Elegance unquestionably plays a big part in the advancement of science. The mathematical simplicity of quantum physics is lovely to behold. But in the hands of geologists, quantum physics has brought to light the glorious, messy, and very inelegant history of our planet.
The Physics and Philosophy of the Bible: How Relativity, Quantum Physics, Plato, and History Meld wi
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To set the stage, recall that fundamental physics can be divided into two sectors with separate but maddeningly incompatible advantages. The gravitational force has, since Einstein's theory of general relativity, been admired for its four dimensional geometric elegance. The quantum, on the other hand encompasses the remaining phenomena, and is lauded instead for its unparalleled precision, and infinite dimensional analytic depth.
The story of the geometric quantum begins at some point around 1973-1974, when our consensus picture of fundamental particle theory stopped advancing. This stasis, known as the 'Standard Model', seemed initially like little more than a temporary resting spot on the relentless path towards progress in fundamental physics, and theorists of the era wasted little time proposing new theories in the expectation that they would be quickly confirmed by experimentalists looking for novel phenomena. But that expected entry into the promised land of new physics turned into a 40-year period of half-mad tribal wandering in an arid desert, all but devoid of new phenomena.
While the Stony Brook history may be less discussed by some of today's younger mathematicians and physicists, it is not a point of contention between the various members of the community. The more controversial part of this story, however, is that a hoped for golden era of theoretical physics did not emerge in the aftermath to produce a new consensus theory of elementary particles. Instead the interaction highlighted the strange idea that, just possibly, Quantum theory was actually a natural and elegant self-assembling body of pure geometry that had fallen into an abysmal state of pedagogy putting it beyond mathematical recognition. By this reasoning, the mathematical basket case of quantum field theory was able to cling to life and survive numerous near death experiences in its confrontations with mathematical rigor only because it was being underpinned by a natural infinite dimensional geometry, which is to this day still only partially understood.
The great Paul Dirac admired general relativity more than any other modern theory (much more than quantum mechanics). He found it as spine-tinglingly inspirational as any great work of music. That is because he valued aesthetic appeal to an extraordinary degree. At a seminar in Moscow in 1956, when asked to summarise his philosophy of physics, Dirac had once scribbled on the blackboard in capital letters, "Physical laws should have mathematical beauty."
In classical physics, a piece of mathematics known as Noether's theorem (named after the mathematician Emmy Noether) associates such observable quantities to symmetries. The arguments involved are non-trivial, which is why one doesn't see them in an elementary physics course. Remarkably, in quantum mechanics the analog of Noether's theorem follows immediately from the very definition of what a quantum theory is. This definition is subtle and requires some mathematical sophistication, but once one has it in hand, it is obvious that symmetries are behind the basic observables. Here's an outline of how this works, (maybe best skipped if you haven't studied linear algebra...) Quantum mechanics describes the possible states of the world by vectors, and observable quantities by operators that act on these vectors (one can explicitly write these as matrices). A transformation on the state vectors coming from a symmetry of the world has the property of "unitarity": it preserves lengths. Simple linear algebra shows that a matrix with this length-preserving property must come from exponentiating a matrix with the special property of being "self-adjoint" (the complex conjugate of the matrix is the transposed matrix). So, to any symmetry, one gets a self-adjoint operator called the "infinitesimal generator" of the symmetry and taking its exponential gives a symmetry transformation. 2ff7e9595c
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