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

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2020.09.27 Sunday ACHK

The square root of the probability, 2

Mixed states, 4.2

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Superposition in quantum mechanics is a complex number superposition.

— Me@2017-08-02 02:56:23 PM

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Superposition in quantum mechanics is not a superposition of probabilities.

Instead, it is a superposition of probability amplitudes, which have complex number values.

Probability amplitude, in a sense, is the square root of probability.

— Me@2020-08-04 03:38:43 PM

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2020.08.04 Tuesday (c) All rights reserved by ACHK

Quantum entanglement, 4

What’s sneaky about quantum mechanics is that the whole system can be in a pure state which when restricted to each subsystem gives a mixed state, and that these mixed states are then correlated (necessarily, as it turns out). That’s what “entanglement” is all about.

The first way things get trickier in quantum mechanics is that something we are used to in classical mechanics fails. In classical mechanics, pure states are always dispersion-free — that is, for every observable, the probability measure assigned by the state to that observable is a Dirac delta measure, that is, the observable has a 100% chance of being some specific value and a 0% chance of having any other value. (Consider the example of the dice, with the observable being the number of dots on the face pointing up.) In quantum mechanics, pure states need NOT be dispersion-free. In fact, they usually aren’t.

A second, subtler way things get trickier in quantum mechanics concerns systems made of parts, or subsystems. Every observable of a subsystem is automatically an observable for the whole system (but not all observables of the whole system are of that form; some involve, say, adding observables of two different subsystems). So every state of the whole system gives rise to, or as we say, “restricts to,” a state of each of its subsystems. In classical mechanics, pure states restrict to pure states. For example, if our system consisted of 2 dice, a pure state of the whole system would be something like “the first die is in state 2 and the second one is in state 5;” this restricts to a pure state for the first die (state 2) and a pure state for the second die (state 5). In quantum mechanics, it is not true that a pure state of a system must restrict to a pure state of each subsystem.

It is this latter fact that gave rise to a whole bunch of quantum puzzles such as the Einstein-Podolsky-Rosen puzzle and Bell’s inequality. And it is this last fact that makes things a bit tricky when one of the two subsystems happens to be you. It is possible, and indeed very common, for the following thing to happen when two subsystems interact as time passes. Say the whole system starts out in a pure state which restricts to a pure state of each subsystem. After a while, this need no longer be the case! Namely, if we solve Schroedinger’s equation to calculate the state of the system a while later, it will necessarily still be a pure state (pure states of the whole system evolve to pure states), but it need no longer restrict to pure states of the two subsystems. If this happens, we say that the two subsystems have become “entangled.”

— December 16, 1993

— This Week’s Finds in Mathematical Physics (Week 27)

— John Baez

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2020.07.19 Sunday ACHK

Consistent histories, 7

In quantum mechanics, the consistent histories (also referred to as decoherent histories) approach is intended to give a modern interpretation of quantum mechanics, …

— Wikipedia on Consistent histories

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It needs to be decoherent in order to be consistent.

— Me@2017-08-08 01:25:54 PM

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decoherent ~ no quantum superposition

consistent ~ classical logic can apply

— Me@2020-05-30 03:52:15 PM

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2020.05.30 Saturday (c) All rights reserved by ACHK

Logical arrow of time, 7.2

Microscopically, there is no time arrow.

— Me@2011.06.23

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No. There is weak force.

— Me@2011.07.22

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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

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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

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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

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2020.03.21 Saturday (c) All rights reserved by ACHK

Bell’s theorem, 7

dools on Nov 21, 2014

When I watched the Leonard Susskind lectures on quantum entanglements he said the whole “communicating faster than light” thing is a bit misleading. The analogy he gives if imagine you have two coins and you ask someone to turn them over so one is heads and the other tails then you give them to 2 people without them knowing which is which, then they go to opposite ends of the universe, and they look at their coins, they instantly know the state of the other coin purely by deduction.

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hasenj on Nov 21, 2014 [-]

That’s what Einstein argued for, and it’s what Bell’s inequality proves to not be the case.

I actually remember seeing a Youtube video of Susskind talking about how “FTL” communication is a hack to try to force the quantum state to conform with our notions about the world. (he didn’t phrase it this way though; just my interpretation).

I don’t know the context it was said, but it seemed to imply that there’s a way of thinking about quantum states completely independent of set theory and our classical notions, and this way of thinking should not require FTL communication between entangled particles.

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evanb on Nov 21, 2014 [-]

While GP’s example is not what happens in QM (Bell’s inequality shows that the state of the two coins is not predetermined-but-secret), it is akin to that example, in the sense that because each person gets a random (though correlated) bit, they cannot transmit information to one another.

Bell’s inequality is a statement about the possible strength of the correlation, rather than about information transmission.

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pfortuny on Nov 21, 2014 [-]

Exactly. It is important to note that there is no information exchange if you do not have any information at all.

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— You can’t get entangled without a wormhole (2013)

— Hacker News

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2020.03.07 Saturday ACHK

Superposition always exists, 2

Decoherence means that the different components in the superposition do not interact with each other, but it does not mean that the components separate to form different macroscopic realities.

