Logical arrow of time, 9.4

The second law of thermodynamics’ derivation (Ludwig Boltzmann’s H-theorem) is with respect to an observer.

How does an observer keep losing microscopic information about a system?

— Me@2017-02-12 07:37:54 PM


This drew the objection from Loschmidt that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism: something must be wrong (Loschmidt’s paradox).

The resolution (1895) of this paradox is that the velocities of two particles after a collision are no longer truly uncorrelated. By asserting that it was acceptable to ignore these correlations in the population at times after the initial time, Boltzmann had introduced an element of time asymmetry through the formalism of his calculation.

— Wikipedia on Molecular chaos


Physical entropy’s value is with respect to an observer.

— Me@2017-02-12 07:37:54 PM


This “paradox” can be explained by carefully considering the definition of entropy. In particular, as concisely explained by Edwin Thompson Jaynes, definitions of entropy are arbitrary.

As a central example in Jaynes’ paper points out, one can develop a theory that treats two gases as similar even if those gases may in reality be distinguished through sufficiently detailed measurement. As long as we do not perform these detailed measurements, the theory will have no internal inconsistencies. (In other words, it does not matter that we call gases A and B by the same name if we have not yet discovered that they are distinct.) If our theory calls gases A and B the same, then entropy does not change when we mix them. If our theory calls gases A and B different, then entropy does increase when they are mixed. This insight suggests that the ideas of “thermodynamic state” and of “entropy” are somewhat subjective.

— Wikipedia on The mixing paradox

— Wikipedia on Gibbs paradox



2021.08.07 Saturday (c) All rights reserved by ACHK

Causal diamonds in time travel

Quantum mechanics is a set of rules that allows an observer to predict, explain, and/or verify observations (and especially their mutual relationships) that he has access to.

No observer can detect inconsistencies within the causal diamonds. However, inconsistencies between “stories” as told by different observers with different causal diamonds are allowed (and mildly encouraged) in general (as long as there is no observer who could incorporate all the data needed to see an inconsistency).

— Raphael Bousso is right about firewalls

— Lubos Motl


There is no “god’s eye view” in physics.

— Me@2021-04-17 03:12:58 PM


Macroscopic time is with respect to an observer. Actually, physics is with respect to an observer.

In the real universe, any observer’s observations must be consistent. When two observers compare their observations, their results must be consistent, because the comparison itself is an observation of an observer.

Time travel in the absolute sense is logically impossible. Let’s assume that it is logically possible.

If a time travel story follows the principle of “an observer’s observations must be consistent”, each character in that story must see a consistent timeline, even if different characters’ timelines may be inconsistent. That is fine as long as such inconsistent observers never meet to compare their results.

If two of such observers choose to meet to compare their results, the action to “meet to compare” itself will render the results consistent. It is similar to the resolution of the twin paradox in special relativity.

There is no “god’s eye view” in physics. Every physical event must be described with respect to an observer. Every physical event, even if the event is “to compare observation results”, must be described with respect to an observer.

— Me@2017-05-10 07:45:36 AM



2021.04.17 Saturday (c) All rights reserved by ACHK

Conscious time, 2

If no one has any kind of date, records, memories, or evidence about the past, retro-diction MAY be the same as prediction. But in such a case, it is by definition not our “past” any more.

— Me@2013-08-08 3:11 PM


If no one has any kind of date, records, memories, or evidence about the past, then consciousness ceases to exist.

We, as conscious beings, cannot exist anymore.

— Me@2021-03-30 4:08 PM



2021.03.30 Tuesday (c) All rights reserved by ACHK

Quantum information makes classical information consistent

Consistent histories, 10 | Cosmic computer, 2


Wrong: Quantum information is inconsistent and classical information is consistent.

Source: Misunderstanding quantum superposition; regarding mathematical superposition as physical superposition, violating logic, such as a particle has gone through both the left slit and right slit at the same time.

