- Email: [email protected]
Aspects of Feynman Graphs
Electronic Notes in Discrete Mathematics 33 (2009) 43–50 www.elsevier.com/locate/endm
Aspects of Feynman Graphs P.Achuthan 1,2 Department of Mathematics Amrita Vishwavidyapeetham (Deemed University) Coimbatore-641 105, India
Narayanankutty Karuppath 3 Department of Sciences (Physics Division) Amrita Vishwavidyapeetham (Deemed University) Coimbatore-641 105, India
Abstract Points (vertices) and lines (edges) can be compared to the particles and waves in nature. Graphical, visual representations of objects and processes possess many properties which are quite advantageous and practically useful. Richard P. Feynman, made copious applications of the graph-theoretic language, ideas and methods for understanding the micro-world of elementary particles and their interactions. Here, we discus certain significant features of Feynman graphs or diagrams in the modern context. We make a novel observation that the CPT theorem is embedded in the space-time concepts of Feynman graphs and hence the success and precision of some the calculations of QED are in fact testament to the correctness and accuracy of this theorem. Keywords: CPT Theorem, Symmetry, QED, Feynman graphs .
1571-0653/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2009.03.007
44
1
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
Introduction
Quantum Mechanics (QM), along with Relativity Theories has so far achieved much success in understanding nature, especially the micro-world. Many outstanding individuals and groups of scientists have contributed in this endeavor. In the galaxy of theoretical physicists of 1930’s, 1940’s and after is Richard Philip Feynman whose remarkable contributions are worthy of investigation and appreciation[8]. From QM and applications of quantum field theory to elementary particle physics Feynman’s works are marvelously substantial. In this paper we dwell on one of the ideas originated by him, namely Feynman Diagram / Feynman Graph which can effectively represent elementary particle interactions, starting from scattering matrix and depictions thereof. Feynman’s work reconciled QM with Einstein’s special theory of relativity. With Dirac’s relativistic wave equation QED was more consistent with Einstein’s work.
2
Scattering
A graph, G = (V, E) is said to be an undirected graph where V = (v1 , v2 , ...) is a non empty set whose elements are called vertices (nodes, points) and E = (e1 , e2 , ...) is another set whose elements are called edges(lines) such that ek is identified with the pairs (vi , vj ) of vertices: vi vj = (vi , vj ) = ek . Directed Graph: A graph G = (V, E) in which every edge in E is directed, is called a directed graph or digraph [3]. We shall use more of digraphs. In an elastic scattering two particles a and b approach each other hit and go separated away without any new particle getting generated. Energy and momentum conservation laws concerning, angular correlation etc. are to be taken care of. Transfer of these dynamic variables could play a major role in scattering experiments out of which we can draw useful conclusions about the nature of the actual process being gone through[9]. We can picture these three possibilities: a + b −→ a + b(straight(s − Channel)) a+a ¯ −→ b + ¯b(exchange(t − Channel)) a + ¯b −→ a + ¯b(cross(u − Channel)) 1
We thank Her Holiness Mata Amriatnandamayi Devi (Amma) for providing facilities for research and her unfailing inspiration. 2 Email: karuppath.gmail.com 3 Email: [email protected]
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
45
We have inelastic scattering, when final, outgoing particles are different from ingoing ones. As a specific example we give the case of Photo production of EtaMeson (549MeV)[1], that is, (1)
γ + p −→ p + η
The three figures below (1a, 1b and 1c) illustrate the same where N is nucleon (=proton), γ = photon, ρ =rho meson, ω = omega meson N * =pion nucleon resonance.
Fig.1a
Fig.1b
Fig.1c Pole Diagrams for η-Meson Photo production
46
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
A fourth order Feynman graph contributing to the Delbruck effect, which is a photon-photon scattering interaction. The interaction between two photons is governed by the Feynman-graph with one electron loop. There are four e-γ vertices. Hence the γ −γ cross-section is fourth order in the fine struc2
2
4
e e ture constant α = c → c = 1/137 There are four e-γ vertices;1,2,3 and 4. The cross section decreases rapidly when the photon energy ω is smaller than electron rest energy mc2 by a factor (ω/mc2 )6 In practice, the γ − γ dissipation is negligible and this is why we can observe optical interference at a macroscopic level. This is an excellent example of Feynman’s approach.
