A few minutes ago a learned about the Scientific American Contest for Fringe Scientists. I have been preparing an essay on Quantism, but unfortunately the deadline expired yesterday. Instead of submitting the full essay, will only post part of it.
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Understanding Why Nobody Really Understands Quantum Mechanics
Daniel Crespin
Universidad Central de Venezuela
dcrespin@gmail.com
During almost nine decades Quantism has made highly successful predictions. But Quantism is controversial and still generates endless debate. The present essay is an attempt to find the causes of this paradoxical situation. The original sin of Quantism is its inattention to conjugate momenta $\phi$ of wave functions $\psi$. But stationary states can still be correctly calculated by Quantism because they have zero momentum. However the time dependent, unitary, quantum evolution equation is physically incorrect. The probabilistic interpretation of states appeared as an ersatz for a good deterministic evolution equation. Unitary evolution and probabilistic interpretation contradict each other and, together with questionable ad-hoc principles, conform the core of Quantism.
A comparison between Classical Mechanics and Quantism reveals that, at its most basic level, Quantism is incomplete.
The first link in the chain of quantum mistakes is the absence of conjugate momenta in (the Hamiltonian formalism of) Quantism. Therefore, the customary quantum states provide an incomplete description of the physical states (of the electron).
Next, because conjugate momenta are absent, the kinetic energy term is also absent from the total energy function (the term $\nabla ^2 \psi$ is in fact internal energy, not kinetic).
With an incomplete energy function, a senseless unitary evolution equation was introduced which does not correspond with the Physics of atoms.
But stationary states necessarily have zero momentum. Therefore the momentum is not required for the calculation of stationary states and stationary energies, and in this task Quantism excels.
However, the time dependent, unitary, quantum evolution equation remains physically incorrect.
Finally, the scaffolding of Quantism is completed by the probabilistic interpretation of states, and supported by ideological constructs (uncertainty, wave-particle duality, special role for the observer, etc.)
To obtain a correct description of bound electrons, momenta should be reintroduced and states have to be renormalized. This is achieved ---for the hydrogen atom, say--- taking as configuration space the projective space $PE$ (definitely not a linear space) associated to the linear space $E$ of real valued wave functions. The correct space of states is then the cotangent bundle $T^*PE$. States are cotangent vectors $(\psi,\phi)$. The normalized square $\psi^2$ of a configuration $\psi$ is interpreted ---an important idea due to Schrödinger--- as charge density. The energy function $f:T^*PE\to \R$ equals the sum of (electrostatic Coulomb) potential, internal (\nabla^2 \psi) and kinetic (\|\phi\|^2).
Comparison of classical Hamiltonians with quantum Hamiltonians make obvious that Quantism disregards wave momenta.
When bound electrons make their continuous transitions between stationary states, the photons ---carriers of radiated and absorbed electromagnetic energy--- should be represented by momenta, but in Quantism they are not. Without momenta it is impossible to obtain correct Hamiltonian evolution equations. This explains the painful failure of the time dependent equation Quantism.
The lack of the kinetic energy term is generally unnoticed because the quantum misnomer \Lq kinetic energy\Rq\ assigned to $\nabla^2\psi$ covers the absence in case of an energy roll call.
As already mentioned, stationary states have zero momentum. Hence the motionless states and their stationary energies can be calculated ignoring momenta. This explains the remarkable computational success of Quantism.
The artificial introduction of unitary evolution equations ---initially by Erwin Schrödinger, later refined by many quantum theorists--- is useless because trajectories of unitary evolution never approach stationary states and in particular none of these trajectories run from one stationary state to another located at a different energy level.
Historically the inadequacy of the unitary evolution equation was improperly solved by Quantism with the invention of the probabilistic interpretation of states. These probabilities supposedly "explain" why systems "prefer" stationary states, and how systems make "discontinuous probabilistic jumps" between such states. This is also called "collapse of the wave packet". But not only probabilistic jumps and collapses convert unitary evolution into a useless decorative fixture, but they violate it. Nevertheless for Quantism it is important to retain both, the victim the and offenders.
The unitary evolution is kept by Quantism perhaps because any theory aspiring to respectability should brandish some mathematical evolution equation. Isaac Newton inaugurated this tendency. Note that all evolution equations ---unitary ones included--- are strict deterministic/causal/continuous laws expressed in the language of Infinitesimal Calculus.
The probabilistic jumps are included in Quantism because they provide a surrogate dynamics required to patch the failure of unitary evolution.
For concreteness assume whenever necessary that the system is the hydrogen atom. The above comments can then be rephrased as follows.
Momenta (photons) were overlooked (Schrödinger; a lapse?), but do not hinder the calculation (Schrödinger) of stationary states and energies. In physical systems transitions exist, called "quantum jumps" (Bohr), connecting stationary states at different energy levels. Evolution equations (all the rage since Newton) were expected from any reputable physical theory. Unaware of the transitions, Schrödinger introduced the artificial unitary evolution. However, unitary evolution contradicts physically undeniable transitions (Bohr; also Schrödinger "verdammte Quantenspringerei"). The even more artificial probabilistic interpretation of states was invented (Born) to explain energy transitions. Some were opposed (Planck, Einstein, de Broglie, Schrödinger). Others, with uncertain reasons, applauded (Heisenberg, Bohr, Born). Then quantum electrodynamics arose (Dirac), sick from birth with probabilistic maladies. The rest is history.
Rife with contradictions and ad-hoc principles, Quantism makes correct calculations of stationary states and energies, maintains an incorrect and useless unitary evolution, adopts the probabilistic jumps to fill the gap left by unitary evolution, and creates a chaos where partial success coexists with paradoxes, contradictions and despair.
The following disturbing fact then arises: Quantism is a mixture of virtuoso calculations providing valid mathematical expressions for stationary states and energies, with crackpot Physics.
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TO BE CONTINUED, TIME AND RESOURCES ALLOWING.
