Thanks m_homm for that excellent description and interesting analogies of the Uncertainty Principle. Without all the math, this is understandable to a layman to physics like myself.
I always thought until this point that the Uncertainty Principle was a "cop-out" in finding the position and momentum of a particle. But your description make it clear.
Looking forward to your second installment.

Here is the second installment. Sorry for being slow, but I've had less time to work on this than I expected.

There is more to say about the uncertainty principle, but first I must talk a bit about the Copenhagen interpretation. Unfortunately, this requires certain mathematical concepts to be crystal clear, so it will be necessary to set up some of these ideas before the main discussion can go forward.

I mentioned earlier that position and momentum are just the two classical aspects of a single piece of physical information, and that this piece of information had many other aspects as well, just as the planet Venus does other things when it is not being the morning or evening star. This is the central idea behind my position on the Copenhagen interpretation, and is also the other radical reinterpretation of physics to which I alluded earlier; it is the road Bohr and Heisenberg did not take.

Fundamentally, it comes down to taking "state space" literally, and thinking like a mathematician instead of like a physicist. To see what I mean by that, let's start with state space. Part of the basic mathematical formalism of quantum mechanics is that the state of a physical system can be described by a vector in an abstract space called a Hilbert space. This is basically just a vector space like ordinary three dimensional space, but may have infinitely many dimensions.

(The fact that it may be infinite dimensional shouldn't be a conceptual roadblock, because the "Hilbert basis theorem" shows that it acts pretty much like a finite dimensional space. You can set up a coordinate system on it and express vectors as series of numbers, just as you can express ordinary 3 dimensional vectors in terms of x, y, and z coordinates. There are just a lot more of them. Just about everything you would want to understand about Hilbert space, you can visualize by thinking about how it would look in 3 dimensions. Also, this is an abstract space, a mathematical construct, so it does not mean that the physical space of the universe is really infinite dimensional.)

To state my first mathematical point most concretely, let's look at a 2 dimensional map, for instance a map of Italy. I can draw a coordinate system on this map, with the origin at Rome, and the x axis pointing due east, and the y axis pointing due north. Then I can give the coordinates of another city, say Naples, as so many km E and so many km S. By giving you these numbers, have I given you the vector that points from Rome to Naples? NO. I have given you a description of the vector in this coordinate system. If I were to choose a different set of axes, still through Rome, but now with the x axis pointing due northeast and the y axis pointing due northwest, I could also give you coordinates for Naples, this time, so many km NE and so many km NW. This would also be a description of the same vector, but in a new coordinate system.

The idea here is that the vector is something real, and the coordinate system is something artificial which we add to the map in order to give coordinates to the vector. Both physicists and mathematicians are aware of this, of course, but physicists (at least in my experience) tend to have a bias in favor of working with coordinates and mathematicians have a bias in favor of working without them.

Why this should be so is another interesting philosophical sidelight to the discussion. "Noumenon" is the word philosophers use to describe the actual reality that lies behind our perceptions, and "phenomenon" describes the observable expression of the underlying noumenon. (For the genetically inclined, you may think of this as similar to the relation of genotype to phenotype.) Physicists have no access to the noumenon, because their theories all must eventually involve measurement of some kind, and these are part of the phenomenon, the appearance of reality. By building ever subtler theories, they hope to trap the noumenon in the web of all its phenomena. Mathematicians, on the other hand, create mathematical constructs by setting up postulates, and then deriving the consequences of those postulates. The postulates are by definition the ultimate reality (noumenon) of the mathematical structures, and the theorems are the phenomena. Mathematicians proceed from noumenon to phenomenon, and physicists (attempt to) proceed the other way.

This means that, to a mathematician, the underlying truth of a vector space comes from its postulates, and the vectors themselves are therefore real things (within the mathematical theory) that exist before a coordinate system is applied to the vector space. In contrast, to a physicist, the coordinate dependent measurements are prior, and from them the coordinate-free reality must be inferred. You can see this in how the theory of tensors is constructed in physical applications: the emphasis is all on how the changes of coordinates must work so that the physical meaning of the quantity will be the same in the new coordinate system as in the old one. Then anything whose appearance changes in the proper way when you change coordinates must be something with an invariant physical meaning; the stable reality is inferred from the way that the various appearances change. Mathematicians, on the other hand, define tensors by certain postulates about how they behave, and then later, when a coordinate system is applied to them, they can say that their abstract tensor looks one way in one coordinate system and another way in a different coordinate system, based on the postulates. They construct the coordinate change properties from the underlying reality instead of constructing the underlying reality from the coordinate change properties.

