*M. S. Sozzi*

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199296668
- eISBN:
- 9780191712074
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296668.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter reviews the general theory of transformations in quantum theory. Topics covered include symmetry and invariance, transformations and symmetries in quantum theory, symmetries and ...
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This chapter reviews the general theory of transformations in quantum theory. Topics covered include symmetry and invariance, transformations and symmetries in quantum theory, symmetries and conservation laws, discrete transformations and inversions, and symmetry violations. Suggested further reading and exercise problems are provided at the end of the chapter.Less

This chapter reviews the general theory of transformations in quantum theory. Topics covered include symmetry and invariance, transformations and symmetries in quantum theory, symmetries and conservation laws, discrete transformations and inversions, and symmetry violations. Suggested further reading and exercise problems are provided at the end of the chapter.

*Chun Wa Wong*

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199641390
- eISBN:
- 9780191747786
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641390.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Einstein's derivation of the Lorentz transformation between space and time coordinates is given. The resulting relativistic kinematics of £ Lorentz spacetime and momentum-energy vectors is readily ...
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Einstein's derivation of the Lorentz transformation between space and time coordinates is given. The resulting relativistic kinematics of £ Lorentz spacetime and momentum-energy vectors is readily applied to practical problems by using their scalar products that are Lorentz-invariant, the same in every Lorentz frame. Dirac went the other way by expanding the Lorentz quadratic invariant operator in the linear wave equation back into their original square-root vectors. The resulting Dirac spinor wave functions give full access to the rich spacetime properties of the wave motion in the square-root space. These properties include their spacetime symmetries, and the breaking of these symmetries by using other types of spinor wave functions. The connection is also made with Cartan's complexified square-root coordinates. Multidimensional arrays of spatial and spacetime vectors can be constructed and used. These objects include dyadics, Cartesian tensors and general tensors. Some of their useful properties are described.Less

Einstein's derivation of the Lorentz transformation between space and time coordinates is given. The resulting relativistic kinematics of £ Lorentz spacetime and momentum-energy vectors is readily applied to practical problems by using their scalar products that are Lorentz-invariant, the same in every Lorentz frame. Dirac went the other way by expanding the Lorentz quadratic invariant operator in the linear wave equation back into their original square-root vectors. The resulting Dirac spinor wave functions give full access to the rich spacetime properties of the wave motion in the square-root space. These properties include their spacetime symmetries, and the breaking of these symmetries by using other types of spinor wave functions. The connection is also made with Cartan's complexified square-root coordinates. Multidimensional arrays of spatial and spacetime vectors can be constructed and used. These objects include dyadics, Cartesian tensors and general tensors. Some of their useful properties are described.

*Steven E. Vigdor*

- Published in print:
- 2018
- Published Online:
- March 2018
- ISBN:
- 9780198814825
- eISBN:
- 9780191852954
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198814825.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Chapter 1 introduces central themes of the book by dealing with the importance of symmetry principles in physics and of small symmetry violations (or “flaws”) in establishing conditions for life. It ...
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Chapter 1 introduces central themes of the book by dealing with the importance of symmetry principles in physics and of small symmetry violations (or “flaws”) in establishing conditions for life. It provides a visual analog of the combined symmetries of parity (P) and charge conjugation (C) via a drawing by M.C. Escher. Escher’s signature represents a small imperfection in the drawing’s otherwise nearly perfect symmetry, inspiring the book’s title. This chapter argues that a well-tuned violation of the combined CP symmetry (i.e., a CP violation) is necessary to establish a preference for matter over antimatter (i.e., matter–antimatter asymmetry), while still allowing for a long-lived universe that resists gravitational collapse. The chapter establishes the book’s focus on the intricate tapestry of inventive experiments physicists are pursuing to expose and quantify the extent of these tiny flaws.Less

Chapter 1 introduces central themes of the book by dealing with the importance of symmetry principles in physics and of small symmetry violations (or “flaws”) in establishing conditions for life. It provides a visual analog of the combined symmetries of parity (P) and charge conjugation (C) via a drawing by M.C. Escher. Escher’s signature represents a small imperfection in the drawing’s otherwise nearly perfect symmetry, inspiring the book’s title. This chapter argues that a well-tuned violation of the combined CP symmetry (i.e., a CP violation) is necessary to establish a preference for matter over antimatter (i.e., matter–antimatter asymmetry), while still allowing for a long-lived universe that resists gravitational collapse. The chapter establishes the book’s focus on the intricate tapestry of inventive experiments physicists are pursuing to expose and quantify the extent of these tiny flaws.

*Steven E. Vigdor*

- Published in print:
- 2018
- Published Online:
- March 2018
- ISBN:
- 9780198814825
- eISBN:
- 9780191852954
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198814825.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This book provides a nonmathematical survey of the past half-century of research in particle physics, nuclear physics, and cosmology bearing on the physical conditions that allow our universe to ...
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This book provides a nonmathematical survey of the past half-century of research in particle physics, nuclear physics, and cosmology bearing on the physical conditions that allow our universe to support the development of structure and the origins of life. These conditions rely on a surprising number of tiny imperfections—deviations from perfect symmetry (i.e., symmetry violations), homogeneity, or predictability—that seem mysteriously fine-tuned. The emphasis here is on the intricate tapestry of elegant experiments that have revealed and quantified these imperfections, as well as on theoretical efforts to understand how the imperfections arose in the infant universe. Among the topics covered are: the dominance of matter over antimatter (i.e., matter–antimatter asymmetry); the existence and intermixing of three generations of quarks and leptons; the stability of hydrogen and synthesis of other elements essential for life; the longevity and energy budget of the universe; the remaining mysteries surrounding dark matter, dark energy, and the postulated inflationary expansion of space in the infant universe; the fundamental role of randomness in quantum mechanics, in generating the first biomolecules and in biological evolution; the apparent perching of the vacuum state in our universe on the edge between stability and meta-stability; and philosophical questions, including the possibility of a multiverse, surrounding the interpretation of a universe that exhibits such fine-tuning. On all of these issues, the book clarifies what we know and how we know it, as distinct from what we speculate and how we might test it.Less

This book provides a nonmathematical survey of the past half-century of research in particle physics, nuclear physics, and cosmology bearing on the physical conditions that allow our universe to support the development of structure and the origins of life. These conditions rely on a surprising number of tiny imperfections—deviations from perfect symmetry (i.e., symmetry violations), homogeneity, or predictability—that seem mysteriously fine-tuned. The emphasis here is on the intricate tapestry of elegant experiments that have revealed and quantified these imperfections, as well as on theoretical efforts to understand how the imperfections arose in the infant universe. Among the topics covered are: the dominance of matter over antimatter (i.e., matter–antimatter asymmetry); the existence and intermixing of three generations of quarks and leptons; the stability of hydrogen and synthesis of other elements essential for life; the longevity and energy budget of the universe; the remaining mysteries surrounding dark matter, dark energy, and the postulated inflationary expansion of space in the infant universe; the fundamental role of randomness in quantum mechanics, in generating the first biomolecules and in biological evolution; the apparent perching of the vacuum state in our universe on the edge between stability and meta-stability; and philosophical questions, including the possibility of a multiverse, surrounding the interpretation of a universe that exhibits such fine-tuning. On all of these issues, the book clarifies what we know and how we know it, as distinct from what we speculate and how we might test it.