Krauss misunderstands energy in general relativity to argue against God

Stephen Hawking has a new book out that claims God did not create the universe (this is probably not news to you). Cosmologist Lawrence Krauss writing in the Wall Street Journal says that Hawking does not go far enough, but his argument is based on a misunderstanding of energy in general relativity. Since we have discussed this recently here I’ll explain why.

[ Before I do, and just for the record, I am an atheists too, but I think science can only say that a God is not necessary. If people still want to believe in one (or more) I don’t claim to be able to argue against them, after all, my philosophy of existence may not be very scientific either. If people make claims which are at odds with science, such as denying evolution, that is a different matter… Feel free to discuss ]

Coming back to the article by Krauss, what he says is that the energy in the universe is zero only when the universe is flat (he means the 3d part of the metric for space is Euclidean, the full 4d metric has curvature). He then says that “Observations of the cosmic microwave background from the Big Bang have unambiguously confirmed that we live in a precisely flat universe.” A more accurate statement would be that the observation confirms that the part of the universe we can observe is flat to within observational errors, but let’s not be picky.

The thrust of his argument then is that the energy of the universe did not have to be zero, but observation confirms that it is. If it was not zero you would then need something to create it, e.g. a God. Since we observe it to be zero we therefore have an observational confirmation that God is not needed. He did not put it in those terms, but that is essentially the point he is trying to make. Feel free to dispute the argument already at this level.

I am going to overlook the weakness of this argument and instead dispute the claim that the energy of the universe is zero only when it is flat. It is a very strange statement from a cosmologists because it is more commonly said that the total energy of the universe is zero for a closed universe, i.e. for positive curvature, not flat space. I agree with this since in a closed universe you can integrate the energy current density over all space and it must be zero because it has zero divergence.

In an open universe the case is not so clear since the volume of space to sum up the energy over is infinite. However, for a perfectly homogeneous and isotropic universe the energy current must itself be zero because the only vector field that respects the symmetry and which has zero divergence is the one that is zero everywhere. In the real universe we can only say that the energy current density is zero on average if the universe is homogeneous on average. In other words we can make a good case that no energy is needed to create the universe whether it is closed, flat or open with negative curvature. So actually Krauss’s argument fails even before you worry about its overall lack of logic.

So how did Krauss get the idea that the energy of the universe is zero only if space is flat? To answer this I’ll start with the equation I gave for energy in cosmology which was

E  =  M c^2 + \frac{\Gamma}{a}  + \frac{\Lambda c^2}{\kappa} a^3  – \frac{3}{\kappa}\dot{a}^2 a = 0

This only covered the case of flat space, but it is easily generalised to curved space like this

E  =  M c^2 + \frac{\Gamma}{a}  + \frac{\Lambda c^2}{\kappa} a^3  – \frac{3}{\kappa}\dot{a}^2 a  – Ka = 0

 Where K is a constant that is zero for flat open space, positive for spherical closed space, and negative for hyperbolic open space. To simplify I’ll ignore the radiation terms and dark energy but you can put them back in if you want.

M c^2  – \frac{3}{\kappa}\dot{a}^2 a – Ka = 0

It is important to remember that the first term is the energy contribution from cold matter in a fixed expanding volume while the other two terms are the negative energy contribution from the gravitational terms. However, people who don’t know the origins of these terms sometimes turn the equation round to look like this

-\frac{\kappa M c^2}{a}  + 3\dot{a}^2  = -\kappa K

This now looks like the Newtonian equation for a particle of fixed mass trying to escape the gravitational pull of an object of mass M with -\kappa K being the total energy. This is the interpretation that Krauss is using. He then says that the energy is zero only in the case K = 0 which is the flat universe. But notice how wrong this interpretation is. Starting from the true energy equation we had to reverse the sign and divide by a. Then the term from the energy of cold matter looked like the Newtonian term for gravitational energy while part of the term for gravitational energy looked like the term for the kinetic energy of a particle and the other part became a constant total energy. Everything was swapped over so the interpretation could not be more wrong!

Apparently Krauss is about to publish his own book based on this false interpretation of the energy equation. I have enjoyed a few of his previous books that I read so I hope this one has some saving grace. At least he is not saying that energy is not conserved in general relativity!


49 Responses to Krauss misunderstands energy in general relativity to argue against God

  1. Ulla says:

    Problems with infinities again. Infinite energies and open universe? Maybe a solution is in the 2-D surfaces linked together into 3-D? The transparency makes us see only the 3-D world?

    I want to make a somewhat off topic comment. It is about intelligent design. I feel it is a big problem that some scientists have a firm believe in some kinds of ‘truths’ and tries to model the universe accordingly. They should always state their beliefe-systems.

    The same can be said about every scientist, maybe. But especially intelligent design is troublesome, because it is implicit.

    Search “Krauss intelligent design” to see what I mean.

  2. krauss says:

    my point was simply that the Newtonian Local total gravitational energy of any object in a flat universe is zero. This is correct, I believe. You are correct that indeed talking about global energy on scales larger than the horizon one must be more subtle… The details could not be discussed in a 700 word oped. As far as your interpretation of the rest of my argrument, I won’t comment.

    • Philip Gibbs says:

      Thank you for taking the trouble to respond to this. I agree that many details may need to be simplified in a short article written for a general audience and that some of the subtleties I mentioned can be glossed over. However the sentence I am disputing is this one which is key to the point you are making:

      “If the positive energy and the negative gravitational energy of the universe cancel out, we end up in a flat universe.”

      I accept that the energy of a flat universe is zero given a reasonable interpretation of what this means. However, you are using the converse of this statement. You are saying that if the energy is zero then it is flat. I disagree because it is in fact zero for closed universes with positive curvature too.

      • Lawrence B. Crowell says:

        The energy for k = 1 is positive. I should have written the so called energy equation as

        (a’/a)^2 = 8πGρ/3 – k/a^2

        with a square on the last term. Here the prime meant “dot” or time derivative. This equation can be derived using Newton’s laws, or the energy of a projectile moving in a gravity field

        1/2mv^2 – GMm/a = E,

        where E is the total energy. We can equate v = da/dt = a’ and that the total mass acting on this particle is contained in a region with volume V = 4πa^3/3 with some homogenous distribution of matter ρ. So we work that out and get

        (1/2)a’^2 – 4πGρa^2/3 = E/m.