Fig.2 Photon-Photon Scattering
3
Rules for Feynman diagrams
The S -matrix (Scattering matrix) can be written as ∞ (−1)n .. dx1 ...dxn (2) S =1+ n! n=1 (3)
T H1 (x1 )...H1 (xN )
(4)
whereH1 (x) = (g/2)[ψ (x), ψ in (x)]φin (x)(1)
in
with T the time ordering symbol. We need to compute the matrix elements of S between two states with a given number of incoming particles. In each term there are creation and annihilation operators taking care of changes in particle numbers. Other operators are ”paired off” to create and annihilate
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
47
virtual particles, each pair giving rise to a factor of the following form: 1 T [φin (x)φin (x1 )] = ΔF (x1 − x), (5) 2 1 (6) T [ψ¯in (x)ψ in (x1 )] = SF (x1 − x), with 2 dkei px 2 1 ΔF (x) = (7) i (2π)4 k 2 + m2 − iE dp(iγp − M )ei px 2 1 (8) SF (x) = i (2π)4 p2 + M 2 − iE (field quantization) A convenient way to visualize the result of a calculation outlined above is obtained with aid of Feynman diagrams. They help us as pictorial representations of particles propagating from one space-time point to another. In such a diagram one represents each particle in the initial or in the final state by ’external line’ and each virtual particle by an ’internal line’. The lines connect various vertices in the diagram where one vertex is introduced for each variable of integration in (9). Each particle like meson or nucleon is distinguished by different kind of lines. After some analysis we find the following rules for the ’translation’ of a Feynman diagram to an analytic formula with which we calculate: a) An internal meson line between points xi and xj corresponds to a factor 1 Δ (xj − xi ) 2 F b) An internal nucleon line from the point x1 to the point xj correspond to a factor of −1 S (xj − xi ) 2 F √ c) An external meson time corresponds to a factor < 0|φin (x)|k >= eik / 2ωV if it describes an incoming meson with momentum k and to a factor √ < k|φin (x)|0 >= eik / 2ωV if it corresponds to a meson with momentum k with the final state. An Analogous results hold for external nucleon lines with the corresponding factors containing solution u(±) we get, √ (9) < 0|ψ in (x)|q >= u(+)(r) (q)eiqx / V for a nucleon in the initial state and (10)
√ in < q|ψ (x)|0 >= overlineu(+)(r) (q)eiqx / V
for a nucleon in the final state, When anti-nucleons are involved we replace the function u(+) by u(−) . d) For a particular interaction we consider here that each vertex corresponds to a factor gF and joins one meson line. In general the number of lines at each vertex is the same as the number of operators appearing in the basic interaction Hamiltonian H1 (x).
48
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
e) The whole expression is multiplied by a certain sign factor. With these prescriptions we can perform integration on xj explicitly. Results contain several δ-functions describing conservation of energy and momentum at each vertex separately. Internal lines are effectively represented by ’energy denominator’ of the form (propagator) K = (Q2 + m2 − iε)−1 (11) for scalar particles where Q is some linear combinations of external energy momentum vectors and internal variables of integration. Feynman’s approach to particle scattering (elastic or inelastic) interactions is simple, elegant and computation-friendly and so is quite popular in general practice. Scattering and Transition amplitude calculations are effectively carried out using the Feynman techniques, the partial representation being done with the help of Feynman graphs. Feynman graphs / Diagrams
Fig.3 Propagators for the propagation of a particle from point P (x0 , t0 ) to point Q (x,t): Diagrams (A), (B) and (C) depict the 0th , 1st and 2nd order contribution to the propagator. H 1 is the vertex function. G is propagator obeying definite rules. x : spacial position parameter, t: time parameter. These can be discussed with particular reference to individual processes.
