Category Archives: physics

Ground states and Annihilation operators

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1. The equation a | 0 \rangle = 0 means that the eigenvalue of a on | 0 \rangle is 0:

a | 0 \rangle = 0 | 0 \rangle

2. The length of the vector a | 0 \rangle is 0:

\langle 0 | a^\dagger a | 0 \rangle = 0

3. The physical meaning is that the probability of the system being at state a | 0 \rangle is 0.
 
In other words, there is no state with an eigen-energy lower than the ground state one.
 
 
4. For the equation a | 0 \rangle = 0 | 0 \rangle, the 0 at the right is a scalar.
 
 
5. For the equation a | 0 \rangle = 0, the 0 at the right is a zero vector – a state vector with length zero.
 
6. | 0 \rangle is a state vector. However, it is NOT the zero vector.

Instead, it is the state vector of the ground state. Its length is 1 unit.

— Me@2015-11-03 03:26:58 PM
 
 
 
2015.11.04 Wednesday (c) All rights reserved by ACHK

Englert–Greenberger duality relation, 2

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The most recent work claims that Afshar’s core claim, that the Englert–Greenberger duality relation is violated, is not true. They re-ran the experiment, using a different method for measuring the visibility of the interference pattern than that used by Afshar, and found no violation of complementarity, concluding “This result demonstrates that the experiment can be perfectly explained by the Copenhagen interpretation of quantum mechanics.”

Numerous experiments have shown, however, that any modification of the apparatus that can determine which slit a particle passes through reduces the visibility of interference at the screen, thereby illustrating the complementarity principle: that light (and electrons, etc.) can behave as either particles or waves, but not both at the same time. An experiment performed in 1987 produced results that demonstrated that information could be obtained regarding which path a particle had taken, without destroying the interference altogether. This showed the effect of measurements that disturbed the particles in transit to a lesser degree and thereby influenced the interference pattern only to a comparable extent.

In other words, if one does not insist that the method used to determine which slit each photon passes through be completely reliable, one can still detect a (degraded) interference pattern.

— Wikipedia on Englert–Greenberger duality relation

— Wikipedia on Double-slit experiment

2015.09.22 Tuesday ACHK

Quantum Indeterminacy

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注定外外傳 1

Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics.

…

Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that errors in measurement may lead to indeterminate outcomes. However, by the later half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.

— Wikipedia on Quantum indeterminacy

Quantum indeterminacy is the inability to predict the behaviour of the system with 100% accuracy, even in principle.

If everything is connected , quantum indeterminacy is due to the logical fact that, by definition, a “part” cannot contain (all the information of) the “whole”.

An observer (A) cannot separate itself from the system (B) that it wants to observe, because an observation is an interaction between the observer and the observed .  

In order to get a perfect prediction of a measurement result, observer (A) must have all the information of the present state of the whole system (A+B). However, there are two logical difficulties.

First, observer A cannot have all the information about (A+B).

Second, observer A cannot observe itself to get (all of) its present state information, since an observation is an interaction between two entities. Logically, it is impossible for something to interact with itself directly. Just as logically, it is impossible for your right hand to hold your right hand itself. 

So the information observer A can get (to the greatest extent) is all the information about B, which is only part of the system (A+B) it (A) needs to know in order to get a prefect prediction for the evolution of the system B.

— Me@2015-09-14 08:12:32 PM

2015.09.15 Tuesday (c) All rights reserved by ACHK

EPR paradox, 5.2

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With a spacelike separation, it makes no absolute sense to say which of Alice and Bob causes the wave function to collapse.

— Me@2012.04.10

2015.06.13 Saturday (c) All rights reserved by ACHK

Truth values in Quantum Mechanics

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It’s pretty much the same mistake that all the “realists” are doing all the time. They are imagining that they can assign truth values to all questions that could be in principle answered by experiments. (In some way, they give answers “No” or “zero” to all the other questions that were actually not addressed by the measurement.) But quantum mechanics prohibits that. If one assigns classical truth values (or real values) to some operators, one can no longer assign truth values (or real values) to “complementary” (not mutually commuting) operators and questions they represent. Instead of the correct statement that “the value of \(N_a\) isn’t determined if one measures \(L\) instead”, they say that it is zero which is just wrong.

— No particle upon a quantum field means no information

— Lubos Motl

2015.06.03 Wednesday ACHK

Bell’s theorem, 6

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E. T. Jaynes pointed out two hidden assumptions in Bell Inequality that could limit its generality. According to him:

1. Bell interpreted conditional probability P(X|Y) as a causal inference, i.e. Y exerted a causal inference on X in reality. However, P(X|Y) actually only means logical inference (deduction). Causes cannot travel faster than light or backward in time, but deduction can.

— Wikipedia on Bell’s theorem

However, it should be interpreted as causal inference because before measurement, there is no Y, such a classical state.