        So what this says is that if we are at the center a = 0, then a galaxy at some distance given by the scale factor a is influenced by all the mass energy inside this volume V = 4πa^3/3. Anything outside of there is by Gauss’ law has no gravitational influence (in EM the Faraday box logic). A little bit of algebra then tells us that

        (a’/a)^2 – 8πGρ/3 = E/ma^2.

        The term E/ma^2 is the constant divided by the scale factor squared and this is the -k/a^2. Again if the energy is negative, this is the same as a suborbital rocket flight or a cannon ball trajectory. For the energy positive we have this is an escape velocity condition, and the projective reaches “infinity” with some finite velocity. If that finite velocity is zero that is the minimal limit for escape velocity. Now the argument is set up so that the k corresponds to the k in the FLRW metric, and for k = 1 this is the suborbital flight, and the scale factor reaches some maximum and then recontracts. The space is a sphere in this case with positive curvature, For k = -1 this is the strange saddle shaped universe which is not finite and the scale factor increases in an unbounded manner. For k = 0 this is the flat universe with zero energy.

        Now the solutions are determined by the nature of ρ, which has different properties for radiation and matter, and for a constant vacuum density. In the latter case we get an exponential solution to the differential equation, which is easily seen, and so the solution is de Sitter for large t. Yet the whole thing was set up to have zero energy. So how does that happen? It is the equation of state with pressure p = -ρ, and the work density done by this turns out to be negative and equal in magnitude to the vacuum energy density.

  3. Lawrence B. Crowell says:

    The energy equation, which can be argued for by Newton’s law as it works out to give the GR result, is

    (a’/a)^2 = 8πGρ/3 – k/a,

    for a the scale factor and k = 1, 0 or -1. For k = 0 we have the flat space which keeps the energy density constant if there is just a vacuum density. Further, as the space is flat we have no problem with there being a creation of energy with the exponential growth of a, for the space is already infinite. The other things which is interesting about this is that the equation of state is w = -1, which gives a pressure p = wρ = -ρ. The energy contribution from this, thinking according to the work energy theorem is just dE = p*da, and if we do a bit of work this turns out to be a negative work density. This negative work density is equal and opposite to the energy density of the vacuum and the total is zero! So the total universe is a big fat nothing or sum(all) = 0.

  4. The best proof that the notion of energy is not well-defined in General Relativity is that there are so many different claims about what the energy of Universe is!

    I have described my own solution of the problem in earlier discussion and discussed also a manner to define non-conserved classical and quantal Poincare charges in asymptotically flat space-time in General Relativity. It is based on the algebra of volume preserving currents with the property that the flow line coordinate extends to a global coordinate. These Beltrami flows have interpretation in terms of local polarization vector and massless momentum vector so that a direct connection with massless fields emerges.

    From what I have understood from a discussion in Lubos Motl’s blog I understand that Hawking’s view about God is badly in need of updating. It is essentially the God allowed by classical deterministic physics. God dictated the initial conditions of Big Bang and lost interest on the Universe after that. This because Godly intervention would break the laws of classical physics. In quantum measurement theory we encounter the same problem: quantum measurement apparently breaks the determinism of Schroedinger equation. Now we cannot however claim that state function collapse or something equivalent with it does not occur. The irrational manner to get rid of the problem is to say that there is no objective reality at all.

    In TGD inspired theory of consciousness can be seen as a generalization of quantum measurement theory in order to overcome this difficulty. It leads to a quantal view about divine as ability to recreate the whole 4-D Universe (or more precisely, their quantum superposition) again and again. This allows to understand biological evolution as something genuine and generalize the concept of evolution. Zero energy ontology means that physical states correspond to pairs of positive and negative energy states so that symmetries and conservation laws do not restrict the free will of quantum jump. Every physical state is in principle reachable from a given physical state by quantum jumps. Free will is completely consistent with the determinism of the laws of classical physics since the free will of quantum jump is outside the space-time and Hilbert space: entire time evolution of Schroedinger equation is replaced with a new one. Consistency with physics does not anymore exclude divine.

    Accepting this view means also a new view about relationship between experienced time and geometric time. They are not one and same thing as should be clear already from the fact that subjective time is irreversible and geometric time reversible. Their identification can however make sense approximately and locally applying to one particular system from which the contents of consciousness of one particular conscious entity is about. Everywhere in 8-D Universe there are space-time sheets about which a contents of sensory consciousness of a particular conscious entity comes from.

    In this framework there is no sense in asserting that consciousness is a kind of 3-D time=constant slice moving towards geometric future. The time slice idea is also in conflict with General Coordinate Invariance since a special time coordinate would be relevant for consciousness. And our conscious experience is not about time=constant snapshot. We have memories- even sensory ones- and the experiments of Libet demonstrated that our volitional act induces neural activity in the geometric past. The contents of our conscious experience is about 4-D space-time region, and the challenge is to understand why our sensory experience is localized to about .1 second wide interval of geometric time in the usual wake up state of consciousness.

    For these reasons I do not find the classical physics view about God selecting initial conditions very interesting. Hawking should find himself more demanding challenges than killing for all practical purposes already dead God of classical mechanics;-)!

    • Lawrence B. Crowell says:

      Matti,

      The continuity equation is a statement on the covariant constancy of the momentum-energy tensor, which is equivalent to saying it is frame dragged along geodesic lines of flow. The problem with energy conservation in a general setting is with projecting out energy by some basis element. The result is then frame dependent and there is not straight forwards way of getting energy conservation. This is why there are these gadgets called pseudo-tensors, which are really sort of local or frame dependent calculational tools.

      I am not sure that consciousness has much to do with this. When it comes to Hawking’s recent statements about religion and the role of God, or absence thereof, it can be said that we are at an age where we can study the universe including its origin without reference to anything outside of physical existence — which includes other “universes.” This does not exactly constitute a proof that God does not exist, or for that matter strong evidence for such nonexistence. It does however, indicate a growing intellectual knowledge base with is eroding away any need for supernatural powers or a God. Quantum mechanics requires initial states just as much as classical mechanics. The no-boundary proposal by Hawking and Hartle was an evaluation of the quantum functionals of the FLRW and de Sitter metrics which illustrated that boundary conditions could be replaced by topologial requirements of fields.

      • Ulla says:

        My opinion is that it is equally right to say there is no God-force, as to say there is. As you said this cannot be ruled out.

