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量子力学

读物

• https://www.feynmanlectures.caltech.edu/III_toc.html
• David Griffiths. Introduction to Quantum Mechanics.
• Gasiorowicz, Stephen, Quantum Physics - the textbook I studied from.
• The Algebra of Grand Unified Theories Baez, Huerta
• JS Bell’s essay “Six possible worlds of quantum mechanics”; it’s in his anthology “Speakable and unspeakable in quantum mechanics”
• 维基百科: Quantum Electrodynamics.
• 维基百科: Standard model (mathematical formulation)
• 维基百科: Quantum field theory

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 c-bits (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 space-time? 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

Quantum physics

• QNLP 2019: Why NLP is quantum-native, and beyond! - Bob Coecke Quantum natural language processing
• John Baez: "Collapse of the wavefunction" does not violate unitary time evolution, which applies to closed systems but not open systems.
• Particles are conditional, fields are unconditional.
• How is "mass" (as the emergent property of the inertia of constrained particles in a moving box) related to "swing" (in jazz rhythm)?
• A system needs to be able to deal with the supersystem above (stepping out, strategizing) and the subsystem below (stepping in, accepting the implementation) and relating the two. This is like the quadratic fetters on the orthogonal polynomials in that you only worry about the levels immediately above and below. In the system level, there is a dialectic regarding the boundary as to whether we step out or step in.
• Supersystem, system, subsystem are the three levels of parsers.
• Does the universe swing between total kinetic energy and total potential (big bang)? And what causes it to swing, for example, back from kinetic to potential?
• Does Gram-Schmidt orthogonalization work for complex inner product spaces?
• What is the connection between imaginary time (the dynamical evolution of the quantum world {$e^{-iHt/ħ}$}) and inverse temperature (the Boltzmann factor {$e^{-\beta H}$}). Partition function {$\textrm{tr}e^{-\beta H}$}.
• How to understand negative energy and negative probability?
• Think of negative probability as removing a possibility, by having it not happen, by having it happen in our mirror universe (presumed to exist by CPT symmetry).
• Given wave function {$x^ke^{-x^2}$}, the chance of being at x=0 is 0 unless it is the ground state.

In measuring an average quality, such as average position:

• You can't average the values you get upon repeatedly measuring one particle because it has taken on a particular state with the first measurement.
• You can't average the values you get by making measurements on an ensemble of particles having the same state because, in general, you don't know what that state is, and even if you did put the particles in such a state, you cannot absolutely maintain them in that state, but only up to a degree of certainty.
• Thus you can average the values that you get by making meaurements on an ensemble of particles that are in rather similar states.
• The axioms {$p \equiv -iħ\frac{\delta}{\delta x}$} and {$E \equiv iħ\frac{\delta}{\delta t}$} and then the energy equation {$E=K+V=\frac{p^2}{2m} + V(x)$} are theoretically more fundamental than Shroedinger's equation, which is empirical, and which is derived by plugging in. Thus Schroedinger's equation simply describes the circumstances for measurement by way of the wave function.
• The factor of 2 for momentum and the factor i for position are both half-way factors, square root factors. i is the square root of negative 1. The factor 2 arises because of differentiating {$p^2$}, perhaps twice?
• There are four dimensions of space-time and accordingly there are four dimensions of the four momentum where the energy {$\frac{E}{c}$} corresponds to time. The principle of least action says that given the two forms of energy, kinetic and potential, {$E=K+V$}, we shift from one to the other as little as possible, thus minimizing K-V. In this sense we can think of time as the dimension in which this conversion occurs, and thus time is like an expressly local dimension, of limited extent, as little as possible. Time is the distance traveled up the potential V. Does that mean that change takes place as slowly and gradually as possible? And can that limitation express the speed of light as a consequence of the principle of least action? And also the shape of the mother function.
• How do we rethink Schroedinger's equation for space time so that energy and momentum are both related in terms of derivatives of the same order? How do we rethink energy as {$-E=iħ\frac{\delta}{\delta t}$}, so that it is compatible with p? So we can write this as {$0=(-E)+K+V$} or {$(-V)=(-E)+K$}. How doe we rethink the operator {$V=-iħ\sum_{j}\frac{\delta}{\delta x_j}$} where multiplying by the potential is the same as acting by the partial derivatives.
• How can the second law of thermodynamics be described in terms of time which is defined as the distance traveled up a potential?
• Symmetry is the laws for the laws (of physics).
• Roger Penrose, Jurekfest, 2019 4:20 "I think probably twister theory has now as little connection with spin networks as loop quantum gravity does with the original aims of spin network theory which was to produce a combinatorial approach to physics where you just ... calculate everything from integers. We've both drifted away from that. I don't know whether we're coming to it back to it or not. I sometimes hope so."