Right: Quantum information is what makes sure that classical information is consistent even when there are indistinguishabilities of some classical cases.


a quantum superposition state

~ a state without classical equivalence because some classical cases are indistinguishable-even-in-principle that they are logically forced into one SINGLE physical state

— Me@2021-02-03 01:46:43 PM


quantum superposition

= mathematical superposition

= physical NON-superposition

= logically one SINGLE physical state

= go-left and go-right are logically indistinguishable due to the “experiment setup is without detector” part of the definition

= the SINGLE state of “both slits are open but no measuring device is installed; so for each photon, we have no which-way information; because there is no which-way DEFINITION”


The definition requirement means that you have to answer

Under what physical phenomenon/phenomena occur(s) that you will say that the photon has gone through the left slit?

In other words, you need to DEFINE “go-left” in terms of at least one potential observable or measurable physical phenomenon. Same for “go-right”.


In the double-slit experiment, if there is no meaning of “the difference between ‘go-left’ and ‘go-right’“, then there is no meaning of “go-left”. (Same for “go-right”.)

In that case, we have only the meaning of “go-through-the-slits (without distinguishing ‘go-left’ and ‘go-right’)“.

We still have that meaning because we can still define

the photon has gone through the board (that consists of those 2 slits)


there is a dot appearing on the final screen almost immediately after a photon is emitted from the source“.

— Me@2021-02-03 07:48:01 AM


quantum information = mathematical information

classical information = physical information


quantum entanglement

~ all measurement results will be consistent

~ all measurement results follow the three basic logic laws (i.e. identity, non-contradiction, excluded middle)

— Me@2021-01-30 09:46:13 AM

— Me@2021-02-03 12:27:07 AM


Quantum theory is:

The minimal mathematical formalism that correctly describes all physical interaction as classical information exchange and all classical information exchange as physical interaction.

Entanglement is:

The condition of interacting with the world through an imaginary interface on which classical information appears.

— What Is Entanglement Anyway?

— Chris Fields


Quantum mechanics is a theory of classical information.

Quantum mechanics explains why all the measurement results are always consistent in spite of the quantum effects, the effects due to the indistinguishabilities of some classical cases.

Quantum mechanics explains why all the measurement results are always consistent in spite of the indistinguishabilities of some classical cases.

— Me@2021-01-28 09:55:56 PM

— Me@2021-02-03 07:48:01 AM



2021.02.03 Wednesday (c) All rights reserved by ACHK

… is in a superposition

Quantum decoherence 5.3.2 | Wheeler’s delayed choice experiment 1.2


For example, in the double-slit experiment, if no detector is installed, the system is in a quantum superposition state.

It is not that each individual photon is in a superposition. Instead, it is that the system of the whole experimental setup is in a superposition.

— Me@2021-01-23 12:57 AM



However, “what ‘the whole experimental setup‘ is” is not 100% objective. In other words, it is a little bit subjective.

“The whole experimental setup”, although largely objective, is partially defined with respect to an observer.

— Me@2021-01-23 12:58 AM


So quantum probability/indistinguishability effect is partly observer-dependent, although the subjectivity is just tiny compared with that of the classical probability in a mixed state.

— Me@2021-01-23 12:59 AM




2021.01.24 Sunday (c) All rights reserved by ACHK

Consistent histories, origin

The square root of the probability, 6


There is no wave function collapse.

For example, in the double-slit experiment, with-detector and without-detector are actually two different physics systems. Different experimental setups provide different probability distributions, encoded in the wave functions. So different experimental setups result in different wave functions.

That is the key to understanding strange quantum phenomena such as EPR. A classical system has consistent results is no magic.

You create either a system with a detector or a system without a detector. With a detector, it will have only distinguishable-at-least-in-definition states, aka classical states. A system with only classical states is a classical system. Then, why so shocked when a classical system has consistent results?