4
Comments on Quantum Mechanics
Wave particle duality, indeterminism, uncertainty principle, correspondence principle, complementarity theory, matter waves can be considered the initial inputs into the theoretical foundations of QM [4]. With their support several experimentally observed phenomena like Photoelectric Effect, Compton Effect, Line Spectra and Atomic Structure etc. could be elucidated. So far there is no clear cut indications to show that the fundamental concepts and
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
49
methods of QM are incorrect or found wanting. Nevertheless we may expect in future more satisfactory theoretical formulations for the fundamental science of matter and radiation. Feynman’s outstanding contributions in the field of fundamental particles and their interactions with techniques such as the Feynman diagram are basically founded on these important steps of development in QM.
5
Symmetry and Feynman Graphs
Certainly, the power of visual representation is immeasurable. In viewing the geometric features of worldly objects symmetry plays a pre-eminent role. There is, indeed, a lot to think in terms of symmetry-related topics in nature. Symmetries are of various kinds, in space, in time and in space-time. Geometrical symmetry has been an active area for long like material symmetry, chirality, utility of imprecision, degrees of symmetry and so on as can be seen by the Nobel Prizes awarded in physics for works related to symmetry. The well known prediction of parity violation by Lee and Yang has been of paramount importance in particle physics[10]. The Physics Nobel Prize-2008 was shared by Nambu for his contributions to spontaneous broken symmetry in sub-atomic physics and Kobayashi with Maskawa for the discovery of the origin of broken symmetry. These have tremendous impact on our knowledge about nature as a whole for which Feynman’s graph theoretic approach for portraying dynamic symmetries has contributed a good deal.
6
Space-Time symmetry and CPT theorem
The wisdom of Dirac, Feynman and others is reflected in the formulation of schemes in representing fundamental (elementary) particles and their basic interactions scattering (elastic or inelastic), production, decay etc. in the space-time of nature. Feynman made the important observation. If in a Feynman diagram particles are represented by forward arrows then antiparticles are the one with backward arrows. Thus an electron moving in forward direction in time is equivalent to a positron traveling backward in time [2,7,5]. The reversal of arrows would mean applying the C, P and T operations on the system (where C,P and T mean the usual Charge conjugation, Parity operation and time reversal operators). The symmetry here would prove the CPT operations. The fact that the experimental and theoretical values with unprecedented precision proves the CPT theorem to be one of the strongest principles in physics. For example calculated and experimental value the well
50
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
known Dirac’s number [6].
7
Summary and Remarks
Of all Feynman’s creations his creations, the simple but most tellingly elegant, effective is the graph or diagram-representation of particles in physics. The vast areas of applications of Feynman’s contributions are fast increasing in varied directions including the nano-science and other areas. We emphasize for the first time that the CPT theorem is implied by the space-time symmetry of Feynman graphs. Several successful and precise calculations in QED and particle physics will vouch for the strength of CPT theorem making the experimental search for its weakness at the nano-level to be superfluous. One may have to look deeper if at all for a CPT violation than the present ones.
References [1] Achuthan, P and T.Chandramohan,Photoproduction of Eta Meson, JMPS, 10 (1976),125-140. [2] Brown,M.Laurie (Ed), ”Feynman’s Thesis- A new approach to Quantum Theory”, World Scientific, 2006. [3] Deo Narsingh,”Graph Theory”, Prentice Hall of India, 2005. [4] Feynman,R.P., ”Feynman Lectures on Physics (vol-3)”, Addison Wesley, 1965. [5] Feynman,R.P., ”Quantum Electrodynamics”, Springer, Frontiers in Physics, 1962. [6] Feynman,R.P., ”QED-The Strange Theory of Light and Matter”, Princeton Univ.Press,N.J., 2004. [7] Feynman, R.P., The Development of space time view of Quantum electrodynamics, Nobel lecture, (1965). [8] James Gleick, ”Genius-The Life and Science of Richard Feynman”, Vintage Books (Random House. Inc.) N.Y.,1993. [9] Gunnar Kallen, ”Elementary Particle Physics”, Addison Wesley, USA (1964) 536-8pp. [10] Morrison, Philip, The Overthrow of parity, Sci.Am, 196 (1957), 45-53.
