and Crystals Well of course the Gauss Bonnet rules are at work here but they are a result of energy changes

Counting Guidelines for Molecules Topological

Idealization Atoms Bonds Molecules

Vertices √ √ √ √ Edges √ √ √ Edges** √ √√ double

bonds √

Surfaces(Faces) √ Crystals √ √ √

Molecules, Bonding and Coordination Number In order to have physically realistic topology that is based on energy and matter some

groundwork is needed for determining ! n( ) that is based upon structural stability, mathematics and physics.

Rules of Identification-Molecules and Crystals • Atoms are located at vertices • Faces (surfaces) are not properties of individual molecules

or atoms • A single bonding electron represents one edge

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• Double electron bonds between atoms are equivalent to two edges

• Atoms are counted as whole atoms (no fractions) in crystals • When evaluating an atomic distribution within a lattice by

ignoring the periodic structure, count bonds and atoms. • No planes or bonds are counted within unit cells

Molecules Methane- CH4 a planar molecule. There is 1 bonding pair of electrons with 2 terminal atoms having 4 of bonding electrons.

Figure 21 A Methane molecule

There are therefore 6 Vertices and 6 edges (1 double bond) resulting in !=0. The

Water H2O

.

Figure 10 A water molecule There are two covalent H:O bonds in the molecule giving A=3 and B= 4 and !(n) = −1

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Water plus nearest neighbors in a liquid

Figure 22 Showing a water molecule with Hydrogen bonds There are still two covalent bonds but bonding to NN molecules adds 4 Hydrogen bonds making A=15 and B=24 with ! n( ) =-9

Seven Types of Molecules having different Geometry, Bonding and Bond Angles

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Table showing Q(n) Formula BeCl2 BCl3 CH4 NH3 H2O # Atoms 3 4 5 4 3 Bonds 4 6 8 6 4 Angles Between Bonding Pairs

180o 120o 109.5o 107o 105o

Shape Linear Trigonal Planar

Tetrahedral Trigonal Pyramid

Bent

Q(n) -1 -2 -3 -2 -1

A Chemical Reaction

CH4+2O2→CO2+H2O

Methane Oxygen Carbon Dioxide Water

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5 Atoms 2 Atoms 3 Atoms 3 Atoms 4 Bonds 2 Bonds 4 Bonds 4 Bonds !(n) = 1 !(n) = 0 !(n) = −1 !(n) = −1 Figure 23 A chemical reaction between methane and oxygen

Crystalline Solids The “edges” of the topological polygon are stable generic bonds that are part of the

cohesive energy of the solid. In these instances the detailed bond types have been ignored and replaced with simply a connection.

Nearest Neighbor Coordination Numbers in a Simple Cubic Crystal Figure 24 shows 6 nearest neighbor atoms surrounding the central red one. There are

also 7 atoms in the coordination polygon. This gives Q(n)=1

NN Cubic Unit Cell Atoms=7 Atoms=8 Bonds=6 Bonds= 12 !(n) = 1 Faces=6 !(n) = 2 Figure 25 A simple cubic crystal structure

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Body Centered Cubic Crystal

A Body Centered Cubic Unit Cell Atoms=9 Bonds= 20 Faces = 6

Q(n)=-5 Distribution around Body atom Atoms =9 Bonds =8

!(n) = 1

Figure 26 A body centered cubic unit cell

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General Nearest Neighbor Coordination 12, 6, 4, and 2

Figure 27 The Euler Characteristic is 1 for all co-ordinations 1,4,8 and 12, a surprising result

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Feynman Diagrams

A topological analysis of some Feynman diagrams was undertaken. The diagrams presently are used to analyze high energy particle interactions and many body effects in quantum systems specifically in terms of quantum chromoodynamics. The character of the diagrams is quite diverse and, other than using them for a particular problem of a sci-entist’s choosing, it’s difficult to say if there’s any method that’s associated with using one set of diagrams over another.

The Feynman diagrams offer the human mind a window into the world of sub atomic particles. The human mind creates a diagram from experimental observations and math-ematical analysis of what it perceives to be particle scattering events. Therefore, Feyn-man diagrams are some kind of an image that has been transcribed by the human mind from the sub-atomic scale to the human scale.

The diagrams have a “Feynman” character to them-they are whimsical, happy and can be exceptionally complicated. Under close inspection there seem to be similar classes of diagrams but it hasn’t been apparent what such similarities might be.

The Feynman diagrams that were analyzed were “Rigged Feynman Diagrams”. Feynman diagrams recognize vertices and edges and give mathematical weight to all of the members. A 2D polygon diagram doesn’t give weight to any of its members and rec-ognizes edges, vertices and (sur)faces. Therefore, a it was necessary to recognize (only) the three members of a set and to find ! C( ) .

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Many Feynman diagrams were chosen at random starting with simple ones and grad-ually getting more complicated-this was OK at first but all that could be done was to make comparisons ...”this one’s just like that one”...which didn’t offer any gain in topo-logical information whatsoever.1

I decided to try using the quantum characteristic in order to analyze the patterns be-cause the patterns clearly had vertices (atoms), edges (trajectories) and surfaces (faces). The criteria that I used is shown below.

I chose to measure the Quantum characteristic (after the Euler characteristic) because it is a topological invariant measure. By measuring it carefully and consistently I was able to obtain a reliable measurement.

