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| Ramond, Weinberg, Planck, Weyl, Levi-Civita, Fermat, Noether, Hilbert, Keyser, Smolin | ||||||
| Hawking Radiation | ||||||
Ramond Wilczek Weinberg Hilbert Noether Ne'emann Keyser |
 When a change occurs in nature,
the quantity of action necessary for this change is the least possible.
The quantity of action is the product of the masses of the bodies by
their speeds and by the distance over which they move.
Maupertuis
It is a most beautiful and awe-inspiring fact that all the fundamental laws of Classical Physics can be understood in terms of one mathematical construct called the Action. It yields the classical equations of motion, and analysis of its invariances leads to quantities conserved in the course of the classical motion. In addition, as Dirac and Feynman have shown, the Action acquires its full importance in Quantum Physics. Ramond  Furthermore,
and now this is the point, this is the punch line, the symmetries
determine the action. This action, this form of the dynamics,
is
the only one consistent with these symmetries [...] This, I
think, is the first time that this has happened in a dynamical theory:
that the symmetries of the theory have completely determined
the
structure of the dynamics, i.e., have completely determined the
quantity that produces the rate of change of the state vector with time. Weinberg
 Thus, the task is,
not so much to see what no one has yet seen; but to think what nobody
has yet thought, about that which everybody sees.
Schrödinger
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Roughly speaking, force is the space derivative of energy and the time derivative of momentum. You can take one more step up the ladder: energy and momentum are both derivatives of action: energy is its time derivative, momentum its space derivative. Wilczek
Increasingly, many of us have come to think that the missing element that has to be added to quantum mechanics is a principle, or several principles, of symmetry. A symmetry is a statement that there are various ways that you can change the way you look at nature, which actually change the direction the state vector is pointing, but which do not change the rules that govern how the state vector rotates with time. The set of all these changes in point of view is called the symmetry group of nature. It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today. Weinberg The magical formula E =
hv from which
the whole of quantum theory is developed, establishes a universal
relationship between the frequency v of an oscillatory process and the
energy E associated with such a process. The quantum of action h is one
of the universal constants of nature. It was first discovered by Planck at the turn of the century in the laws of black body radiation; that is, radiation which is enclosed in a cavity and is in thermodynamic equilibrium with matter of a definite temperature, which by emission and absorption causes an exchange of energy between the various frequencies contained in the radiation. Since this equilibrium is independent of the particular nature of the matter involved, Planck considered, as a kind of schematic matter, a system of linear oscillators of all possible frequencies. A charge oscillating with frequency v interacts with the electromagnetic field by emitting and absorbing radiation of the same frequency. Planck assumed that the exchange of energy took place in integral multiples of an energy quantum [...] Weyl
How
far does the laser move in color space?
 Nature is thrifty in all its actions. (Maupertuis) |
action |
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If the initial point P0 and the final point P1 of the path of a ray of light are fixed, the time taken by the ray to go from P0 to P1 along a line s will obviously be expressed by the integral
Levi-Civita
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She did some good work in invariant theory at Erlangen and was invited to Gottingen by Klein ... and David Hilbert (1862-1943). It was as a result of working with the latter, especially after his involvement with general relativity, that she set on the investigation of the role of symmetry groups in physics in the most general terms. She read her two theorems in 1918, and Klein stressed their standing as extending the Erlanger program to physics. In her first theorem, she showed how the invariance of the action (or of the Lagrangian, Hamiltonian, or, in more modern terms, of the scattering matrix, path integral, etc.) under the action of a finite Lie group implied the conservation of a set of "charges" corresponding to the group's infinitesimal generator algebra. Ne'emann
Consider the field of the data of sense—a field of universal interest—and fundamental. We are here in the domain of sights and sounds and motions among other things [...] Do the colors constitute a group?[ ...] Let us pass from colors to figures or shapes—to figures or shapes, I mean, of physical or material objects—rocks, chairs, trees, animals and the like—as known to sense perception [...] And what of sounds —sensations of sound? Are sounds combinable? Is the result always a sound or is it sometimes silence? If we agree to regard silence as a species of sound—as the zero of sound—has the system of sounds the property of a group? Keyser
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The connection between symmetries and conservation laws is one of the great discoveries of twentieth century physics . But I think very few non-experts will have heard either of it or its maker — Emily Noether, a great German mathematician. But it is as essential to twentieth century physics as famous ideas like the impossibility of exceeding the speed of light. Smolin
 Emmy
Noether In
the judgement of the most competent living mathematicians, Fraulein
Noether was the most significant creative mathematical genius thus far
produced since the higher education of women began. Einstein
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