Ramond, Weinberg, Planck, Weyl, Levi-Civita, Fermat, Noether, Hilbert, Keyser, Smolin
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.
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.
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.
a derived quantity instead of a fundamental one, our whole set
of ideas about uncertainty will be altered: is the
quantity that occurs in the Heisenberg uncertainty relation connecting
the amount of uncertainty in a position and in a momentum. This
uncertainty relation cannot play a fundamental role in a theory in
which itself is not a
fundamental quantity. I
think one can make
a safe guess that uncertainty relations in their present form will not
survive in the physics of the future.
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.
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?
Thus the colors with their various qualities and intensities fulfill the axioms of vector geometry if addition is interpreted as mixing; consequently, projective geometry applies to the color qualities.
It is just like the mathematics of the addition of vectors, where (a, b, c) are the components of one vector, and (a', b', c') are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians. In fact, Schrödinger wrote a wonderful paper on color vision in which he developed this theory of vector analysis as applied to the mixing of colors.
Physicists speak of the world as being made of particles and force
fields, but it is not at all clear what particles and force fields
actually are in the quantum realm. The world may instead consist of
bundles of properties, such as color and shape
Duality in mathematics is not a theorem, but a “principle”. It has a simple origin, it is very powerful and useful, and has a long history going back hundreds of years. Over time it has been adapted and modified and so we can still use it in novel situations. It appears in many subjects in mathematics (geometry, algebra, analysis) and in physics. Fundamentally, duality gives two different points of view of looking at the same object. There are many things that have two different points of view and in principle they are all dualities.
Very recently, in the last two years,Witten and his collaborators (mainly Kapustin and Gukov, two young Russians) have managed to deduce what is required for the geometric Langlands program from non-abelian dualities in physics. The kind of dualities they use are close to the dualities in Donaldson’s theory and to the dualities in the mirror symmetry, and are based (at least) on the electric- magnetic duality. The original dualities in physics are those between electricity and magnetism.
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.
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.
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 [...]
How far does the laser move in color space?
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
since m, as we have just said, is the reciprocal of the velocity. Now the line actually followed by the light is the one which makes this integral a minimum, and therefore satisfies the condition
This variational equation, which sums up the whole of geometrical optics, is known as Fermat's principle.
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.
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
In general, it has long been suspected that gauge theory, especially in the large N limit, may be equivalent to a string theory. Moreover, it is suspected that such gauge/string equivalence may be the key to understanding the aspects of quantum Yang-Mills theory that are currently out of reach. By now, in many examples of large N gauge theories such as N = 4 Yang-Mills theory, gauge/string equivalence is relatively well-established, using the AdS/CFT correspondence first proposed in [Maldecena PDF].
I think it’s actually very difficult to see what advance in the near term could make the gauge theory interpretation of geometric Langlands accessible for mathematicians. That’s actually one reason why I’m excited about Khovanov homology. My approaches to Khovanov homology and to geometric Langlands use many of the same ingredients, but in the case of Khovanov homology, I think it is quite feasible that mathematicians could understand this approach in the near future if they get excited about it. I believe it will be more accessible. If I had to bet, I think I have a decent chance to live to see gauge theory and Khovanov homology recognized and appreciated by mathematicians, and I think I’d have to be lucky to see that in the case of gauge theory and the geometric Langlands correspondence — just a personal guess. A lot of things that number theorists like have appeared in physics, and some have even appeared in my own work. Plenty has been found to show that the physics theories that we work on as string theorists are interesting in number theory.
This uncertainty relation cannot play a fundamental role in a theory in which ħ itself is
not a fundamental quantity. I think one can make a safe guess that uncertainty relations in
their present form will not survive in the physics of the future. ~Dirac