What if the Bohmian model turned out to be correct?

[Schrodinger's Dog]

My problem with the realness of the wave function is not that it uses imaginary numbers, it's that "realness" isn't a scientific principle in the first place. Science doesn't know how to judge what is real, it only knows how to describe it. For example, we may have a hard time imagining the square root of -1, but we can easily imagine the concepts of magnitude and phase of some cycle-- and that's all one needs to have "complex numbers". The use of the square root of -1 is a mathematical convenience, not an essential part of a wave function, so I don't think we can rule on its realness on that basis.

The best we can do, if so inclined, is rule on the measurability of a concept, and there are weird applications in things like superconductivity where wave functions might seem to get pretty close to what one might think of as real. But that doesn't matter, I argue, because we don't sit in judgement of that, we only judge the value of our theories. It is too easy to mistake familiarity for understanding for us to start claiming that an electron is real but its wave function isn't, so I don't see exactly what you mean by "pictorial terms".

I agree but what if what we are describing we don't have the ability to describe given our limitations? If we could would that make us able to describe it in deterministic terms or quantum terms? I think probably it is quantum, and that's why we fail to describe it; but as a virtual laymen with some physics knowledge but not as much as a graduate, I am of course only speculating on the ideas of others.

Ie when we talk about the tensors or equations what we are really doing is fudging it based on a lack of knowledge of what is really going on? Is that clear? As you say we are not in a position to make claims beyond that which we know. But if we were?

 

[samalkhaiat]

The real and only challenge for the so-called Bohm's mechanics is the following exercise;
Derive Bohm's equations from the (p-representation) Schrodinger equation

<br /> i \partial_{t}\Phi (p,t) = \frac{p^{2}}{2m} \Phi (p,t) + \int d^{3} \bar{p} V(p , \bar{p}) \Phi ( \bar{p},t)<br />

If one manages to do this, then one can say that the mathematical structure of Bohm's is equivalent to that of Schrodinger's. The point is this; we can choose to write (and solve) Schrodinger equation in the x-rep., the p-rep. or the whatever-representation, but we can not do the same thing with Bohm's equations, i.e., while QM is a representation-free theory, Bohm's is not. However, leaving this defficiency aside, Bohm's mechanics is still able to reproduce all the results of the non-relativistic QM.
In the relativistic domain, Bohm's approach does not function at all.

It seems to me that Bohm's mechanics stands somewhere between QM and CM (abit of each worlds). I also believe that Biology (which I do not know much about) lies in the same domain! Bohm's concepts such as "Nonseparability" and "Self-organisation" are (I believe) of great importance for Biological systems. It would be the greatest triumph of physics if BM turns out to be a good dynamical framework for describing protein-making processes. I believe one day equations of physics WOULD DO to biology what Schroginger equation DID to chemistry. Would those equations be Bohm's? Biophysicists should try their luck! I would be a very lucky man if my idea (speculation) turns out to be correct

regards

sam

 

[Ken G]

Well isn't i just a mathematical trick that links geometry with the imaginary plane. Is it really what we are representing, ie by pictorial Bohr meant is it a snapshot or photograph of what really happens.

OK, I understand what you mean by "pictorial" now, and the issue of the "realness" of the wave function, but I still don't see the argument involving i, because yes it is just a mathematical trick. It would not seem to be hard to write the Schrodinger equation without i by breaking it into two parts (parts that would have no point in being called "real" and "imaginary", as none of the predictions of quantum mechanics rely on that distinction).

http://plato.stanford.edu/entries/qm-copenhagen/

I realize this is a philosophical treatise (so is CI to an extent) but I think it sums up his theory well enough, it may be out of date, but it's something I read.

The article is good, and causes me to see just how much I agree with Bohr's view of the meaning of quantum mechanics-- and why it's not really what gets called the CI at all.

 

[Ken G]

I agree but what if what we are describing we don't have the ability to describe given our limitations?

Not a hypothetical question-- that seems to be just the case. The question is, should this state of affairs surprise us? Our technology for studying the world has vastly outstripped our intellect and experience, it is amazing we hang on as well as we do.

If we could would that make us able to describe it in deterministic terms or quantum terms?

Either, both, whatever works in whatever situation.

Ie when we talk about the tensors or equations what we are really doing is fudging it based on a lack of knowledge of what is really going on?

