Group Theoretical Foundations of Quantum Mechanics

The conformal group is the invariance group of geometry (which is not understood), the largest one. Physical applications are implied, as discussed, including reasons for interactions. The group structure as well as those of related groups are analyzed. An inhomogeneous group is a subgroup of a homogeneous one because of nonlinearities of the realization. Conservation of baryons (protons can't decay) is explained and proven. Reasons for various realizations, so matrix elements, of the Lorentz group given. The clearly relevant mass level formula is compared with experimental values. Questions, implications and possibilities, including for differential equations, are raised.

WHY GOD COULD NOT CREATE THE UNIVERSE WITH A DIFFERENT DIMENSION EVEN IF IT WANTED TO or perhaps anything else. Perhaps the universe must be the way it is. It seems that what is omnipotent is mathematics, elementary arithmetic, just counting. Yet even mathematics is not powerful enough to create a universe-there are just too many conditions, conflicting. Existence is impossible. Beyond that for there to be structure is quite inconceivable. But the universe does exist, there are galaxies, stars, even the possibility of life. That life is possible merely allows it to exist but only with the greatest good fortune does it actually occur. Intelligence is vastly less likely, ability and technology far more improbable. That we are, what we are, seem so strange, inconceivable, that we are left merely with wonder-and, as we seem unable to realize, the need for the deepest care, responsibility and gratitude. life with the most wondrous part of the universe, the ability to think, to know, to create, to wonder-and thus the demand that we use our most awesome gifts to protect them, to protect and preserve the world in which they exist, and the life, likely so rare if not unique in the universe, which has received these astounding favors of chance, that has been given by nature its most exalted constituents. What we are requires that we enhance what we are, what we are part of, to see, understand and be grateful.

Geometry through its fundamental transformations, the Poincare group, requires that wavefunctions belong to representations. Massless and massive representations are very different and their coupling almost impossible. Helicity-1 gives electromagnetism, helicity-2 gives gravitation; no higher helicities are possible. Basis states, thus the fundamental fields, are the potential and connection. General relativity is derived and is the unique theory of gravity, thus the only possible quantum theory of gravity. It is explained why it is. Because of transformations trajectories must be geodesics. Momenta are covariant derivatives and must commute. Covariant derivatives of the metric are zero.

Quantum mechanics, its properties including wavefunctions, complex numbers and uncertainty, are necessary and completely reasonable and understandable, with no weirdness. Classical physics is impossible. Much uncertainty comes from Fourier analysis. Waves and particles and collapse of wavefunctions are meaningless. Their seeming appearance in analyzed. Reasons and limitations of superposition are considered. Gravitation is an example of nonlinearity. All objects interact so nonlinearity is universal. How quantum mechanics then fits in is shown. Dirac's equation comes from Poincare group. Physics is necessarily impossible in any space but that with dimension 3+1. Spin-statistics is a property of rotation groups.

"This is a thought-provoking book. Mirman is a crusader for the advancement of a no-nonsense approach to physics. He does not request your complete agreement, but your willingness to think, which is good. I enjoyed the ride." David J. Santos, Mathematical Reviews In-depth analysis of the necessity for quantum mechanics and for its properties. All are (easily) understandable and unavoidable. Geometry, group theory and consistency require wavefunctions (group basis vectors) and that nature be probabilistic. Absolute values for experiments follow. Probability is defined. Reasons for confusion and misunderstandings are analyzed; much are based on misuse of (what is often not) language. These are extirpated. It is shown why (not only) the Einstein, Podolsky, Rosen paradox, uncertainty principles, the vacuum, are so thoroughly misunderstood. Quantum mechanics is local (with no spookiness) and requires the vacuum be empty. Reality is mathematical.

Conformal groups illustrate and emphasise how rich group theory is, something usually not recognised, and they also emphasise how fundamental geometry is in physics (and conversely). Reasons, and implications for physics, are explored as a start to the study of the vast fields of mathematics and physics suggested and required. These groups have non-linear transformations, ones with singularities. Other aspects also show richness beyond what is usually realised, giving possibilities of many new insights and applications, to physics, mathematics, group theory, geometry, (non-linear) differential equations, special functions, and likely elsewhere, including optics, squeezing, quantum communication, quantum computation, cryptography, and so on. There are also suggestions about relations to the well-known elementary -- particle mass-level formula, which is reviewed. And representations of the groups for dimension greater than two have Clifford algebras as entries in their matrices. The formalism developed for quantum field theory, as a foundation for the study of conformal field theory, itself has important applications. For example while the obvious reason that the proton cannot decay -- baryon number thus lepton number must be conserved -- is almost trivial, the formalism allows a mathematical proof. Other insights provided by this and the conformal group can also be seen. How much of physics does geometry (through its transformation group) determine? As we see, a very large part. Might the geometry in which physics takes place, in which we exist, completely determine the physics within it, and conversely? These studies are motivated by that, and so attempt to raise questions, to show possibilities, to stimulate. These subjects are indeed stimulating.

Excision of errors and confusion about quantum mechanics -- and stimulation of thoughtful and adventurous readers are pre-eminent rationales of this entire work; these requiring definitions and analysis of underlying concepts of quantum mechanics, of quantum field theory -- why probability is given by the absolute square, what wavefunctions are and are not and why, and many others -- and also examination of some from the philosophy of science. People's beliefs about quantum mechanics are often just the reverse of what fundamental principles give, seen most spectacularly with the EPR 'paradox'. The puzzles, the mystical, the bizarre, come merely from negligence, from blunders, including the outlandish belief that the universe must be explained using classical physics. Careless, unthinking physicists, and gullible journalists who naively accept their confusion as statements about nature, cause so much misunderstanding and nonsense about physics. Among the many examples considered are the non-existence in quantum mechanics of waves and particles, so of wave-particle duality; the reason that general relativity must be the quantum theory of gravity; the mystery of the cosmological constant: why people believe in it though it would be obvious to a high school student that there cannot be any, it must be zero; the absurdity (and wild incorrectness) of much of the discussion about the vacuum; the required locality of quantum mechanics and the impossibility of action-at-a-distance; and many others. Many blunders, not only about physics, come from abuse of language, the use of words, phrases, sentences without content, with con- notation but no denotation, of names --- quantum mechanics, particles, waves, and so on -- that deceive and misrepresent, of questions that ask nothing. It is not only in physics that answers to questions without meaning smother and hide.