Does the Born Rule conflict with determinism in quantum mechanics?

[rodsika]

I'm a layman. I understand the basic of the Hilbert Space and can imagine what are basis vectors and eigenvalues (allowable values) and observables (properties like spin, position, momentum, etc) and know operators are operations that involve observables. I also know the basic of the Schroedinger Equation and what it does. However I don't know how to perform differentiation or Integration. Just general idea of it all in verbal form. Now I want to focus on the Born Rule. References out there are just so complex with so many equations. I want to know the basic idea.

I know the Born Rule involves getting particular values of say the position eigenvalues. Is the Rule just like a cooking recipe how to mix certain operations to get the values of the eigenstates whatever they are. Or is Born Rule related primarily with how to create the randomness in the first place. Supposed, just for sake of discussions, determinism is beneath quantum mechanics as proposed by one of the best theoretical physicists today, George t' Hooft in the paper "Determinism Beneath Quantum Mechanics" in http://arxiv.org/abs/quant-ph/0212095 What will happen to Born Rule (BR) since BR involves randomness? Will BR disappears or remain as is as Born Rule involves just operations to get certain values of the position eigenstates (without regard to whether they are acquire randomly or has hidden determinism as George proposed)?

 

[JesseM]

The Born rule says that upon measurement, an arbitrary quantum state vector of a system is "projected" onto one of the basis vectors corresponding to that measurement. If you're familiar with the basics of vector addition in 2D space you can think of it in terms of a diagram like this (from this page):

Here the blue vector is being projected onto both the vertical and horizontal axis of a 2D coordinate system (unit vectors in the vertical and horizontal directions would be a set of "basis vectors" in this 2D space). The horizontal green vector is the projection of the blue vector onto the horizontal basis vector, and the vertical green vector is the projection of the blue vector onto the vertical basis vector. We can also say that the blue vector can be expressed as the sum of the two green vectors (see vector decomposition). Similarly in QM, if you have a basis then an arbitrary quantum state vector can be expressed as a weighted sum of basis vectors.

Notice that in the above diagram the green vectors have different lengths, and if you rotated the angle of the blue vector while keeping its length constant, the lengths of those green vectors would change (for example if the blue vector was exactly vertical, the vertical green vector would be identical to the blue vector, while the horizontal green vector would have a length of zero). Similarly if you decompose a quantum state vector into a bunch of projections parallel to the different basis vectors, all these projections would have different "amplitudes", which instead of real numbers are complex numbers. And remember that each of the basis vectors is associated with a different measurement outcome (the "eigenvalue" associated with that "eigenvector" for a given measurement operator). So what the Born rule says is that when you make a measurement, the state vector will instantly transform at random into one of the many possible projections along different basis vectors (or "collapse" onto one of them as it's often said), and the probability that it will transform into any given projection is proportional to the square of the amplitude of that projection (really it's the amplitude multiplied by its own complex conjugate, so for example if the amplitude is 0.8+0.6i then you'd multiply it by 0.8-0.6i to get a probability of 0.64 - 0.36 = 0.28...and there may be an additional step where you have to "normalize" all the different products of the amplitudes and their complex conjugates, to make sure the probabilities add up to 1). And of course, the probability of a given measurement outcome is just the same as the probability that the state vector transforms into the projection along the corresponding basis vector. So, a common way of summarizing the Born rule is that it tells you the probability of a given measurement outcome is given by the square of the amplitude associated with that outcome.

 

[rodsika]

Thanks for this clear and intuitive illustration about the Born Rule. I have some questions about Hilbert Space. I can imagine how a single particle can be described by a Hilbert Space. But how about two particles. Do you just add the Hilbert Space of two particle together, supposed the Hilbert space of one particle has 1 million basis vector axis. The second has another million. So you just append the second to the first.. just like that? And if you have a billion particles. You must add their basis vector axes together? But could all this be described by just one main vector or ray in Hilbert Space that represent all the things at once?

 

[The_Duck]

Close; to combine two particles into a single system you "multiply" their Hilbert spaces. It's easiest to consider a simple example like the spin states of two spin-1 particles. Each particle's spin state is located in a Hilbert space with only three basis vectors; call them 1, 0, and -1, representing the three possible values of spin angular momentum around the z axis for a spin-1 particle. If we have two particles the Hilbert space has size 3x3 = 9; the basis vectors can be represented as:

(1, 1), (1, 0), (1, -1)
(0, 1), (0, 0), (0, -1)
(-1, 1), (-1, 0), (-1, -1)

where the first number in a pair gives the angular momentum of the first particle around the z axis and the second number in a pair gives the angular momentum of the second particle around the z axis. Hopefully you can see why this is considered a kind of "multiplication" of the two Hilbert spaces for the individual particles.

