Q: anyone know the formalism of the transactional interpretation of QM ?

[vanesch]

Hello all,

I would like to learn more about the formalism of the transactional interpretation in QM. Apparently, the formalism is NOT the same as in unitary QM (but the claim is that the empirical predictions are equivalent). As such, it is not so much an *interpretation* but an equivalent, but different, physical theory. However, the few documents I can find
(namely, by its founder: http://www.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html )
do not seem to be very explicit on it. His paper
http://www.npl.washington.edu/npl/int_rep/gat_80/
does not seem to address the relationship between a multi-particle Hilbert space description (in standard QM) and his single-particle "squared" Schroedinger equation.

In other words, I do not find anywhere, what mathematical structure is seen as the "state space" of nature, and how it is related to standard QM (in order to justify the claim of empirical equivalence). Most of the writings seem to try to make the reader accept the idea of "backwards causation", but I would like to see simply how this idea is put in music wrt to unitary quantum theory.

My point behind it is to find out if there is a natural way in which this view can avoid my body to end up in a superposition ; in what way he can avoid any MWI scenario.

For instance, how do you view the state of an Argon atom in this picture...

 

[slyboy]

As far as I know, the transactional interpretation has never been worked out for a multiparticle system. It is hard to do because Cramer's ideas rely heavily on thinking about the wavefunction as a wave in ordinary 3-d space rather than configuration space. Thus is is better to think of the transactional programme as an "idea for an interpretation" rather than an "interpretation" because we would expect the latter to at least work for all of nonrelativistic quantum mechanics.

 

[Careful]

In other words, I do not find anywhere, what mathematical structure is seen as the "state space" of nature, and how it is related to standard QM (in order to justify the claim of empirical equivalence). Most of the writings seem to try to make the reader accept the idea of "backwards causation", but I would like to see simply how this idea is put in music wrt to unitary quantum theory.

My point behind it is to find out if there is a natural way in which this view can avoid my body to end up in a superposition ; in what way he can avoid any MWI scenario.

For instance, how do you view the state of an Argon atom in this picture...

Hi Vanesch,

I am not sure whether your question is the right one. First of all, I would not like it to be equivalent to unitary QM since otherwise your zombies seem to come around the corner again. Second, it is not fair to wonder about what such theory would have to say about Argon, since we really do not know what QM has to say about that either. Anyway, I think we should look up the papers and then see for ourselves (quite some work). If you would find some http references, please post them here.

Cheers,

Careful

 

[vanesch]

As far as I know, the transactional interpretation has never been worked out for a multiparticle system. It is hard to do because Cramer's ideas rely heavily on thinking about the wavefunction as a wave in ordinary 3-d space rather than configuration space. Thus is is better to think of the transactional programme as an "idea for an interpretation" rather than an "interpretation" because we would expect the latter to at least work for all of nonrelativistic quantum mechanics.

Yes that was exactly my feeling when I read the few references I could find on it. But I thought that somewhere, the whole idea WAS worked out.

 

[vanesch]

I am not sure whether your question is the right one.

Put this in your head: my questions are ALWAYS the right ones :tongue: :rofl:

First of all, I would not like it to be equivalent to unitary QM since otherwise your zombies seem to come around the corner again.

Ok, but then it is not to be sold as "an interpretation" of quantum theory, or even an "empirically equivalent" theory. I thought it was kind of like Bohmian mechanics is wrt QM: a different formalism, but of which (under certain conditions) empirical equivalence can be demonstrated.
I was wondering how this formalism got around the strict unitarity.

On the other hand, the idea of constructing a DIFFERENT theory, using backwards causation (maybe your wet dream) is another discussion. But then it will (IMO) not be equivalent to QM in all respects.

 

[vanesch]

Second, it is not fair to wonder about what such theory would have to say about Argon, since we really do not know what QM has to say about that either.

I don't understand what you mean ?
Something like this read ?
http://www.chem.ucla.edu/dept/Faculty/scerri/pdf/Howabinitio.pdf

 

[Careful]

** Put this in your head: my questions are ALWAYS the right ones :tongue: :rofl: **

:rofl: :rofl: Oh yeh!

