Wednesday, September 13, 2017

ER=EPR

ER stands for Einstein-Rosen bridge or the so-called wormehole solution of Einstein's equations which connects spacetime points which are very far away. So this side of the equation represents general relativity.
Whereas EPR stands for Einstein-Podolsky-Rosen (and one should add Bohm and Bell) quantum correlation between two entangled particles or the so-called Bell pair. So this side of the equation represents quantum mechanics.
These two effects seem both to challenge the locality of spacetime, i.e. our inability to send signals faster than the speed of light. However, it is generally argued that EPR correlations can not be used to send information faster than the speed of light while the ER bridge is generally a not traversable Lorentzian wormehole.
Thus quantum entangled particles are connected by a wormhole. Or more precisely, the ER bridge between two black holes is actually created by a quantum EPR correlations between the microstates of the two black holes. This is a conjecture put forward original by Susskind and Maldacena in 2013. See also Israel in 1976. In this conjecture it is seen that the ER bridge is a special kind of EPR correlations in which quantum entanglement can be described by a weakly interacting gravity theory which is precisely Einstein theory here. Conversely, it is hopped that every EPR correlated state (even the singlet state of two spins) is connected by some ER bridge which is not necessarily the one given by general relativity. Thus the geometry of spacetime and gravity is determined by quantum entanglement.
Some of the early work leading to this conjecture I mention van Raamsdonk in 2010 which states that a maximally extended AdS black hole, which is a non-traversable wormhole, is dual via the celebrated AdS/CFT correspondence to a pair of maximally entangled thermal conformal field theories.
This conjecture is also in some sense a competing proposal for the resolution of the information loss problem (although a much more profound one) to the AMPS firewall.
Recall that a Hawking particle $B$ must be entangled with the early part $R_B$ of the Hawking radiation and with the particle $A$ which went behind the horizon (complementarity). This is however forbidden by monogamy. The AMPS resolution is via a loss of entanglement by breaking the entanglement between $B$ and $A$. Since $A$ is a high energy mode the breaking will release a firewall. However, the mechanism of this breaking is unknown. The interior of a black hole (just inside the event horizon) is thus not smooth.
However, in the ER=EPR conjecture the entanglement between $A$ and $B$ is possible via a wormhole, i.e. an ER bridge. In fact the ER=EPR conjecture states that the Hawking radiation is "connected by very quantum ER bridges to itself and also to the black hole horizon". This can also be seen by collecting all Hawking radiation and then collapsing it into a second black hole. The two black holes are then connected by an ER bridge and the interior of the black hole is seen to be smooth.

References:
1-Einstein, Rosen."The Particle Problem in the General Theory of Relativity".
2-Einstein, Podolsky, Rosen. "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?".
3-Maldacena, Susskind. "Cool horizons for entangled black holes"
4-Israel, Thermo field dynamics of black holes.
5-van Raamsdonk. "Building up spacetime with quantum entanglement".
6-Susskind. "Copenhagen vs Everett, Teleportation, and ER=EPR".
7-Susskind."ER=EPR, GHZ, and the consistency of quantum measurements"

Saturday, September 9, 2017

Experimental free will


In 1965 Kornhuber (physician) and Deecke (his student) discovered the famous so-called readiness potential (RP) which is a large and solw potential in the brain (motor cortex) which precedes voluntary movement, thus it is an event-related potential in the brain...
Then in 1983 Libet (neuroscientist) discovered that the RP potential begins in the brain almost 350 micro seconds before the conscious decision to move...
Libet asked his experimental subjects to report exactly the time W when they decided to initiate a particular voluntary movement. Then he compared this time W with the time of the onset of the readiness potential RP and he found that the readiness potential precedes the conscious decision to move with at least 350 micro seconds..
What does this mean?
On the face of it, this simply means that Libet's subjects have consciously and freely decided to initiate a movement only a long time after the preparatory action in the brain has already started...
Simply put: the decision to move is not the cause of the motion!!!!
As a consequence, if conscious decisions are not the cause of actions then our conscious free will is not free at all, and it might even be an illusion..
This counter-intuitive conclusion led Libet himself (who rejected the obvious and possibly the inevitable consequences for free will) to propose the so-called the veto response hypothesis which states that although our conscious decision is not the cause of the action it can still veto the action before it occurrence...
However, this is not satisfactory at all, and a theory more consistent with this experimental finding is the hypothesis of compatibilism (as opposed to libertarian free will and determinism)..
The veto response hypothesis is effectively a dualistic theory of a libertarian free will executed by a nondeterministic agent..
Whereas in compatibilists view, free will is not absolute freedom but the unfettered ability to act when constraints are absent. In other words, freedom really rely on the absence of external constraints rather than requiring being outside the causal chain..
The experiments of Libet were repeated by many other groups with the same results..And several systematic critiques of this sort of experiments are also put forward..In conclusion the results stand up to scrutiny...
See the reference which is an excellent concise description of the main experimental findings and the philosophical positions together with an extensive list of references...
Reference:
"Benjamin Libet’s work on the neuroscience of free will" by
William P. Banks and Susan Pockett, in
"The blackwell companion to consciousness", Velmans and Schneider.

