MSI RTX 3060 Ti VENTUS 2X 8G OCV1 LHR

Visualizing higher dimensions, 2.2 | Remove time from physics, 2

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Mathematics is local (left brain).

Physics is global (right brain).

— Me@2017-06-22 06:16:59 PM

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Mathematical processes, i.e. the calculations, are local.

Physical intuitions before a calculation and the interpretations after are global.

— Me@2023-01-13 07:45:24 PM

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However, in an opposite sense, physics is local and mathematics is global.

— Me@2023-01-14 08:13:17 PM

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Geometry is global.

Space is what we can see at once.

Dynamics is local.

Time is what we cannot see at once.

— Me@2017-02-07 10:11:34 PM

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… math is what you get when you remove time from physics.

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2023.01.14 Saturday (c) All rights reserved by ACHK

Black hole mass can’t be

A singularity doesn’t have mass. Mass is a property of an object that exists in time. A (spacelike, e.g. Schwarzschild) singularity is not an object that exists in time. A singularity is a moment in time when time ends along with mass. Furthermore, a black hole does not have a center. The geometry of the Schwarzschild spacetime inside the horizon is an infinitely long 3-cylinder with a quickly shrinking circumference. Also, no black hole solution is valid inside the horizon, because all solutions assume a static metric, but it is not static inside the horizon.

— safesphere

— May 20, 2019 at 10:38

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And if you wanted to say that the whole mass M is obtained from the singularity, you won’t be able to get a good calculation because the integral over the singularity would be singular. Moreover, the space and time are really interchanged inside the black hole (the signs of the components grr and gtt get inverted for r < 2GM) so the exercise is in no way equivalent to a simple 3D volume integral of M \delta(x) \delta(y) \delta(z). The Schwarzschild singularity, to pick the "simplest" black hole, is a moment in time, not a place in space. It is the final moment of life for the infalling observers. In a locally (conformally) Minkowski patch near the singularity with some causally Minkowskian coordinates t,x,y,z and r = |(x,y,z)|, the Schwarzschild singularity looks like a t=t_f hypersurface, not as r=0.

— Black hole mass can't be visualized as a property of the black hole interior

— Lubos Motl

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2022.11.08 Tuesday ACHK

Coherent light source

superkuh 17 hours ago [–]

Coherence isn’t what you think it is. It is not an “alignment in phase” of the sinusoid, like all the lay diagrams show. It isn’t even being the same frequency. In the early days of quantum physics the light sources were mercury arc lamps (muliple freqs) that achieved coherence by shining through tiny pinholes.

https://web.archive.org/web/20220820182938/http://amasci.com/miscon/coherenc.html

Coherence is being a point source. Stars, except for our sun since it is too close, are coherent light sources.

— Hacker News

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2022.10.27 Thursday ACHK

Swampland

In physics, the term swampland refers to effective low-energy physical theories which are not compatible with string theory, in contrast to the so-called “string theory landscape” of compatible theories. In other words, the swampland is the set of consistent-looking theories with no consistent ultraviolet completion in string theory.

Developments in string theory suggest that the string theory landscape of false vacua is vast, so it is natural to ask if the landscape is as vast as allowed by consistent-looking effective field theories. Some authors, such as Cumrun Vafa, suggest that is not the case and that the swampland is in fact much larger than the string theory landscape.

— Wikipedia on Swampland (physics)

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Recall that Vafa’s Swampland is a giant parameter space of effective field theories that cannot be realized within a consistent theory of quantum gravity i.e. within string/M-theory. Only a tiny island inside this Swampland, namely the stringy Landscape, is compatible with quantum gravity. String/M-theory makes lots of very strong predictions – namely that we don’t live in the Swampland. We have to live in the special hospitable Landscape.

— Vafa, quintessence vs Gross, Silverstein

— The Reference Frame

— Luboš Motl

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2022.10.21 Friday ACHK

For all, 10

No observer can observe and get all the information of the current state of the whole universe.

Since the definition of the “universe” is “everything”, any observer must be part of the universe. Also, in the universe, any observer has at least one thing it cannot observe directly—itself.

Therefore, no observer can observe the whole universe in all details.

— Me@2022.09.30 07:57:46 PM

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Can a part of a painting represent all the information of the whole?

No.

(Kn: Yes, if excluding itself.)

That is exactly my point.

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“Yes only if that part does not contain that part itself” is equivalent to “no”.