        When you look at all properties that God has, you will very soon find out that they are a quatum state. So in that way there is no material God. He is only in our experiencies of that quantum world, and if we think our experiencies and perceptions are telling us something essential about our surroundings, we also must accept our perceptions of ‘God’ as real.

        The quantum world is ‘supernatural’ and ‘ethernal’. That’s why I prefer to think there is a ‘God’ in form of an analogy of the reality only.

        The quantum world is coherent, entangled, and that is nothing but consciousness-ingredients. When we talk of consciousness, the assumption is usually ‘made by man in his brain’ but that is totally wrong. Also Earth can in a small degree be said to be conscious, in fact everything that can react is conscious. Also molecules and atoms. Consciousness is a qualia charachter from reaction and entanglement?

        Information (and consciousness) brings us to the holography, and holography can be understood as a quantum character. The circle is closed.

      • Lawrence B. Crowell says:

        Ulla,

        To be honest I avoid a lot of what might be called quantum woo-woo. Quantum mechanics is actually very linear and perfectly rational, It really is a sort of logical system with meets and joins being defined by additions of states (superpositions) and multiplications of projectors and density operators, as the analogue of OR and AND in set theory logic with union and intersection. Thigs become strange because there is a macroscopic world which has classical properties. It is the emergence of the classical world that is strange.

      • Ulla says:

        Well, it was Mattis wiev you asked for, not mine.

        Otherwise I thought quantum world was non-local. Look for Planck scale geometry.

        “Quantum mechanics is actually very linear and perfectly rational” fits badly in that picture? Quantum world have no causation in ordinary way. The math you use to describe that non-local, non-timal, and entangled, coherent world would be very interesting to know. Every reaction needs time and decoherence, that is classical world.

  5. HHu says:

    The below are some food for thought:

    “As a physicist, that is, a man who had devoted his whole life to a wholly prosaic science, the exploration of matter, no one would surely suspect me of being a fantast. And so, having studied the atom, I am telling you that there is no matter as such! All matter arises and persists only due to a force that causes the atomic particles to vibrate, holding them together in the tiniest of solar systems, the atom. Yet in the whole of the universe there is no force that is either intelligent or eternal, and we must therefore assume that behind this force there is a conscious, intelligent Mind or Spirit. This is the very origin of all matter.” (Planck, as cited in Eggenstein 1984, Part I; see “Materialistic Science on the Wrong Track”).

    “Every one who is seriously involved in the pursuit of science becomes convinced that a Spirit is manifest in the laws of the universe – a Spirit vastly superior to that of man, and one in the face of which we with our modest powers must feel humble. In this way the pursuit of science leads to a religious feeling of a special sort, which is indeed quite different from the religiosity of someone more naive.” (Einstein 1936, as cited in Dukas and Hoffmann, Albert Einstein: The Human Side,
    Princeton University Press, 1979, 33).

    “The first gulp from the glass of natural sciences will turn you into an atheist, but at the bottom of the glass God is waiting for you.” [“Der erste Trunk aus dem Becher der Naturwissenschaft macht atheistisch, aber auf dem Grund des Bechers wartet Gott.”] (Heisenberg, as cited in Hildebrand 1988, 10).

    Schroedinger claims that science is a creative game with rules, which are designed by God himself: “In the presentation of a scientific problem, the other player is the good Lord. He has not only set the problem but also has devised the rules of the game – but they are not completely known, half of them are left for you to discover or to deduce.” “The uncertainty is how many of the rules God himself has permanently ordained, and how many apparently are caused by your own mental inertia, while the solution generally becomes possible only through freedom from its limitations. This is perhaps the most exciting thing in the game.” (Schroedinger, as cited in Moore 1990, 348).

    Source: Tihomir Dimitrov SGJ V1(3) http://scigod.com/index.php/sgj/issue/view/3

  6. Ulla says:

    “The energy of the universe did not have to be zero, but observation confirms that it is. If it was not zero you would then need something to create it, e.g. a God. Since we observe it to be zero we therefore have an observational confirmation that God is not needed. ”

    This sounds very peculiar to an layman. The assumption is a material/energetic God, that also is symmetric. He creates and destroys with the same rate. But in fact creation is 70 % of him, so he must destroy fiercly when he does. Is black holes such a solution? Can a black hole give rise to the vacuum energy? Hawking is the expert.

    Time is again the missing link? If this was true there is no time and no creation, it has always been like this. Still we believe there has been a creation in the Big Bang and inflanation. How and when has that creative process stopped? We say the universe is expanding, so the content must be diluted by a mysterious vacuum negative pressure? The earliest galaxies was small, today they are big. Are they more diluted today? If a black hole is in the center is sucks. This is not dilution? Only if the sucking force is weker than the diluting force. But it was on an average.

    Entropy is another peculiar fact. Everything goes to the enthropic death (decoherence) and that says there has been a coherent state in the beginning. Coherence is also information. What keeps the information together and give us new information? Information are destroyed? or cannot be destroyed? The thermodynamic laws are wrong or not complete?

    The third is the negative pressure and the adiabatic process. This happens only in closed situations. The way from a black hole to the vacuum energy is long. How does it work? And how is the ‘holes’ explained? There is no adiabatic process? An imprint from another Universe?
    Too much ad hoc ideas?

  7. Lawrence,

    energy conservation has of course nothing to do with consciousness. I am not personally advertising God but for me free will (not completely free of course) is real. As anyone implicitly assumes when telling about his daily activities.

    The reason for the irrational denial of free well has been very banal: there is n’o mathematically consistent description for it in the world view of classical mechanics. Situation changes in quantum theory. Non-determinism is an empirical fact and the only rational way to proceed is an attempt to describe if in a mathematically consistent manner.

    Copenhagen interpretation or some other “interpretation” is of course not enough. There is something very badly wrong with the basic ontology and this relates to the relationship between experienced time and geometric time and to the corresponding causalities. The causality of field equations is not the causality of volition. It is difficult to imagine a more fascinating puzzle for an ambitious and creative theoretician that the challenge to develop the rules of so called quantum measurement theory to a genuine theory without logical paradoxes. That this topic has been a taboo tells about power of groupthink.

  8. Still a comment to Lawrence. We should not forget that quantum measurement theory is the second half of quantum theory. It is extremely irrational to forget this half and the deep problems associated with it. The only way to make a progress is to honestly admit the problems and try to solve them instead of appeal to authority. This is just the opposite to what you call quantum woo-woo.