Thursday, September 7, 2017

Monogamy and Fierwall (AMPS)

If an experimenter throws the first spin of an entangled pair of spins into a black hole, the spin will pass through the horizon with nothing exceptional happening in accordance with the equivalence principle, but the two spins become then entangled, with no chance for the spin thrown in to escape again the black hole. The state of the spin outside the black hole is mixed and this is the final state after the complete evaporation of the black hole. If this is really what is happening then information is lost.
However, under the assumption of unitarity,  the information carried out by the spin which went in must get out at some time  before the complete evaporation of the black hole (Page time when half of the black hole  is evaporated). The final state must then become purified.

Hawking radiation is precisely of this sort. In other words, it is formed of pairs of entangled particles created around the horizon with one of each pair passing through the horizon whereas the other one goes to infinity. The principle of black hole complementarity requires that the infalling observer sees the bits of information going inside the black hole whereas the external observer sees them in the (late) Hawking radiation. Thus the infalling observer sees nothing special in the horizon whereas the external observer sees the horizon as a hot membrane which re-radiates. This principle is therefore a strong form of non locality since the same bit of information in the Hilbert space is seen at different places.

Thus the assumption of unitarity forces us to the conclusion that  a particle $B$ from the late Hawking radiation (after Page time)  must be entangled with the earlier part $R_B$ of Hawking radiation. But from locality of quantum field theory this particle $B$ must be entangled with a particle $A$ inside the horizon. But the principle of monogamy of quantum entanglement forbids that the same system be entangled with two different other systems at the same time. In other words, the principle of black hole complementarity and the principle of monogamy of quantum entanglement are in conflict.

Thus we have to give either:
  • (a) Purity: Unitarity of quantum mechanics- Thus black hole information is carried out with the late Hawking radiation. This effect is measured by an external observer.
  • (b) Effective Field Theory (EFT)-Quantum Field Theory in Curved Backgrounds (Locality)- In other words, we suppose that semiclassical gravity is valid outside the horizon, i.e. small curvature near and outside the horizon. Thus the external observer will see the horizon as a physical membrane (stretched horizon).
  • (c) No Drama: The equivalence principle-Nothing special happening as we fall across the horizon, i.e. small curvature near and outside the horizon, and hence the absence of strong quantum gravity effects. This effect is measured by an infalling observer.

We would have expected (Polchinski own expectation) that  EFT, in the guise of its underlying locality assumption, breaks down and complementarity is then sufficient. But the situation seems more complicated.

This new paradox is claimed to be resolved by the firewall of Almheiri, Marolf,  Polchinski,  and Sully (AMPS). The firewall arises from a loss of entanglement resulting from the following contradiction:
  • First from (a) any particle of the late Hawking radiation (after page time) is fully entangled with some subsystem of the early radiation because the early particles have much more states. It is an eigenstate of $bb^{\dagger}$.

  • However, by assumption (b), we can propagate (essentially freely) any late Hawking mode back from infinity (where the unitarity assumption should hold) to the horizon (where the no drama assumption is supposed to hold). The mode becomes highly blue shifted, i.e. it has a very high energy. In other words, we obtain an early mode of the Hawking radiation with very high energy which is encountered by the infalling observer.

  • But from (c) we conclude that this same Hawking mode is maximally entangled with a modes inside the horizon. The field is in the ground state $a^{\dagger}_{\omega}a_{\omega}=0$ and the mode is not an eigenstate of $bb^{\dagger}$.

These three statements are contradictory. Note that (a) and (c) lead effectively to cloning!! but fortunately this is forbidden by the monogamy of quantum entanglement.

The most conservative resolution (qualified as such by AMPS) is by breaking up the entanglement between modes inside and outside the horizon, i.e. by giving up the equivalence principle, and thus releasing an enormous energy given by  a firewall of Planck-energy particles just inside the horizon.