 

[rodsika]

I'd like to understand how the position eigenvectors are depicted. For example for a gas of electrons in a container containing billions and billions of particles. You mean each basis vector would be composed of (1, 2, 3 up to Nth value depending on numbers of particles)? But a basis vector just represents the possibility of being in a certain position.. so what does it mean to put the information of all those billions of particles in just one one basis vector representing the possibility of being in a certain position in space?

 

[JesseM]

I'd like to understand how the position eigenvectors are depicted. For example for a gas of electrons in a container containing billions and billions of particles. You mean each basis vector would be composed of (1, 2, 3 up to Nth value depending on numbers of particles)? But a basis vector just represents the possibility of being in a certain position.. so what does it mean to put the information of all those billions of particles in just one one basis vector representing the possibility of being in a certain position in space?

It's similar to the concept of a phase space in statistical mechanics, where every possible configuration of N particles is represented as a single point in a 6N dimensional space (for each particle there are 3 axes for its position and 3 for its momentum). For a simplified model, imagine you just represented to represent the positions of 3 particles in 1 dimension, then you could make a 3D phase space and represent them by a single point in this space, where the x-coordinate of this point would represent the position of the first particle, the y-coordinate would represent the position of the second particle, and the z-coordinate would represent the position of the third particle. The more dimensions you have in a space, the more information can be contained by a single point (or single vector) in that space--and Hilbert space is potentially infinite-dimensional!

 

[rodsika]

It's similar to the concept of a phase space in statistical mechanics, where every possible configuration of N particles is represented as a single point in a 6N dimensional space (for each particle there are 3 axes for its position and 3 for its momentum). For a simplified model, imagine you just represented to represent the positions of 3 particles in 1 dimension, then you could make a 3D phase space and represent them by a single point in this space, where the x-coordinate of this point would represent the position of the first particle, the y-coordinate would represent the position of the second particle, and the z-coordinate would represent the position of the third particle. The more dimensions you have in a space, the more information can be contained by a single point (or single vector) in that space--and Hilbert space is potentially infinite-dimensional!

But there is a difference between Phase space and Hilbert space. A phase space has many real numbers on one axis such as 1 to 10. While one basis vector in Hilbert Space represents only one number or possibility. So I wonder how to model the positions of say 10 electrons in a container using Hilbert space. Pls. don't use example in phase space but directly in Hilbert Space. I think the 10 electrons can be represented by 30 dimensions (10 x 3 dimensions of x, y, z), right? But each basis vector in hilbert space has only one position value compares to phase space which has infinite value in one basis or axis. So how can 30 dimensions be the same requirements for both. Where did I analyze it wrong? Don't use simplifed example as I want to know how it work in more complex example. Thanks.

 

[JesseM]

But there is a difference between Phase space and Hilbert space. A phase space has many real numbers on one axis such as 1 to 10. While one basis vector in Hilbert Space represents only one number or possibility. So I wonder how to model the positions of say 10 electrons in a container using Hilbert space.

Since position is continuous, only an infinite-dimensional Hilbert space will do. If you're just interested in a discrete quantity like spin along a particular axis (where the values for each particle are spin-up or spin-down), you can use a finite-dimensional Hilbert space.

I think the 10 electrons can be represented by 30 dimensions (10 x 3 dimensions of x, y, z), right?

No, even for 1 electron in 1 dimension, for position you'd need a Hilbert space with infinite dimensions, see the last paragraph here.

 

[rodsika]

Close; to combine two particles into a single system you "multiply" their Hilbert spaces. It's easiest to consider a simple example like the spin states of two spin-1 particles. Each particle's spin state is located in a Hilbert space with only three basis vectors; call them 1, 0, and -1, representing the three possible values of spin angular momentum around the z axis for a spin-1 particle. If we have two particles the Hilbert space has size 3x3 = 9; the basis vectors can be represented as:

(1, 1), (1, 0), (1, -1)
(0, 1), (0, 0), (0, -1)
(-1, 1), (-1, 0), (-1, -1)

where the first number in a pair gives the angular momentum of the first particle around the z axis and the second number in a pair gives the angular momentum of the second particle around the z axis. Hopefully you can see why this is considered a kind of "multiplication" of the two Hilbert spaces for the individual particles.

The_Duck, in your example here, the basis vector seemingly contains two values, the (1,1) for example. It confused me a bit. But I think you are just showing it to represent the two particles.. but in actual one basis vector in Hilbert Space for position can only contain one value, right? Please confirm. Thanks.

 

[rodsika]

Since position is continuous, only an infinite-dimensional Hilbert space will do. If you're just interested in a discrete quantity like spin along a particular axis (where the values for each particle are spin-up or spin-down), you can use a finite-dimensional Hilbert space.