I would say it's worse than we lack the knowledge to know what's going on, we lack the very language that could provide meaning to the expression "what's really going on". It seems to me science is above all a set of rules for creating a language, and we should follow those rules instead of being beguiled into using a language that we cannot make scientific. I believe that is also what Bohr was saying, as I interpret it.

 

[Schrodinger's Dog]

Not a hypothetical question-- that seems to be just the case. The question is, should this state of affairs surprise us? Our technology for studying the world has vastly outstripped our intellect and experience, it is amazing we hang on as well as we do.

Either, both, whatever works in whatever situation.

I would say it's worse than we lack the knowledge to know what's going on, we lack the very language that could provide meaning to the expression "what's really going on". It seems to me science is above all a set of rules for creating a language, and we should follow those rules instead of being beguiled into using a language that we cannot make scientific. I believe that is also what Bohr was saying, as I interpret it.

The first question was meant to be rhetorical, accidentally put a question mark in I think.

Uncannily that's exactly how Bohr put it. If we do not even have the language to describe the quantum, then how do we expect to describe it. Paraphrasing again.

The real and only challenge for the so-called Bohm's mechanics is the following exercise;
Derive Bohm's equations from the (p-representation) Schrodinger equation

<br /> i \partial_{t}\Phi (p,t) = \frac{p^{2}}{2m} \Phi (p,t) + \int d^{3} \bar{p} V(p , \bar{p}) \Phi ( \bar{p},t)<br />

Not related to Bohm per se but have you seen a 3d solution (technically 4d I suppose) to the shcrödinger equation, I'll fish out the paper if you're interested; maths degree level though, so was a bit above me, having only recently studied calculus. I guess that's as close to pictorial as you're likely to get atm? It was a link in the tutorial section IIRC supplied by HOI. I hope you won't mind if I pass up trying to solve that for the time being.

If one manages to do this, then one can say that the mathematical structure of Bohm's is equivalent to that of Schrodinger's. The point is this; we can choose to write (and solve) Schrodinger equation in the x-rep., the p-rep. or the whatever-representation, but we can not do the same thing with Bohm's equations, i.e., while QM is a representation-free theory, Bohm's is not. However, leaving this defficiency aside, Bohm's mechanics is still able to reproduce all the results of the non-relativistic QM.
In the relativistic domain, Bohm's approach does not function at all.

Well QM isn't entirely relativistic, so perhaps we're missing the language there too.

It seems to me that Bohm's mechanics stands somewhere between QM and CM (abit of each worlds). I also believe that Biology (which I do not know much about) lies in the same domain! Bohm's concepts such as "Nonseparability" and "Self-organisation" are (I believe) of great importance for Biological systems. It would be the greatest triumph of physics if BM turns out to be a good dynamical framework for describing protein-making processes. I believe one day equations of physics WOULD DO to biology what Schroginger equation DID to chemistry. Would those equations be Bohm's? Biophysicists should try their luck! I would be a very lucky man if my idea (speculation) turns out to be correct

regards

sam

I think a good analogy to biology would be to explain the homochirality of DNA and amino acids. One being left the other right handed. Seems a wonderfully odd way of working life into the equation.

 

[Demystifier]

In my previous posts, I tried to explain why I have doubts about your specific arguments (that are supposed to prove that BI and CI may have different predictions for relativistic quantum theory) as follows:
"I see the travel-back-in-time parts of trajectories somewhat differently (as an anti-particle moving ahead in time), and am not quite sure Kopenhagen and Bohm give different predictions for that case." "...Whether neutral particles can be called their own anti-particles, is not important, IMO. It is important, though, that creation and annihilation of pairs of neutral particles is possible, as far as I understand. So for time-space trajectories with travel-back-in-time sections, it is possible that at some point in time they describe three particles, not one. Therefore, acting on one of those particles, you cannot change the past for the other two particles, only the future."

I agree, this is also a logically possible interpretation. However, the predictions of such an interpretation would significantly differ from those of the conventional interpretation:
1. According to your interpretation, neutral particles (e.g. photon) would also have antiparticles that are NOT identical to particles, contradicting the conventional interpretation.
2. According to your interpretation, pair creation would be possible even without field interactions, contradicting the conventional interpretation. (This is directly testable.)

Further, your interpretation is mathematically less elegant than mine. Your interpretation requires a preferred time even for a 1-coordinate wave function. Mine does not.