(By the way, this is all from personal experience. I majored in both Mathematics and Physics separately as an undergraduate. In fact, I simply told each department that I was majoring in their subject but never informed them that I was also majoring in the other one. Apparently I wasn't supposed to do this, because when I graduated, I got two separate diplomas and a rather petulant letter from a university bureaucrat telling me that I wasn't allowed to have done that, but it was now too late to stop me. Anyway, this is the difference in emphasis between the mathematicians' approach and the physicists' approach, which I observed by doing the two majors side-by-side. I'm not saying that all mathematicians and all physicists think in these ways, just that there is a noticable degree of bias.)

Now, what does this have to do with the Copenhagen interpretation? My position is as follows: Both the Copenhagen interpretation and the uncertainty principle arise from the same interpretive error, and that error is our failure to recognize at a gut level that our state vector coordinate systems are not part of reality. The preference for coordinates over abstract vectors biases us towards making this mistake, and both our education in classical physics and our everyday experience reinforce that bias so strongly that it can be difficult to shake off the conviction that it is right. But it is a mistake nonetheless.

In detail then, what is this mistake? Let's first look at this mistake in the context of a very simple system, such as the map of Italy, then look at how a very similar, but subtler mistake was recognized and corrected by special relativity, and finally, proceed to the case of the Copenhagen interpretation, which fails to correct the mistake.

Suppose I show you my map of Italy upside down. Even with no text printed on it, you immediately know it is upside down, because it looks weird that way. But there is no inherent right way to map Italy. The peninsula itself does not come with a "this end up" arrow printed on the ground. The sense of weirdness comes entirely from our familiarity with seeing things one way and our unfamiliarity with seeing them any other way. We are disoriented (literally: the word "disoriented" originally meant that you didn't know which way was east) and it is hard to resist the urge to turn it the "right" way up in order to "see it better." The point here is that culturally based expectations can generate a quite strong sense that one of two ways is "right" or "real" and the other way is "wrong" or "fake." How much stronger than that, then, is the conditioning that we absorb from our entire education and our entire experience of the everyday world? It can be very difficult to force your mind to accept a new perspective.

When I turn the map right way up, Venice is directly above Rome, and Naples is 45 degrees down and to the right from Rome. Let's put a coordinate system on this map with the "y" axis pointing north and the "x" axis east. You could say that the vector from Rome to Venice was "pure" y, but the vector to Naples was a "superposition" of -y and x. Now suppose I turn the map 45 degrees counterclockwise, so that northeast is at the top, and I draw a new coordinate system on the map with the "y" line straight up the map (i.e. due northeast) and the "x" directly horizontal (due southeast). On this coordinate system, Venice does not appear to be directly above Rome, but somewhat above and to the left of it. In this coordinate system, you could say that the vector to Venice was now a mixture of -x and y, while the vector to Naples was now pure x.

What is this map discussion telling us? Three things: conditioned bias can give you a compelling but false feeling that one way is right, coordinate systems can be changed arbitrarily without affecting reality, and most importantly, the concepts of "pure" directions (due North, for instance) and "superposed" directions are an artifact of the coordinate system. The first point is obvious from experience, and the second point is just Alfred Korzybski's famous saying that "the map is not the territory." The third point is the one to keep in mind: the idea that some vectors are pure and others are superpositions is false. These concepts do not apply to the vectors themselves (the noumenon) but to their expressions in particular coordinate systems (the phenomenon). In fact, for any given vector, I can always just draw a coordinate system with one axis along that vector and the other axis perpendicular to it, to make the vector pure in that coordinate system. For every vector, there is a coordinate system in which it is pure (i.e. has only one nonzero component).

This discussion of the map may seem trivial, but that is intentional. I want to show what the error is in a simple case where it is extremely obvious that it is an error. For the map, it is completely clear that changing the coordinate system will not actually move the city of Naples! Who would think otherwise? The separation between the map and the reality, and between the vectors themselves and their coordinate system representations should also be completely clear in the map example. Finally, it should also be crystal clear that the the property of a vector being "pure" is not a feature of reality. Now we can turn to the next case, special relativity, in which all these same statements are widely acknowledged to be true, but are much less obvious than for the map.

In special relativity, there are various puzzling effects, such as mass increase, time dilation, length contraction, and shifts in simultaneity. These are all aspects of the general Lorentz transformation, which is basically a coordinate transformation. What was new in special relativity was that the coordinate system encompassed both space and time. Instead of viewing the world as having three spatial dimensions and a separate time dimension, the space and time dimensions were treated as parts of a single whole called "spacetime." This is a profound difference, in that it allows for the direction of time to change. Out of the entire 4 dimensional spacetime, one direction is singled out to be time, and three other directions are chosen to form the three space axes, x, y, and z. When an observer begins to move, the direction of time is altered for that observer, so that his time axis is now a different line in spacetime, and the three dimensional slice of spacetime that looks like space to him also changes.