Quantum mechanics is “strange”, but not “that strange”. It is not so strange that it is unexplainable.

— Me@2021-01-20 07:11 PM

— Me@2021-01-22 08:48 AM



2021.01.22 Friday (c) All rights reserved by ACHK

The square root of the probability, 4.3

Eigenstates 3.4.3


The indistinguishability of cases is where the quantum probability comes from.

— Me@2020-12-25 06:21:48 PM


In the double slit experiment, there are 4 cases:

1. only the left slit is open

2. only the right slit is open

3. both slits are open and a measuring device is installed somewhere in the experiment setup so that we can know which slit each photon passes through

4. both slits are open but no measuring device is installed; so for each photon, we have no which-way information


For simplicity, we rephrase the case-3 and case-4:

1. only the left slit is open

2. only the right slit is open

3. both slits are open, with which-way information

4. both slits are open, without which-way information


Case-3 can be regarded as a classical overlapping of case-1 and case-2, because if you check the result of case-3, you will find that it is just an overlapping of result case-1 and result case-2.

However, case-4 cannot be regarded as a classical overlapping of case-1 and case-2. Instead, case-4 is a quantum superposition. A quantum superposition canNOT be regarded as a classical overlapping of possibilities/probabilities/worlds/universes.

Experimentally, no classical overlapping can explain the interference pattern, especially the destruction interference part. An addition of two non-zero probability values can never result in a zero.

Logically, case-4 is a quantum superposition of go-left and go-right. Case-4 is neither AND nor OR of the case-1 and case-2.


We can discuss AND or OR only when there are really 2 distinguishable cases. Since there are not any kinds of measuring devices (for getting which-way information) installed anywhere in the case-4, go-left and go-right are actually indistinguishable cases. In other words, by defining case-4 as a no-measuring-device case, we have indirectly defined that go-left and go-right are actually indistinguishable cases, even in principle.

Note that saying “they are actually indistinguishable cases, even in principle” is equivalent to saying that “they are logically indistinguishable cases” or “they are logically the same case“. So discussing whether a photon has gone left or gone right is meaningless.


If 2 cases are actually indistinguishable even in principle, then in a sense, there is actually only 1 case, the case of “both slits are open but without measuring device installed anywhere” (case-4). Mathematically, this case is expressed as the quantum superposition of go-left and go-right.

Since it is only 1 case, it is meaningless to discuss AND or OR. It is neither “go-left AND go-right” nor “go-left OR go-right“, because the phrases “go-left” and “go-right” are themselves meaningless in this case.

— Me@2020-12-19 10:38 AM

— Me@2020-12-26 11:02 AM


It is a quantum superposition of go-left and go-right.

Quantum superposition is NOT an overlapping of worlds.

Quantum superposition is neither AND nor OR.

— Me@2020-12-26 09:07:22 AM

When the final states are distinguishable you add probabilities:

\displaystyle{P_{dis} = P_1 + P_2 = \psi_1^*\psi_1 + \psi_2^*\psi_2}


When the final state are indistinguishable,[^2] you add amplitudes:

\displaystyle{\Psi_{1,2} = \psi_1 + \psi_2}


\displaystyle{P_{ind} = \Psi_{1,2}^*\Psi_{1,2} = \psi_1^*\psi_1 + \psi_1^*\psi_2 + \psi_2^*\psi_1 + \psi_2^*\psi_2}


[^2]: This is not precise, the states need to be “coherent”, but you don’t want to hear about that today.

edited Mar 21 ’13 at 17:04
answered Mar 21 ’13 at 16:58


— Physics Stack Exchange


\displaystyle{ P_{ind} = P_1 + P_2 + \psi_2^*\psi_1 + \psi_2^*\psi_2 }

\displaystyle{ P_{\text{indistinguishable}} = P_{\text{distinguishable}} + \text{interference terms} }

— Me@2020-12-26 09:07:46 AM


interference terms ~ indistinguishability effect

— Me@2020-12-26 01:22:36 PM



2021.01.05 Tuesday (c) All rights reserved by ACHK

The square root of the probability, 4.2

Eigenstates 3.4.2


The difference between quantum and classical is due to the indistinguishability of cases.