In addition, I do not understand your last statement. What kind of "action on one of those particles" you have in mind? In a deterministic theory (and Bohmian mechanics is deterministic) an "action" (whatever that means in a deterministic theory) equally influences both the future and the past.

 

[akhmeteli]

I agree, this is also a logically possible interpretation. However, the predictions of such an interpretation would significantly differ from those of the conventional interpretation:
1. According to your interpretation, neutral particles (e.g. photon) would also have antiparticles that are NOT identical to particles, contradicting the conventional interpretation.

I did not say that antiparticles of neutral particles are not identical to particles. If you believe what I said implies that, could you explain how?

2. According to your interpretation, pair creation would be possible even without field interactions, contradicting the conventional interpretation. (This is directly testable.)

I am not saying that "pair creation is possible even without field interactions". However, there are always field interactions, so there is always pair creation. I am not sure this contradicts the conventional interpretation - such processes take place, say in QED, on the level of virtual particles - a "dressed" propagator includes loops. If, however, you have in mind a comparison with a one-particle quantum theory, then experimental differences between Bohm's interpretation and such a theory would not be very interesting, as long as the experimental results agree, say, with QED.

Further, your interpretation is mathematically less elegant than mine. Your interpretation requires a preferred time even for a 1-coordinate wave function. Mine does not..

I am not sure my interpretation requires a preferred time, not any more than special relativity. It is important whether one event is inside or on the future (past) light cone of the other event, or whether the events are spacelike. These three (or five:-) ) possibilities do not depend on preferred time.

In addition, I do not understand your last statement. What kind of "action on one of those particles" you have in mind? In a deterministic theory (and Bohmian mechanics is deterministic) an "action" (whatever that means in a deterministic theory) equally influences both the future and the past.

The "action" is some measurement that is performed in an experiment aiming to find differences between predictions in two different interpretations. As for influencing the past, I agree that Bohm's interpretation is nonlocal, furthermore, I wrote previously that you may be right stating that predictions of Bohm's interpretation for the relativistic case might be different from those of CI. However, as I said, I don't see convincing specific arguments in favor of this statement in your paper. I am not saying that you don't offer specific arguments, I am saying they fail to convince me. Specifically, I am not enthusuastic about "ignoring the dotted trajectories" in your paper.

Another thing. Many people believe that it is a drawback of the Bohmian interpretation that it (apparently) gives the same predictions as the standard quantum theory. It does not matter here whether this opinion is correct or wrong. However, proponents of BI who agree with this statement may be too eager to find some differences between the predictions. If they erroneously find such differences and experiments do not confirm their predictions, BI's status will suffer dramatically and unfairly. Again, I am not sure I am enthusiastic about BI, but if it is to be "kicked", better if it is kicked for its real faults, not imaginary ones.

 

[Demystifier]

I am not saying that "pair creation is possible even without field interactions".

Even if you are not saying it, from your interpretation it follows that pair creation without field interactions is possible. Therefore, the predictions resulting from your interpretation are necessarily different from those of the conventional information, without regard whether you say this or not. Of course, just as with my interpretation, in most practical cases these differences cannot be easily observed, so your interpretation does not seem to be in conflict with existing experiments. Nevertheless, your interpretation, just like mine, can, in principle, be distinguished from the conventional interpretation.

 

[Demystifier]

Another thing. Many people believe that it is a drawback of the Bohmian interpretation that it (apparently) gives the same predictions as the standard quantum theory. It does not matter here whether this opinion is correct or wrong. However, proponents of BI who agree with this statement may be too eager to find some differences between the predictions. If they erroneously find such differences and experiments do not confirm their predictions, BI's status will suffer dramatically and unfairly. Again, I am not sure I am enthusiastic about BI, but if it is to be "kicked", better if it is kicked for its real faults, not imaginary ones.

I understand your point. I am not saying that my version of relativistic Bohmian mechanics is the only possible one. Nevertheless, this version seems so simple and natural and elegant to me, that, if an experiment would rule out this particular version, I would no longer find the Bohmian approach so appealing and promising. But that's just me.

 

[Demystifier]

Specifically, I am not enthusuastic about "ignoring the dotted trajectories" in your paper.