**
Ok, but then it is not to be sold as "an interpretation" of quantum theory, or even an "empirically equivalent" theory. **

I never said it was (and as you know I strongly despise marketing words) and neither am I interested in the philosophical aspect. But it seems entirely plausibe that you can make it empirically equivalent.

**I thought it was kind of like Bohmian mechanics is wrt QM: a different formalism, but of which (under certain conditions) empirical equivalence can be demonstrated. **

My god, let's hope not, since that would be entirely useless.

**I was wondering how this formalism got around the strict unitarity. **

I don't know, but you need to get around strict unitarity. In a certain sense - as I told you many times - you do that too when you are assigning consciousness to a particular base state (the superposition of which is the actual state you are considering). Since I want a one world picture, unitarity has to fly out of the window.

** On the other hand, the idea of constructing a DIFFERENT theory, using backwards causation (maybe your wet dream) is another discussion. But then it will (IMO) not be equivalent to QM in all respects. **

As I said, let's read the papers prior to spelling out opinions. It is not my wet dream of constructing a theory which gives up the arrow of time in an appropriate sense: I merely anticipate that starting from local realism is perfectly safe, given the fact that you can get higher correlations by merely dropping the arrow of time (while keeping locality and realism safe).

 

[Careful]

I don't understand what you mean ?
Something like this read ?
http://www.chem.ucla.edu/dept/Faculty/scerri/pdf/Howabinitio.pdf

What does this tell me?? Your electrons in these models are not allowed to radiate so (a) it is not said at all that these orbitals are stable (I have been told that some are not actually) (b) whether in case they would be - they correspond to the lowest energy states of the full Hamiltonian (c) these computations are done in HF approximation or something alike. I mean if you do not allow your electrons to radiate, then one could figure out a classical variant of the Bohr model to ``explain´´ the atom :rofl: :rofl: :rofl:

In other words: who guarantuees me that the Hamiltonian of the full problem does not have a continuous spectrum and that no electrons can drop on the nucleus ?
Cheers,

Careful

 

[Careful]

By the way, as I said I have still problems with the ``local´´ character of your MWI. As an example, take a setup with spatially separated observers 1,2 performing spin measurement with axes at relative angle x.

The singlet state is a linear combination of the form (sorry I am too lazy to put in the correct coefficients but this is hardly the point):

a |up> |up> + b |up>|down> + c |down>|down> + d|down>|up>

and suppose number ONE performs an up/down measurement, then locality would assume that there is one conscious observer for |up> and one for |down>. However, this would imply that the consciousness (zombie) of ONE is still in TWO universes instead of one! As far as local physics is concerned, this is all the information to be obtained for each observer, hence the two fold degeneracy is a non local effect: the number of zombie copies depends upon the experiment to be performed by observer 2.

Anyway, you keep your full material dynamics unitary, but only a small nonunitary part of it is visible to your consciousness, hence your observations satisfy non-unitary laws as illustrated above. It might be that I have expressed this before in another way; in that case sorry for the repetition. Unitarity should be some time averaged statistical property of the outcomes of non-unitary processes (that is also how we percieve it), and not some ``instaneously´´ preserved quantity of ``would be´´ experiments (I have certainly said that before). At least, when you want to preserve the arrow of time :-) and that is why I think this retrocausality idea is interesting (if you accept QM as it stands).

Cheers,

Careful

 

[vanesch]

By the way, as I said I have still problems with the ``local´´ character of your MWI. As an example, take a setup with spatially separated observers 1,2 performing spin measurement with axes at relative angle x.

The singlet state is a linear combination of the form (sorry I am too lazy to put in the correct coefficients but this is hardly the point):

The singlet state will of course only contain b and d non-zero

EDIT: oops, sorry, didn't realize the relative angle...

So let us assume a general (non-singlet) state:

a |up> |up> + b |up>|down> + c |down>|down> + d|down>|up>

and suppose number ONE performs an up/down measurement, then locality would assume that there is one conscious observer for |up> and one for |down>.

What you have is:

Initial state:

|ONE*> |TWO*> (a |up> |up> + b |up>|down> + c |down>|down> + d|down>|up>)

Now, you want observer ONE to do a measurement on the (first I presume) state. The asterix indicates the "perceived" state from the viewpoint of the observer in question. In the initial state, they are both of course ignorant (they didn't interact with the system yet).