— Me@2020-12-26 01:25:03 PM


Statistical effects of indistinguishability

The indistinguishability of particles has a profound effect on their statistical properties.


The differences between the statistical behavior of fermions, bosons, and distinguishable particles can be illustrated using a system of two particles. The particles are designated A and B. Each particle can exist in two possible states, labelled \displaystyle{ |0 \rangle } and \displaystyle{|1\rangle}, which have the same energy.

The composite system can evolve in time, interacting with a noisy environment. Because the \displaystyle{|0\rangle} and \displaystyle{|1\rangle} states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states. (This is discussed in the article on quantum entanglement.) After some time, the composite system will have an equal probability of occupying each of the states available to it. The particle states are then measured.

If A and B are distinguishable particles, then the composite system has four distinct states: \displaystyle{|0\rangle |0\rangle}, \displaystyle{|1\rangle |1\rangle} , \displaystyle{ |0\rangle |1\rangle}, and \displaystyle{|1\rangle |0\rangle }. The probability of obtaining two particles in the \displaystyle{|0\rangle} state is 0.25; the probability of obtaining two particles in the \displaystyle{|1\rangle} state is 0.25; and the probability of obtaining one particle in the \displaystyle{|0\rangle} state and the other in the \displaystyle{|1\rangle} state is 0.5.

If A and B are identical bosons, then the composite system has only three distinct states: \displaystyle{|0\rangle |0\rangle}, \displaystyle{ |1\rangle |1\rangle }, and \displaystyle{{\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle +|1\rangle |0\rangle)}. When the experiment is performed, the probability of obtaining two particles in the \displaystyle{|0\rangle} is now 0.33; the probability of obtaining two particles in the \displaystyle{|1\rangle} state is 0.33; and the probability of obtaining one particle in the \displaystyle{|0\rangle} state and the other in the \displaystyle{|1\rangle} state is 0.33. Note that the probability of finding particles in the same state is relatively larger than in the distinguishable case. This demonstrates the tendency of bosons to “clump.”

If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state \displaystyle{{\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle -|1\rangle |0\rangle)}. When the experiment is performed, one particle is always in the \displaystyle{|0\rangle} state and the other is in the \displaystyle{|1\rangle} state.

The results are summarized in Table 1:

— Wikipedia on Identical particles



2020.12.30 Wednesday (c) All rights reserved by ACHK

The square root of the probability, 4

Eigenstates 3.4


quantum ~ classical with the indistinguishability of cases

— Me@2020-12-23 06:19:00 PM


In statistical mechanics, a semi-classical derivation of the entropy that does not take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive (is not proportional to the amount of substance in question). This leads to a paradox known as the Gibbs paradox, after Josiah Willard Gibbs who proposed this thought experiment in 1874‒1875. The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. A related paradox is the “mixing paradox”. If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, the paradox is averted.

— Wikipedia on Gibbs paradox



2020.12.27 Sunday (c) All rights reserved by ACHK

Logical arrow of time, 6.4.2

Logical arrow of time, 6.1.2


The source of the macroscopic time asymmetry, aka the second law of thermodynamics, is the difference between prediction and retrodiction.

In a prediction, the deduction direction is the same as the physical/observer time direction.

In a retrodiction, the deduction direction is opposite to the physical/observer time direction.


— guess —

If a retrodiction is done by a time-opposite observer, he will see the entropy increasing. For him, he is really making a prediction.

— guess —


— Me@2013-10-25 3:33 AM


A difference between deduction and observation is that in observation, the probability is updated in real time.


each update time interval ~ infinitesimal


In other words, when you observe a system, you get new information about that system in real time.