Aspects of Feynman Graphs P.Achuthan 1,2 Department of Mathematics Amrita Vishwavidyapeetham (Deemed University) Coimbatore-641 105, India
Narayanankutty Karuppath 3 Department of Sciences (Physics Division) Amrita Vishwavidyapeetham (Deemed University) Coimbatore-641 105, India
Abstract Points (vertices) and lines (edges) can be compared to the particles and waves in nature. Graphical, visual representations of objects and processes possess many properties which are quite advantageous and practically useful. Richard P. Feynman, made copious applications of the graph-theoretic language, ideas and methods for understanding the micro-world of elementary particles and their interactions. Here, we discus certain significant features of Feynman graphs or diagrams in the modern context. We make a novel observation that the CPT theorem is embedded in the space-time concepts of Feynman graphs and hence the success and precision of some the calculations of QED are in fact testament to the correctness and accuracy of this theorem. Keywords: CPT Theorem, Symmetry, QED, Feynman graphs .
1571-0653/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2009.03.007
44
1
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
Introduction
Quantum Mechanics (QM), along with Relativity Theories has so far achieved much success in understanding nature, especially the micro-world. Many outstanding individuals and groups of scientists have contributed in this endeavor. In the galaxy of theoretical physicists of 1930’s, 1940’s and after is Richard Philip Feynman whose remarkable contributions are worthy of investigation and appreciation[8]. From QM and applications of quantum field theory to elementary particle physics Feynman’s works are marvelously substantial. In this paper we dwell on one of the ideas originated by him, namely Feynman Diagram / Feynman Graph which can effectively represent elementary particle interactions, starting from scattering matrix and depictions thereof. Feynman’s work reconciled QM with Einstein’s special theory of relativity. With Dirac’s relativistic wave equation QED was more consistent with Einstein’s work.
2
Scattering
A graph, G = (V, E) is said to be an undirected graph where V = (v1 , v2 , ...) is a non empty set whose elements are called vertices (nodes, points) and E = (e1 , e2 , ...) is another set whose elements are called edges(lines) such that ek is identified with the pairs (vi , vj ) of vertices: vi vj = (vi , vj ) = ek . Directed Graph: A graph G = (V, E) in which every edge in E is directed, is called a directed graph or digraph [3]. We shall use more of digraphs. In an elastic scattering two particles a and b approach each other hit and go separated away without any new particle getting generated. Energy and momentum conservation laws concerning, angular correlation etc. are to be taken care of. Transfer of these dynamic variables could play a major role in scattering experiments out of which we can draw useful conclusions about the nature of the actual process being gone through[9]. We can picture these three possibilities: a + b −→ a + b(straight(s − Channel)) a+a ¯ −→ b + ¯b(exchange(t − Channel)) a + ¯b −→ a + ¯b(cross(u − Channel)) 1
We thank Her Holiness Mata Amriatnandamayi Devi (Amma) for providing facilities for research and her unfailing inspiration. 2 Email: karuppath.gmail.com 3 Email: [email protected]
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
45
We have inelastic scattering, when final, outgoing particles are different from ingoing ones. As a specific example we give the case of Photo production of EtaMeson (549MeV)[1], that is, (1)
γ + p −→ p + η
The three figures below (1a, 1b and 1c) illustrate the same where N is nucleon (=proton), γ = photon, ρ =rho meson, ω = omega meson N * =pion nucleon resonance.
Fig.1a
Fig.1b
Fig.1c Pole Diagrams for η-Meson Photo production
46
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
A fourth order Feynman graph contributing to the Delbruck effect, which is a photon-photon scattering interaction. The interaction between two photons is governed by the Feynman-graph with one electron loop. There are four e-γ vertices. Hence the γ −γ cross-section is fourth order in the fine struc2
2
4
e e ture constant α = c → c = 1/137 There are four e-γ vertices;1,2,3 and 4. The cross section decreases rapidly when the photon energy ω is smaller than electron rest energy mc2 by a factor (ω/mc2 )6 In practice, the γ − γ dissipation is negligible and this is why we can observe optical interference at a macroscopic level. This is an excellent example of Feynman’s approach.