But you must agree that measurement involves a (unitary) change of the wave function, so the trajectories must be different from those without the measurement. Are you familiar with the theory of quantum measurements in Bohmian mechanics? Do you understand how an effective wave function collapse takes place due to interaction with the measuring apparatus? If not, then you do not understand the essence of BM.

 

[Ken G]

Can you summarize that essence, with the intention of interesting me (or others) in it enough to go look it up and dig deeper?

 

[akhmeteli]

Even if you are not saying it, from your interpretation it follows that pair creation without field interactions is possible.

Not really, for the simple reason that, as I said, there are always field interactions, even if they are somewhat hidden in one-particle theories.

Therefore, the predictions resulting from your interpretation are necessarily different from those of the conventional information, without regard whether you say this or not.

Not really, because there can be no predictions of the conventional interpretation (I guess that's what you meant?) for the case of the absence of field interactions - again, for the simple reason that there are always field interactions.

... your interpretation, just like mine, can, in principle, be distinguished from the conventional interpretation.

It may be so, but I am not aware of any convincing specific arguments.

 

[akhmeteli]

But you must agree that measurement involves a (unitary) change of the wave function, so the trajectories must be different from those without the measurement.

Whether I must agree with that or not, I fail to see what this is supposed to prove. If you believe this proves that the predictions of BI and CI differ in the relativistic case, I just cannot imagine how your argument can prove that, as the argument is, on the face of it, equally correct or wrong both for the relativistic and the nonrelativistic case, and I believe you agree that in the nonrelativistic case the predictions of BI and CI coincide.

Are you familiar with the theory of quantum measurements in Bohmian mechanics? Do you understand how an effective wave function collapse takes place due to interaction with the measuring apparatus? If not, then you do not understand the essence of BM.

No offence, but my knowledge or lack thereof is irrelevant, unless you believe I wrote something terribly wrong because of my ignorance. If you do believe that, could you please indicate what it is? So far you just said that my interpretation, while also logical, predicts something different from what CI does (I don't exactly disagree, but neither do I see reasons to agree) and does not appeal to you (and this is a matter of opinion, not of me being ignorant or knowledgeable). Again, all I'm saying is your arguments fail to convince me, and I tried to explain why.

 

[Demystifier]

Can you summarize that essence, with the intention of interesting me (or others) in it enough to go look it up and dig deeper?

I have presented a short review in the Appendix of
http://xxx.lanl.gov/abs/quant-ph/0208185 [Found.Phys.Lett. 17 (2004) 363]
For more details see also the references cited at the beginning of this Appendix.

 

[Demystifier]

Whether I must agree with that or not, I fail to see what this is supposed to prove.

It is supposed to explain why the dotted trajectories are ignored, or better to say, why they are modified by the position measurement, in a manner that forbids them to join the dashed trajectories. As it is essential, I will not comment other points until we clear this up.

 

[Demystifier]

Not really, for the simple reason that, as I said, there are always field interactions, even if they are somewhat hidden in one-particle theories.

The predictions that depend on the value of the interaction constant cannot be the same as those that do not depend on it. I do not see how the interaction (except the mass and wavefunction renormalization that cannot explain particle creation) could be "hidden" in one-particle theories. Do you?

 

[Schrodinger's Dog]

http://arxiv.org/abs/quant-ph/0303156

S. Goldstein, working with D. Dürr, R. Tumulka, and physicist N. Zanghi of the University of Genoa in Italy.

http://arxiv.org/abs/0707.3487

W. Struyve and H. Westman.

2 recent papers on Bohm interpretation.

This may also be of interest.

http://arxiv.org/abs/hep-th/0610032

A. Valentini

Here's a brief excerpt from the article that set me to thinking about Bohmian mechanics. Like I said at the start I'm not convinced like many people but it did get me to looking into this model.

http://www.newscientist.com/article/mg19726485.700-quantum-randomness-may-not-be-random.html"

Quantum randomness may not be random

AT ITS deepest level, nature is random and unpredictable. That, most physicists would say, is the unavoidable lesson of quantum theory. Try to track the location of an electron and you'll find only a probability that it is here or there. Measure the spin of an atom and all you get is a 50:50 chance that it is up or down. Watch a photon hit a glass plate and it will either pass through or be reflected, but it's impossible to know which without measuring it.