After ONE does his measurement interaction (assuming TWO didn't do anything), we get an entanglement between the first state and the state of ONE:

|TWO*> (a |ONEup> |up>|up> + b |ONEup> |up> |down> + c |ONEdown> |down> |down> + d |ONEdown> |down> |up>) ;

which, upon rewriting in the H_ONE x H_rest form (Schmidt form), becomes:

|ONEup> (|TWO*> (a |up> |up> + b |up>|down>) )
+ |ONEdown> (|TWO*> ( c |down>|down> + d|down>|up>) )

The Hilbert norm for ONEup is a^2 + b^2 while the Hilbert norm for ONEdown is c^2 + d^2, so, according to the Born rule, the asterix is now assigned to one of both (say, |ONEup>):
|ONEup*> (|TWO*> (a |up> |up> + b |up>|down>) )
+ |ONEdown> (|TWO*> ( c |down>|down> + d|down>|up>) )

The same state, from the TWO point of view, must be written in H_TWO x H_rest', and becomes:

|TWO*> (a |ONEup> |up>|up> + b |ONEup> |up> |down> + c |ONEdown> |down> |down> + d |ONEdown> |down> |up>)

It is still in a single product form. TWO didn't interact. So nothing happens to its asterix. However, in TWO's world view, ONE is in a superposition of ONEup and ONEdown (but cannot know it, because knowing it would mean that TWO *interacts* with ONE, and hence gets entangled too).

However, this would imply that the consciousness (zombie) of ONE is still in TWO universes instead of one!

The "number of universes" is observer-dependent, because it depends upon a different slicing-up of H into H_observer x H_rest. This is what most people do not see, and object to MWI that a remote process on Betelgeuse will split *my* universe. No. As long as I don't get entangled with any of it, this is an internal affair within the H_rest element I'm in product with. In other words, a unitary operator that acts only on H_rest does not make "my universe" split, nor does a unitary operator that acts only on H_observer. It is only a unitary operator that acts upon both non-trivially that will split "my" universe ; in order to do so, however, this operator has to correspond to a LOCAL interaction between myself and something nearby.

As far as local physics is concerned, this is all the information to be obtained for each observer, hence the two fold degeneracy is a non local effect: the number of zombie copies depends upon the experiment to be performed by observer 2.

The "number of terms" is a bad quantity. What counts is the total hilbert length of each DISTINCT state of each observer (in the H_observer x H_rest expansion), and that hilbert length is invariant under a unitary operator acting on only one of the spaces, in casu H_rest).

Anyway, you keep your full material dynamics unitary, but only a small nonunitary part of it is visible to your consciousness, hence your observations satisfy non-unitary laws as illustrated above.

Yes, that's the trick :-)

Unitarity should be some time averaged statistical property of the outcomes of non-unitary processes (that is also how we percieve it), and not some ``instaneously´´ preserved quantity of ``would be´´ experiments (I have certainly said that before). At least, when you want to preserve the arrow of time :-) and that is why I this retrocausality idea is interesting (if you accept QM as it stands).

I don't know what unitarity SHOULD be. My only point is that what I sketch is, IMO, the view on QM which sticks closest to the current, unitary formalism, and that it (although admittedly very weird) is a logically consistent viewpoint - something what cannot be said of several other viewpoints on QM. I think that it is difficult to argue AGAINST this viewpoint and in favor of strict unitarity.

 

[vanesch]

In other words: who guarantuees me that the Hamiltonian of the full problem does not have a continuous spectrum and that no electrons can drop on the nucleus ?

This happens sometimes, no ? Electron capture and a proton -> neutron + neutrino emission nuclear transmutation.
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/radact2.html

 

[Careful]

**The singlet state will of course only contain b and d non-zero

EDIT: oops, sorry, didn't realize the relative angle... **

Good, you saved yourself from a to the power of

** The "number of universes" is observer-dependent, because it depends upon a different slicing-up of H into H_observer x H_rest. This is what most people do not see, and object to MWI that a remote process on Betelgeuse will split *my* universe. No. As long as I don't get entangled with any of it, this is an internal affair within the H_rest element I'm in product with. **

Forget what I said : three days ago I still went trough it in my head (where I assume the measurement induces a bifurcation of the universe on the future lightcone) and knew it was entirely consistent. It is just that I am so much opposed to this idea that I tend to ``forget´´ my previous thoughts

