"A physical system is said to possess a symmetry if one can make a change in the system such that, after the change, the system is exactly the same as it was before. We call the change we are making to the system a symmetry operation or a symmetry transformation. If a system stays the same when we do a transformation to it, we say that the system is invariant under the transformation." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"Although nature suggests a pathway to a mathematical description of everything, it has thus far eluded a final or complete grand mathematical synthesis. […] Mathematics is therefore inspired by nature. But it does not have to conduct experimental observations to proceed. The worlds of mathematics and theoretical physics are therefore distinct - they have different 'mission statements'. Whereas theoretical physics maps the properties of the nature we experience, mathematics builds a map of all possible 'natures' that logic permits to exist." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"Art imitates nature
through the incorporation of discrete symmetries, and indeed, reflection symmetry
can be found throughout nature."
"Eternal constancy of
the laws of physics is a symmetry. What we see as we look back in time, or we
peer through telescopes out into space, or we look through our powerful
microscopes (particle accelerators), is the same system of laws of physics
governing the whole universe at all times and at all places. These are the
basic symmetries of the structure of our universe and its contents and, at a
deeper level, the symmetries of the laws that govern the universe themselves.
Indeed, the symmetries we uncover are the basic principles that define our laws
of nature and the laws of physics, hence those that control our universe."
"General relativity
explains gravitation as a curvature, or bending, or warping, of the geometry of
space-time, produced by the presence of matter. Free fall in a space shuttle
around Earth, where space is warped, produces weightlessness, and is equivalent
from the observer's point of view to freely moving in empty space where there
is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a
straight-line motion over small distances. But it becomes a curved trajectory
when viewed at large distances. This is what produces the closed elliptical
orbits of planets, with tiny corrections that have been correctly predicted and
measured. Planets in orbits are actually in free fall in a curved space-time!"
"In the physics of complex systems we can introduce a statistical concept, a measure of randomness, called entropy. In a quiet equilibrium, like hot onion soup sitting in a thermos bottle with no escaping heat, the entropy remains constant in time. However, in violent nonequilibrium processes, like shattering glass or explosions, the entropy always increases. Essentially, entropy, as a measure of randomness, will always increase when a very ordered initial condition leads to a very disordered final state through the normal laws of physics. The fact that entropy at best stays the same in equilibrium, or increases in all other processes, is called the second law of thermodynamics." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"Mathematicians have evolved a systematic way of thinking about symmetries that is fairly easy to grasp at the outset and a lot of fun to play with. This almost magical subject is known as group theory. […] Group theory is the mathematical language of symmetry, and it is so important that it seems to play a fundamental role in the very structure of nature. It governs the forces we see and is believed to be the organizing principle underlying all of the dynamics of elementary particles. Indeed, in modem physics the concept of symmetry serves as perhaps the most crucial concept of all. Symmetry principles are now known to dictate the basic laws of physics, to control the structure and dynamics of matter, and to define the fundamental forces in nature. Nature, at its most fundamental level, is defined by symmetry." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"Quantum-mechanical effects appear in physical systems that are exceedingly small. A small system means very tiny objects with very tiny amounts of energy, moving around over very short time intervals. Quantum effects show up dramatically once we arrive at length scales the size of the atom, about one ten-thousandth of a millionth of a meter. In fact, we simply cannot understand an atom without quantum mechanics. This is not to say that nature itself suddenly 'switches off'' classical mechanics and 'switches on' quantum mechanics when we enter this new submicroscopic realm. Quantum mechanics is always valid and always holds true at all scales of nature. Rather, quantum effects gradually become more and more pronounced as we descend into the world of atoms. Quantum mechanics is the ultimate set of rules, as far as we know, that governs how nature works" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"So, a scientist's definition of symmetry would be something like this: symmetry is an invariance of an object or system to a transformation. The invariance is the sameness or constancy of the system in form, appearance, composition, arrangement, and so on, and a transformation is the abstract action we apply to the system that takes it from one state into another, equivalent, one. There are often numerous transformations we can apply on a given system that take it into an equivalent state." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"Suppose that while
flipping a coin, a small black hole passed by and ate the coin. As long as we
got to see the coin, the probabilities of heads and tails would add to one, but
the possibility of a coin disappearing altogether into a black hole would have
to be included. Once the coin crosses the event horizon of the black hole, it
simply does not meaningfully exist in our universe anymore. Can we simply
adjust our probabilistic interpretation to accommodate this outcome? Will we
ever encounter negative probabilities?"
"Symmetry is ubiquitous. Symmetry has myriad incarnations in the innumerable patterns designed by nature. It is a key element, often the central or defining theme, in art, music, dance, poetry, or architecture. Symmetry permeates all of science, occupying a prominent place in chemistry, biology, physiology, and astronomy. Symmetry pervades the inner world of the structure of matter, the outer world of the cosmos, and the abstract world of mathematics itself. The basic laws of physics, the most fundamental statements we can make about nature, are founded upon symmetry." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"The space and time of
the universe that we humans inhabit contain symmetries. These are almost
obvious yet subtle, even mysterious. Space and time form the stage upon which
the dynamics - that is, the motion and interactions of the physical systems,
atoms, atomic nuclei, protozoa, and people - are played out. The symmetries of
space and time control the dynamics of the physical interactions of matter.
"The subject of mathematics has its own identity, on the other hand, and, in contrast to physicists, mathematicians attempt to create a roadmap of all possible logical systems that could consistently exist, whether they ultimately have anything to do with nature or not. Yet nature provides the basis of abstraction that gives birth to mathematics.
"The symmetries that we sense and observe in the world around us affirm the notion of the existence of a perfect order and harmony underlying everything in the universe. Through symmetry we sense an apparent logic at work in the universe, external to, yet resonant with, our own minds. [...] Symmetry gives wings to our creativity. It provides organizing principles for our artistic impulses and our thinking, and it is a source of hypotheses that we can make to understand the physical world." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"Theoretical physics borrows from mathematics (or, if there's none to borrow, they invent new mathematics) in order to create a mathematical roadmap of things that can happen in the real world, in nature. It strives to explain all of the many different phenomena observed in the universe, perhaps ultimately seeking one elegant and economical logical system. However, physicists usually settle for lesser triumphs, in which many physical systems with common and comprehensible behaviors are successfully described. This description is always created in the abstract language of mathematics." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"We have come, in our time, to systematize our understanding of the rules of nature. We say that these rules are the laws of physics. The language of the laws of nature is mathematics. We acknowledge that our understanding of the laws is still incomplete, yet we know how to proceed to enlarge our understanding by means of the 'scientific method' - a logical process of observation and reason that distills the empirically true statements we can make about nature." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
"We often wonder why the more complex systems seem to indicate a preferred direction of time, or an arrow of time, whereas their elementary counterparts do not. […] This has to do with the if-then nature of physics questions. Anything we observe involves laws of motion but also particular initial conditions. […] The initial conditions are what make a situation look peculiar when we time reverse it." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)
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