  9. Lawrence B. Crowell says:

    I do not know if quantum mechanics has much to do in a direct manner with consciousness. I tend not to think there is much about quantum mechanics that is conscious. There are also a number of observations which make the hypothesis of quantum consciousness questionable as well. For one if the brain were quantum mechanical then our visual field might be act as a beam splitter, and two eyes would act as an interferometer. Tests for this are completely negative. So there is nothing so far which is verifiable that links consciousness to quantum physics.

    Free will is something we certainly sense that we have. I don’t think one way or the other about the matter at this point. Free will might in the end be a sort of illusion, and at this time I think there is no conclusive evidence that it does exist. To say that it does exist is to say “it must be so” on the basis of one’s hopes. That is not a particularly good way to proceed with things. Of course it might exist, but it is an elusive thing we so far have not been able to pin point.

    Measurement is a form of entanglement. A measurement of a system is an entanglement of a state |ψ> = sum_nc_n|n> with a detector or some other system in a state |φ> so that

    |ψ>|φ> = sum_n c_n|n>|φ> – -> sum_n c_n|n>|φ_n>,

    where the arrow indicates the process of entanglement. We might consider the state of a system is in some superposition of states, of the sort

    |ψ> = |ψ’> + |ψ”> = sum_n(b_n + c_n)|n>

    then the 1 = is equal to

    1 = sum_n (|b_n|^2 + |c_n|^2 + (b*_nc_n + c*_nb_n)).

    The term b*_nc_n + c*_nb_n is the overlap or inferference term due to the superposition of amplitudes. In the double slit experiment these might equal b_n = exp(inkx) and c_n = exp(ink’x), where the index n might denote a discrete setting for a position on the screen. It is then not hard to see how this leads to a cosine result.

    Now we entangle this system with an other state that assumes the state |φ(b)> if the system is in the state |ψ’> and |φ(c)> if the system is in the
    state |ψ”>. So the entangled state is

    |ψ>|φ> sum_n(b_n|φ(b)> + c_n|φ(c)>|n>,

    and we compute the modulus squared to get

    1 = sum_n (|b_n|^2 + |c_n|^2

    + (b*_nc_n + c*_nb_n)).

    The difference now is that since = = 0 the interference term has been removed. The interference term has been replaced by an entanglement.

    The trace over the pointer states defines the entanglement entropy of this system. I construct that density matrix as

    ρ = |ψ> + |ψ”>)(<ψ'| + + c_m|φ(c)>)(b_n<φ(b)| + c_n<n|.

    So there is the total density matrix. The experimentalist is interested in what the entangled states are telling her. So she looks at the expectation values of just those states — they are the "instrument readouts." So take the trace of this density matrix with respect to these states

    Tr ρ = + =

    sum_{m,n}(b*_m + c*_m)(b_n + c_n)|m><n|.

    sum_{m,n}(b*_m + c*_m)(b_n + c_n)|m><n|(x)(1/2)I

    Where in doing this I did not not normalize things above and I have made up for that oversight which is where the (1/2)I comes from. There are then equal probabilities 1/2 in this 2×2 matrix which represent the 50% probabilities of finding the photon in either of the two slits. The entanglement entropy is found as S = -k*(1/2 log(1/2) + (1/2)log(1/2)) = klog(2). This is the maximal with regards to knowledge of the system, and without a measurement we have S = 0 and so the total information available is S_max – S' = log(2) – 0 = log(2).

    Of course all I did here was to trace over the pointer states, and we are left with probabilities. This is not a dynamical theory for why one outcome obtains and the other does not. Further, a macroscopic pointer involves a whole lot of states, say a mole of states in a photo-electric tube. So now there is an additional loss of information concerning which states of the instrument I am entangled with. There is now a whole set of such states, a decoherence set, where the entanglement of the system have been given over to. The inability to track that leads to classical-like entropy.

  10. Lawrence B. Crowell says:

    PS, In the last equation I don’t know what happened with the first line, but the + = should not be there. Also I forgot to say that the term (x) means “otimes.”

    • Ulla says:

      They are clever, those Romans, said Asterix.

      You found no simpler way to tell this? It is quite selfevident. But tells us nothing about our question. You can better? ‘God’ is in fact entanglement 🙂

  11. Ulla says:

    This is quite a bit off topic, but I want to say this:

    I have worked hard against the plans for building of new nuclear plants in Finland. In my mind it is pure madness. This question is also about energy, but of a completely different kind, manmade energy. Can manmade energy be linked to the Universal energy balance? Maybe? If we can get energy from our magnetic field around Earth. Or from Sun. This energy is always only a loan.

    Tonight was a documentar on TV about ‘the final solution’, a ‘shining cage’ underground, named Onkalo. The documentar really put this question into frames of such an caliber that it make you freeze.

    Why is this nuclear power used? To get energy. If say Chinese and India want the same living standard as us say after 20 years there need to be built three new nukes EVERY DAY. This will guarantee energy for these states in another 20 year. Why not longer? Because then the uranium is depleted, only sources with too low levels to use are left. I know the nuclear industry claims otherwise, but there is no facts today supporting them.

    Nuclear power is claimed to diminish the greenhouse climate, but in reality it is about the same bad waste as coal, say some experts, because uranium mining need so much energy. So this argument fails.

    Put in a time-frame this little advantage seems very ridiculous. Nuclear waste is dangerous in 100000 years. If we look back this is when the humans went out of Africa. Neanderthals was extinct about 40000 y ago. The pyramids was built 4000 y ago. If we divide 100000 y in 10 we get 10 periods as long as the whole human civilization from when humans began to use simple tools.

    There is no way to guarantee the waste safety for so long time. Best is to seal it underground and totally forget about it !!!!

    But there must be built many life dangerous underground cities, many, many.

    All this only because humans today want to live comfortable in a few years. For our comfort our children and their children etc. will live with life danger in eternal times. And we call this civilization.

    Thanks.

  12. Some clarifications to Lawrence. What you explain are standard algebraic facts about density matrix and do not relate to the problem of quantum measurement theory.

    First of all, nondeterminism of quantum world is a solid fact despite the fact that it is more than often put under the rug. One can discuss about whether this non-determinism relates to free will but for a thinker genuinely interested on the mystery of consciousness this hypothesis is a very natural starting point.

