概论
数学
发现
Andrius Kulikauskas
 ms@ms.lt
 +370 607 27 665
 My work is in the Public Domain for all to share freely.
用中文
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量子力学
 In what sense does {$\frac{E}{m}=c^2$} relate to {$\frac{E}{h}$} and express the quantization of energy? And how does it relate to the {$\frac{1}{r^2}$} law and the surface area of a sphere?
Uncertainty
 Does Heisenberg's uncertainty principle allow for momentary forces? appearing out of nowhere?
 In what sense does the combination of energy and time allows for uncertainty in the energy, and also the "borrowing" of energy, even in the form of mass of particles and antiparticles, it if can be paid back?
Infinities
 There is conservation of energy but potential energy is infinite. So we're dealing with sums of infinity. Is this related to renormalization?
读物
Collapsing the wave function
 Consider the wave function collapsing, time and time again, and consider the infinitesimal limit of those time differences.
 Energy is conserved precisely when time becomes irrelevant, for example, when the uncertainty in measuring time becomes very large. But energy is not at all conserved when time becomes absolutely precisely defined. The collapse of the wave function occurs when time is absolutely precisely defined. And that happens when the corresponding event is take as the reference point in time. Thus collapsing the wave function is setting precisely t=0 internally. And it is also defining a real event in the external world. When time is thus absolutely defined, then phase space  position and momentum  can be defined relative to that absolute, albeit with a combined uncertainty, nevertheless allowing for measurement. Thus t, x, p, E form a foursome.
 Wave function collapse  arisal of coordinate systems  is not because a system is "big and heavy" but because there arises a difference between what a measurement would mean, yielding different results between one end and another end of the system, thus indicating a coordinate system.
 Quantum world can't have gravity  gravity (and mass?) only exist upon collapse of the wave function.
 The Schroedinger equation has continuity. Discreteness enters in with the act of measurement, with the collapse of the wave function, with the breaking of symmetry between observer and observed.
Superposition
 Classical mechanics states are based on sets and cbits (coins). Quantum mechanic states are based on vector spaces (thus ordered lists) and qubits.
Entanglement
 Mathematical entanglement  One of {$e\pi$} and {$e + \pi$} is transcendental and the other is rational. But currently we don't know which is which.
Lie theory
 N.Mukunda: {$QM=e^{iCM}$} relation between quantum mechanics and classical mechanics. But how is this exponential related to the other exponential relating Lie groups and Lie algebras?
Heisenberg's uncertainty principle
 Heisenberg's uncertainty principle deals with units of angular momentum, ΔxΔp, ΔEΔt. Special relativity deals with units of velocity: x/t, E/p. This suggest a polar decomposition into (complex) angular momentum and (real) velocity. Note that special relativity deals with straight lines or geodesics, whereas quantum mechanics deals with rotating a detector. Thus the observed moves straight but the observer moves around.
 ΔxΔp is a box of slack described by symplectic geometry.
 Heisenberg's uncertainty principle describes the slack in a vacuum  is this related to the slack in gravitational waves? which is the relocation of slack in spacetime? How much slack there is in the vacuum depends on time, so also space (and Planck's constant)?
 Heisenberg's uncertainty principle deals with measurement discrepancies, whereas relativity deals with measured values.
 Einstein relates absolute measurements of X and T, or E and P. Heisenberg relates discrepancies of ΔX and ΔP, ΔT and ΔE. Interesting coupling.
Planck's constant
 Vasil Penchev iš Bulgarijos  Planck's constant yra atomic unit for connecting rotation between real and imaginary axes (perhaps as in my chains for Lie algebra root systems).
 Susskind: Turning the detector upside down shifts the answer from up to down. So this expresses a global symmetry. Qubit has a sense of directionality. Orientation is being distinguished. Some qubits have a sense of spatial orientation.
 Quantum mechanics: energy is not continuous but discrete. Noncontinuity suggests jolts  what is needed for causality. Particle and wave descriptions are necessary to relate continuity and discreteness for causality.
 Quantum mechanics superposition of states may be based on borrowing of energy. When the energy debt is too high, as when it gets entangled with an observer, then the debt has to be paid. Otherwise, the debt can be restructured in complicated ways.
 The wave equation is defined on phase space but in such a way that it is understood in terms of a superposition of waves for position, yielding a "blob"  a wave packet, and an analogous superposition of waves for momentum, and the two are related by the Fourier transform, by the Heisenberg uncertainty principle.
 Wave function Smolin says is ensemble, I say bosonic sharing of space and time.
 What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets?
 Which state is which amongst "one" and "another" is maintained until it is unnecessary  this is quantum entanglement.
 Geometric unity I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we'll understand entanglement a lot better.
 The classical Lie groups relate counting backwards and forwards. So how does that relate to changes in energy levels, up and down?
 What does it mean (Harris, 2.65) that the lowering operator is the hermitian conjugate (or adjoint) of the raising operator, and how does that relate to adjunctions?
 Suppose you can never know the state that a system is in  it is in all states. But suppose you can know, if there is entanglement, what you will find in one particular state given another, so that definite knowledge is relative. Now suppose there was a constituent that entered a "fixed point" (either through consciousness, or by definition as a particle, etc.) Then it has a single state, thus a definite state, and it can communicate and propagate that definiteness through entanglements.
 Conceptual mistake on page 7: "the electron interferes with itself". A single electron in the double slit experiment does not interfere with itself (because then we would report only a part of an electron or sometimes twice an electron) but rather its probability interferes with itself (and so the electron is found accordingly). So the electron's probability is real, experimentally. Whereas the electron itself may not be real.
 Rydberg atom  one or more electrons have a very high principal quantum number