There are many similarities to the map of Italy example here, and also some important differences. The direction of the time axis may change in spacetime, just as the direction of the y axis may change on the map. Also, just as on the map, but even more so, we have a strong feeling that one of these axis choices is "more right" than the other. For instance, suppose observer A stands on the shore, while observer B on a ship goes 100km north in one day. Imagine (if you can) drawing a line through the 4 dimensional spacetime from observer A today to observer B tomorrow. From A's point of view, there is a change of both position and time, so this vector would appear as a mixture of space and time displacements. From B's point of view, with his space coordinate system attached to the ship, he has stayed at the origin of his coordinate system the whole time, but aged one day, and so to him the same spacetime vector from A today to B tomorrow would appear to be purely a change of time. This vector is B's time axis. Clearly, A and B disagree on whether the same physical vector is or is not pure time. Since the vector cannot really be different from itself, the disagreement can only be the result of using different coordinate systems. This means that the concept of being pure time cannot be attached to the vector itself, but only to its expression in a particular coordinate system. The direction of time is therefore not a feature of reality.

However, it is hard to shake off the feeling that observer A is "really" right about the direction of time, just as it is hard to get over wanting to turn the map so that north is straight up. In fact, it is much harder with relativity than with the map because we experience time so very differently from space that it is quite a stretch of the imagination to see how they could be aspects of one and the same thing. This is a point where special relativity differs from the map of Italy: there is no inherent difference between the x and y axis directions. You could walk in the x direction or in the y direction just by choosing to do so. You cannot walk in the time direction, so time really is essentially different from space. Also, for the map you can choose the directions of x and y at will, but in relativity you cannot choose the directions of time and space. They appear to be chosen for you in some way. The choice varies as your velocity varies, but you have no direct control over it. By changing your velocity, you can cause a certain change of coordinates specified by the Lorentz transformation, but that is as much control as you have. The combination of lack of ability to choose the coordinates for yourself, coupled with the very different perceptions we have of time and space, make it much more difficult to accept that they are parts of the same whole.

In special relativity we can see that there is a real difference between time and space, in the sense that one of the 4 coordinates behaves quite differently from the other three. However, which exact direction in spacetime is the time axis depends on who is looking and how they are moving. This reminds me of the old latin saying, mors certa sed hora in certa (death is certain, only the hour is uncertain). The vector from A today to B tomorrow is prefectly certain, and the distinction between time and space is also perfectly certain, but exactly how much of that vector is space and how much is time is uncertain, because it varies with the observer's motion. It should be clear now how this is like the coordinate system on the map of Italy: there is a strong bias towards looking at things in the usual way and regarding that as the "right" way; the concept of a vector being "pure" is misapplied, because it applies to the coordinate system choice, not to the vector itself; and finally, all of the confusion in both cases stems from mistaking the coordinate system for a part of reality, which it is not. Arguments about whether the vector from A today to B tomorrow is "really" pure time or not are just as silly as arguments about whether the direction from Rome to Naples is "really" pure x or not, and for the exact same reason.

Classical physics assumed that time and space were completely independent of each other, which is the same as saying that the choice of time axis is fixed and can never vary. This is a clear case of confusing the coordinate system with reality. Special relativity cures this problem by correctly separating the coordinate system from the underlying spacetime, which therefore becomes a single 4 dimensional whole. This is not really any harder to understand than that tilting the map of Italy does not tilt the real Italy, but it is harder to accept because all of our experience is at such slow velocities that we are living within the same coordinate system all our lives. Since it is a permanent feature of our experience, it is very easy to mistake it for a permanent feature of the universe. The truth is that space is merely my three dimensional slice of spacetime that looks like space to me, and time is merely my line through spacetime that looks like time to me. One of the best things ever to happen to special relativity was the coining of the word "spacetime," because it forces space and time together into one thing on a verbal level, which helps anchor the somewhat slippery ideas of relativity to a nice graspable noun. Never underestimate the power of language: if you give something a name, people will immediately start to treat it as real. The word "spacetime" itself makes it easier to think about special relativity. If quantum mechanics had done the same thing with position and momentum, we would not now have either the uncertainty principle or the Copenhagen interpretation.

I'm afraid I must stop here for now, so that I can post this. Sorry it is so long, and sorry also that I still haven't got to the Copenhagen interpretation itself, but I felt is was important to make the argument clearly and in detail. You can probably already see where the trend is leading. I promise to finish the next installment ASAP.

More later.

--Stuart Anderson

@ALL
I'm going to state some opinions with little scientific basis.