No, even for 1 electron in 1 dimension, for position you'd need a Hilbert space with infinite dimensions, see the last paragraph here.

Ok. I want to know something. In the Stern-Gerlach experiment, when you choose a certain axis for example X in one run, all the axis in subsequent runs will choose the X axis. It's like the instrument is able to bias the axis chosen to be X. How come?

 

[JesseM]

The_Duck, in your example here, the basis vector seemingly contains two values, the (1,1) for example. It confused me a bit. But I think you are just showing it to represent the two particles.. but in actual one basis vector in Hilbert Space for position can only contain one value, right? Please confirm. Thanks.

A single position eigenvector for a multiparticle system corresponds to a set of positions for all the particles (thought The_Duck's example involved spin, not position). I'd forgotten the rules for how eigenvalues work in this case, but http://ls.poly.edu/~jbain/philqm/philqmlectures/05.MultiparticleSystems.pdf says that when you form a "product space" for a multiparticle system by multiplying the Hilbert spaces for the individual particles as The_Duck mentioned, then the eigenvalues associated with eigenvectors in the product space are themselves just the products of the eigenvalues for the individual single-particle eigenvectors.

 

[JesseM]

Ok. I want to know something. In the Stern-Gerlach experiment, when you choose a certain axis for example X in one run, all the axis in subsequent runs will choose the X axis. It's like the instrument is able to bias the axis chosen to be X. How come?

I think you're referring to the idea that if we choose to measure along the x-axis and get a certain result such as spin-up, then subsequent measurements along the same axis will give the same result of spin-up. I've forgotten the details but because of the way the wavefunction evolves in these experiments, if a particle's state vector is a spin-x eigenvector, the state vector will remain the same spin-x eigenvector as long as it's not "disturbed" by the measurement of an observable which fails to commute with spin on the x-axis, such as spin on the y-axis (I think this requires the right experimental conditions though, other magnetic fields along the path of the particle besides the ones generated by the Stern-Gerlach apparatus would I think make this no longer work). If you measure spin on the x-axis and get spin-up, then measure spin on the y-axis, it collapses the state vector in such a way that it's no longer an eigenvector of spin-x, which means a subsequent measurement of spin-x has a nonzero probability of collapsing it onto the spin-down eigenvector.

 

[rodsika]

I think you're referring to the idea that if we choose to measure along the x-axis and get a certain result such as spin-up, then subsequent measurements along the same axis will give the same result of spin-up. I've forgotten the details but because of the way the wavefunction evolves in these experiments, if a particle's state vector is a spin-x eigenvector, the state vector will remain the same spin-x eigenvector as long as it's not "disturbed" by the measurement of an observable which fails to commute with spin on the x-axis, such as spin on the y-axis (I think this requires the right experimental conditions though, other magnetic fields along the path of the particle besides the ones generated by the Stern-Gerlach apparatus would I think make this no longer work). If you measure spin on the x-axis and get spin-up, then measure spin on the y-axis, it collapses the state vector in such a way that it's no longer an eigenvector of spin-x, which means a subsequent measurement of spin-x has a nonzero probability of collapsing it onto the spin-down eigenvector.

I want to understand generally how the collapse agent work. Supposed Many Worlds were not true. What do you think is the agent that can collapse the wave function?

 

[JesseM]

I want to understand generally how the collapse agent work. Supposed Many Worlds were not true. What do you think is the agent that can collapse the wave function?

If many worlds were not true I think my second choice would be something like Bohmian mechanics, which also doesn't require a collapse. The idea that collapse could be "real" just seems sort of silly to me given what's known about decoherence, since we know if we have an experiment where a quantum system interacts with a bunch of measuring-device like systems which leave some sort of records of their results, and we model all these interactions using deterministic wavefunction evolution and only at the very end do we apply "collapse", the records will show something that looks precisely like a series of earlier "collapses" even though in our model it was just decoherence. The idea that you could have this weird hybrid of "real collapse" and decoherence which, upon collapse of measurement records yields records that look precisely identical to those caused by real collapse, just seems way to contrived and coincidental to be plausible to me.

Still, if you want to see some models that treat collapse as a real objective event, check out objective collapse theory, along with the more extensive Stanford Encyclopedia article on collapse theories.

 

[The_Duck]

The_Duck, in your example here, the basis vector seemingly contains two values, the (1,1) for example. It confused me a bit. But I think you are just showing it to represent the two particles.. but in actual one basis vector in Hilbert Space for position can only contain one value, right? Please confirm. Thanks.

My notation was pretty confusing :/ . Vectors in Hilbert space are not numbers, and do not "contain" numbers; they are abstract mathematical things. Instead of writing "1" to represent a vector I should have used a more common notation $$| m_z = 1 \rangle$$ is commonly used "Dirac notation" for a vector in the Hilbert space of angular momentum states. Here $$m_z$$ is the variable name commonly used to mean the spin angular momentum of the particle about the z axis.