    A person who has learned the materialistic view wants of course to be loyal to his teachers and tries to pretend that free will as an illusion. Very few of them can continue this for longer than one minute without failing to express what they really believe. For this reason it remains for me a complete mystery how someone can see the refusal to accept this dogma as a wishful thinking. The basic motivation for this belief was the belief that classical physics, which is a strictly deterministic theory, describes the physical world. For a century we have known that this belief was wrong. Why should we continue to pretend since this belief forces the conclusion that consciousness is something totally passive and leads to empty statements like “consciousness is a property of brain state” telling absolutely nothing about consciousness? Or should we also pretend that also consciousness is an illusion (as one finnish colleague publicly claimed!) – that we are not conscious at all- to get rid of the nasty problem. And what on hell this kind of circular statement could mean? And what about the consequences?: in the world without free will there is no point in talking about ethics, values and moral.

    To me it is absolutely clear that we must challenge the dogmas. Standard quantum theory is not enough. Its basic problem is that the observer remains still an outsider although one is forced to admit that the physical reality is not immutable. The best scientific manner to proceed is to ask how to extend quantum measurement to a theory of consciousness, where observer- we might call it self- is not anymore an outsider.

    My last point. It is misleading to say that quantum measurement is a form of entanglement. Quantum measurement is preceded by entanglement, quantum measurement reduces it, and leads in an ideal situation to maximally unentangled state for which density matrix reduces to 1-D projector. This is what is needed to explain why the rules apply. One can develop various “interpretations” but all these interpretations make things more complicated and fail to solve the problem.

    P.S. We should not forget that the very notion of hyper-determinant about which we had a lengthy discussion and to which some people want to assign somewhat magic meanings is used to describe various degrees of entanglement when the number of tensor factors is larger than two. We tend to forget the meaning and purpose when we concentrate too much on algebra.
    By the way, we should not forget that the very notion of hyper-determinant about which we had a lengthy discussion and to which some people want to assign somewhat magic meanings is used to describe various degrees of entanglement when the number of tensor factors is larger than two.

  13. Sorry for a repetition at the end of the posting.

  14. Lawrence B. Crowell says:

    Quantum mechanics is perfectly deterministic. Quantum mechanics is really nice, linear and involves wave equations which have solutions over nice linear vector spaces. What makes it strange is it does not conform to our expectations of things from a macroscopic perspective. What we call indeterminism

    Entanglement is the dynamics of measurement. The pointer state example I worked out is just one step in the process, where in an actual measurement the entanglement might jump to a mole of such entanglement steps. In that case we lose track of things. If those mole of states in this entanglement are closely spaced then we coarse grain over them and make einselection arguments. This really is not that different from classical mechanics and the structure of the H-theorem in statistical mechanics.

    Hyper-determinants will naturally describe 4-Qbit entanglements, and from there it seems natural that higher structures could be established. We had an interesting discussion going a few days ago, but it seems to have languished.

    At this time I don’t see much role for quantum mechanics in large scale biological systems. Some macromolecules have surface quantum physics, and there are exciton processes, such as e-hole transport with photosynthesis. Yet a quantization on the large of something like a brain with up to 10^{15} processes per second taking place distributed over 1100cm^3 volume seems implausible.

    As for nuclear power, my sense is it should be minimized. However, I don’t think it can be abandoned. Renewable energy sources are best exploited as far as we can, but they have short falls, and nuclear power might be the only thing which will fill those gaps.

    • Ulla says:

      Why do you think Life is based on carbon? The only reasonable explanation is in QM.

    • Ulla says:

      In the nuclear question I talk of the future, Lawrence. What do you think your grand-grand-grand children will say about our thinking? Greedy, very greedy?

  15. Lawrence B. Crowell says:

    Ultimately everything is quantum mechanical. However, this does not mean the dynamics of the system can be encoded in superposition of states or in a grand entanglement of states. This does work well enough for 4 states, say two states of one system and the other two states (pointer states) which enter into an entanglement with the first. Once you begin to increase the number of possible states the combinatorics explode on you. So a brain with some n-moles of atoms and over 10^{15} dendrite connections has a huge number of possible entanglement configurations and a large entanglement entropy. So the observable eigenvalues which are stable against quantum noise or a random reservoir of states define a classical(like) configuration of the system.

    The quantum brain conjecture has been for the most part abandoned. There is little reason to think there is some quantum wave function associated with consciousness. As things stand the nature of consciousness is some sort of epiphenomenon based on processes which are largely classical. Quantum processes are only local to molecular processes, but which do not define a whole coherent state, but rather a statistical distribution of states.

    As for nuclear power and the future, the problem goes far beyond nuclear energy. In a nutshell our species amounts to 7 billion ground apes exponentially rampaging out of control. One property of intelligent life is the ability to remove any constraint upon that species of life through a conscious understanding and solution of problems. It probably started some point with Australopithecus or Homo erectus when they learned to take themselves off the menu by throw rocks at threatening predators, say by chasing off that offending leopard, and then using stone tools to put more on their menu. So our first evolutionary step was in putting ourselves on the top of the food chain. Our evolutionary ancestors then learned to manipulate fire, which permitted us to us more energy than what is required for our metabolic needs. From there it raced forwards on ever faster exponential trends. So here we are, a species with our size and energy demands that number far beyond anything ever seen in the evolutionary history of this planet.

    If there is any role our species has in this universe it might be to understand the universe up to the limits of what is comprehendible and observable. This touches on unknown matters of what is consciousness and whether it has some role in the universe. Yet it is pretty clear that we are on an energy-entropy course that can’t persist endlessly, whether we fire up nuclear power plants or not. The basic problem is that we are consuming the planetary stores of energy, materials and biological wealth at a far faster rate than they can be naturally restored through the open thermodynamic processes of the planet. We further replace those stores with chemical forms of entropy, which we call garbage or pollution. Right now a Belgium’s worth of real estate is being converted to desert every year, so the long term trend is already clear. Our species is most likely heading for a mass-collapse and potentially self-extermination, where the uncertainty is not “IF,” but more “WHEN and HOW.”

    • HHu says:

      Hi Lawrence,

      You state “The quantum brain conjecture has been for the most part abandoned. There is little reason to think there is some quantum wave function associated with consciousness. As things stand the nature of consciousness is some sort of epiphenomenon based on processes which are largely classical. Quantum processes are only local to molecular processes, but which do not define a whole coherent state, but rather a statistical distribution of states.”

