This book provides an in-depth and accessible description of special relativity and quantum mechanics which together form the foundation of 21st century physics. A novel aspect is that symmetry is given its rightful prominence as an integral part of this foundation. The book offers not only a conceptual understanding of symmetry, but also the mathematical tools necessary for quantitative analysis. As such, it provides a valuable precursor to more focused, advanced books on special relativity or quantum mechanics. Students are introduced to several topics not typically covered until much later in their education.These include space-time diagrams, the action principle, a proof of Noether's theorem, Lorentz vectors and tensors, symmetry breaking and general relativity. The book also provides extensive descriptions on topics of current general interest such as gravitational waves, cosmology, Bell's theorem, entanglement and quantum computing. Throughout the text, every opportunity is taken to emphasize the intimate connection between physics, symmetry and mathematics.The style remains light despite the rigorous and intensive content. The book is intended as a stand-alone or supplementary physics text for a one or two semester course for students who have completed an introductory calculus course and a first-year physics course that includes Newtonian mechanics and some electrostatics. Basic knowledge of linear algebra is useful but not essential, as all requisite mathematical background is provided either in the body of the text or in the Appendices. Interspersed through the text are well over a hundred worked examples and unsolved exercises for the student.

This is a textbook that derives the fundamental theories of physics from symmetry. It starts by introducing, in a completely self-contained way, all mathematical tools needed to use symmetry ideas in physics. Thereafter, these tools are put into action and by using symmetry constraints, the fundamental equations of Quantum Mechanics, Quantum Field Theory, Electromagnetism, and Classical Mechanics are derived. As a result, the reader is able to understand the basic assumptions behind, and the connections between the modern theories of physics. The book concludes with first applications of the previously derived equations. Thanks to the input of readers from around the world, this second edition has been purged of typographical errors and also contains several revised sections with improved explanations.

A pedagogical introduction to the modern applications of groups, algebras, and topology for undergraduate and graduate students in physics.

This is the third volume in the Single Monad Model of the Cosmos series. The second volume introduced the Duality of Time Theory, which provided elegant solutions to many persisting problems in physics and cosmology, including super-symmetry and matter-antimatter asymmetry. In addition to uniting the principles of Relativity and Quantum theories, this theory can also explain the psychical and spiritual domains; all based on the same discrete complex-time geometry. Super-symmetry, and quantum gravity, are realized only with the two complementary physical and psychical worlds, while the spiritual realm is governed by hyper-symmetry, which mirrors the previous two levels together, and all these three realms mirror the ultimate level of absolute oneness that describes the symmetry of the divine presence of God and His Beautiful Names and Attributes. This "ULTIMATE SYMMETRY" is a modern scientific account of the same ancient mystical, and greatly controversial, theory of the "Oneness of Being" that is often misinterpreted in terms of "pantheism", but it is indeed the concluding gnostic knowledge of God and creation. Otherwise, how can we understand the origin of the cosmos, with both or either of its corporeal and incorporeal realms, without referring to its Originator! In the literal sense, ultimate or perfect symmetry may seem to be trivial, because it means that all possible transformations in such a symmetric system are invariant. The system we are talking about here is the whole Universe that we are watching and experiencing its immense and sometimes shattering changes every moment of time. Yet many great philosophers, such as Parmenides and Ibn al-Arabi, maintained their firm belief that reality is unchanging One and existence is timeless and uniform, while all apparent changes are mere illusions induced by or in our sensory faculties. Nevertheless, since we are living inside it, this illusion is as good as reality for us. Therefore, we still need to explain how the Universe is being formulated. Only when are able to transcend beyond the current chest of time, we shall discover that we were living a dream, and we shall be able to see the whole Universe as unchanging symmetry. The Single Monad Model and the resulting Duality of Time Theory provide the link between this apparent dynamic multiplicity of creation and the ultimate metaphysical oneness. In fact, the complex-time geometry concludes that we are imagining the reality because we are observing it from a genuinely imaginary time dimension. Since the ultimate reality is One, we cannot view it from outside, because there is none! Thus, as we quoted in the Introduction, in the Book of Theophanies, Ibn al-Arabi ascribes to God as saying: Listen, O My beloved! I am the conclusive entity of the World. I am the center of the circle (of existence) and its circumference. I am its simple point and its compound whole. I am the Word descending between heaven and earth. I have created perceptions for you only to perceive Me. If you then perceive Me, you perceive yourself. But don't ever crave to perceive Me through yourself! It is through My Eyes that you see Me and see yourself. But through your own eyes you can never see Me! This Theophany of Perfection summarizes the Ultimate Symmetry between the single point and the encompassing space. It also summarizes the instantaneous process of creation, or re-creation, which is breaking this symmetry into the two arrows of time, that produce particles and anti-particles, and then restoring it through each subsequent annihilation. This reunion is also the fundamental cause of motion, which is formulated as the Principle of Love that leads to the stationary action that is the initial assumption of most physics theories including Relativity and Quantum Field theories.

This is a book about representing symmetry in quantum mechanics. The book is on a graduate and/or researcher level and it is written with an attempt to be concise, to respect conceptual clarity and mathematical rigor. The basic structures of quantum mechanics are used to identify the automorphism group of quantum mechanics. The main concept of a symmetry action is defined as a group homomorphism from a given group, the group of symmetries, to the automorphism group of quantum mechanics. The structure of symmetry actions is determined under the assumption that the symmetry group is a Lie group. The Galilei invariance is used to illustrate the general theory by giving a systematic presentation of a Galilei invariant elementary particle. A brief description of the Galilei invariant wave equations is also given.

The first two chapters of the book deal, in a detailed way, with relativistic kinematics and dynamics, while in the third chapter some elementary concepts of General Relativity are given. Eventually, after an introduction to tensor calculus, a Lorentz covariant formulation of electromagnetism is given its quantization is developed. For a proper treatment of invariance and conservation laws in physics, an introductory chapter on group theory is given. This introduction is propedeutical to the discussion of conservation laws in the Lagrangian and Hamiltonian formalism, which will allow us to export this formalism to quantum mechanics and, in particular, to introduce linear operators on quantum states and their transformation laws. In the last part of the book we analyze, in the first quantized formalism, relativistic field theory for both boson and fermion fields. The second quantization of free fields is then introduced and some preliminary concepts of perturbation theory and Feynmann diagrams are given and some relevant examples are worked out.

Structured as a dialogue between a mathematician and a physicist, Symmetry and Quantum Mechanics unites the mathematical topics of this field into a compelling and physically-motivated narrative that focuses on the central role of symmetry. Aimed at advanced undergraduate and beginning graduate students in mathematics with only a minimal background in physics, this title is also useful to physicists seeking a mathematical introduction to the subject. Part I focuses on spin, and covers such topics as Lie groups and algebras, while part II offers an account of position and momentum in the context of the representation theory of the Heisenberg group, along the way providing an informal discussion of fundamental concepts from analysis such as self-adjoint operators on Hilbert space and the Stone-von Neumann Theorem. Mathematical theory is applied to physical examples such as spin-precession in a magnetic field, the harmonic oscillator, the infinite spherical well, and the hydrogen atom.