If I say that the spins state of a particle is $$| m_z = 1 \rangle$$, I mean that when you measure the spin of the particle about the z axis you will measure angular momentum $$\hbar \cdot 1 = \hbar$$. It turns out that this is information is enough to calculate the expected results of all other spin measurements along any other axis, which is why it is acceptable notation to name the spin state $$| m_z = 1 \rangle$$. It is possible for a particle to be in a "superposition" of several spin states, which is represented as a /weighted sum/ of Hilbert space vectors, for example a possible state is

$$\sqrt{\frac{1}{3}}| m_z = 1 \rangle + \sqrt{\frac{2}{3}} |m_z = 0 \rangle$$

Here the spin angular momentum about the z axis is not certain: if you measure it there is a probability $$\sqrt{\frac{1}{3}}^2 = \frac{1}{3}$$ you to get the value $$\hbar \cdot 1 = \hbar$$ and a probability of 2/3 to get the value 0.

Any possible spin state of one spin-1 particle can be represented as some weighted sum of the following three "basis vectors":

$$| m_z = 1 \rangle$$, $$| m_z = 0 \rangle$$, $$| m_z = -1 \rangle$$.

Since any state vector can be built from these three basis vectors, we say that the Hilbert space of spin states for a spin-1 particle is 3-dimensional.

Now, suppose you want to talk about the combined spin state of two particles. This can be written as the "direct product" of two single-particle spin states, for instance:

$$| m_z = 1 \rangle \otimes | m_z = 0 \rangle$$

This is a state with the following properties: the first particle has spin angular momentum $$\hbar \cdot 1 = \hbar$$ along the z axis, and the second particle has spin angular momentum 0 around the z axis. For two-particle spin states there is a set of 3x3 = 9 basis vectors (the same as in my last post, but now in Dirac notation):

$$| m_z = 1 \rangle \otimes | m_z = 1 \rangle$$
$$| m_z = 1 \rangle \otimes | m_z = 0 \rangle$$
$$| m_z = 1 \rangle \otimes | m_z = -1 \rangle$$
$$| m_z = 0 \rangle \otimes | m_z = 1 \rangle$$
$$| m_z = 0 \rangle \otimes | m_z = 0 \rangle$$
$$| m_z = 0 \rangle \otimes | m_z = -1 \rangle$$
$$| m_z = -1 \rangle \otimes | m_z = 1 \rangle$$
$$| m_z = -1 \rangle \otimes | m_z = 0 \rangle$$
$$| m_z = -1 \rangle \otimes | m_z = -1 \rangle$$

So the Hilbert space of spin states of two spin-1 particles is the "direct product" of the Hilbert spaces of single-particle spin states, and is 9-dimensional.

Now, throughout this post I've been ignoring position, assuming that the only degree of freedom for the particle is its spin, perhaps because it's fixed in place. If you want to describe the state of a particle that can move around you first construct a Hilbert space of position states, which is infinite-dimensional. Let's say the particle can only move along the x axis. Then the basis of position states is composed of vectors that we might name like

$$|x = 5 \rangle$$

which would be a position state of a particle located at the position x=5. Then if we want to talk about a moving particle with spin, its states are in the Hilbert space that is the direct product of the Hilbert space of position states and the Hilbert space of spin states, which is composed of vectors that look like

$$|x = 7 \rangle \otimes |m_z = -1 \rangle$$

which would be the state of a particle located at x = 7 with spin angular momentum $$-\hbar$$ along the z axis.

 

[rodsika]

The_Duck, thanks for the details, you would make a good quantum textbook writer :)

 

[rodsika]

A single position eigenvector for a multiparticle system corresponds to a set of positions for all the particles (thought The_Duck's example involved spin, not position). I'd forgotten the rules for how eigenvalues work in this case, but http://ls.poly.edu/~jbain/philqm/philqmlectures/05.MultiparticleSystems.pdf says that when you form a "product space" for a multiparticle system by multiplying the Hilbert spaces for the individual particles as The_Duck mentioned, then the eigenvalues associated with eigenvectors in the product space are themselves just the products of the eigenvalues for the individual single-particle eigenvectors.