Where does this randomness come from? Before quantum theory, physicists could believe in determinism, the idea of a world unfolding with precise mathematical certainty. Since then, however, the weird probabilistic behaviour of the quantum world has rudely intruded, and the mainstream view is that this uncertainty is a fundamental feature of everything from alpha particles to Z bosons. Indeed, most quantum researchers celebrate the notion that pure chance lies at the foundations of the universe.

Until now, that is. A series of recent papers show that the idea of a deterministic and objective universe is alive and kicking. At the very least, the notion that quantum theory put the nail in the coffin of determinism has been wildly overstated, says physicist Sheldon Goldstein of Rutgers University in New Jersey. He and a cadre of like-minded physicists have been pursuing an alternative quantum theory known as Bohmian mechanics, in which particles follow precise trajectories or paths through space and time, and the future is perfectly predictable from the past. "It's a reformulation of quantum theory that is not at all congenial to supposedly deep quantum philosophy," says Goldstein. "It's precise and objective - and deterministic."

If these researchers can convince their peers, most of whom remain sceptical, it would be a big step towards rebuilding the universe as Einstein wanted, one in which "God does not play dice". It could also trigger a search for evidence of physics beyond quantum theory, paving the way for a better and more intuitive theory of how the universe works. Nearly a century after the discovery of quantum weirdness, it seems determinism may be back.

 

[Demystifier]

Here's a brief excerpt from the article that set me to thinking about Bohmian mechanics. Like I said at the start I'm not convinced like may people but it did get me to looking into this model.

http://www.newscientist.com/article/mg19726485.700-quantum-randomness-may-not-be-random.html"

A larger part of this paper recently published in New Scientist is copied here:
http://www.groupsrv.com/science/post-2760759.html

 

[Schrodinger's Dog]

A larger part of this paper recently published in New Scientist is copied here:
http://www.groupsrv.com/science/post-2760759.html

Hehe that neatly avoids the legal complications by passing the buck. Thanks Demystifier.

 

[akhmeteli]

It is supposed to explain why the dotted trajectories are ignored, or better to say, why they are modified by the position measurement, in a manner that forbids them to join the dashed trajectories. As it is essential, I will not comment other points until we clear this up.

Now that you mentioned the dashed trajectories, I understand that I mixed up the dotted and the dashed trajectories, saying that "I am not enthusuastic about "ignoring the dotted trajectories" in your paper." I should have said that "I am not enthusuastic about the phrase "The dashed one is unphysical because it is assumed that only one particle exists" in your paper." I do apologize for this mix-up. However, this does not mean that now I find the arguments convincing. Specifically, I am not ready to accept the assumption that one-particle relativistic quantum equations, such as the Klein-Gordon equation, describe just one particle (however natural and even tautological that assumption may look, on the face of it), as I believe that travel-back-in-time trajectories describe pair creation.
Also, I am not sure that it is important whether the dotted lines join the dashed lines after measurement.

 

[akhmeteli]

The predictions that depend on the value of the interaction constant cannot be the same as those that do not depend on it. I do not see how the interaction (except the mass and wavefunction renormalization that cannot explain particle creation) could be "hidden" in one-particle theories. Do you?

Actually, I do. The mass and wavefunction renormalization that you mention are a direct result of particle creation. Maybe they cannot "explain" particle creation, but for the same reason that an effect cannot "explain" its own cause. Anyway, particle creation does take place in one-particle theories, if only through wave function/mass renormalization. On the other hand, a particle mass is created (whether partially or totally, is not important here), by the particle's field, and thus depends "on the value of the interaction constant". So there are always field interactions, whether explicit or implicit. And again, if an experiment yields a result incompatible with the predictions of, say, the Dirac equation, but compatible with those of QED, that will not be very exciting.

 

[Demystifier]

1. Specifically, I am not ready to accept the assumption that one-particle relativistic quantum equations, such as the Klein-Gordon equation, describe just one particle (however natural and even tautological that assumption may look, on the face of it), as I believe that travel-back-in-time trajectories describe pair creation.

2. Also, I am not sure that it is important whether the dotted lines join the dashed lines after measurement.

1. In my view, these are merely two ways to say the same thing. As long as there is only ONE CONTINUOUS CURVE in spacetime, it does not matter whether we call it one particle that can move backwards in time, or many particles that can be created and destructed.