References:
1-'t Hooft, On the quantum structure of a black hole.
2-'t Hooft, The black hole interpretation of string theory.
3-Sussking,Thorlacius, Uglum, The Stretched Horizon and Black Hole Complementarity.
4-Susskind, Lindesay, An introduction to black holes, information and the string theory revolution: The holographic universe.
5-Almheiri, Marolf, Polchinski, Sully,  Black holes: complementarity or firewalls?.
6-Polchniski, http://www.preposterousuniverse.com/blog/2012/09/27/guest-post-joe-polchinski-on-black-holes-complementarity-and-firewalls/


Monday, September 4, 2017

What is information loss in black holes?


Laws of Nature


First we start by stating the three fundamental laws of nature (so-called by susskind in his book):

  • -Information conservation (unitarity of quantum mechanics): The information is defined as the difference between the coarse-grained Boltzmann thermodynamic entropy and the fine-grained Von Neumann entanglement entropy. This is a conserved quantity.

  • -Equivalence principle (general relativity).

  • -Quantum zerox principle (linearity of quantum mechanics): This forbids the duplication of quantum information.


Quantum Field Theory in Black Hole Backgrounds


We consider now the gravitational collapse of a star and the formation of a real black hole. According to Hawking we should consider quantum field theory in the background geometry of this black hole.
The Hilbert space of initial massless particles $|\psi_{\rm in}\rangle$ is associated with the null rays or quanta coming in from the light-like infinity $J_-$ corresponding to $r=\infty$ in the past.
These incoming particles will interact via Feynman diagrams of the quantum field theory in the black hole background and some of them will be absorbed by the black hole and reach the singularity ${\cal S}$ and others will be scatter off the black hole to escape to the light-like infinity $J_+$ corresponding to $r=\infty$ in the future.
The Hilbert space of final massless particles $|\psi_{\rm out}\rangle$ is thus a tensor product of the states on the light-like infinity $J_+$ and the states on the singularity ${\cal S}$. This is the assumption of locality of quantum field theory (operators on ${\cal S}$ commute with operators on $J_+$ since ${\cal S}$ and $J_+$ are space-like separated).
The final state $|\psi_{\rm out}\rangle$ is related to the initial state $|\psi_{\rm in}\rangle$ by a unitary scattering matrix $S$, viz \[|\psi_{\rm out}\rangle=S|\psi_{\rm in}\rangle.\] This is the assumption of unitarity of quantum mechanics. Both states are pure, i.e. vectors in a Hilbert space.
However, with respect to the outside observer at the infinity $J_+$, the final state can only be given by a density matrix since she can not access the states on the singularity ${\cal S}$. This density matrix represents a mixed state given explicitly by tracing out the states on ${\cal S}$, namely \[\rho_{\rm out}={\rm Tr}_{\cal S}|\psi_{\rm out}\rangle\langle \psi_{\rm out}|.\]
The Schrodinger equation can not lead to an evolution from a pure state $|\psi_{\rm in}\rangle$ to a mixed state $|\rho_{\rm out}\rangle$.
As an example we take an entangled pair of particles created on $J_-$, then scattering off the black hole, one of the particles escapes to $J_+$ whereas the other one passes through the horizon and reaches the singularity ${\cal S}$.

What happens if the black hole evaporates?


Here there are six positions:
  • -Information loss: Hawking (and the majority of relativists) state that the purity of the state will not be restored, i.e. the final density matrix $\rho_{\rm out}$ is correct, and thus pure states can indeed evolve into mixed states, and the Schrodinger equation breaks down.
    Thus the relation between the final density matrix $\rho_{\rm out}$ and the initial density matrix $\rho_{\rm in}=|\psi_{\rm in}\rangle\langle\psi_{\rm in}|$, given by the so-called dollar matrix, is not in general given in terms of the scattering matrix $S$ by the Schrodinger equation written in the form
    \[\$_{m_1m_2,n_1n_2}=S_{m_1n_1}S^*_{n_2m_2}
    .\] This means that information (the information that has been traced out) is lost inside the black hole completely when it fully evaporates.
  • -Unitarity: The majority of field and string theorists maintain that the mixed state $\rho_{\rm out}$ gives only a coarse-grained description of what happens. The final state is a maximally entangled pure state. The black hole evaporates completely but information is conserved throughout since there is always a unitary map from the initial (incoming shell) to the final (outgoing radiation) states. The Hawking radiation carries out information in subtle quantum correlations between late and early particles.
    However, this solution implies a breakdown of the semi-classical description
    and the machinery of effective field theory.
  • -Recovery: Information is only recovered at the end of evaporation when the
    singularity at $r = 0$ becomes a naked singularity. This contradicts the
    principle of information conservation with respect to the observables at $J_+$
    which states that by the time (Page or retention time) the black hole
    evaporates around one half of its mass, the information must start coming
    out with the Hawking radiation.
  • -Remnants: Evaporation stops at a Planck-mass remnant which contains all the information with extremely large entropy. Thus we end up with an entanglement entropy which exceeds the Bekenstein-Hawking value.
  • -Brick wall: The horizon is like a brick wall which cannot be penetrated. This contradicts the equivalence principle in an obvious way.
  • -Duplication: The horizon duplicates the information by sending one copy outside the horizon (as required by the principle of information conservation) while sending the other copy inside the horizon (as required by the equivalence principle). This is, however, forbidden by the linearity of quantum mechanics or the so-called quantum xerox principle.
In summary, for all the proposals above some law of nature (one of the three laws stated above) must break down at least for some observables. The unitarity option remains the best one since it only requires the breaking of the effective field theory approach.

