Group Theory in Particle, Nuclear, and Hadron Physics

Syed Afsar Abbas

• Publisher: CRC Press
• Year: 2016
• ISBN:
  9781315354446
• Edition: 1
• Topic: Matematica
• Subtopic: Aritmetica

This user-friendly book on group theory introduces topics in as simple a manner as possible and then gradually develops those topics into more advanced ones, eventually building up to the current state-of-the-art. By using simple examples from physics and mathematics, the advanced topics become logical extensions of ideas already introduced. In addition to being used as a textbook, this book would also be useful as a reference guide for graduates and researchers in particle, nuclear and hadron physics.

Chapter 1

Basic Symmetry Concepts

1.1 Symmetries Everywhere

1.2 Elementary Concepts of Symmetry

1.3 Kepler’s Construction: A Mistake?

1.4 Symmetries and Conservation Laws

1.5 Appendix A: Mathematics as the Language of Nature

1.6 Solutions of Problems

1.1 Symmetries Everywhere

Prior to embarking on our odyssey to comprehend the abstract expressions and finer intricacies of group theory, we shall first analyze the simpler concepts of symmetry.

Now, humans are endowed with an intuitive understanding of what “symmetry” denotes. All around we are literally surrounded by objects – some display no features of symmetry whatsoever, while others exhibit clear evidence of symmetry that is intrinsically recognized by humans [1]. In fact, during the course of our evolution as a species, symmetry has grown intimately intertwined with the very concept of beauty itself [2]. Whatever is symmetric in nature appears beautiful to the human eye.

Little wonder then, that symmetry manifests itself in all fields of human creative enterprise, be it in art, architecture, music or poetry. A few prominent examples from architecture would suffice to illustrate this point:

- From about 2500 B.C. to 500 B.C. the Cyclopean Ziggurats of Ur and the Gargantuan Towers of Babylon in Ancient Iraq which transfixed all travellers of yore were gloriously symmetrical structures testifying to the architectural genius of Greater Middle Eastern civilization.

- The Blue Mosque of Istanbul in Turkey (1609 – 1617), also known as the Sultan Ahmet Mosque, was built by the master Turco-Islamic architect Mehmet during the magnificent Ottoman Caliphate. A splendid structure towering over the mighty Bosphorus, it is the only Istanbul mosque containing six symmetrically placed minarets – in marked contrast to the more common quartet.

- The Peruvudaiyar Kovil Temple in Thanjavur, Tamil Nadu (1010 AD) is the tallest temple in the world. A masterpiece of the highly distinctive and monumentalist Dravida school of architecture, it exhibits countless striking symmetries, right from the large-scale architecture of its colossal Gopuram to the intricate carvings decorating its wonderful walls. This masterpiece demonstrates the hoary antiquity of the indigenous Shaivite religion, as well as the astonishing continuity of the present Dravidian civilization.

- The Russian Cathedral of St. Basil the Blessed in the Red Square of Moscow is a most remarkable and eye-catching structure of unique beauty. The bright enlivening colours which appear interwoven into an intricate tapestry of geometrical symmetry has struck visitors as one of the most breathtaking and awe-inspiring creations of the Oriental Slavo-Byzantine civilization [2].

- Finally, mention must be made of the glorious Taj Mahal of Agra in Oudh [3], constructed by the powerful rulers of the Later Timurid Caliphate, who are occasionally mistakenly termed as “Mughals”. Often considered the most beautiful structure ever created by man, various layers of symmetry are evident right from the opulent Islamic domes towering into the Awadhi sky, through the whirling Perso-Arabic inscriptions adorning the translucent marble walls, down to the precious Kashmiri sapphires, Rajastani emeralds and Golcunda diamonds which once adorned its splendid surface.

In the realm of decorative arts, one may mention the works of the prolific Nordic-Dutch lithographer Maurice Escher [4]. Including dazzling outpourings of geometrical splendour exhibiting a perplexing interplay of different symmetries and optical illusions such as “Air and Water” (1938) or “Ascending and Descending” (1960), his Netherlandish productions have a special appeal for physicists and mathematicians.

The performing arts also exhibit a marked fondness for symmetry. For example, poets have always waxed eloquent about the beauty of symmetry and have always turned lyrical regarding the rhythm of rhyme. For example, a poem written by Su Dongpo (eleventh century) in China, consisting of eight vertical lines of seven characters each. The poem can be read forward and backward each time with correct rhyme and meter. In this context, one must remember that the hoary Chinese language is read commencing from right, top-down a column, and then down along the second column and so forth.

Musicians too are not far behind in seeking beauty from symmetry. For example, the Crab Canon of the Nordic-German musician J. S. Bach (seventeenth century) is a violin duet where each voilin’s music is a time reversed-version of that of the other violin.

So clearly symmetry and its intrinsic beauty has been the focus of much human creative endeavour among artists. The same can be said of scientists too. The first motivation, more as an act of intellectual satisfaction, is for seeking beauty for its own sake

Let us recollect the Nordic-British poet John Keats’ final lines in his 1819 poem “Ode on a Grecian Urn”

Beauty is truth, truth beauty, that is all
Ye know on earth, and all ye need to know.

A further quote from Paul Dirac serves to emphasise the fascination physicists have with symmetry [5]: “It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress.”

In fact, the search for symmetry in nature – with symmetry being sought for its own sake – has been found to be an extremely fruitful enterprise by scientists. First we study symmetry as a scientific concept, and thereafter move over to group theory, which is the mathematical language that incorporates symmetry in a consistent manner.

1.2 Elementary Concepts of Symmetry

To understand symmetry, let us consider figures drawn on a page. Symmetry of a figure exists if the shape it represents does not change in size or layout when subjected to a rotation or reflection within itself. These changes are termed transformations. A “no-change” is indeed a transformation of a special kind itself, and is denominated as an identity. Just as multiplication by unity does not alter the value of a quantity, so in the case of symmetry, the value of symmetry does not change when the identity participates.

In Figure 1.1(a),(b), the figure (denoting a neck – tie) before and after the transformation appears the same. Hence, it is symmetric under the reflection transformation.

Note that a rotation by 6π3=2π returns Figure 1.1(c) to the original configuration. So the cycle is repeated. All the more reason that the “identity” is a transformation. Note that points 1, 2, 3 are external labels (not existing on the figure itself) and hence the figures under all the above three transformations Figure 1.1(c),(d),(e) – appear the same, or are in fact identical. Hence these entities are symmetric under these transformations.

Figure 1.2(a) provides a rough idea of symmetry. Essentially, symmetry involves some entity. This entity retains its identity when a...