— Me@2016-08-20 03:30:26 PM

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2022.10.01 Saturday (c) All rights reserved by ACHK

Photons in expanding space

If a photon (wave package) redshifts (stretches) travelling in our expanding universe, is its energy reduced?

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Since you say you’re talking about what happens locally (in a small volume), I’ll answer from that point of view. The usual formulation of energy conservation in such a volume is that energy is conserved in an inertial reference frame. In general relativity, there are no truly inertial frames, but in a sufficiently small volume, there are reference frames that are approximately inertial to any desired level of precision. If you restrict your attention to such a frame, there is no cosmological redshift. The photon’s energy when it enters one side of the frame is the same as the energy when it exits the other side. So there’s no problem with energy conservation.

The (apparent) failure of energy conservation arises only when you consider volumes that are too large to be encompassed by a single inertial reference frame.

To be slightly more precise, in some small volume V=L^3 of a generic expanding Universe, imagine constructing the best possible approximation to an inertial reference frame. In that frame, observers near one edge will be moving with respect to observers near the other edge, at a speed given by Hubble’s Law (to leading order in L). That is, in such a frame, the observed redshift is an ordinary Doppler shift, which causes no problems with energy conservation.

If you want more detail, David Hogg and I wrote about this at considerable (perhaps even excessive!) length in an AJP paper.

— Photons in expanding space: how is energy conserved?

— answered Aug 16, 2011 at 15:31

— Ted Bunn

— Physics Stack Exchange

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2022.09.11 Sunday ACHK

Quantum as potential, 2

Only measurement results (aka physical phenomena) form the physical reality.

Quantum Mechanics is a theory of measurement results.

Quantum Mechanics is a theory of reality.

Quantum Mechanics is not a theory of unobservables (undefined-observables).

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Quantum mechanics is a story of reality, not a story of story.

— Me@2022-07-27 10:38:32 AM

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2022.07.29 Friday (c) All rights reserved by ACHK

Quantum as potential

Realist view is wrong.

Before measurement, there are quantum potentials only.

quantum ~ potential

Note that it is NOT the “quantum potential” in the Bohm interpretation.

— Me@2016-08-21 06:13:49 PM

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A wave function encodes the probabilities of different potential measurement results of a physical experiment. It is not a physical wave.

Quantum superposition is NOT a superposition of realities.

Physics should consider only measurement results and their probabilities. Only measurement results are realities.

No measurement result, aka physical phenomenon, is in a superposition.

For example, in the double-slit experiment, the measurement results are (the locations of) the dots on the final screen. Every dot location is not in a superposition.

— Me@2022-07-25 06:43:05 PM

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2022.07.25 Monday (c) All rights reserved by ACHK

Schrodinger’s cat, 3.5

This description is wrong.

Quantum superposition is exhibited in fact in many directly observable phenomena, such as interference peaks from an electron wave in a double-slit experiment.

— Wikipedia on Quantum superposition

Whatever you observe, it is not a superposition.

If no left-right detector is allowed, the moving-left/right variable is an unobservable. What you observe, instead, is the dot on the final screen.

In other words, the observable is the final position of a particle when it reaches the final screen. And the final screen itself acts as the detector for that observable.

Superposition is unobservable due to the lack of definition (of the distinction between different states) of the corresponding physical variable, due to the fact that no corresponding detector, such as the left-right detector, is allowed in the experimental design.

physical phenomena

~ observable events

Superposition

~ logically unobservable, since not yet defined

~ not yet defined, since logically undefinable

~ logically undefinable, since no corresponding detector is allowed

A superposition state is not an observable state. In other words, it is not a physical state. Then what is the point of considering it?

Although a superposition state is not a physical state, it is a mathematical state that can be used to calculate the probabilities of different possible physical-states/observable-events.

— Me@2022-07-06 06:00:55 PM

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2022.07.07 Thursday (c) All rights reserved by ACHK

Schrodinger’s cat, 3.4

A macroscopic system (such as a cat) may evolve over time into a superposition of classically distinct quantum states (such as “alive” and “dead”).

— Wikipedia on Quantum superposition

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The components of a superposition must be indistinguishable states.

A superposition is neither an AND state nor an OR state.

AND or OR are only possible for more than one state.

AND or OR are only possible for at least 2 (distinguishable) states.

The cat is not in a superposition state of “alive” and “dead”.

Instead, it is in a mixed state of “alive” and “dead”.

A mixed state is an OR state (of at least 2 distinguishable states).