Black hole complementarity ('t Hooft and Susskind et al)

This is another complementarity principle (such as the wave-particle duality and the entanglement-coherence, the position-momentum and the energy-time relations) which is postulated by 't Hooft and Susskind et al to be satisfied by black holes so that both general relativity and quantum field theory principles are respected.
Recall that complementarity or duality means that two properties are dual or complementarity to each other if, among other things, the two properties can not be observed simultaneously..
Recall also that black hole threatens one of the three principles of nature: equivalence principle, information conservation and linearity of quantum mechanics. It is always seen by an observer (the external or the infalling) that one of these laws at least is going to be violated.
Black hole complementarity states that no single observer will ever witness a violation of a law of nature. This principles assumes unitarity and no information loss.
Thus the external observer will see the infalling matter heating up the stretched horizon (not the event horizon but a physical membrane above the event horizon a Planck distance away), which then reradiates back the infalling information in the form of Hawking radiation through unitary quantum mechanics.
At every instant the Von Neumann fine grained entropy of the radiation field can not exceed the entropy of the black hole due to entanglement. For the external observer the information, before re-emission, is seen uniformly spread out over the stretched horizon.
The infalling observer however sees something different. Because of the equivalence principle there is nothing special happening at the horizon and the infalling information or matter will actually pass through and reach the singularity. The information is seen by this observed as localized on the stretched horizon.
Thus the information is both reflected at (external observer) and passed through (infalling observer) the event horizon. These two stories are complementary not contradictory since the two observes can not communicate: There is an infinite time dilation at the horizon and thus it takes an infinite amount of time to communicate with the horizon and furthermore the infalling observer can not signal the external observer from behind the horizon.

In some sense the complementarity principle resolves the violation of quantum xerox or no cloning theorem. Thus, information is either inside the black hole (with respect to infalling observer) or outside the black hole (with respect to external observer). 

Explicitly, the three postulates of black hole complementarity are given by the three principles applicable with respect to the external observer: 1) unitarity, 2) semi classical approximation and 3) Hawking-Bekenstein law.  The conservation of information (no-cloning or xerox theorem) and the equivalence principle are also implicitly assumed.

The AMPS firewall paradox is the information loss paradox in which the resolution given by the principle of black hole complementarity is shown to be in conflict with the principle of entanglement monogamy (sub additivity property of entropy). More precisely, unitarity and field theory are shown to be in conflict with the equivalence principle. The ER=EPR is a more fundamental approach which states the AMPS firewall is not really there.

Monogamy and Fierwall (AMPS)


If an experimenter throws the first spin of an entangled pair of spins into a black hole, the spin will pass through the horizon with nothing exceptional happening in accordance with the equivalence principle, but the two spins become then entangled, with no chance for the spin thrown in to escape again the black hole. The state of the spin outside the black hole is mixed and this is the final state after the complete evaporation of the black hole. If this is really what is happening then information is lost.
However, under the assumption of unitarity,  the information carried out by the spin which went in must get out at some time  before the complete evaporation of the black hole (Page time when half of the black hole  is evaporated). The final state must then become purified.

Hawking radiation is precisely of this sort. In other words, it is formed of pairs of entangled particles created around the horizon with one of each pair passing through the horizon whereas the other one goes to infinity. The principle of black hole complementarity requires that the infalling observer sees the bits of information going inside the black hole whereas the external observer sees them in the (late) Hawking radiation. Thus the infalling observer sees nothing special in the horizon whereas the external observer sees the horizon as a hot membrane which re-radiates. This principle is therefore a strong form of non locality since the same bit of information in the Hilbert space is seen at different places.