      This is not true! I invite you to read http://arxiv.org/abs/quant-ph/0208068 and there is also a journal called NeuroQuantology devoted to the subject. JCER will be running a Focus Issue “The Current Landscape (and Future Directions) of Theoretical & Experimental Quantum Mind/Brain/Consciousness.” Actually, I just wrote a commentary for something else which states “The importance of this and other experimental results such as those of Persinger’s group and our own is obvious: quantum effects play important roles in brain/cognitive functions despite of the denials and suspicions of the naysayers and skeptics.”

      So, it is advisable to be open-minded on the quantum brain issue. Thank you.

      Huping

    • Ulla says:

      Lawrence,
      When you say like that about the energy question I must be of the same opinion. But I must also point out that we have always a choice. If we see and understand it is unwise, why not use that knowledge?

      Or do you seriously claim that our world would be so much otherwise without nuclear force? I remember when the first oil crisis came people got frightened, and they started to think at alternatives. Then they saw the prize wasn’t too high any more, and what did they do? THEY STOPPED DEVELOPING NEW TECHNIQUE. I claim that if we had no nukes we would instead have another alternative source.

      Free will? It is to not following the standards. To take the jump at the side and find new paths.

      Today we understand we must have new technique.

  16. Ulla says:

    You disappoint me. This is only the classical trash. Life has found ways to overcome the decoherence and maintain coherence. How do your brain function? In a very decoherent manner? What you say looks like that 🙂 Only words.

    http://en.wikipedia.org/wiki/Soliton A topological soliton, or topological defect, is any solution of a set of partial differential equations that is stable against decay to the “trivial solution.” Soliton stability is due to topological constraints, rather than integrability of the field equations. The constraints arise almost always because the differential equations must obey a set of boundary conditions, and the boundary has a non-trivial homotopy group, preserved by the differential equations. Thus, the differential equation solutions can be classified into homotopy classes. There is no continuous transformation that will map a solution in one homotopy class to another. The solutions are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include the screw dislocation in a crystalline lattice, the Dirac string and the magnetic monopole in electromagnetism, the Skyrmion and the Wess-Zumino-Witten model in quantum field theory, and cosmic strings and domain walls in cosmology.

    Look at carbon, oxygen, water, hydrogen… and the impact they have on solitons. The networks they are doing. The energy gaps, superconductions at high temperature etc.

  17. Lawrence B. Crowell says:

    A soliton is not a quantum wave function. Of course a quantum wave function can be determined by field amplitudes that are non-linear, but the wave function is linear. A field theory can have an action that is nonlinear,

    S = ∫dtd^3x(1/4)F_{ab}F^{ab}, for F_{ab} = ∂_bA_a – ∂_aA_b + g[A_a, A_b],

    so that ψ ~ e^{iS}, but this wave function is completely linear. The potentials A in a second quantization act on a Fock space in a perfectly linear sense. That is the one thing strange about quantum mechanics — it is linear to the point of being “old fashioned.” Yet the strangeness of quantum mechanics is it does not fit well with our macroscopic sense of things.

    I am terribly agnostic about the matter of quantum consciousness. When the idea first surfaced, largely with Penrose’s “Emperor’s New Mind,” the idea seemed interesting, though I always thought it difficult to connect it with quantum gravity. Yet the idea suffers from some problems with coherence issues and so far I have not seen an argument on how a biological system can exhibit quantization on the large, such as with condensate physics or over complete coherent photon states with lasers.

    • HHu says:

      Hi Lawrence,

      It seems that you are materialist. So, what’s your opinion about Planck’s statement:

      “As a physicist, that is, a man who had devoted his whole life to a wholly prosaic science, the exploration of matter, no one would surely suspect me of being a fantast. And so, having studied the atom, I am telling you that there is no matter as such! All matter arises and persists only due to a force that causes the atomic particles to vibrate, holding them together in the tiniest of solar systems, the atom. Yet in the whole of the universe there is no force that is either intelligent or eternal, and we must therefore assume that behind this force there is a conscious, intelligent Mind or Spirit. This is the very origin of all matter.” (Planck, as cited in Eggenstein 1984, Part I; see “Materialistic Science on the Wrong Track”).

      I am inclined to believe that: (1) Consciousness is both transcendent and immanent (here I borrow the notions from certain philosophy of Hinduism), that is, the transcendental aspect of consciousness produces and influences reality as the interactive output of consciousness and, in turn, reality produces and influences immanent aspect of consciousness as the interactive input to consciousness; and (2) Human consciousness is a limited or individualized version of this dual-aspect consciousness such that we have limited free will and limited observation/experience which is mostly classical at macroscopic levels but quantum at microscopic levels.

      As a limited transcendental consciousness, we have through free will the choice of what measurement to do in a quantum experiment but not the ability to control the result of measurement. That is, the result appears to us as random. On the other hand, at the macroscopic level, we also have the choice through free will of what to do but the outcome, depending on context, is sometimes certain and at other times uncertain. Further, as a limited immanent consciousness, we can only observe the measurement result in a quantum experiment which we conduct and experiences the macroscopic environment surrounding us as the classical world.

      I did a book review on “Quantum Enigma – Physics Encounters Consciousness” See here: http://www.theassc.org/files/assc/2674.pdf

      Huping

      • Lawrence B. Crowell says:

        I don’t particularly know what consciousness is. So far the inner subjective experience of a mind is not observable in an exterior or objective manner. So we really at this point are in a bit of the dark on just what this is. A lot of fMRI work has identified certain experiences reported by individuals with measurable neurological activity. I am not able to comment much more in depth on this because I have little or no expertise in this area.

        The odd thing about quantum mechanics is not quantum waves and the like. What is odd is the existence of a classical or macroscopic world which does not conform to it. I remember learning QM for the first time and finding it in part to be an example of linear algebra — easy stuff in a way. What is very strange is how it is there is a classical world with observable quantities that are “hard” or stable.

    • Ulla says:

      Lawrence,

      A soliton isn’t a quantum wave function? Really?

      What is your opinion then about quantum Hall effect, quasiparticles, meson oscillation – no quantum wave function? Kaon oscillation is such a function? This is parity, chirality. This is essential for life.

      A linear non-local wave function, that one you must explain to me. I thought non-locality was non-abelian? How can it then be linear?

      You say: I have not seen an argument on how a biological system can exhibit quantization on the large, such as with condensate physics – I have.

