Objectives & Background
When one begins any physics course, at the graduate or undergraduate level, one is not given the feeling that it is a coherent theory resting upon a few principles and experimental observations. Nor is one clearly given the gaps and outstanding problems in each domain of knowledge. Rather one is given more of the approach “Learn this collection of equations, problems, and solutions, and the truth will somehow emerge.” I will try to do better - to show both the trees and the forest.
The objective of this course is to lay out the laws of fundamental physics with a very strong emphasis on the theory of continuous groups and the associated mathematics, and to attempt to identify the assumptions and foundational structure of each domain of physics. The presentation will be more of an operational approach rather than one that is mathematically rigorous. It is better to first understand the motivation and to learn the operational techniques, and then to study a more formal axiomatic mathematical approach. This same approach is shown in excellent form in Messiah’s book on Quantum Mechanics where a few chapters introduce the student to the experimental and historical emergence of quantum mechanics. Messiah then lays out the mathematical and the physical foundations and proceeds to explore the consequences with applications in various domains of physics.
Certainly each domain of physics is not ‘complete’. But what we can do for the student is to try to emphasize the assumptions, rules, results, and shortcomings. We can also try to show the integration of the various fields in a clearer way. We will do this using Lie groups and algebras and related areas of mathematics.
In fact, mathematics, as a language for our physical world, is not yet complete. We still need to use words and even then we still need more new concepts, structures, and interpretations such as set theory, logic, and even areas of mathematics that are not yet on a firm foundation. The hope that all of mathematics can rest on a firm foundation of logic has been proven impossible. This fact constitutes our first encounter with incomplete information or uncertainty – a concept that will reoccur throughout these lectures and which we will revisit extensively in the last lectures where we link Lie Algebras with information theory. With Wigner, we are amazed that our universe is described so well by our mathematics. Have we somehow encapsulated our intuition of the world in our mathematical structures? It will become apparent very quickly to the student that they have been studying Lie algebras and Lie groups in some contexts without even recognizing them, usually in the form of differential equations and special functions. This course will pull together that past work into a cohesive structure.
Let me suggest some references. For general reading on a leisurely day we suggest the four volume set: The World of Mathematics. For up to date articles, Scientific American, The American Journal of Physics, and Physics Today. Find some free time and read through the current year of each of these, then the previous year for about 5 years. Science and Nature provide more difficult articles. For foundational reference, one should refer to standard calculus level physics texts such as Holliday & Resnick or Serway (which are naturally assumed to be known by the student as a requirement for this course). Feynman’s lectures on Physics are also excellent. One should likewise refer to solid Calculus texts such as Taylor’s Calculus and Analytical Geometry and Kaplan’s Advanced Calculus both of which are also assumed for this course. A good knowledge of all of these texts are prerequisites for this course. Texts for the course include but are not limited to the following: We will provide a general mathematical background following Birkhoff & MacLane’s Survey of Modern Algebra. A solid text on Mathematical Physics is the one by Arfken & Weber. Other core texts include Goldstein’s Classical Mechanics, Mesiah’s Quantum Mechanics, Hammermesh’s Group Theory, Schweber’s Advanced Quantum Mechanics, Gelfand & Neimarks books on the Lorentz group and Jauch & Rohrlichs book on relativistic electromagnetic theory. The entire series of texts from Landau and Liftshtz are excellent. We will use perhaps 10 different additional references for our work on group theory and Lie algebras.
Some remarks on units are in order. The notation for this course is: SI units (MKS) with right hand rules and coordinate systems. An indefinite time-space metric +--- is used throughout. Systems that use ‘ict’ to hide the indefinite metric of relativity should be avoided at all costs as they confuse the mathematical structure and cause extensive problems when one moves to relativistic quantum mechanics. Both Dirac and index notation are used for vectors & tensors.
Many new exciting areas of mathematics are emerging including fractals, non-integral dimensionality, wavelet analysis, and complexity theory all of which have made important contributions in the last few years. Other new areas involve the structure of computational science itself. These are all very divergent and nontraditional approaches. We need to keep an open mind about new tools that we can use to understand, model, and predict the universe and its component structures. For example look at the book “A New Kind of Science” by Steven Wolfram.
All of our scientific numerical values are expressed in terms of certain fundamental scientific units – mass, length, time, electrical current, - the standard units and concepts. In essence, the unit is used to provide a reference scale to the value measured, for example, how many units of meters will fit between this point and another point. Are there other units? Certainly there are: information, charm, strangeness, and isospin. Often we think of taking Plank’s constant, h, and the velocity of light, c, to be one in order to naturally ‘fix’ the scale of the universe but we need one more unit. Here, one might take G to be 1 to complete the scale of observables corresponding to real numbers (as opposed to the others which are quantized as integers). Although this determines the fundamental set, it has the problem of singling out the gravitational force as special when in fact it is the weakest and least well incorporated into the quantum theory of observables. We can wonder if the fundamental units are one-to-one with the fundamental observables which are in turn one-to-one with the fundamental observables in a Lie algebra. But this path is not clear. But obviously, units provide part of the ‘meta’ information about an observable.
Information itself as an observable is defined as the log of a probability, as is entropy according to the works of Boltzmann & Shannon. Human senses are the logs of physical quantities (frequency, loudness, touch, & brightness) thus allowing us a wide range of responses. It is as though evolution figured out how to bring in the maximum information to our brain by taking the log of the physical quantity!! But how do we measure the information that is obtained in the quantum measurement of an observable. What is the definition of information and how is it related to uncertainty and entropy. What are the tradeoffs in information among observables and can this always be expressed as commutation relations in a Lie algebra of observables? Also, how is this related to numerical uncertainty and logical uncertainty in the mathematical domain? Can an improved mathematical theory of numerical uncertainty and information theory be used to help us with related foundational questions in physics and in particular physics defined in terms of Lie algebras? Is information (or equivalently entropy) an observable in a Lie algebra like momentum and position (the space-time algebra) or an observable in the internal symmetry group with charge and charm? Is it even a linear operator? Can the theory of Lie algebras give us insight into the concepts of information and order, and of entropy and disorder? How might such theories of information relate to quantum theory and the uncertainty principle?