Thus the assumption of unitarity forces us to the conclusion that  a particle $B$ from the late Hawking radiation (after Page time)  must be entangled with the earlier part $R_B$ of Hawking radiation. But from locality of quantum field theory this particle $B$ must be entangled with a particle $A$ inside the horizon. But the principle of monogamy of quantum entanglement forbids that the same system be entangled with two different other systems at the same time. In other words, the principle of black hole complementarity and the principle of monogamy of quantum entanglement are in conflict.

Thus we have to give either:
  • (a) Purity: Unitarity of quantum mechanics- Thus black hole information is carried out with the late Hawking radiation. This effect is measured by an external observer.
  • (b) Effective Field Theory (EFT)-Quantum Field Theory in Curved Backgrounds (Locality)- In other words, we suppose that semiclassical gravity is valid outside the horizon, i.e. small curvature near and outside the horizon. Thus the external observer will see the horizon as a physical membrane (stretched horizon).
  • (c) No Drama: The equivalence principle-Nothing special happening as we fall across the horizon, i.e. small curvature near and outside the horizon, and hence the absence of strong quantum gravity effects. This effect is measured by an infalling observer.

We would have expected (Polchinski own expectation) that  EFT, in the guise of its underlying locality assumption, breaks down and complementarity is then sufficient. But the situation seems more complicated.

This new paradox is claimed to be resolved by the firewall of Almheiri, Marolf,  Polchinski,  and Sully (AMPS). The firewall arises from a loss of entanglement resulting from the following contradiction:
  • First from (a) any particle of the late Hawking radiation (after page time) is fully entangled with some subsystem of the early radiation because the early particles have much more states. It is an eigenstate of $bb^{\dagger}$.

  • However, by assumption (b), we can propagate (essentially freely) any late Hawking mode back from infinity (where the unitarity assumption should hold) to the horizon (where the no drama assumption is supposed to hold). The mode becomes highly blue shifted, i.e. it has a very high energy. In other words, we obtain an early mode of the Hawking radiation with very high energy which is encountered by the infalling observer.

  • But from (c) we conclude that this same Hawking mode is maximally entangled with a modes inside the horizon. The field is in the ground state $a^{\dagger}_{\omega}a_{\omega}=0$ and the mode is not an eigenstate of $bb^{\dagger}$.

These three statements are contradictory. Note that (a) and (c) lead effectively to cloning!! but fortunately this is forbidden by the monogamy of quantum entanglement.

The most conservative resolution (qualified as such by AMPS) is by breaking up the entanglement between modes inside and outside the horizon, i.e. by giving up the equivalence principle, and thus releasing an enormous energy given by  a firewall of Planck-energy particles just inside the horizon.








 

References:


1-'t Hooft, On the quantum structure of a black hole.
2-'t Hooft, The black hole interpretation of string theory.
3-Sussking,Thorlacius, Uglum, The Stretched Horizon and Black Hole Complementarity.
4-Susskind, Lindesay, An introduction to black holes, information and the string theory revolution: The holographic universe.
5-Almheiri, Marolf, Polchinski, Sully,  Black holes: complementarity or firewalls?.
6-Polchniski, http://www.preposterousuniverse.com/blog/2012/09/27/guest-post-joe-polchinski-on-black-holes-complementarity-and-firewalls/


 

Friday, September 1, 2017

The three musketeers of quantum mechanics!!!


The three fundamental theorems of quantum philosophy are:
-The no hidden variable theorems in particular Kochen-Specker theorem (1967): No hidden variable description of quantum mechanics is possible.
These theorems depend on the no- contextuality requirement: The results of a given measurement which are predicted by the underlying state (wave function and hidden variables) do not depend on what other measurements are being performed on the system.
In the contextual hidden variable theories (such as Bohm's) the result of a given measurement depends on the state and on the other measurements being performed on the system.
-Bell's theorem (1965): hidden variables theories can only be non-local. This is the most fundamental of all these theorems.
-The Greenberger-Horne-Zeilinger (GHZ) theorem (1989): This theorem is a generalization of Bell's theorem which is mid-way between the algebraic no hidden variable theorem (combinatorial considerations) of Kochen and Specker and the statistical hidden variable theorem (multi-particle considerations) of Bell. This situation is termed Bell without statistics by Price. The GHZ theorem involves a maximally entangled tripartite system as opposed to the maximally entangled bipartite system considered in Bell's theorem. As Bell's theorem the GHZ theorem rules out local hidden variable theories. Both Bell and GHZ rely on the absence of advanced action.
These three major theorems are the most difficult objections to the ignorance interpretations of quantum mechanics which assumes that quantum mechanics is incomplete and thus should be supplemented by hidden variables. These theorems show that any hidden variable description of quantum mechanics must be both contextual and nonlocal.

