If I were to tell you that we have not the faintest idea how the universe came into being out of nothing, I would not be telling any of you anything you don't already know full well. How can we get "something" from nothing? The ancient Greek philosophers said, "Nothing from nothing." The ancient Hebrew's started with Yaweh. Every society has its own creation myth for the origin of "the world", life, its people.
I am about to try to state a creation myth. I do not believe it. But I do not not believe it either. There will be just enough sense in what I will say, that the right skeptical response seems to be, "Well......maybe."
Paul Dirac, famous physicist, famously told one young student that his theory, "Was not even wrong!". It is a wonderful line and, I hope a true tale. But in some defense I say back, "Any idea at all cannot be worse than no idea."
I think, however, we are safe. I do not envision armies of the converted rising to defend my speculations below.
It all has to do with some odd characteristics of "Possibilities". I am going to try to build a prototheory in which a sudden explosion of possibilities underlies the creation of the universe out of nothing but "The Possible."
I begin with the fact that Quantum Mechanics has, as one consistent interpretation of the famous Schrodinger equation, as I have discussed in previous blogs, that what is "waving" are ontologically real possibilities.
A first clue that we might want to take an ontologically real "possible" seriously arises from the 19th Century American philosopher, Charles Sanders Pierce. Pierce noted that there were three categories of "the the Modal: Actual, Possible, and Probable. It all is related to "The Law of the Excluded Middle" in logic:
Consider the statement, ("Princess Diana died in a car crash." AND "Princess Diana did not die in a car crash.") This can be schematized as (A AND NOT A). Taken together, "A AND NOT A" is a contradiction, forbidden by the Law of the Excluded Middle.
Now flip a coin 10,000 times and conisder the statement, ("The probability of 4723 heads is 0.214 AND the probability of 4723 heads is NOT 0.214"). Again this is a logical contradiction forbidden by the Law of the Excluded Middle.
Now consider the two slit experiment at the statement, (It is possible that the photon went through the left slit AND it is possible that the photon did not go through the left slit."). This statement is not only not a contradiction, but on the Copenhagen interpretation of Quantum Mechanics, the statement is true!
As Pierce pointed out, Actuals and Probabilities obey the law of the Excluded Middle.
Possibilities do not obey the Law of the Excluded Middle.
This fact alone is an important clue that an interpretation of quantum mechanics in terms of an ontologically real "possible" is a legitimate interpretation, and can be taken as one line of evidence for an ontologically real "possible."
Pause to take this in. Empedocles said that only Actuals exist in the world. Aristotle seemed to say both Actuals and Potentia were ontologically real. The early 20th Century philosopher and mathematician, Alfred North Whitehead, believed that both Actuals and Possibles were ontologically real. He held that Actuals give rise to Possibles which give rise to Actuals which give rise to Possibles."
In an earlier post, I pointed out that with the interpretation of quantum mechanics in which quantum possibilities, like Feynman's sum over all possible histories, are ontologically real, a very important feature arises: Constructive and destructive interference of the Schrodinger wave to give the famous interference light and dark band pattern in the two slit experiment. If the "possibility waves" are ontologically real, then interference must be interpreted as ontologically real interactions among ontologically real possibilities, just as real possibilities "out in the world." As real possibilities they can partially or completely block one another in destructive interference, when Possibility peaks coincide in space with Possibility troughs, yielding a dark band on the photodetector screen. Conversely, these ontologically real possibilities can augment one another when wave peaks coincide or wave troughs coincide.
Then, on this interpretation of quantum possibilities, in a sum over the history of all possible pathways photons can take to the light detector via the two slits, those ontologically real possibilities "out there" can interact with one another.
It is, in fact, a mind bending idea. But let's hold onto it.
This interpretation suggests that the Possible is real.
But first, let's back track to Newton. His laws, say for billiard balls on a table, are a predicted succession of only Actuals. The Actual positions and momenta of the balls exactly determine, via integration of his laws of motion, the next Actual position of the balls. There are no possibles, except perhaps in the weak sense of the forward and backward time trajectory of the deterministic Newtonian system.
With Einstein's wonderful General Relativity and the four dimensional block universe of space-time, there are only world lines of events that weave through the block universe. In this block universe, time itself disappears. This disappearance is called the problem of time in General Relativity. But even more strongly, there are no Possibles at all. All is purely Actual.
It seems deeply interesting to me that Einstein's General Relativity, widely regarded as the highest culmination of classical physics, deals only with Actuals. Yet Einstein received his Nobel Prize for the photoelectric effect, a major step towards modern Quantum Mechanics which can deal with possibles.
It is commonly realized that fitting General Relativity together with Quantum Mechanics is very hard because GR is a strongly non-linear theory and Quantum Mechanics is a linear theory. I'll return to this below, for the linearity of Quantum Mechanics is the heart of this blog.