Fig.2 Photon-Photon Scattering
3
Rules for Feynman diagrams
The S -matrix (Scattering matrix) can be written as ∞ (−1)n .. dx1 ...dxn (2) S =1+ n! n=1 (3)
T H1 (x1 )...H1 (xN )
(4)
whereH1 (x) = (g/2)[ψ (x), ψ in (x)]φin (x)(1)
in
with T the time ordering symbol. We need to compute the matrix elements of S between two states with a given number of incoming particles. In each term there are creation and annihilation operators taking care of changes in particle numbers. Other operators are ”paired off” to create and annihilate
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
47
virtual particles, each pair giving rise to a factor of the following form: 1 T [φin (x)φin (x1 )] = ΔF (x1 − x), (5) 2 1 (6) T [ψ¯in (x)ψ in (x1 )] = SF (x1 − x), with 2 dkei px 2 1 ΔF (x) = (7) i (2π)4 k 2 + m2 − iE dp(iγp − M )ei px 2 1 (8) SF (x) = i (2π)4 p2 + M 2 − iE (field quantization) A convenient way to visualize the result of a calculation outlined above is obtained with aid of Feynman diagrams. They help us as pictorial representations of particles propagating from one space-time point to another. In such a diagram one represents each particle in the initial or in the final state by ’external line’ and each virtual particle by an ’internal line’. The lines connect various vertices in the diagram where one vertex is introduced for each variable of integration in (9). Each particle like meson or nucleon is distinguished by different kind of lines. After some analysis we find the following rules for the ’translation’ of a Feynman diagram to an analytic formula with which we calculate: a) An internal meson line between points xi and xj corresponds to a factor 1 Δ (xj − xi ) 2 F b) An internal nucleon line from the point x1 to the point xj correspond to a factor of −1 S (xj − xi ) 2 F √ c) An external meson time corresponds to a factor < 0|φin (x)|k >= eik / 2ωV if it describes an incoming meson with momentum k and to a factor √ < k|φin (x)|0 >= eik / 2ωV if it corresponds to a meson with momentum k with the final state. An Analogous results hold for external nucleon lines with the corresponding factors containing solution u(±) we get, √ (9) < 0|ψ in (x)|q >= u(+)(r) (q)eiqx / V for a nucleon in the initial state and (10)
√ in < q|ψ (x)|0 >= overlineu(+)(r) (q)eiqx / V
for a nucleon in the final state, When anti-nucleons are involved we replace the function u(+) by u(−) . d) For a particular interaction we consider here that each vertex corresponds to a factor gF and joins one meson line. In general the number of lines at each vertex is the same as the number of operators appearing in the basic interaction Hamiltonian H1 (x).
48
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
e) The whole expression is multiplied by a certain sign factor. With these prescriptions we can perform integration on xj explicitly. Results contain several δ-functions describing conservation of energy and momentum at each vertex separately. Internal lines are effectively represented by ’energy denominator’ of the form (propagator) K = (Q2 + m2 − iε)−1 (11) for scalar particles where Q is some linear combinations of external energy momentum vectors and internal variables of integration. Feynman’s approach to particle scattering (elastic or inelastic) interactions is simple, elegant and computation-friendly and so is quite popular in general practice. Scattering and Transition amplitude calculations are effectively carried out using the Feynman techniques, the partial representation being done with the help of Feynman graphs. Feynman graphs / Diagrams
Fig.3 Propagators for the propagation of a particle from point P (x0 , t0 ) to point Q (x,t): Diagrams (A), (B) and (C) depict the 0th , 1st and 2nd order contribution to the propagator. H 1 is the vertex function. G is propagator obeying definite rules. x : spacial position parameter, t: time parameter. These can be discussed with particular reference to individual processes.