My background in QM is from the best layman book called Deep Down Things. I learned there that the Schroedinger equation is simply the wave-mechanical statement that the sum of the kinetic and potential energies at any given point is just equal to the total enregy. Once Psi(x) is known, one can solve for the object kinetic energy, speed and direction, probability of finding the object in any point in space, etc. But the book never mention about Hilbert Space. So I can't quite connect the two. The Hilbert Space has infinite basis vectors so we can surely say it doesn't occur in real spacetime but just a mathematical formalism of arranging data. But a quantum wave depicted in the wave functions looks like a 3D wave that you can locate in space and time. Can we say it's Fourier components are like the basis vectors in Hilbert Space? But fouriers components are 3D, similar to how you can separate the Fourier components of music. But since the vector in Hilbert Space represents the wave function. What are the equivalent of basis vectors in the Wave function? Can we just eliminate the idea of Hilbert Space and use plain language of wave function so that all the Fourier components are located in 3D space (instead of the infinite dimensions in Hilbert Space) so you can visualize the wave easier?

 

[rodsika]

What makes it confusing is because in Bohmian Mechanics all the properties are in the wave function itself with real ontology (unlike in Hilbert Space which is just imaginary).. such as mentioned in the following wikipedia article:

"In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space. For example, measuring position is considered to be a measurement of the position operator. This relationship between physical measurements and Hilbert space operators is, for standard quantum mechanics, an additional axiom of the theory. The de Broglie–Bohm theory, by contrast, requires no such measurement axioms (and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory). In particular, the usual operators-as-observables formalism is, for de Broglie–Bohm theory, a theorem.[14] A major point of the analysis is that many of the measurements of the observables do not correspond to properties of the particles; they are (as in the case of spin discussed above) measurements of the wavefunction."

Anyway. In ordinary Copenhagen. Both wavefunction and Hilbert Space are fictitious. All properties and states of the system are put in Hilbert space. As I undertand it. The wave function can't accommodate all the states so the rest have to be put in Hilbert space. But what the properties that are put in the wave function only?

 

[JesseM]

My background in QM is from the best layman book called Deep Down Things. I learned there that the Schroedinger equation is simply the wave-mechanical statement that the sum of the kinetic and potential energies at any given point is just equal to the total enregy. Once Psi(x) is known, one can solve for the object kinetic energy, speed and direction, probability of finding the object in any point in space, etc. But the book never mention about Hilbert Space. So I can't quite connect the two. The Hilbert Space has infinite basis vectors so we can surely say it doesn't occur in real spacetime but just a mathematical formalism of arranging data. But a quantum wave depicted in the wave functions looks like a 3D wave that you can locate in space and time. Can we say it's Fourier components are like the basis vectors in Hilbert Space? But fouriers components are 3D, similar to how you can separate the Fourier components of music. But since the vector in Hilbert Space represents the wave function. What are the equivalent of basis vectors in the Wave function? Can we just eliminate the idea of Hilbert Space and use plain language of wave function so that all the Fourier components are located in 3D space (instead of the infinite dimensions in Hilbert Space) so you can visualize the wave easier?

Actually the "wave function" is exactly the same thing as the "state vector" in Hilbert space (see here for example), if you see it depicted visually in 3D space what they're really showing are just the probabilities assigned to different points in space by the state vector/wave function. Historically I think in the early days of QM people may have just been thinking about the position wave function and imagined it as a kind of density wave in 3D space, not sure exactly when it was first realized that it should be treated as a single vector that assigned probabilities to other quantities as well as position, maybe it was the same time that Born realized the amplitudes could be interpreted in terms of probabilities.

 

[rodsika]

If many worlds were not true I think my second choice would be something like Bohmian mechanics, which also doesn't require a collapse. The idea that collapse could be "real" just seems sort of silly to me given what's known about decoherence, since we know if we have an experiment where a quantum system interacts with a bunch of measuring-device like systems which leave some sort of records of their results, and we model all these interactions using deterministic wavefunction evolution and only at the very end do we apply "collapse", the records will show something that looks precisely like a series of earlier "collapses" even though in our model it was just decoherence. The idea that you could have this weird hybrid of "real collapse" and decoherence which, upon collapse of measurement records yields records that look precisely identical to those caused by real collapse, just seems way to contrived and coincidental to be plausible to me.

Still, if you want to see some models that treat collapse as a real objective event, check out objective collapse theory, along with the more extensive Stanford Encyclopedia article on collapse theories.

　

I spent sometime reading this Objective Collapse thing. I also spent a lot of time reading the google and amazon pages of Peter Byrne Many Worlds of Hugh Everett III. In the latter part of the book. He gave the impression that Zeh and others thought the measurement problem was solved by how decoherence able to pick out the Preferred Basis. But you and others emphasized one still has to use the Born Rule. But can't we argue that once the environment chose the Preferred basis. It's automatic that the wave function must collapse into them? It's like in a roller coaster.. then you were on top. It's inevitable that you would fall down. Similarly when environment picked out the Preferred Basis. Wave function must via Born Rule fall into the values resulting in collapse. This is like falling down on top of roller coaster. Anyway. For standard usage. Which of the following is true?