But beyond non-linearity and linearity, there can be a metaphysical difference between General Relativity, with no Possibilities, and Quantum Mechanics which at least has a consistent interpretation in terms of Ontologically real Possibilities. The two cannot match one another, if this claim is true, and these fundamental ontological differences may be a deep reason for difficulties finding a theory of quantum gravity.
So: I begin with the assumption of an ontologically real Possible. In my creation myth, The POSSIBLE WAS, before a single universe emerged.
The next thing I want us to do is to start without any laws at all. I want the laws of the universe to emerge. Indeed, I want the laws of the universe to emerge by a kind of abiotic selective advantage, out of the Possible, all on their own, and naturally.
Next, let's notice what might be two huge clues:
First, consider all the known subatomic particles in the Standard Model. It is a famous fact that these form a mathematical group. That is, they reflect the symmetries of an underlying mathematical structure with the property that each particle can stay the same, can convert to some other particles, can revert back to the initial particle and most importantly, this entire process gives rise to exactly the same set of particles.
Why should this be true? One can imagine particles giving rise to jets of ever new particles forever. Why should the particles form a self recreating group?
I will suggest that any such group has an enormous abiotic selective advantage in an early universe or pre-universe. compared to particles that jetted off in streams of ever new particles. In biological terms, the group of particles is a "self maintaining set" in the minimal sense that, once formed, the set recreates only itself. Let the jets of particles jet away, the self maintaining set "gets to exist", even as quantum objects. More if the particles in the group can ever multiply, so particle number in the pre-universe or early universe was not conserved, a group becomes the abiotic analogue of a "replicator". It produces more of exactly itself in The Possible.
Second, Quantum Mechanics has two magical properties. It is a linear wave equation and the square of the amplitudes of the all the waves, representing all the possibilities, add exactly to 1.0. The latter property means that a global property of the amplitudes is exactly conserved. Each property confers what I am again going to call enormous "abiotic selective advantage" on such a set of Possibilities.
The first linear property of the Schrodinger wave equation, say of an electron in a box, or potential well, has as mathematical solutions what are called eigenfunctions, showing the space-time pattern of amplitudes for the position of the electron in the potential well. But in fact, mathematically, there are for a linear theory two further magical properties. There are an essential infinity of eigenfunction solutions to the Schrodinger equation for the electrons in the potential well.
More strikingly, since the theory is linear, all the infinite possible sums and differences of any pair of eigenfunction solutions, are also possible solutions of the Schrodinger wave equation.
Thus there are vastly, indeed, infinitely, many possible solutions to the Shrodinger partial differential equation. The possibilities of the Schrodinger equation can diversify wildly, yet their squared amplitudes sum to 1.0 so a feature of their total amplitudes is exactly conserved. In The Possible, solutions, or possibilities, derived from the Schrodinger equation can explode yet, in total, via the sum to 1.0 of their squared amplitudes, the ontologically real possibility amplitudes do not disappear. In a pre-universe, such possibilities have enormous abiotic selective advantage compared to possibilities for which these properties do not hold.
Of course there is no normal biological selection and competition, but if we can think about the total number of possibilities in the Possible, the Schrodinger equation real possibilities would do very well compared to possibilities that remain few in number.
These abiotic selective advantages will be the basis for my hoped for natural emergence of both the group property of our particles and something like the linear Schrodinger equation linking the behaviors of those particles.
For those of us not familiar with eigenfunctions, we can be helped by a familiar guitar string. It's ends are fixed. It can vibrate in its harmonic mode, or any octive above that to infinite frequencies in classical physics. These patterns of vibrations are the eigenfunctions of the equations for the guitar string. Just as in quantum theory, the sums and differences of these different string vibrations correspond to different proportions of the diverse harmonics of the base tone.
In short, a first magical property of the linear Schrodinger equation is that it yields an infinite spray of Possibilities. We'll see below that this does not seem to be true of most possibilities and that fact is central to my creation myth.
And again, the other amazing property of the Schrodinger equation is that the square of the absolute values of the amplitudes of the ontologically real possibility waves, sum exactly to 1.0. So as Max Born first pointed out, these squared amplitudes can be interpreted as the probability (probability, not mere possibility) that if the electron is measured in the potential well, the probability of its location being detected in such and such a spot and moment is as given by the squared amplitude for this possibility.
Thus, for my creation myth, we have two remarkable features of Quantum Mechanics and the Standard Model which unites the strong, weak and electromagnetic forces, but not gravity. The particles form a self maintaining group. The probabilities of the Schrodinger equation, by always summing to 1.0, maintain themselves. Both exactly.
Why?
Let's turn to what we know in real life about possibilities in biological evolution and our practical life as, I hope, free willed agents.