Wednesday, August 30, 2017

The many-mind interpretation of quantum mechanics


The many-mind interpretation is a very close relative of the many-world interpretation which involves the following crucial modification/twist: The split or branch of the World into parallel branches when a quantum measurement is performed is shifted to a split or branch of the Mind into parallel minds. In both interpretations, it is assumed that quantum mechanics as it stands is a complete theory of nature and that there is no collapse of the wave function under measurement and this is what is accounted for by the splitting into branches.
This means in particular that the fundamental law is given by the Schrodinger equation alone, and the relationship between branching and relative frequencies, which is ill defined a priori in both pictures, should be given for consistency by the Born rule.
The most important versions of the many-mind interpretations are:
  • -Albert and Loewer theory. This is an in intrinsically dualistic theory which assumes in the words of Albert "that every sentient physical system there is is associated not with a single mind but rather with a continuous infinity of minds". However, in this theory, there is no supervenience of brane states and mind states.
  • -Lockwood theory: In this theory there is a complete supervenience of the physical and mental.
  • -There are other versions of the many-mind interpretation which can be found referenced for example in Hemmo and Pitowsky paper. In the following we will only discuss Albert-Loewer and Lockwood theories following Hemmo and Pitowsky.
Before we begin we mention few other implications of the Albert-Loewer theory which is the most important one for me here because of its dualistic character. Firstly, as an epistemological implication of the Albert-Loewer theory is the observation that our current experience could be fully compatible with the fact that the Universe has always been in the vacuum state. This seem to me to be also an ontological implication. Another implication of the Albert-Loewer theory is that quantum nonlocality is removed completely from the physical and delegated to the mental world. In fact all many-world and many-mind interpretations are no collapse models and they avoid nonlocality by claiming that Bell correlations (predicted by Bell's theorem) are not fully objective correlations but they are observer-dependent. Price critique of these claims reach the conclusion that these no collapse models do not really eliminate nonlocality but they simply explain it.
The system $S$ under consideration is assumed to be composed of a single electron. We are interested in the measurement of the $z$ component of the spin. The measurement apparatus is denoted $M$ and the observer is denoted $O$. The total system $S+M+O$ is initially prepared in the state \[|\Psi_0\rangle=(\alpha|-\rangle+\beta|+\rangle)\otimes|\psi_0\rangle\otimes|\phi_0\rangle.\] The state $|\psi_0\rangle$ is the initial state of the apparatus and $|\phi_0\rangle$ is the initial state of the observer's brain. The complete state $|\Psi_0\rangle$ is supposed to obey the Schrodinger equation only. In other words, we assume that there is no collapse.
The measurement interaction between the system $S$ and the measurement apparatus $M$ creates a one-to-one correlation between the states of up and down spins of the electron and the pointer states $|\psi_{\pm}\rangle$ of the apparatus. Hence the system $S$ and the measurement apparatus $M$ become entangled, i.e. the measurement interactions results in taking the above state $|\Psi_0\rangle$ to the combination \[|\Psi_1\rangle=(\alpha|-\rangle\otimes|\psi_+\rangle+\beta|+\rangle\otimes|\psi_-\rangle)\otimes|\phi_0\rangle.\] Next we assume that the brain states corresponding to all possible outcomes of all possible experiments form a preferred basis in the brain's Hilbert space. These states correspond to those mental states associated with the conscious perception of the outcomes of the experiments. Let us denote the two brain states associated with the conscious perception of the states of up and down spins of the electron by $|\phi_{\pm}\rangle$. Then the interaction between the measurement apparatus $M$ and the observer $O$ will take the state $|\Psi_1\rangle$ to the final state \[|\Psi_f\rangle=\alpha|-\rangle\otimes|\psi_+\rangle\otimes |\phi_+\rangle+\beta|+\rangle\otimes|\psi_-\rangle\otimes|\phi_-\rangle.\] The measurement has no definite result and thus this theory (called the bare theory by Albert) is not complete and it should then be supplemented by extra ingredients.
The many-mind interpretation of Albert and Loewer is a no collapse interpretation in which we suppose that the bare theory is complete with respect to the physics including the brain. However, regarding the relationship of the brain states $|\phi_{\pm}\rangle$ to the mental states of the observer $O$ we also assume the following two postulates:
  • Each brain state $|\phi\rangle$ is associated at all times with an infinity of nonphysical minds.
  • The minds do not obey the Schrodinger equation but evolve in time in a stochastic way with a probability given by the Born rule.
Thus we start with an infinity of minds associated with the initial brain state $|\phi_0\rangle$. In some sense the minds are degenerate described all by the single brain state $|\phi_0\rangle$. Each mind then evolves in a stochastic way to either the state $|\phi_+\rangle$ with the Born probability $|\alpha|^2$ or to the state $|\phi_-\rangle$ with the Born probability $|\beta|^2$. Thus the state $|\Psi_f\rangle$ should be replaced by \[|\Psi_f(m,n)\rangle=\alpha|-\rangle\otimes|\psi_+\rangle\otimes |\phi_+(m)\rangle+\beta|+\rangle\otimes|\psi_-(n\rangle\otimes|\phi_-\rangle.\] The notation $|\phi(m)\rangle$ means that the brain state $|\phi\rangle$ corresponds to and is index by the subset $m$ of the set of minds. In other words, the description of the post measurement state includes the quantum states of the system $S$ and of the apparatus $M$ and the subsets of the set of minds in the $+$ and $-$ branches of the superposition.
This interpretation is truly probabilistic since before the divergence of the minds into the branches of the state $|\Psi_1\rangle$ it is fully random which branch each mind will actually follow. The probability is given by the quantum mechanical Born rule. The requirement of an infinite number of minds is put forward in order to avoid 1) the so-called mindless hulk problem and also in order to avoid 2) Bell's nonlocality.
More importantly is the fact that this interpretation is dualistic in the sense that only subsets of the set of minds (and not individual minds) supervene on the brain states. Thus any $m-$mind can be exchanged with any $n-$mind leaving the physics invariant.
The other issue concerns the relationship between branching and relative probability which is a major problem in the many-world interpretation. This is solved by simply assuming the Born rule as shown originally by Everett in $1957$. It can then be proved that the probability of each branch on a given tree is given by the quantum mechanical Born rule and that each individual mind performs a classical random walk on this tree with this probability. This does not mean that there exists a non-contextual classical probability distribution which can assign the correct probability to all branches at once in accordance with the violation of Bell's inequality.
Indeed, the probability of the branching must be conditional on the measurement performed. If the probability were predetermined then the minds will act as hidden variables and they will necessarily violate Bell's inequality, i.e. we have a non-local hidden variables theory. Hence in order to avoid this nonlocality we must assume a random distribution of the minds which is conditional on the given measurement.
In Lockwood interpretation there is a complete supervenience of the (continuous infinity of) minds on the brain states. Also the minds are supposed to be not stochastic. Thus the final post measurement state is given by $|\Psi_f\rangle$ and not $|\Psi_f(m,n)\rangle$. In other words, subsets of minds are indexed by brain states as opposed to brain states being indexed by subsets of minds. Thus the fraction of minds in the branch $+$ is proportional to $|\alpha|^2$ whereas the fraction of minds in the branch $-$ is proportional to $|\beta|^2$. On the other hand, the dynamics of minds, which is not random in this case, is not clear and some possibilities are discussed for example in Hemmo and Pitowsky.