Since you gain new knowledge of the system in real time, the probability assigned to that system is also updated in real time.

— Me@2020-10-13 11:27:59 AM



2020.12.04 Friday (c) All rights reserved by ACHK

Consistent histories, 8

Relationship with other interpretations

The only group of interpretations of quantum mechanics with which RQM is almost completely incompatible is that of hidden variables theories. RQM shares some deep similarities with other views, but differs from them all to the extent to which the other interpretations do not accord with the “relational world” put forward by RQM.

Copenhagen interpretation

RQM is, in essence, quite similar to the Copenhagen interpretation, but with an important difference. In the Copenhagen interpretation, the macroscopic world is assumed to be intrinsically classical in nature, and wave function collapse occurs when a quantum system interacts with macroscopic apparatus. In RQM, any interaction, be it micro or macroscopic, causes the linearity of Schrödinger evolution to break down. RQM could recover a Copenhagen-like view of the world by assigning a privileged status (not dissimilar to a preferred frame in relativity) to the classical world. However, by doing this one would lose sight of the key features that RQM brings to our view of the quantum world.

Hidden variables theories

Bohm’s interpretation of QM does not sit well with RQM. One of the explicit hypotheses in the construction of RQM is that quantum mechanics is a complete theory, that is it provides a full account of the world. Moreover, the Bohmian view seems to imply an underlying, “absolute” set of states of all systems, which is also ruled out as a consequence of RQM.

We find a similar incompatibility between RQM and suggestions such as that of Penrose, which postulate that some processes (in Penrose’s case, gravitational effects) violate the linear evolution of the Schrödinger equation for the system.

Relative-state formulation

The many-worlds family of interpretations (MWI) shares an important feature with RQM, that is, the relational nature of all value assignments (that is, properties). Everett, however, maintains that the universal wavefunction gives a complete description of the entire universe, while Rovelli argues that this is problematic, both because this description is not tied to a specific observer (and hence is “meaningless” in RQM), and because RQM maintains that there is no single, absolute description of the universe as a whole, but rather a net of inter-related partial descriptions.

Consistent histories approach

In the consistent histories approach to QM, instead of assigning probabilities to single values for a given system, the emphasis is given to sequences of values, in such a way as to exclude (as physically impossible) all value assignments which result in inconsistent probabilities being attributed to observed states of the system. This is done by means of ascribing values to “frameworks”, and all values are hence framework-dependent.

RQM accords perfectly well with this view. However, the consistent histories approach does not give a full description of the physical meaning of framework-dependent value (that is it does not account for how there can be “facts” if the value of any property depends on the framework chosen). By incorporating the relational view into this approach, the problem is solved: RQM provides the means by which the observer-independent, framework-dependent probabilities of various histories are reconciled with observer-dependent descriptions of the world.

— Wikipedia on Relational quantum mechanics



2020.09.27 Sunday ACHK

Logical arrow of time, 7.2

Microscopically, there is no time arrow.

— Me@2011.06.23


No. There is weak force.

— Me@2011.07.22


Myth: The arrow of time is a consequence of CP-symmetry violation.

The weak nuclear interactions violate the CP symmetry which is equivalent to saying that they violate the T symmetry. Is it the reason why eggs don’t unbreak? Of course not. There are two basic ways to see why. First, the weak interactions much like all other interactions preserve the CPT symmetry – there is extensive theoretical as well as experimental evidence supporting this assertion. And the CPT symmetry would be enough to show that eggs break as often as unbreak. More precisely, eggs break as often as mirror anti-eggs unbreak. ;-)

— Myths about the arrow of time

— Lubos Motl


Weak force’s T-symmetry-violation has nothing to do with the time arrow.

In other words, microscopic time arrow has nothing to do with the macroscopic time arrow.

— Me@2020-03-21 07:56:01 PM


About T-violation and the arrow of time: the simple answer is that the weak interactions are perfectly unitary, even if they are not T-invariant. They don’t affect the entropy in any way, so they don’t help with the arrow of time.