      The Heimburg group in Copenhagen has done much work with this question. http://en.wikipedia.org/wiki/Soliton_model

      See as instance ‘On soliton propagation in biomembranes and nerves’ or go to his homepage http://www.membranes.nbi.dk/publications.html

      Only for a start.

      Or look at Mattis homepage. Plenty to read. Matti is a wise man 🙂 Kea has also something interesting going on. http://pseudomonad.blogspot.com/

      Then look at graphene and condensed matter physics. Begin with http://faraday.fc.up.pt/cfp/Members/evcastro. Then go to resonanses, Liboff as instance, and quantum phase transitions.

      Although you have atheistic sympathies it must not mean you avoid looking at other explanations. I have read religion although I am ‘atheist’ too in religious eyes, and in those texts are much worth reading. As things about consciousness. It is impossible to explain some experiencies without it, or as a connection to quantum physics. To be a blind atheist is no better than to be a blind theist. In fact I prefer to think at ‘God’ as a quantum state.

      Take your brain as example. How do you explain it doesn’t work in decoherent manner?

      • Lawrence B. Crowell says:

        Sorry, but in these cases the wave function is completely linear. This is even if the gauge system or underlying structure is nonlinear. People have tried, including Weinberg, to derive a nonlinear quantum mechanics in order to treat these problems more effectively. These efforts always crash into a mess. Further, a recent experiment:

        Ruling Out Multi-Order Interference in Quantum Mechanics
        Urbasi Sinha, Christophe Couteau, Thomas Jennewein, Raymond Laflamme, and Gregor Weihs
        Science 23 July 2010: 418-421.

        illustrates the linearity of quantum physics to great accuracy. Of course this makes quantization of nonlinear system very difficult. with gravitation it is further difficult because of the signature or noncompactness of the group space.

        When it comes to consciousness and its relationship to neurophysiology I really have to wait until some clear data comes in.

      • Ulla says:

        http://en.wikipedia.org/wiki/Wave_equation

        The basic wave equation is a linear differential equation and so it will adhere to the superposition principle. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the Fourier transform breaks up a wave into sinusoidal components.

        The wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves…

        And those are solitons. I cannot see the problem.

        http://www.ma.utexas.edu/users/uhlen/solitons/notes.pdf

        “When it comes to consciousness and its relationship to neurophysiology I really have to wait until some clear data comes in.”
        You can look at the Danish scientists. They have plenty of clear data. Good luck!

      • Ulla says:

        Lawrence,
        Wolfram has a non-linear wavefunction

        http://demonstrations.wolfram.com/NonlinearWaveEquations/

        with solitons too.

      • Ulla says:

        Cap 9. http://arxiv.org/PS_cache/quant-ph/pdf/0205/0205092v8.pdf
        Quote: However, the Växsjö model predicts, Khrennikov (2004),
        broader spectrum of evolutions in the Hilbert space (induced by evolutions of contexts). We could not go deeply into mathematical details and only remark that in general the contextual dynamics C → C(t) can induce nonlinear evolutions in H

      • Lawrence B. Crowell says:

        The sine-Gordon equation does have a semi-classical quantization, where there are two states, a soliton and an anti-soliton, where these two states are in a linear superposition in a quantum wave function. This model is dual under S-duality to the quartic Thirring model of fermions. In this way the S-G equation has an interpretations in S-matrix theory. In these types of models you can talk about an underlying soliton system, eg Skymrion theories, in a quantum mechanical format.

        The quantum wave is linear, even though the underlying system (classical Lagrangian etc) is nonlinear. This is what is central to quantum mechanics. In these cases the quantum wave function is over linear states which are representations of solitons.

    • HHu says:

      Hi Lawrence,

      You state: “I don’t particularly know what consciousness is. So far the inner subjective experience of a mind is not observable in an exterior or objective manner. So we really at this point are in a bit of the dark on just what this is. A lot of fMRI work has identified certain experiences reported by individuals with measurable neurological activity. I am not able to comment much more in depth on this because I have little or no expertise in this area.”

      I see that you are reluctant on the subject. But thanks and it’s good enough. I just have the following observations to add:

      Indeed, in many fields of the mainstream sciences today, the study, and even the mere mention of consciousness, are still taboo, and the physicists’ version of a theory of everything does not include consciousness. The irony is if we cannot understand ourselves, and refuse to do so, we cannot hope to fundamentally understand the world surrounding us. Clearly, one cannot call one’s theory a theory of everything if everything is not included. But should the reasoning be that in order to fundamentally understand the external world we need also (or we must first) understand how consciousness works?

      In my view, there are no legitimate excuses for physicists and other scientists not exploring the mystery of consciousness and its connections to quantum physics.

      Huping

  18. Sphere Coupler says:

    Against my better judgment I have decided to comment in this highly interesting conversation in hopes that something I say may be useful. If my words are unwelcome then disregard and accept my apologies.

    I full well understand that disturbance of any system is needed to understand a particular phenomena, in this case via the atheist/ theist genre. It is my personal experience that it is illogical to face these problems from either perspective, for example, I am neither atheist, nor theist for that determination is irrelevant, it is not hidden nor is it relevant to the pursuit of understanding. It is the question that is irrelevant.
    God from a historical view has always been allocated to that which we do not understand to soothe the developing mind of man and when such knowledge has been deduced, well then, God moves on to another yet unresolved problem. If such problems were approached from the view of irrelevancy then progress may be made sooner yet without immediate satisfaction and identification. The dynamical space in which we live most likely will not be describable from one perspective only, they however may overlap and in this way be continues.
    Take for example the human method of I am right and you are wrong, always a competition, where One theory rivals and kills the less explanatory theory, while the dead theory has potential to lead in a differing direction it will be lost until the winning theory has been exhausted. The next step in evolution must combine competing theories in overlapping parameters. It is the dynamics of several differing viewpoints that will prevail.

    Take for example the phenomena of lightning, for many, many years we observe lightning to be the transference of electron flow, but this is yet just another illusion, I now know that lightning is the result of renormalization and this starts from the point of disconnection and is in no way an indication of direction of electron flow.

    Of course that’s just crazy…Right?

    I do not know if my words will be relevant to those reading and do not wish to interrupt such fine debate, my goal is only to lend a different perspective.

    Sphere Coupler
    9/13/10

  19. “Once you begin to increase the number of possible states the combinatorics explode on you. So a brain with some n-moles of atoms and over 10^{15} dendrite connections has a huge number of possible entanglement configurations and a large entanglement entropy.”