I have a problem with Copenhagen. I have a problem with long term superposition. I think Schrodinger's cat died, or didn't, long before anyone looked.

I have this notion that quantum effects are limited within regions of space-time. Kind of a quantum fuzziness. Think of fuzziness moving through time, but continually resolving itself. It's always fuzzy at the moment, but leaves a trail of "whatever happened". Everything at the quantum level is a mess of interacting probabilities ever changing in spacetime. The end effect is the illusion of stability, continuity, and consistency. Most of the time (on average?) the most probable things happen, so here we are in a stable reality.

Roger Penrose says some interesting things:
"Penrose hypothesised that the transition between macroscopic and quantum begins on the scale of dust particles, which could exist in more than one location for as long as one second "

@mr_homm

By this did you mean that the forward direction of time is not a feature of reality? There is forward and backward, and they are different. We experience time in a forward "cause and effect" sense probably due to our physical dependence on the Second law of thermodynamics.

Even if you buy into Cramer and Transactional interpretations, forward and backwards are different.

Looking forward to more from you.

@meBigGuy:

I meant only that the direction of the time axis through spacetime can be skewed. So in that sense, the direction of time is changed, similarly to taking the y axis in my map example and turning it a few degrees away from notrh. What I said is literally true, because there are many different directions through spacetime that can be the time axis for different observers, but along each axis time does flow forward, never backward.

I think the miscommunication arose because I forgot and used the technical meaning of "direction" in a general interest discussion. The technical usage has the word "sense" for describing "forward or backward along a line" and "direction" for the orientation of the whole line in space. But everyday language uses the word "direction" for both of these ideas, and so the meaning was not clear. More formally, I should say that the Lorentz transformation varies the direction of the time axis, but cannot reverse the sense of time flow along that axis.

(Actually, if the relative speed is > c, then the sense of time flow for one observer would reverse relative to another observer. This is usually taken as one of the proofs that faster than light travel is impossible, since it would violate causality.)

About Cramer: I know this guy, very tangentially. He's a physics professor at the same university I work at, and many years ago when I was a student, I was the grader for his daughter Katherine's physics homework. She didn't go into physics herself, but into publishing, At the last time I heard, which was when I ran into her at a science fiction convention a few years back, she is one of the more important people in the science fiction publishing business. Her father is a science fiction writer as well as a physicist, so I suppose it runs in the family. I never had a class from him directly, but I've talked to him. He's a very interesting guy, full of interesting ideas, and he has the most distinctive laugh I've ever heard, which starts off like a loud donkey-bray, and then always ends in three rapid snorts. His transactional interpretation ideas are very interesting, but I haven't looked at them in a while, so they've gone a bit fuzzy in my memory. I don't recall them allowing for time reversal though, but I could be wrong.

As to Penrose, his idea (well, he has a LOT of ideas, but this particular one) is that any kind of interaction with a particle will collapse the wave function, and there are particles all over the place, so all the wave functions are collapsing all the time, which keeps the universe looking classical. So for him, the fate of Schrodinger's cat is already decided, since the molecules of the box itself function as an observer by interacting with the cat, collapsing the wave function immediately.

My take on it (see next installment, whenever I manage to finish it) will be a bit different, because in my opinion there was never any quantum fuzziness in the first place, nor any wave function collapse. These ideas are both artifacts of our clinging to a classical "coordinate system" in our description of the particles.

Thanks for an interesting discussion so far!

--Stuart Anderson

Trying to follow the interesting discussion above, and note that Physorg.com recently posted an article "The Worlds lowest noise laser: Researchers outsmart quantum physics".

In trying to go beyond the article, I found this link:

http://www.aei.mpg.de/english/research/tea...Proj/index.html

The Index lists a number of projects worldwide that are attempting to measure gravity waves.

The Max Planck Institute for Gravitational Physics, is one of the participating members of the GEO600 ground based detector, with arms some 600 meters long. The article in Physorg shows a picture of a model that uses red laser and green laser said to get an improvement of three times.

It appears to my thinking that the authors make the claim that by distributing within the crystal that is excited by a red lase, green laser energy, the normally random "noise" is reduced since it is, maybe separated and and more concentrated.

Have not found any supporting essays. But since some String Theories I've read assign the EM forces as a transverse to the strings and Gravity forces longitudinal, I wonder if there is information as to how they would aim their device which is 600 meters long.

Is this another Michelson Morely Experiment that did not detect a difference for the speed of light?

Maybe an effort to get accurate 100 Hertz per second gravity waves at very low amplitudes?

Best, rm

http://front.math.ucdavis.edu/author/R.Schnabel

Try this link to the author of the above GEO600, I just downloaded his and two other author's papers, it is 16 pages long, has many detail drawing.

Best, rm

Hi mr_homm,