Jumping to the end-it was possible to measure the features of the Feynman diagrams and to determine the important topological invariant the quantum (Euler) characteristic. The results are shown in the following table

Summary of Measurements Made on Feynman Diagrams

Total number analyzed =78 Q(N) Number Counted %

-2 1 1.28 -1 1 1.28 0 6 7.69 1 30 38.5 2 17 21.8 3 6 7.69 4 3 3.84 5 1 1.28

THIS ANALYSIS IS DIFFERENT FROM THAT WHICH IS USED ON FEYNMAN DIAGRAMS! This analysis was done in order to find if it was pos-sible to determine the quantum characteristic for “Rigged” Feynman diagrams.

Questions and Answers

Can a reliable analysis be done on “Rigged Feynman Diagrams” The answer is yes indeed a reliable analysis can be done. A topological invariant can be measured

What’s it worth? That’s not clear yet

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Is there something new that can be done with Feynman diagrams and topological invariants that have been measured? Yes I think so. The surfaces have to be incorporated into the Feynman diagrams. The pre-sent equations have to be looked at etc. It won’t be simple because although the diagrams look simple, the mathematics that accompany them is not. All I can say is that I’ve started to work on everything.

Elements of Feynman Diagrams

The three fundamental features of a Feynman diagram • particles-photons • particles=electrons • vertices-the exchange of eenergy at a vertex

A primitive vertex is where phase changes occur

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Figure A1 This diagram describes the electromagnetic force between two electrons-there’s an emission of a virtual photon.

Identification of Measured Features (above) Item Value Atoms 2 Trajectories 5 Surfaces 4 Q© 1

=# of interacting particles 2

n total number of terms responsible for phase shifts -

2

m∑

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Diagram Number #1 Atoms-V 2 Trajectories -E 5 Surfaces -F 4 Quantum Characteristic 1 Figure A2

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Diagram Number #2 Atoms-V 4 Trajectories -E -9 Surfaces -F 6 Quantum Characteristic 1 # particles 9 v=4 Deficit 287.7

Figure A3

Trait/# 1 2 3 4 A 4 4 4 4 T 8 6 7 8 S 5 5 5 5 Q 1 3 2 1 n 2 2 2 2 Angle Deficit 312 312 312 312

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Figure A4

Trait/Number 1 2 3 4 A 4 4 4 4 T 9 9 9 9 S 6 6 6 6 Q 1 1 1 1 n 7 7 8 8 Angle Deficit 131.9 131.9 131.9 131.9

Figure A5

Diagram Number Atoms-V 3 Trajectories -E -6 Surfaces -F 4 Quantum Characteristic 1 n 2

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Angle deficit 54.16

Figure A6

Trait/# 1 2 3 4 5 A 2 2 2 4 4

T 5 5 2 7 8 S 4 4 3 5 5 Q 1 1 3 2 1 n 2 2 2 2 3 156.1 156.1 156.1 156.1 156.1 Figure A7

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Diagram Number Atoms-V 6 Trajectories -E 13 Surfaces -F 8 Quantum Characteristic 1 n 11 Angle deficit 110.5

Figure A8

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Traits/# 1 A 7 T 11 S 6 Q 2 n=2 6.373

Figure A9

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Trait #1 A 6 T 13 S 8 Q 1 n=6 79.38

Figure A10

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Trait/# #1 A 9 T 17 S 11 Q 1 n=10 97.73

Figure A11

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Trait/# 1 2 3 4 5 6 A 6 5 6 5 6 6 T 11 10 10 9 11 11 S 5 6 6 6 6 6 Q 0 1 2 2 1 1 Trait/# 7 8 9 10 11 12 A 6 6 5 6 6 6 T 9 10 9 10 10 13 S 6 5 6 6 6 6 Q 3 1 2 2 -2 -1 Trait/# 13 14 15 16 17 18 A 5 6 5 6 6 6 T 11 10 9 11 10 9 S 6 6 5 5 5 5 Q 0 2 1 0 1 2

Figure A 12

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Item/# #1 2t Atom 8 2W Trajectory 17 2g Surface 10 5b Q(n) 1 1 l, H n=14 26.3

Figure A13

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Item/# 1 Atoms 6 Trajectories 11 Surfaces 6 Q 1 m=2 n=6

Figure A 14

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Histories and Many Body Theory

Trait/# 1 2 3 4 5 6 A 2 2 3 4 4 10 T 5 5 7 8 7 12 S 4 4 5 5 5 9 Q 1 1 1 1 2 7

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Figure A15

Trait/# 1 2 3 4 5 6 A 0 2 4 6 8 T 2 3 5 7 9 S 3 3 4 5 6 Q 1 2 3 4 5

Figure A16

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Trait/ # #1 #2 #3 #4 #5 A 2 2 2 2 3 T 5 3 3 5 5 S 5 4 4 4 5 Q 2 3 3 1 3 Trait/# #6 #7 #8 #9 #10 A 2 2 2 2 2 T 5 3 3 5 5 S 5 4 4 4 4 Q 2 3 3 1 1

Figure A 17

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Item/# 1 2 3 4 5 6 7 8 9 A 2 2 4 5 4 4 6 6 5 T 3 3 4 8 6 6 9 7 8 S 1 1 4 5 4 4 5 5 5 Q 0 0 4 2 2 2 2 4 2

curvature quantum curvature and feynman diagrams

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