A bit more carefully: if you did want to explain the arrow of time using microscopic dynamics, you would have to argue that there exist more solutions to the equations of motion in which entropy grows than solutions in which entropy decreases. But CPT invariance is enough to guarantee that that’s not true. For any trajectory (or ensemble of trajectories, or evolution of a distribution function) in which the entropy changes in one way, there is another trajectory (or set…) in which the entropy changes in precisely the opposite way: the CPT conjugate. Such laws of physics do not in and of themselves pick out what we think of as the arrow of time.

People talk about the “arrow of time of the weak interactions,” but ask yourself: in which direction does it point? There just isn’t any direct relationship to entropy.

— Sean Carroll



2020.03.21 Saturday (c) All rights reserved by ACHK

Black hole information paradox, 4

So we seem to have a direct contradiction between [QM and unitarity] and [GR and causality]. Both of these principles, unitarity and causality, cannot be exactly correct because a contradiction arises from their explosive mixture.


As most quantum mechanicians have known from the very beginning, it is the unitarity, a principle of quantum mechanics, that wins in the battle and remains universally valid.

On the other hand, causality becomes an approximate principle that is only valid in the limit of infinitely large causal diamonds. In the presence of black holes, the internal causal structure is modified by quantum phenomena and the information can “tunnel” out of the black hole.


It shouldn’t be so surprising that unitarity survives completely while causality doesn’t. After all, the basic postulates of quantum mechanics, including unitarity, the probabilistic interpretation of the amplitudes, and the linearity of the operators representing observables, seem to be universally necessary to describe physics of any system that agrees with the basic insights of the quantum revolution.

On the other hand, geometry has been downgraded into an effective, approximate, emergent aspect of reality. The metric tensor is just one among many fields in our effective field theories including gravity. In string theory, there are, in some sense, infinitely many such fields besides the metric tensor – the whole “stringy tower”. The metric tensor doesn’t have to exist as a good degree of freedom at the Planck scale or in other extreme conditions. We know many other fields that are only good enough at low energies – e.g. the pion field.

— Black hole information puzzle

— Lubos Motl



2020.02.09 Sunday ACHK

Spacetime interval

Two contrasting viewpoints on time divide prominent philosophers.

One view is that time is part of the fundamental structure of the universe – a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is sometimes referred to as Newtonian time.

The opposing view is that time does not refer to any kind of “container” that events and objects “move through”, nor to any entity that “flows”, but that it is instead part of a fundamental intellectual structure (together with space and number) within which humans sequence and compare events.

This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, and thus is not itself measurable nor can it be travelled.

— Wikipedia on Time


Special relativity declares a similar law for all motion: the combined speed of any object’s motion through space and its motion through time is always precisely equal to the speed of light.

— The Fabric of the Cosmos: Space, Time, and the Texture of Reality

— Brian Greene


Space is relative, in the sense that the space interval, \Delta d, (aka distance) between two events can have different values for different observers.

\displaystyle{ \begin{aligned}   \Delta {d} &= \sqrt{{\left(\Delta {x}\right)}^{2}+{\left(\Delta {y}\right)}^{2}+{\left(\Delta {z}\right)}^{2}} \\    \end{aligned} }

Time is relative, in the sense that the time interval, \Delta t, (aka duration) between two events can have different values for different observers.

Spacetime is absolute, in the sense that the spacetime interval, (\Delta s)^2, between two events cannot have different values for different observers.

\displaystyle{  \begin{aligned}  (\Delta s)^{2}  &= (\Delta ct)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2} \\  &= (\Delta ct)^{2}-(\Delta d)^{2} \\  \end{aligned} }

— paraphrasing The Fabric of the Cosmos

— Me@2020-01-26 12:46:41 AM



2020.02.03 Monday (c) All rights reserved by ACHK