4
Comments on Quantum Mechanics
Wave particle duality, indeterminism, uncertainty principle, correspondence principle, complementarity theory, matter waves can be considered the initial inputs into the theoretical foundations of QM [4]. With their support several experimentally observed phenomena like Photoelectric Effect, Compton Effect, Line Spectra and Atomic Structure etc. could be elucidated. So far there is no clear cut indications to show that the fundamental concepts and
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
49
methods of QM are incorrect or found wanting. Nevertheless we may expect in future more satisfactory theoretical formulations for the fundamental science of matter and radiation. Feynman’s outstanding contributions in the field of fundamental particles and their interactions with techniques such as the Feynman diagram are basically founded on these important steps of development in QM.
5
Symmetry and Feynman Graphs
Certainly, the power of visual representation is immeasurable. In viewing the geometric features of worldly objects symmetry plays a pre-eminent role. There is, indeed, a lot to think in terms of symmetry-related topics in nature. Symmetries are of various kinds, in space, in time and in space-time. Geometrical symmetry has been an active area for long like material symmetry, chirality, utility of imprecision, degrees of symmetry and so on as can be seen by the Nobel Prizes awarded in physics for works related to symmetry. The well known prediction of parity violation by Lee and Yang has been of paramount importance in particle physics[10]. The Physics Nobel Prize-2008 was shared by Nambu for his contributions to spontaneous broken symmetry in sub-atomic physics and Kobayashi with Maskawa for the discovery of the origin of broken symmetry. These have tremendous impact on our knowledge about nature as a whole for which Feynman’s graph theoretic approach for portraying dynamic symmetries has contributed a good deal.
6
Space-Time symmetry and CPT theorem
The wisdom of Dirac, Feynman and others is reflected in the formulation of schemes in representing fundamental (elementary) particles and their basic interactions scattering (elastic or inelastic), production, decay etc. in the space-time of nature. Feynman made the important observation. If in a Feynman diagram particles are represented by forward arrows then antiparticles are the one with backward arrows. Thus an electron moving in forward direction in time is equivalent to a positron traveling backward in time [2,7,5]. The reversal of arrows would mean applying the C, P and T operations on the system (where C,P and T mean the usual Charge conjugation, Parity operation and time reversal operators). The symmetry here would prove the CPT operations. The fact that the experimental and theoretical values with unprecedented precision proves the CPT theorem to be one of the strongest principles in physics. For example calculated and experimental value the well
50
P. Achuthan, N. Karuppath / Electronic Notes in Discrete Mathematics 33 (2009) 43–50
known Dirac’s number [6].
7
Summary and Remarks
Of all Feynman’s creations his creations, the simple but most tellingly elegant, effective is the graph or diagram-representation of particles in physics. The vast areas of applications of Feynman’s contributions are fast increasing in varied directions including the nano-science and other areas. We emphasize for the first time that the CPT theorem is implied by the space-time symmetry of Feynman graphs. Several successful and precise calculations in QED and particle physics will vouch for the strength of CPT theorem making the experimental search for its weakness at the nano-level to be superfluous. One may have to look deeper if at all for a CPT violation than the present ones.
References [1] Achuthan, P and T.Chandramohan,Photoproduction of Eta Meson, JMPS, 10 (1976),125-140. [2] Brown,M.Laurie (Ed), ”Feynman’s Thesis- A new approach to Quantum Theory”, World Scientific, 2006. [3] Deo Narsingh,”Graph Theory”, Prentice Hall of India, 2005. [4] Feynman,R.P., ”Feynman Lectures on Physics (vol-3)”, Addison Wesley, 1965. [5] Feynman,R.P., ”Quantum Electrodynamics”, Springer, Frontiers in Physics, 1962. [6] Feynman,R.P., ”QED-The Strange Theory of Light and Matter”, Princeton Univ.Press,N.J., 2004. [7] Feynman, R.P., The Development of space time view of Quantum electrodynamics, Nobel lecture, (1965). [8] James Gleick, ”Genius-The Life and Science of Richard Feynman”, Vintage Books (Random House. Inc.) N.Y.,1993. [9] Gunnar Kallen, ”Elementary Particle Physics”, Addison Wesley, USA (1964) 536-8pp. [10] Morrison, Philip, The Overthrow of parity, Sci.Am, 196 (1957), 45-53.