Wave Function Collapse = Preferred basis selection + Born Rule

Wave Function Collapse = Born Rule

That is. Is the standard terms wave function collapse simply an execution of Born Rule or does the term also include the choosing of preferred basis?

 

[JesseM]

　

I spent sometime reading this Objective Collapse thing. I also spent a lot of time reading the google and amazon pages of Peter Byrne Many Worlds of Hugh Everett III. In the latter part of the book. He gave the impression that Zeh and others thought the measurement problem was solved by how decoherence able to pick out the Preferred Basis. But you and others emphasized one still has to use the Born Rule. But can't we argue that once the environment chose the Preferred basis. It's automatic that the wave function must collapse into them? It's like in a roller coaster.. then you were on top. It's inevitable that you would fall down.

Well, there's no moment when the environment exactly selects such a basis since the interference terms in the density matrix don't quite go to zero even if they become very tiny. But the bigger problem I see with trying to have an "objective collapse" theory like this is that where you draw the boundary between "system" and "environment" is somewhat arbitrary so I don't think this your description would lend itself easily to a more well-defined theory that has precise rules for what subsystems of the universe experience "objective collapse" at what times.

　Similarly when environment picked out the Preferred Basis. Wave function must via Born Rule fall into the values resulting in collapse. This is like falling down on top of roller coaster. Anyway. For standard usage. Which of the following is true?

Wave Function Collapse = Preferred basis selection + Born Rule

Wave Function Collapse = Born Rule

That is. Is the standard terms wave function collapse simply an execution of Born Rule or does the term also include the choosing of preferred basis?

The Born rule can only be applied when you've picked what property is being "measured", then the state collapses onto one of the eigenvectors of that operator, which define a particular basis.

 

[rodsika]

Well, there's no moment when the environment exactly selects such a basis since the interference terms in the density matrix don't quite go to zero even if they become very tiny. But the bigger problem I see with trying to have an "objective collapse" theory like this is that where you draw the boundary between "system" and "environment" is somewhat arbitrary so I don't think this your description would lend itself easily to a more well-defined theory that has precise rules for what subsystems of the universe experience "objective collapse" at what times.

The Born rule can only be applied when you've picked what property is being "measured", then the state collapses onto one of the eigenvectors of that operator, which define a particular basis.

For example, the setup is an electron double slit experiment and the property position is being measured. The eigenvectors are those possible positions in the interference screen right? But you said "operator, which define a particular basis." In the double slit expement. What are the "particular basis" for example? Since environment chooses the basis, then it is the interference screen that chooses it? If wrong basis was chosen, what would happen to the positions of the screen for example, are you saying the position can become behind the screen if wrong basis chosen. Thanks.

 

[JesseM]

For example, the setup is an electron double slit experiment and the property position is being measured. The eigenvectors are those possible positions in the interference screen right? But you said "operator, which define a particular basis." In the double slit expement. What are the "particular basis" for example? Since environment chooses the basis, then it is the interference screen that chooses it?

There's two different senses of "choosing a basis" that we're talking about here--according to the Born rule measurement collapses the system onto one of the basis vectors of whatever variable is being measured at the moment of measurement, and the environment can choose a basis prior to the moment of measurement if there is decoherence going on, like in the example where the buckyball is traveling through a hot gas. This environment-chosen basis has nothing to do with what you measure at the end of the experiment, for example if your final measurement which "collapsed" the buckyball's state was a measurement of momentum rather than position, it would still be true that the hot gas had caused the buckyball to have acted like it was in a statistical mixture of different position eigenstates at each moment prior to measurement. So it would be as if there were detectors continually measuring its position as it traveled, but you don't actually know what the results of those measurements were, so you can't incorporate the details of its positions into your calculation of the probability you will get different outcomes with your final momentum measurement. Another way of thinking about it is to imagine a very large series of trials with identical starting conditions, where on each trial the buckyball's position was being continually measured as it traveled (giving different measured paths on different trials), and at the end of each one the same momentum measurement was performed--the average frequency of different momentum results over all these trials should be about the same as the probability of different momentum results on a single trial where the electron was traveling through a hot gas which caused continual decoherence.