References:
1-Albert, D. and Loewer, B. [1988]: ‘Interpreting the Many Worlds Interpretation’,Synthese, 77, pp. 195–213.
2-Lockwood, M. [1996a]: ‘Many Minds Interpretations of Quantum Mechanics’, British Journal for the Philosophy of Science, 47, pp. 159–88.
3-Hemmo,M and Pitowsky, I.[2003]: 'Probability and Nonlocality in Many Minds Interpretations of Quantum Mechanics'.
4-Price, H. [1996]: 'Time's Arrow and Archimedes Point: New Directions for the Physics of Time'.
5-David Albert, Quantum Mechanics and Experience.

Tuesday, August 29, 2017

Aristotle on Time

This goes as follows:

-Change is the actuality of what is potentially (thus it can be said that a stage of change is before or after another stage of change)..
-Time is the number of change (not any particular change) with respect to the before and after (thus although there are particular changes taking place at many different places there is only a single universal time)...
-Time is number and thus it is counting by a consciousness (what Aristotle calls Soul), i.e. There is no time without consciousness...
-The now follows the thing in motion just as time follows motion...In other words, the interdependence that exists between the thing in motion and motion holds between the now and time..
-The now, which is a boundary between past and present, is therefore always the same (since the now is derived from the thing in motion and hence it binds time together and makes it a unity) and is also always different (since time although is made continuous by the now is also divided by it)...
-Time is also a measure of change not only a number of change.


This is very profound indeed!!!

Monday, August 14, 2017

Time's Arrow and Archimedes' Point: An Executive Reading.