 

[Careful]

This happens sometimes, no ? Electron capture and a proton -> neutron + neutrino emission nuclear transmutation.
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/radact2.html

I do not understand why you mention this :
(a) this is a transition in the weak interactions while I was merely asking for the stability of the electron foton interaction (remember this was one of the primary reasons to dismiss classical atomic models).
(b) to my knowledge neither the weak, nor the EM interactions have been fully taken into account for a complicated bound state like an Argon atom (you can treat the electrons and the nucleus nonrelativistically - that ``simplifies´´ things a lot), work in the interaction picture, and try to compute the scattering matrix perturbatively (actually you should not even do that). The usual way to do this is to start from states which are solutions to the free Hamiltonian (which has a *continuous* spectrum) |atom> * |radiation>, consider the action of the interaction Hamiltonian and try to find eigenstates of the FULL Hamiltonian with the same energy using the *formal* Born series (I am unaware of general theorems about the mathematical sanity of this procedure). This procedure gives us (normalizable?) heavily entangled electron/radiation free eigenstates. To know the physics of the electrons, we have to compute momenta of postion and momentum observables. AFAIK, no detailed *nonperturbative* analysis of this problem has been done even not for the Helium atom (if I were wrong here, then I would appreciate references).

A full relativistic treatment within the context of electroweak theory is of course even more complex (actually I do not know of the existence of proper bound states which one needs to consider this problem).


Cheers,

Careful

 

[vanesch]

I do not understand why you mention this :
(a) this is a transition in the weak interactions while I was merely asking for the stability of the electron foton interaction (remember this was one of the primary reasons to dismiss classical atomic models).

Ah. Ok, there are many procedures (as the one you mention) which are unproven in all rigor, right. But if you accept that you can start from the free hamiltonian solutions |atom> x |EM field> AS A BASIS of the Hilbert space (in the sense that the interaction is not going to add degrees of freedom), can't we simply say that, if "atom" is in its ground state, and |EM> is in its ground state, that the (relatively small) interaction term will not succeed in introducing any other term |atom*>, because all these other terms have much higher (free) energy ?

 

[Careful]

Ah. Ok, there are many procedures (as the one you mention) which are unproven in all rigor, right. But if you accept that you can start from the free hamiltonian solutions |atom> x |EM field> AS A BASIS of the Hilbert space (in the sense that the interaction is not going to add degrees of freedom), can't we simply say that, if "atom" is in its ground state, and |EM> is in its ground state, that the (relatively small) interaction term will not succeed in introducing any other term |atom*>, because all these other terms have much higher (free) energy ?

No, I don't believe anything of that and I guess the situation is going to be identical to the one in classical mechanics. The argument people usually hold is that : ``there is a discrete energy gap between one energy level and the next, therefore you need a jump´´ pretending as if the interactions preserve the product form of the state. In exactly the same way I could devise a classical model (without radiation) with a discrete number of stable ``attractors´´ (attractor is here to be understood in a time averaged sense) and say that the small Lorentz term will keep the electron within the same attractor. Blatant nonsense of course (unless there is some other mechanism to compensate for this). Interactions are turning the naive product state into a highly entangled one so that the electron is really in no particular energy orbit at all. One needs to do the full calculation of the momenta of position and energy, and it might very well be that we arrive at the classical misery.

In this sense the paper of SED about how a background radiation field is to produce the ground state of the H atom illuminates the role played by the background radiation : it has to be precisely so that that it compensates the Lorentz term. No such condition arises from the QM calculations as far as I see. So, here you have a nice research topic : try to confirm (or falsify :-)) QM by studying this problem rigorously.

Cheers,

Careful

 

[Careful]

EDIT to my previous post:
My main worry was/and partially is whether the ``quantum force´´ can keep an atom stable even if the electrons are allowed to radiate: hence whether the full Hamiltonian has a well defined ground state/ and what happens to the traditional electron energy levels. This is by *no means* a trivial question/ however there exist some very recent results about this which show that non-relativistic atoms have a well defined ground state:
http://arxiv.org/PS_cache/math-ph/pdf/0307/0307046.pdf
and
math-ph/0401004

Cheers,

Careful