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Just like when a 100-soldier army’s marching gets interrupted, the decoherent soldiers do not form a single army anymore, because their actions become out of sync.

However, they do not become 100 armies either.

Instead, they form a group of 100 random people in the street.

Although now they are out of sync with each other, all original soldiers still exist, forming the (new) average result; all or most of them have become part of the environment.

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But it is an analogy only. It has an important distinction.

In quantum superposition, we discuss the relationships between different component states of the superposition. Those states exist not in physical space, but in a mathematical space.

In the army analogy, we discuss the relationships between the actions between different material items (solders in this case). Those material items exist in physical space.

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The unselected eigenstates do not cooperate with other particles to form macroscopic realities.

Although the spirit of the statement is correct, the statement itself is incorrect in multiple senses.

First, an eigenstate is a quantum state. It interferes with other eigenstates, not other particles.

Second, although the “unselected” eigenstates seem to disappear, they actually still exist; they entangles with the environment, which includes the apparatus and measurement devices of that experiment.

— Me@2013.01.01

— Me@2020-02-26 06:49:46 AM

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In “decoherence means that the different components do not interact with each other”, the meaning of “interact” is not defined yet.

The word should probably be “interfere”, instead of “interact”.

— Me@2020-02-25 10:44:23 PM

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interference ~ superposition with pattern

Decoherence means that the phase differences between different components in a superposition are not constants anymore. It does not mean that there is no superposition anymore.

Superposition is always there.

What disappears is the interference pattern, not the superposition.

— Me@2019-09-20 06:48:55 AM

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2020.02.26 Wednesday (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.

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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.

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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

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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

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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

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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

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2020.02.03 Monday (c) All rights reserved by ACHK

Compare results

Within a universe, any two observers can, at least in principle, compare results.

When they compare, they will have consistent results for any observables/measurables.

— Me@2018-02-06 08:48:24 PM

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being entangled ~ being consistent, with respect to any two observers, when they compare the results

— Me@2018-02-05 10:11:45 PM

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2020.01.04 Saturday (c) All rights reserved by ACHK

Two dimensional time 5.2.3

The first time direction is uncontrollable; the second is controlled by making choices, traveling through different realities. Future is a set of parallel universes.

— Me@2017-12-15 10:59:49 AM

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The first time direction, which is along the timeline, is uncontrollable, because one can only travel from the past to the future, not the opposite.

The second direction, which is across different timelines, is controlled by making choices, forming different realities.

— Me@2019-12-21 11:03:23 PM

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2019.12.22 Sunday (c) All rights reserved by ACHK

Two dimensional time 5.2.2

time direction ~ direction of change

multiple time directions ~ multiple directions of change

— Me@2019-12-22 04:38:47 PM

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the first dimension of time ~ direction of change

the second dimension of time ~ direction of change of changes

— Me@2019-12-22 04:46:47 PM

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2019.12.22 Sunday (c) All rights reserved by ACHK

Classical physics

Quantum Mechanics 6

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As Gene and Sidney Coleman have pointed out, the term “interpretation of quantum mechanics” is a misnomer encouraging its users to generate logical fallacies. Why? It’s because we should always use a theory, or a more accurate, complete, and universal theory, to interpret its special cases, to interpret its approximations, to interpret the limits, and to interpret the phenomena it explains.

However, there’s no language “deeper than quantum mechanics” that could be used to interpret quantum mechanics. Unfortunately, what the “interpretation of quantum mechanics” ends up with is an attempt to find a hypothetical “deeper classical description” underneath the basic wheels and gears of quantum mechanics. But there’s demonstrably none. Instead, what makes sense is an “interpretation of classical physics” in terms of quantum mechanics. And that’s exactly what I am going to focus in this text.

Plan of this blog entry

After a very short summary of the rules of quantum mechanics, I present the widely taught “mathematical limit” based on the smallness of Planck’s constant. However, that doesn’t really fully explain why the world seems classical to us. I will discuss two somewhat different situations which however cover almost every example of a classical logic emerging from the quantum starting point:

  1. Classical coherent fields (e.g. light waves) appearing as a state of many particles (photons)

  2. Decoherence which makes us interpret absorbed particles as point-like objects and which makes generic superpositions of macroscopic objects unfit for well-defined questions about classical facts

— How classical fields, particles emerge from quantum theory

— Lubos Motl

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There is no interpretation problem for quantum mechanics. Instead, if there is a problem, it should be the interpretation of classical mechanics problem.

— Lubos Motl

— paraphrased

— Me@2011.07.28

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2019.12.14 Saturday (c) All rights reserved by ACHK

Introduction to Differential Equations

llamaz 1 hour ago [-]

I think the calculus of variations might be a better approach to introducing ODEs in first year.

You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of \displaystyle{\frac{dy}{dx} = 0}, you end up with an ODE (viz. the Euler-Lagrange equation).

This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system.

Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you’ve motivated ODEs and given the students a reason to solve ODEs with constant coefficients.

— Hacker News

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2019.10.02 Wednesday ACHK