— Me@2012-04-07 2:35:55 PM

2015.05.23 Saturday (c) All rights reserved by ACHK

Quantum Gravity

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We sometimes say that string theory is the only consistent theory of quantum gravity. It’s the only game in town. This is an observation mostly based on various types of circumstantial evidence. Whenever you try something that deviates from string/M-theory, you run into inconsistencies. Sometimes you don’t run into inconsistencies but something else happens. Many good ideas that were thought to be “competitors” to string theory were shown to be just aspects of some (usually special) solutions to string theory (noncommutative geometry, CFT, matrix models, and even the Hořava-Lifshitz class of theories have been found to be parts of string theory), and so on. And decades of attempts to find a truly inequivalent competing theory have utterly failed. That’s not a complete proof of their absence, either, but it is evidence that shouldn’t be completely ignored.

But that doesn’t mean that the statement that every consistent theory of quantum gravity has to be nothing else than another approach to string/M-theory is just an expression of vague feelings, a guesswork, or a partial wishful thinking. We don’t have the “most complete proof” of this assertion yet – this fact may be partly blamed on the absence of the completely universal, most rigorous definition of both “quantum gravity” and “string theory”. But there exist partial proofs and this paper is an example.

— Every theory of quantum gravity is a part of string theory: a partial proof

— A successful test in \(AdS_3\)

— Lubos Motl

2015.05.06 Wednesday ACHK

Ideal clock 1.3

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An ideal clock is a clock (i.e., recurrent process) that makes the most other recurrent processes periodic.

— Wikipedia on Clock

The ideal clock itself is a meta-clock — a property of a set of clocks.

— Me@2015-04-16 9:35 AM

If you compare the accuracies of two clocks (A and B) by comparing each of them with a third clock (C), then I can ask, “How can you know that C is more accurate than both A and B?”

By defining the ideal clock as a meta-clock, we avoid infinite regress.

— Me@2015-04-27 11:36:59 AM

2015.04.29 Wednesday (c) All rights reserved by ACHK

Ideal clock 1.2

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A clock is a recurrent process and a counter.

A good clock is one which, when used to measure other recurrent processes, finds many of them to be periodic.

An ideal clock is a clock (i.e., recurrent process) that makes the most other recurrent processes periodic.

— Wikipedia on Clock

The ideal clock itself is a meta-clock — a property of a set of clocks.

— Me@2015-04-16 9:35 AM

2015.04.27 Monday (c) All rights reserved by ACHK

機遇再生論 1.4

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「

『機遇再生論』的大概意思是，所有可能發生的事情，例如重生，只要等足夠長的時間，總會發生。

」

但是，即使避開了「無限」，用了「足夠長」，仍然會有其他問題。「足夠長」這個詞語雖然不算違法，但是十分空泛，空泛到近乎沒有意義。

試想想，怎樣才為之「足夠長」呢？

.

以前在本網誌中提及過，凡是科學句子，都一定要有「可否證性」。因為凡是科學句子，都對世界有所描述，所以必為「經驗句」，不是「重言句」。凡是「經驗句」，必定有機會錯。換而言之，無論正確的機會率有多高，都不會是百分百。

因此，要測試某一句說話，是不是「科學句子」，你可以檢查一下，它有沒有「可否證性」。「可否證性」的意思是，如果一句「科學句子」有意義，你就可以講得出，至少在原則上，它在什麼情況下，為之錯。

例如，

「

甲在過身之後，一千億年內會重生。

」

是句「科學句」（經驗句），因為你知道在什麼情境下，可以否證到它 —— 如果你在甲過身後，等了一千億年，甲還未重生的話，那句就為之錯。

但是，

「

甲在過身之後，只要等足夠長的時間，必會重生。

」

則沒有任何科學意義。

— Me@2015.04.08

.

.

2015.04.15 Wednesday (c) All rights reserved by ACHK

機遇再生論 1.3

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「

『機遇再生論』的大概意思是，所有可能發生的事情，例如重生，在無限長的未來時間中，必會發生。

」

機遇再生論原始版本，有問題的字眼中，除了「所有」之外，還有「無限」。「無限」通常都是一個違法詞語。「無限」引起的問題，以前論述過，現不再詳談。請參閱「無限」系列的文章。

你可以嘗試移除「無限」這個詞語，只把「無限」的意思中，有意義的部分保留：

「

『機遇再生論』的大概意思是，所有可能發生的事情，例如重生，只要等足夠長的時間，總會發生。

」

但是，即使避開了「無限」，用了「足夠長」，仍然會有其他問題。「足夠長」這個詞語雖然不算違法，但是十分空泛，空泛到近乎沒有意義。

試想想，怎樣才為之「足夠長」呢？

— Me@2015.04.08

.

.

2015.04.09 Thursday (c) All rights reserved by ACHK

Learn Physics by Programming in Haskell

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Learning functional programming and partially applying functions to other functions and such helped me understand tensors a lot better, since that’s basically what contraction is doing. It’s nice to see that the approach can be taken further.

— Snuggly_Person

I also think Haskell and some similar languages (especially Idris) have a great conceptual synergy with physics.

In physics too we strive to express things in ways that strip out extraneous details as much as possible. Haskell really embraces this concept in the sense that you write functions essentially by writing equations. You don’t describe all the mechanical steps to produce an output, you just write down the ‘invariant content’ of the function.

— BlackBrane

2015.03.29 Sunday ACHK