3.7 billion years ago, life emerged on the earth or jumped here via space. In any case, over these eons, species have come to be, created opportunities, or niches, for other species to come into existence and make a living, have gone extinct, and a rolling wave of becoming. Each species creates possibilities, adaptive opportunities, for other species, which in turn create opportunities and also block other opportunities for other species to come into existence. Thus, I have written about the non-prestatable emergence by Darwinian preadaption of the swim bladder from the lungs of lung fish. But once there were swim bladders, we can imagine bugs that could only live in swim bladders arising in evolution. It is not fanciful. A very small bacterium lives only in the lungs of sheep. So the coming into unforeseeable existence of some species and organs creates the possibilities for other species to come into existence. And presumably the same process blocks the coming into existence of still other species. Given the wolf, a similar predator cannot easily come to occupy its niche.
In evolution, selective opportunities - or selective possibilities - arise, are selected, enable some other further possibilities, swim bladder to swim bladder bug, but block other possibilities, wolf blocking evolution of near wolves from alternative founder species.
Now lets try our real life. You go to your lawyer about founding a business, business plan all worked out. You start talking. You say, "But of course the plan is reasonable. I've assessed the risks as I must. They check out. But of course, if X, which is quite unlikely to happen, did happen, that would ruin or at least lower the likelihood of this part of my plan. On the other hand, if Y then occurred, now possible because X occurred, it would tend to wipe X out, so my plan would be safe. On the other hand...."
Your lawyer, looking at her watch, says, "Enough, we can go down these alleys of ever more remote possibilities until Doomsday. We don't know, let's cut back to the short term and get real."
We all know this experience. In short, in our real lives, opportunities seem to have likelihoods and to enable or block one another. More we are aware that as we extend to possibilities further in the future that tend to be enabled by or blocked by earlier opportunities in our planning imagination, they become ever more unlikely, precisely because the become ever more "contingent". X only happens if Y does not happen, but could, and Z occurs first to make X more possible, and...
Both in human life and planning and in the evolution of the biosphere, possibilities, like the quantum wave constructive and destructive interferences, enable or block one another.
But something critical is different about these possibilities compared to Quantum Mechanics, which thanks to its linearity gives rise to an infinite set of eigen-function solutions of possible behaviors for the Schrodinger equation for the electron in the potential well, and also the infinite set of the the sums and differences of these solutions are further possibilities, and more magically, the square of the amplitudes of all these possibilities sum to exactly 1.0 and is conserved.
No, in normal common variety possibilities, they do not give rise to an infinite, or at least vast spray of new possibilities, nor do they have a known measure, which, summed over all the possibilities and their likelihoods, let's call those likelihoods "amplitudes", can be squared or some other simple constant mathematical operation acting simultaneously on all the likelihoods to sum to exactly 1.0.
In short, our familiar possibilities are not like quantum possibilities at all.
A brief comment about "bifurcation theory" will be useful. Consider a bowl with a dip in its bottom, honey on the side and a marble rolling down the bowl to the bottom of the dip. This bowl is a "potential well." It has a single minimum, the dip in the bowl. Now imagine an outside control "parameter" P, for piston, that can move the dip upward into the bowl. At some point the dip disappears and a little bump appears in the bottom of the bowl. Now the bowl suddenly loses its single minimum, and the marble will roll into the "mexican hat" well around the bump. If, for simplicity, we tip the bowl a bit so the Mexican hat has a lowest point in its "well", the marble will roll there. OK, we have just seen a "bifurcation", one solution of the ball's behavior disappeared, the dip in the bowl, and a new behavior, a steady state of the marble on the lower side of the Mexican hat well, appeared. If the piston goes down again so the dip reappears, the marble will again roll into the dip.
The point about this is that as the piston moves, say very slowly, old possibilities disappear, (the dip minimum), and new ones appear, (the low point on the Mexican hat well), then the new ones can change further, in our case, disappearing and the dip reappearing.
These bifurcation appearances and disappearances are well known in ordinary and partial differential equations, and may bear a similarity to the possibilities in our life appearing and disappearing. Note there is no infinity of possibilities here. I will try to use this below.
We do not have yet any clear idea how to mathematize the likelihood - amplitudes - of normal possibilities which, in some sense, match how the biosphere evolves and the world in which we human live. But it is important to point out that those amplitudes, if we try to mathematize them, would very likely be non-linear and, ifcoupled, interact non-linearly. And they would probably have a modest number of solutions and have bifurcations.
Why nonlinearly? Well, we're just creating a creation myth. However, it is mathematically true that there are vastly many nonlinear partial differential equations. Linear partial differential equations like the Schrodinger wave equation are a "set of measure zero" in the space of all mathematical partial differential equations. So if one picked a pot full of partial differential equations as a zeroth order trial mathematical model of our familiar possibilities propagating in time and space, almost all would almost certainly be non-linear.