Indefinite and definite - Bosons and fermions

• Undefined - no bathroom stall, defined - bathroom stall.
• A fermion is specified by its qualities. A boson is not specified by its qualities. Yet they may both be defined by their qualities.
• Bosons are the same contradiction and fermions are distinct contradictions.

Bosons express the undefined with four different gauges:

• U(0)? gravity - affine - contradictory - body
• U(1) light - projective - mind
• U(2) weak force - conformal - emotions
• U(3) strong force - symplectic - will

Time

• Asymmetry of time is smuggled in statistical mechanics through the notion of probability whereby a system is approximated by a comparable finite collection of states. The problem is that the latter is simply a subsystem of the former. Thus we have static causality of the system upon the subsystem. This is where the irreversibility sneaks in.

Angular momentum

• Angular momentum is the relationship of a local contradiction with itself, thereby breaking symmetry, and then relating to the outside. Thus we have two levels of relationship as with the foursome.
• Do particle spin types express divisions of everything?
• 0 is 1, 1/2 is 2,... 3 is 7

Twin universe

• Is our universe's twin a prerequisite of its slack?

Physics

• Physics has to be compatible with human imagination and human intuition. But the history of physics shows that human intuition can be wrong.

Standard model

• 6 quarks, 6 leptons = 12 topologies?
• 4 gauge bosons = 4 representations of nullsome
• 1 Higgs boson = God or what?

Thomas asked, Why are there 3 + 1 dimensions of time and space? What could my philosophy say?

• The relevant structure would be the eightfold way: 1 + 3 + 3 + 1. It suggests that (external) space and (internal) time are two different ways of relating a relative framework with an absolute framework. For space, the absolute framework (God) is beyond the relative framework. There are three independent dimensions (being, doing, thinking) which relate the relative to the absolute. So the relative precedes the absolute. For time, the absolute framework (good) is within the relative framework (the three-cycle) and prior to it. Time is the dimension of slack within the three-cycle which goes on infinitely and for which forwards and backwards are qualitatively different directions.

Petri Muinonen I'm reading your doctoral thesis as I have a physics model under development where particles are described as combinatoric aggregates of logarithmic poles (singularities in 4D). They form negative simplex vertices around center of positive ones (convention). See my Youtube channel petrimuinonen

• Neutrino oscillations under Logarithmic pole model v1.0 - a candidate for a Theory of Everything
• Petri: From your diary "This suggested to me that the collapse must occur when a coordinate system is introduced and thereby separates the observer from the observed. ... at a certain point ... which collapses the wave function.... Thus this would be the place to look for a relation between general relativity and quantum mechanics."
• This makes perfectly sense to me. My understanding is that we are talking about division algebra, quaternions, and a negative 4-distance expressed in quaternion terms (squared separation of 4D locations, outside light cone) collapses into a vectorial quaternion (corresponding to a purely spatial separation) There is infinite number of roots to that, all space-like. Will have to explain this better later!
• You might also want to check Julian Barbour's concept of shape dynamics
• Phase transition models personal growth experience https://www.quantamagazine.org/mathematicians-prove-symmetry-of-phase-transitions-20210708/
• A particle is a singularity, a local contradiction, and the origin for a coordinate system.
• Phase of quantum particle is the inherent slack given by the locality of its contradiction.
• Geometric Unity - A Theory of Everything (Eric Weinstein)
• The collapse of the wave function resets the clock of time for that system. And it diffuses by the phase factor from there.
• Relativistic invariance of the Dirac operator.
• What are the invariant subspaces of the shift operator?

Fivesome

• All of physics (and decision-making) comes from the fivesome.
• Global causality: not every global cause has had its global effect. Local causality: every local effect has had its local cause.

Coordinate spaces

• Understand how the Einstein equations of general relativity equate geometric changes with physical quantities (mass, momentum, energy). Then show that in the extended Feynman diagrams, gravity deals with the kinematics of the edges, not the potential energy of the vertices. And consider how different relations of two coordinate spaces, system and subsystem, are different geometries for general relativity.