 

[rodsika]

There's two different senses of "choosing a basis" that we're talking about here--according to the Born rule measurement collapses the system onto one of the basis vectors of whatever variable is being measured at the moment of measurement, and the environment can choose a basis prior to the moment of measurement if there is decoherence going on, like in the example where the buckyball is traveling through a hot gas. This environment-chosen basis has nothing to do with what you measure at the end of the experiment, for example if your final measurement which "collapsed" the buckyball's state was a measurement of momentum rather than position, it would still be true that the hot gas had caused the buckyball to have acted like it was in a statistical mixture of different position eigenstates at each moment prior to measurement. So it would be as if there were detectors continually measuring its position as it traveled, but you don't actually know what the results of those measurements were, so you can't incorporate the details of its positions into your calculation of the probability you will get different outcomes with your final momentum measurement. Another way of thinking about it is to imagine a very large series of trials with identical starting conditions, where on each trial the buckyball's position was being continually measured as it traveled (giving different measured paths on different trials), and at the end of each one the same momentum measurement was performed--the average frequency of different momentum results over all these trials should be about the same as the probability of different momentum results on a single trial where the electron was traveling through a hot gas which caused continual decoherence.

Ok thanks. Say, what is Einstein describing when he wrote to Born and said:

"Let Psi(1) and Psi(2) be solutions of the same Schroedinger equation... When the system is a macrosystem and when Psi(1) and Psi(2) are 'narrow' with respect to the macrocoordinates, then in by far the greater number of cases this is no longer true for Psi = Psi(1) + Psi(2). Narrowness with respect to macrocoordinate is not only independent of the principles of quantum mechanics, but moreever, incompatible with them." (The translation from Born (1969) quoted here is due to Joos (1986).)

 

[zenith8]

What will happen to Born Rule (BR) since BR involves randomness? Will BR disappears or remain as is as Born Rule involves just operations to get certain values of the position eigenstates (without regard to whether they are acquire randomly or has hidden determinism as George proposed)?

Valentini et al have a paper out showing how the Born rule arises naturally as the end equilibrium point of a deterministic process - so it doesn't necessarily involve randomness.. See http://arxiv.org/abs/1103.1589" .

 

[JesseM]

Ok thanks. Say, what is Einstein describing when he wrote to Born and said:

"Let Psi(1) and Psi(2) be solutions of the same Schroedinger equation... When the system is a macrosystem and when Psi(1) and Psi(2) are 'narrow' with respect to the macrocoordinates, then in by far the greater number of cases this is no longer true for Psi = Psi(1) + Psi(2). Narrowness with respect to macrocoordinate is not only independent of the principles of quantum mechanics, but moreever, incompatible with them." (The translation from Born (1969) quoted here is due to Joos (1986).)

Hmm, I can't really understand that quote, where is it from?

 

[rodsika]

Hmm, I can't really understand that quote, where is it from?

It's from Zurek paper http://arxiv.org/abs/quant-ph/0105127 at the footnote of page 2. Zurek said: "Therefore, there is no a priori reason for macroscopic objects to have definite position or momentum. As Einstein noted localization with respect to macrocoordinates is not just independent, but incompatible with quantum theory."

How can he said that. Why is meant by that?

Anyway I have a separate question for you that has been bugging me for the whole day and reading many references just to find the answer. Do you think the Preferred Basis problem only occur in Many Worlds Interpretation? For Copenhagen Interpretation, does it also occur? In the electron passing thru gas before reaching the slits, any decoherence can only remove the interference pattern in the final screen, here is the preferred basis important or not important as choosing spin or momentum basis won't change the fact that the interference pattern would be suppressed in the final stage at the screen. About this final measurement on the screen just as Born Rule is applied. For a given quantum state, what determines the orthogonal set of projectors (Preferred Basis) to which the Born rule assigns the probabilities? Is this completely understood already or still a mystery? What's your own opinion?

 

[rodsika]

There's two different senses of "choosing a basis" that we're talking about here--according to the Born rule measurement collapses the system onto one of the basis vectors of whatever variable is being measured at the moment of measurement, and the environment can choose a basis prior to the moment of measurement if there is decoherence going on, like in the example where the buckyball is traveling through a hot gas. This environment-chosen basis has nothing to do with what you measure at the end of the experiment, for example if your final measurement which "collapsed" the buckyball's state was a measurement of momentum rather than position, it would still be true that the hot gas had caused the buckyball to have acted like it was in a statistical mixture of different position eigenstates at each moment prior to measurement. So it would be as if there were detectors continually measuring its position as it traveled, but you don't actually know what the results of those measurements were, so you can't incorporate the details of its positions into your calculation of the probability you will get different outcomes with your final momentum measurement. Another way of thinking about it is to imagine a very large series of trials with identical starting conditions, where on each trial the buckyball's position was being continually measured as it traveled (giving different measured paths on different trials), and at the end of each one the same momentum measurement was performed--the average frequency of different momentum results over all these trials should be about the same as the probability of different momentum results on a single trial where the electron was traveling through a hot gas which caused continual decoherence.

Jesse. I was assuming the preferred basis problem includes how the screen chooses the position basis, is there no mystery here already?