2. Well, if they don't join, then you have a problem.
Assume first that they do. Then, even with your many-particle interpretation, the positions of all these particles are determined by ONLY ONE initial particle position. This differs from the n-particle state in the usual sense, which (in the Bohmian interpretation) requires n initial particle positions.
Now assume that they don't. If these two parts of the curve are not joined, then how do you know that they actually belong to the same curve? If you simply say that they do not belong to the same curve, which indeed is in the spirit of your many-particle interpretation, then how do you know that one particle must be accompanied with another one?

I hope you see that the arguments in 2. show that a single-particle interpretation has certain advantages (even if you are still not completely convinced).

 

[Demystifier]

The mass and wavefunction renormalization that you mention are a direct result of particle creation.

In a sense you are right, but the theoretical origin of this "particle creation" differs significantly from "particle creation" in relativistic Bohmian mechanics. The former is related to virtual particles that make sense only in the perturbative method of calculation based on Feynman diagrams. The latter makes sense only in the Bohmian interpretation, irrespective of the method of the calculation. If two things look similar (though not identical), but have different theoretical origins, then it does not seem very likely that they are actually the same.

 

[Demystifier]

And again, if an experiment yields a result incompatible with the predictions of, say, the Dirac equation, but compatible with those of QED, that will not be very exciting.

As far as we know, all effects of QED on the single-particle Dirac equation is a renormalization of the parameters of the Dirac equation. An example is a correction of g, which for free Dirac equation is g=2. The QED correction of this value is one of the most remarkable triumphs of QED.

 

[Demystifier]

Akhmetely, one additional remark. At the beginning, I was also hoping, using similar arguments that you do, that motions backwards in time could be related to genuine particle creation. Nevertheless, by using arguments that I presented above, I have concluded that it was not possible (or at least very unlikely).

However, a truly amazing result was when I recently realized that the Bohmian motions backwards in time ARE related to genuine particle creation - in string theory.
See my preprints arXiv:hep-th/0702060 and arXiv:0705.3542 (I am not giving the direct links because ZZ does not allow to do that for papers that are not yet published). See in particular Fig. 1 in the first paper that summarizes various views of particle creation.

 

[akhmeteli]

As far as we know, all effects of QED on the single-particle Dirac equation is a renormalization of the parameters of the Dirac equation.

I guess you mean the free Dirac equation? Because the Lamb shift does not look like a renormalization of parameters of the Dirac equation.

An example is a correction of g, which for free Dirac equation is g=2.

I am not sure g is a parameter of the free Dirac equation. So what parameter is renormalized in this case?

 

[Demystifier]

Akhmetely, see also the most convincing argument that motions backwards in time of particles cannot be sufficient to explain particle creation. It certainly cannot explain the creation of new kinds of particles, that is, kinds of particles that were not present in the initial state. For example, there are no particle trajectories of electrons that could explain the creation of photons.
Note also that this problem is elegantly avoided in string theory, because in string theory different kinds of particles are nothing but different states of the same object - the string. Therefore, a string can continuously transit from an electron to a photon. A particle cannot do that. See also Fig. 1 mentioned in my previous post.

 

[Demystifier]

I guess you mean the free Dirac equation? Because the Lamb shift does not look like a renormalization of parameters of the Dirac equation.

I am not sure g is a parameter of the free Dirac equation. So what parameter is renormalized in this case?

You are right: strictly speaking g has an operational physical meaning only when the interaction with the magnetic field is also present. Nevertheless, when the magnetic field is treated classically, g (or more precisely the magnetic moment mu) can be viewed as a parameter of the Dirac equation in a classical magnetic field, but a parameter that can be expressed in terms of other parameters (mass and charge). But this is not really important for our main discussion, is it?

 

[akhmeteli]

You are right: strictly speaking g has an operational physical meaning only when the interaction with the magnetic field is also present. Nevertheless, when the magnetic field is treated classically, g (or more precisely the magnetic moment mu) can be viewed as a parameter of the Dirac equation in a classical magnetic field, but a parameter that can be expressed in terms of other parameters (mass and charge). But this is not really important for our main discussion, is it?

I fully agree, it is not important for the discussion.

I cannot reply to your other post(s) now, I'll try to do it in the evening (Pacific time). Take care.

 

[Demystifier]

I cannot reply to your other post(s) now, I'll try to do it in the evening (Pacific time). Take care.

I am looking forward!