Double slit experiment

• Consider particles going through one at a time.
• The particle impacts the screen at a single point but a diffraction pattern builds up.
• Can interpret this as follows:
  • There is a diffraction pattern for a single particle (hard to imagine, acts probabilistically as a point). Repeating the experiment manifests the probability (this is easy to imagine).
  • Alternatively, there is a single particle that takes its own definite path and affects a single point on the screen (this is easy to imagine). Then there is a diffraction pattern that results from the knowledge of the particles (this is hard to imagine). Yet if time is symmetric, then why is this a problem. It is only a problem if we think of time moving forwards.
• Can imagine that the mirror is an experiment between the two sides of the double-split, the incoming particles and the screen they affect.
• What would it mean to reverse time, to consider both directions?
• Is there an adjunction here between what is hard (free) and easy (forgetful)? In either direction?
• Natural law requires self-interference so that you get an equation in order to have a mathematical law. A measurement is self-interference within a subsystem. And for that measurement we need to open the subsystem before and after. So a measurement takes place in time. How does this relate to Goedel?
• Quantization: It appears at a single place on the screen. But it also goes through both slits so that it self interferes.

Quantum physics

• Ramybė sąlygiškas buvimas nebuvime. Quantum foam.
• How would we quantize angular momentum instead of energy? It is naturally periodic.
• Time has direction when there is a charge so it can be reversed with opposite charge. Marked opposite?
• Thomas Gajdosik
• Andrea had a fellow student Alekas Mazuliauskas at Stony Brook University.
• Sidney Coleman. Lectures on Quantum Theory
• Energy is quantized because measurement takes place over time t.
• Quantization of action gives it periodicity thus it is angular momentum.
• In a magnetic field, electrons and positrons spiral in opposite directions. Thus they go in reverse directions timewise.
• Measurement expresses the questions we ask Nature. The uncertainty principle expresses constraints on the answers Nature provides us.

Quantum mechanics courses

• https://www.edx.org/learn/quantum-physics-mechanics

Spin

• Spin Phenomena Interdisciplinary Center
• Think about the algebra of spins. For example Up and Down, Right and Left. Ignoring square roots of two, we have: U = R+L, D = R-L, R = U+D, L = U-D. How to represent that geometrically?

Quantum mechanics

• Single world interpretation.
  • Wave functions carries all information as a superposition.
  • Doing a measurement collapses the information measurement relates global and local. If a different measurement would have been done, then a different collapse would have occurred.
  • Measurement has a reason Why (motivated).
• Measuring the slit. Focusing the election to choose changes the possibilities - keeps it from interfering with itself.
• Particle-antiparticle answers yes or no (true or false). Here knowledge is in the scope of nothing, it is very strict. Does the scope grow?

Knowing universe interpretation

• The universe knows what will happen in an experiment (it is predetermined) and so the interference is happening directly (not via possibilities) by all the particles in the experiment.
• Thomas supportive of the knowing universe interpretation because it respects the experimental basis for randomness, the universe as a nonstupid medium. Moreover, it supports a purposeful universe and a meaningfulness for investigation.
  • Nature's language gets limited (yes-no with spin)
• Quantize action
  • Action represented as angular momentum
• Poincare group
  • 4 boosts (dimensions), 6 rotations (pairs of dimensions)
• Affine time (no origin - clock - loop - define steps), coordinate systems are spatial (projective, conformal, symplectic) and finite
  • Time provides the steps. Time is cut but not extended. Space is grown through time.
• Quantum of global action
• Feynman diagrams - powers of the field

The spectral theorem

• Symmetric matrix, real eigenvalues - the underlying basis is orthogonal - so you can diagonalize with a rotation.
• Self-adjoint, Hermitian matrix - complex values, real eigenvalues - underlying basis is orthogonal in the complexes.
• Unitary operator - same works for an infinite dimensional space.

Quantum potential - Bohm and Basil Hiley

• Quantum potential from looking at the real component we get an extra potential term. I imagine this is the extra energy that is stored when observations are forced to take on quantum states.
• Not everything can be made explicit at the same time. When a particle has perfectly specified position then it has no momentum and vice versa.
• Bohm theory. Chains of unfolding and enfolding. Compare with universal covering space and folding into equivalences.