This is my own personal summary of chapter 5 of the book "Time's Arrow and Archimedes' Point" by Huw Price. It is in my view one of the best books on the subject of time written from a mixture of physical and philosophical perspectives with highly non-trivial ramification concerning the problem of interpretation of quantum mechanics and the problem of causation.

Summary:


Chapter 1 is an outline of the contents of the book...
Chapters 2, 3 and 4 introduce the arrows of thermodynamic, radiation and cosmology..
Chapter 5 is the most important chapter of the book (in my view) in which the author formulate his proposal for a unified picture of time, causality and the quantum.
Summary of chapter 5:
There are two intuitive principles underlying much of classical physics..
From the one hand, we have the principle of T-symmetry which is the symmetry under the reversal of the time direction since all physical laws show no distinction between past and future...
From the other hand, we have the principle of innocence which states that innocence precedes experience, i.e. particles are correlated in the future and not in the past..Hence, properties of physical bodies are independent of each other unless and until they interact..
The principle of innocence is effectively the principle of independence of incoming influence or PI which states that properties of interactive systems are independent before interactions can take place..
The principle of innocence and the PI principle are intimately related to the law of conditional independence (O.Penrose and I.C.Percival), the principle of the common cause (Reichenbach), and the fork asymmetry (analytical philosophy)....
These principles all include a time asymmetry described by the fork asymmetry, i.e. V-shaped (open to the future) correlations are dominant while Lambda-shaped (open to the past) correlations are nonexistant...
At the macroscopic level all time asymmetry contained in these principles are described by the asymmetry of thermodynamic (second law)...This asymmetry is due to the boundary conditions and not to the physical laws themselves which are perfectly T-symmetrical...
Thus, at the macroscopic level PI and innocence are both compatible with T-symmetry since all the asymmetry is only due to the special initial conditions..
However, at the microscopic level the principle of innocence becomes the principle of micro-innocence and the above arguments stop to apply with the same force...
The macroscopic innocence principle is the statement that "no entropy-reducing correlations" exist...While microscopic innocence is the statement that there are no microscopic preinteractive correlations... Thus although particles which have interacted in the recent past are clearly not independent, it is only supposed that particles which will interact in the near future are independent...This is micro-innocence and it is clearly an asymmetric principle...
The empirical evidence for the macro-innocence principle is ample given by the countless manifestations of the second law of thermodynamics..And it can be explained at the fundamental level by asymmetric boundary conditions...
However, there seems to be no observational evidence for the principle of micro-innocence...As a consequence, micro-innocence can not be reconciled with T-symmetry since it can not be reduced to asymmetric boundary conditions...
In summary, micro-innocence requires an asymmetry in correlations produced by microscopic interactions while macro-innocence supported by observations requires an asymmetry in entropy changing correlations...The principle of micro-innocence can not then be explained by asymmetric boundary conditions and thus it is intrinsically in conflict with the principle of T-symmetry...
Thus we have to sacrifice either T-symmetry or innocence..Symmetry is always favoured and furthermore in this case observational support is not in favour of micro-innocence..Thus by giving up micro-innocence we are stating that interactions in the future are no less a source of correlations than interactions in the past..Putting it differently, physical bodies who are going to enter an interaction in the future are not independent in the same way that physical bodies who had just entered an interaction in the past are not independent...
Furthermore, by giving up micro-innocence the physical world is expected to be exactly the kind we find in quantum mechanics..In other words, quantum mechanics seems to describe the kind of world we ought to have expected had we rejected micro-innocence in the first place...
Both Bell's theorem (quantum nonlocality) and no-hidden variable theorems (quantum indeterminism) rely on the assumption of micro-innocence...In other words, it seems to be partly because of micro-innocence that quantum mechanics lead to this nonclassical behaviour...Turning this argument on its head it can be seen that quantum mechanics provides in some sense the empirical confirmation that micro-innocence must fail..In fact, abandoning micro-innocence pays big dividends concerning the interpretation of quantum theory..
Thus, if micro-innocence fails then systems who will interact in the future are really not independent in the same way that systems who had interacted in the past are not independent...This leads us immediately to the possibility of backward or advanced causation...Thus causation seems to be threatened here..But causation itself is T-asymmetric..Fortunately, micro-innocence and the arrow of causation turn out to be separate issues...The asymmetry and temporal orientation of causation is in fact subjective and perspectival..Whereas micro-innocence and the related backward causation are thoroughly objective proposals which are consistent with this subjective and perspectival understanding of causation...
The perspectival approach to causation is developed in chapters 6 and 7 whereas the backward causation approach to quantum mechanics is developed in chapters 8 and 9...

Time's Arrow and Archimedes' Point