— Me@2022-07-03 11:02:24 AM

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2022.07.03 Sunday (c) All rights reserved by ACHK

Schrodinger’s cat, 3.3

The modern view is that this mystery is explained by quantum decoherence.

Quantum decoherence is useful, but NOT necessary.

It is useful for the self-consistency checking of quantum mechanics.

Some microscopic states are expressed as (mathematical) superpositions of macroscopic-indistinguishable-if-no-measuring-device-is-allowed states.

Two states are called “macroscopic-distinguishable” only if they result in two different physical phenomena. In other words, the distinction must be observable.

an eigenstate

~ an observable (at least in principle) state

~ a physical state

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a superposition state

~ an unobservable (even in principle) state

~ a mathematical (but not physical) state

We define microscopic states and events in terms of macroscopic states and events. A consistent theory must be able to deduce (explain or predict) macroscopic states and events from those microscopic states and events.

— Me@2022-06-15 07:40:37 PM

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2022.06.29 Wednesday (c) All rights reserved by ACHK

Eigenstates 3

an eigenstate

~ an state identical to the overall average

In analogy, in the equation

\displaystyle{\frac{12+13+14}{3}=13},

the number 13 appears both on the left (as one of the component numbers) and on the right (as the overall average).

In this sense, the number 13 is an “eigenstate”.

— Me@2016-08-25 01:36 AM

— Me@2022-06-28 08:19 PM

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An eigenstate has a macroscopic equivalence.

— Me@2016-08-29 06:10:21 PM

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An eigenstate is a microstate that has a corresponding macrostate.

An eigenstate is a mathematical state which is also a physical state.

An eigenstate is an observable state.

— Me@2022-06-28 07:36:40 PM

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2022.06.28 Tuesday (c) All rights reserved by ACHK

Schrodinger cat’s misunderstanding

Schrodinger’s cat, 3.2

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In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger’s cat, which highlighted this dissonance between quantum mechanics and classical physics.

The main point of the Schrödinger’s cat thought experiment is NOT to prove that there should also be superposition for macroscopic objects. Instead, the main point of the thought experiment is exactly the opposite—to prove that regarding a superposition state as a physical state leads to logical contradiction.

— Me@2022-06-15 07:19:36 PM

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2022.06.22 Wednesday (c) All rights reserved by ACHK

Schrodinger’s cat, 3.1

It is natural to ask why ordinary everyday objects and events do not seem to display quantum mechanical features such as superposition. Indeed, this is sometimes regarded as “mysterious”, for instance by Richard Feynman. In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger’s cat, which highlighted this dissonance between quantum mechanics and classical physics.

The modern view is that this mystery is explained by quantum decoherence. A macroscopic system (such as a cat) may evolve over time into a superposition of classically distinct quantum states (such as “alive” and “dead”). However, the state of the cat is entangled with the state of its environment (for instance, the molecules in the atmosphere surrounding it). If one averages over the quantum states of the environment—a physically reasonable procedure unless the quantum state of all the particles making up the environment can be controlled or measured precisely—the resulting mixed quantum state for the cat is very close to a classical probabilistic state where the cat has some definite probability to be dead or alive, just as a classical observer would expect in this situation.

Quantum superposition is exhibited in fact in many directly observable phenomena, such as interference peaks from an electron wave in a double-slit experiment. Superposition persists at all scales, provided that coherence is shielded from disruption by intermittent external factors. The Heisenberg uncertainty principle states that for any given instant of time, the position and velocity of an electron or other subatomic particle cannot both be exactly determined. A state where one of them has a definite value corresponds to a superposition of many states for the other.

— Wikipedia on Quantum superposition

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It is natural to ask why ordinary everyday objects and events do not seem to display quantum mechanical features such as superposition. Indeed, this is sometimes regarded as “mysterious”, for instance by Richard Feynman.

Superposition is not “mysterious”. It is “mysterious” only if you regard “a superposition state” as a physical state.

Only observable states are physical states. Any observable, microscopic or macroscopic, is NOT a superposition.

A superposition is NOT observable, even in principle; because the component states of a superposition are physically-indistinguishable mathematical states, aka macroscopically-indistinguishable microscopic states.

(Those component states, aka eigenstates, are observable and distinguishable once the corresponding measuring device is allowed.)

They are indistinguishable because the distinction is not defined in terms of the difference between different potential experimental or observational results.

Actually, the distinction is not even definable, because the corresponding measuring device is not allowed in the experimental design yet.