    What is the cause for the existence of n-mole of atoms and n-numbers of cells and/or clusters. The problem lies within the fundamental understanding of how matter operates as a function of antimatter in space/time. If particles (electrons) change their locality from one atom to another, it does not necessarily convince me that the very same electrons can be tracked in a matrix continuous manner. What causes the existence of temporal strings of matter and anti-matter? This simply assumes that particles do not have a constant mass as a function of continued time although they appear to have. It’s like a balanced equation, but the equation itself do not explain the mechanism. Indeed just the end state. It was accepted for many that it is not important to know the mechanism, but it appears that it is. Microscopic world can be compressed further, and the more we go to negativity the larger the space we can define. Function precedes structure. Open systems can’t measure entropy nor “negentropy” The origin of mass and negativity separation can serve as reference because we have always looked for the reference, like we refer to zero in simple coordinate system, but zero is just a whole, nothing, where one could in theory “jump” into another local space. Then, there is no continuity because the symmetry is always broken. Non-linear quantum mechanics can be computed when mass is compressed by breaking time into n-dimensional system as long as you define it in a dt and mass change is approximated as a function of negativity flux.

  20. Ulla says:

    To Lawrence and Radoslav

    And communication?
    …connections has a huge number of possible entanglement configurations and a large entanglement entropy.
    S-matrix? Background? Emergent space? Qubit entangled Hilbert space? The more we go to negativity the larger the space?

    Remember the string analogy as a black hole talked of earlier.
    Duff again, http://arxiv.org/PS_cache/hep-th/pdf/0612/0612036v3.pdf

    Entropy is relative, a ratio always, so we can talk of entropy also in open systems? Entropy is about decoherence – entanglement axis.

    Jump into another local space through a zero? How do you, Radoslav, handle the ‘negativity’ and infiniteness? Linear equations?

  21. Lawrence B. Crowell says:

    Phil,

    I might have found a problem with your paper at:

    http://prespacetime.com/index.php/pst/article/viewFile/81/73

    On page 905 you have an equation

    k^a_{;am} – k^a_{;ma} = k^aR_{am},

    where the R_{am} appears to be the Ricci tensor. The problem is that this is symmetric on indicial interchange. I wonder if this equation is supposed to read as,

    k^a_{;am} – k^a_{;ma} = k^aR_{bcam}U^bV^c,

    for U and V noncommuting or nonholonomic vectors.

    • Philip Gibbs says:

      The identity comes a more general identity that defines the curvature tensor:

      k_{a;bc} – k_{a;cb} = k^d R^{dabc}

      then you contract over a and b to get the Ricci Tensor

      I know the identity looks like it cant be right because it seems to be antisymmetric in a and m on the left and symmetric in a and m on the right, but since the a is a contracted index this is not a problem.

      • Lawrence B. Crowell says:

        Agreed, I have gotten myself tripped on this before, which seems to be a snag like consistently mispelling a word. Of course on the other hand I have derived this as well.

  22. Ulla,

    The more we compress the time as a function of molecular mass, the larger the space where negativity occupies within an open system. Starting with a set of linear equations and transform them into partial when using the whole Zero, entropy, or 1 “negentropy” to jump in another local space. “the S-matrix relates the infinite past to the infinite future in one step, without being decomposable into intermediate steps corresponding to time-slices.” Because the electrons position and momentum is undefinable at every time-slice due to the nature of anti-matter property causing matter change as a function of space, space is negativity potential, not a cube,prism or any other geometric object, then time is change of mass, in other words if mass didn’t change, there will be no time. This is why resonance theory said its everywhere and got rid of the time. But this is only working on a tiny local space (a single molecule). You don’t have to go to infinity because systems within systems are infinity as long as you approximate the mass change, one can “jump” thus avoiding infinity. From the above, it will follow that general relativity is conservation of energy within expanding mass, BECAUSE MASS GETS INTO SMALLER AND SMALLER PIECES, THE ABUNDANCE OF MUONS. On sun the opposite exist synthesis. Then it is very clear that Sun must Shrink (hypothesis) Can you measure that?

    • Ulla says:

      This is so very difficult to understand.

      Matti has said time is a function of the electron in 0,1 s interval, important for or perceptions. The other cutoff is the proton mass. This gives us the hydrogen as basic to our classical world.

      S-matrix in your text is the same as the zero energy ontology, with a negative energy in past and positive in future. You talk of entropy (1), zero, and negativity (-1). This has the structure of a bose-einstein condensate in your picture? No intermediate step? You want to have this step looking like an antimatter-matter oscillation? In this case the proton would be a function of space, and also its changes in size (as the muonic hydrogen)? Space is negativity potential without geometry, you say. Here hyperdetrminants etc. have been discussed as space-energy structures. In your case space has no underlying structure? What is the potential then?

      The change is essential to you, the same as the quantum jump, the oscillations. Resonances is the result? But resonances needs equalities.

      Systems within systems are infinities = subentities. The jump/change get rid of infinities? This is done by quantizition? Bound entanglements?

      Then you say: general relativity is conservation of energy within expanding mass, BECAUSE MASS GETS INTO SMALLER AND SMALLER PIECES, THE ABUNDANCE OF MUONS. On sun the opposite exist synthesis.
      This one you must explain more to me.

      I thin really the energy see must have substructures, otherwise also our obvious structures would be hard to create. Energy is uneven distributed in microscale? The expansion is not even either, so something (else) than an adiabatic process must give it.

  23. Radoslav Bozov says:

    In response to Gibbs and Lawrence.

    The problem is in defining variables relative to mass dynamics. I don’t think that total photon number is conserved, because you can’t treat the universe as a closed system. Universe expands when mass compresses relative to subatomic particle interaction with string/wave emergence as a function of a constant broken symmetry. It implies that photon conservation onto spinning particles is relative to string conversion into waves. Therefore, universe expands because mass changes and carbon carbon spatial communication through negativity is a function of energy conservation. Energy is conserved within mass and that all Einstein’s equation implies. General relativity is powerful reference theory on expanding space. But space can expand only when matter changes. Your assumption that photons can be totally independent of mass is wrong because photons conservation in space is a function of negativity and negativity is a function of mass conservation.

  24. Lawrence B. Crowell says:

    Radoslav,

    I have to confess that what you are saying makes little sense to me. I am not sure just what it is that you are trying to communicate.

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