Do you think the Preferred Basis problem only occur in Many Worlds
Interpretation? For Copenhagen Interpretation, does it also occur? In the
electron passing thru gas before reaching the slits, any decoherence can only
remove the interference pattern in the final screen, here is the preferred basis
important or not important as choosing spin or momentum basis won't change the
fact that the interference pattern would be suppressed in the final stage at the
screen? About this final measurement on the screen just as Born Rule is applied.
For a given quantum state, what determines the orthogonal set of projectors
(Preferred Basis) to which the Born rule assigns the probabilities? Is this
completely understood already or still a mystery? What's your own opinion?

 

[rodsika]

Jesse, A week ago you said the following "In Copenhagen the choice of what to measure determines what basis the quantum state will "collapse" onto a basis vector of." But after some research lately, I think it's not that simple. Preferred basis problem seems to occur too in Copenhagen (in the final measurement) as well as in Many Worlds. Do you agree or disagree? I found out the following paper which refutes your idea.

http://www.lps.uci.edu/barrett/publications/preferred%20basis%20problem%20and%20quantum%20mechanics%20of%20everything.pdf

"The recurring problem of choosing which observable to make determinate in a solution to the quantum measurement problem is the preferred-basis problem. The problem is that we do not know what determinate physical property would make our most immediately accessible physical records determinate.

The preferred-basis problem is often presented as a complaint that any choice of a preferred physical observable looks mathematically ad hoc. The argument is that since each complete physical observable corresponds to an orthonormal basis of the Hilbert space used to represent a physical system, and since any complete orthonormal basis is as good as any other for representing the state of the system, from the mathematical perspective, any particular choice of a basis is purely conventional. But the preferred-

basis problem is not just a matter of one not wanting one’s choice of a preferred observable to look ad hoc. There are simple matters of physical fact at stake here: one’s choice of the determinate physical observable is what determines what determinate physical facts there can be in a physical world described by the theory. So if one takes there to be an objective matter of fact concerning what physical facts there are, one cannot

take this choice to be simply a matter of personal preference, convenience, or convention. And among the physical facts one wants determinate are those facts concerning the values of our measurement records; but, again, we do not know what determinate physical property would in fact make our most immediately accessible measurement records determinate. This is something that ultimately depends on the relationship between mental and physical states and on exactly how we expect our best physical theories to account for our experience, and this is presumably something physicists would prefer not to address in formulating a satisfactory version of quantum mechanics."

 

[JesseM]

Jesse, A week ago you said the following "In Copenhagen the choice of what to measure determines what basis the quantum state will "collapse" onto a basis vector of." But after some research lately, I think it's not that simple. Preferred basis problem seems to occur too in Copenhagen (in the final measurement) as well as in Many Worlds. Do you agree or disagree? I found out the following paper which refutes your idea.

How does it refute it? Can you summarize in your own words what you think the Barrett's argument is, and why you think it refutes what I said? The Copenhagen interpretation only requires that the method I described involving the Born rule and wavefunction "collapse" should work in practice for making predictions, and in practice isn't it always clear what property an experimenter has chosen to measure? From this sentence on p. 3, it seems like Barrett is trying to argue that there's a problem if we want to do away with the concept of "collapse" due to measurement and explain everything in terms of unitary evolution (the deterministic evolution of the wavefunction between measurements):

Further, if one insists on the applicability of the standard unitary dynamics for all interactions and insists that there is only one post-measurement mental record, then one must add something to the description of the physical state on which the mental record might be taken to supervene. While this parameter is commonly referred to as a hidden variable, it is the determinate value of this new parameter together with the standard quantum mechanical state that will explain an observer’s determinate measurement results.

So I would think Barrett isn't really talking about Copenhagen at all (which doesn't involve any hidden variables, and doesn't say unitary dynamics should apply for measurement interactions), but about alternatives to it which try to eliminate the idea that anything special happens on measurement. The part about the preferred basis problem which you quotes, which appears at the bottom of p. 7, seems to be about a preferred basis problem in hidden-variables theories like Bohmian mechanics--Bohmian mechanics assumes position is the variable that in reality has a well-defined value at all times, with other observables like momentum not being true properties of the particle but just results of how the particle and measuring device interact, causing the positions of dials or other macroscopic readers on the measuring device (pointer states) to show readings consistent with the predictions of Copenhagen QM (see quantum observables from the Stanford Encyclopedia article on Bohmian mechanics). I think Barrett is arguing that this choice has a certain arbitrariness, perhaps one could come up with a Bohm-like interpretation where it was momentum rather than position that had a well-defined value at all times, and as Barrett argues earlier on p. 7:

But because Bohm’s theory makes only one classical physical determinate, and because we do not know whether mental records are in fact fully determined by particle positions, an assessment of whether Bohmian mechanics in fact explains why we experience what we do is ultimately contingent on the yet-to-be-determined relationships between mental and physical states.