— Me@2022-06-15 11:51:22 AM

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2022.06.16 Thursday (c) All rights reserved by ACHK

Physically-indistinguishable mathematical states

… that’s not totally correct, because a macroscopic state, even in principle, cannot be a superposition of macroscopic eigenstates.

— Me@2012.12.31

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A macrosopic state is an actual physical state.

A macrosopic state, by definition, cannot be a superposition of different macroscopic states. A superposition must be of different macroscopically-indistinguishable microscopic states.

In other words, a physical state, by definition, cannot be a superposition of different physically-distinguishable physical states. A superposition must be of different physically-indistinguishable mathematical states.

— Me@2022-05-17

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2022.05.17 Tuesday (c) All rights reserved by ACHK

數學教育 7.5.1

Genius 4.2.1 | A Fraction of Algebra, 2.1

這段改編自 2010 年 4 月 24 日的對話。

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另外,他提的另一個,有關學習數學的要點是,即使假設你在大學中,學到的數學,在日常生活中沒有用,單單是為獲取,那些嶄新的元素概念本身,就已經能夠令你有超能力;令你有一些,常人沒有的思考工具、比喻語言。

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(安:但是,這個講法可能有一個問題。

雖然,你剛才列舉了數個例子,來示範如何將高深數學,間接應用到人生處世,但是,一般人未必有那種能力。所以我想問,你又是如何去跨過這個難關呢?)

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什麼難關?

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(安:去翻譯那些抽象數學概念,到其他範疇,或者日常生活。)

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那不是「難關」。你的意思是,一般人也沒有那個能力,而我有。所以,那是超能力;我當年一定是,用了一些秘技,才獲取之。

天才之道,點滴累積。其實並沒有所謂的「秘技」。只要一步一步地,學習數學,就自然建構出,一個相對接近完整的數學思考體系,生成「翻譯抽象數學概念到其他範疇」等能力。

所以,我猜想你的疑問是,其實我所講的「點滴累積」,或者「一步一步地」,雖然理想上是,基本的要求,但是現實中是,大部人也做不到。那就代表著,大部人可能也會遇到,一個共通的「難關」。那個「難關」究竟是什麼?我又是如何克服它,而做到「一步一步地」「點滴累積」的呢?

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你問題的最簡化版本是:「如何學習數學,開創人生?」

有起碼以下三個先決條件:

1. 對數學(及其他學問人生),有極大興趣;

2. 遇到合理的老師和書籍:

重點是,數學概念或運算上的主要步驟,亳無違漏。支節可免,但主旨必須。細節可以無師自通,大節必靠前人指點。平地自己行,斜地靠梯級。平地可跳步,梯不可跳級。

3. 極超大量的背誦和練習:

數學是理科,所以其背誦方法,不是「死背」零碎隨機的資料,而是「生背」息息相關的訊息。融匯貫通地背誦的唯一方法是,極超大量的操練。

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你所講的「難關」,就是以上的第二點。老師有分好老師和差老師。大部分也是差老師。而差老師再分兩類:不懂數學和不懂教學。

— Me@2022.05.02 11:48 PM

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2022.05.03 Tuesday (c) All rights reserved by ACHK

Dynamical pictures

Comparison of pictures

The Heisenberg picture is closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly correspond to classical Poisson brackets). The Schrödinger picture, the preferred formulation in introductory texts, is easy to visualize in terms of Hilbert space rotations of state vectors, although it lacks natural generalization to Lorentz invariant systems. The Dirac picture is most useful in nonstationary and covariant perturbation theory, so it is suited to quantum field theory and many-body physics.

Summary comparison of evolutions

— Wikipedia on Dynamical pictures

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2022.04.05 Tuesday ACHK

Huygens’ principle, 1.1

Why are there no backward secondary wavefronts?

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A secondary wave source is of different nature from a primary wave source. Consider an one dimensional transverse wave on a string:

Primary wave source is activated by the force from above, and then the wave propagates to both directions. Secondary wave source is activated by the particle on the left.

Primary wave source energy is from outside the string. Secondary wave source energy is from an adjacent string molecule.

When the secondary source S reaches its maximum, although it drags both P and Q, they are in different situations. At that moment, while the instantaneous velocity of Q is upward, that of P is downward.

Note that the red arrow at P is the force on P by S. It is not the net force on P.

— Me@2022-03-09 11:14:07 PM

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2022.03.30 Wednesday (c) All rights reserved by ACHK