I am about to propose that such non-linear partial differential equations for possibles would be expected to give rise to blocking and enabling of one another of these possibles, more or less as we are familiar with in our everyday and evolutionary experiences of possibilities.
The next point to consider is mathematical, and a difficult area concerning partial differential equations. Some partial differential equations are known to have a set of solutions, eigenfunctions. This set is the spectrum of the partial differential equation. But some partial differential equations are known NOT to have solutions, hence do not have eigenfunctions and a spectrum of solutions. The relative density of arbitrary linear and arbitrary non-linear partial differential equations which have solutions, eigenfunctions and a spectrum of solutions is, I feel confident, not yet known.
More some nonlinear partial differential equations have only one or a few solutions and undergo bifurcations in which old solutions disappear, new ones appear, further new ones can appear, then some can disappear. This again is rather like possibilities in our ordinary life.
I am going to hope the mathematicians one day prove that arbitrary linear partial differential equations are far more likely to have solutions, eigenfunction spectra, than do arbitrary non-linear partial differential equations. One day, we may know. If non-linear partial differential equations often do not have solutions, or better, have only a few solutions and have bifurcations as is already known for some nonlinear partial differential equations, Then such nonlinear partial differential equations propagate no possibilities at all if they have no solutions, or only a few bifurcating solutions otherwise!
But a further point that seems likely, and may well be known, is that linear partial differential equations often will allow all possible infinitely different sums and differences of solutions to be further solutions. Such equations generate infinitely many possibilities.
By contrast, why should linear sums and differences of a set of solutions to some non-linear partial differential equation also be solutions? If not, for those, probably more rare, nonlinear partial differential equations that even have solutions, they cannot generate the infinite set of all possible sums and differences of solutions as further solutions.
Then if we can imagine "mathematizing" possibilities by arbitrary non-linear and linear partial differential equations, (in some unknown dimensions of the Possible before time and space), linear partial differential equations are special in that those that do have a spectrum of solutions also have all the possible sums and differences of those solutions as further solutions. So such linear partial differential equations are expected to generate vastly more possibilities than, I hope, non-linear partial differential equations.
Therefore, this may be a hint that a subset of linear partial differential equations that have vastly many possible solutions may have, as suggested above, the abiotic "proliferative advantage" in a "Possible" with a welter of vastly many arbitrary nonlinear and the many fewer linear partial differential equations in my attempt to even imagine mathematizing propagating possibilities.
Then just perhaps, the linear Schrodinger partial differential equation is "the winner" in this space of "The Possible." Its possibilities proliferate wildly and it "wins". If so, the start of quantum mechanics emerges on its own by a rough but natural abiotic natural selection.
But the Schrodinger equation operates on photons, electrons and other quantum particles and degrees of freedom in the Standard Model. But here I have noted another clue above: why do the particles of the Standard Model form a GROUP, all transforming into one another? Why don't particles generate jets of ever new particles?
Could this Group property, of obvious selective advantage in a soup of possible types of particles since it recreates itself, possibly emerge on its own in the Possible?
Just maybe.
Thus, one more preamble then my creation myth. Some years ago Walter Fontana, then at the Santa Fe Institute, did a wonderful computer experiment. Lisp is a computer language. Lisp expressions can act on Lisp expressions to yield Lisp expressions. Fontana populated his computer with 60,000 random Lisp expressions. Random pairs of expressions bumped into one another, one was chosen at random to act on the other. Fontana iterated the process for a long time.
He also created "selective conditions". If the total number of lisp expressions in the computer pot became larger than 60,000, he randomly threw out some Lisp expressions down to 60,000. So he was selective for Lisp expressions that got themselves formed easily.
Here is what he found. First, he saw a very long sequence of ever new Lisp expressions, then began to see some of the same Lisp expressions. In due course, a Lisp expression arose that could copy any Lisp expression, including itself. This copier Lisp expression took over the computer pot and became the only expression. Note that the copier is a self maintaining Lisp expression.
Then Fontana tired of copiers and just disallowed them and reran his experiment. He got a wonderful results: He got a collectively autocatalytic set of Lisp expressions" that each made one another. This collectively autocatalytic set of lisp expressions is also an "identity operator" in the vast space of Lisp expression. A second wondrous property of Fontana's collectively autocatalytic sets is that they formed a mathematical algebra. It is not a group, for it lacks an identity operator in which a Lisp expression stays the same, and more importantly it does not have an inverse. That is expression A acting on B gives expression C. But B acting on C does not, in general, give A.
Fontana's algorithmic chemistry, or, in Santa Fe, Alchemy, demonstrates that random rules can evolve to form a self identity set, and in his case it can also reproduce.
I'm ready for my creation myth: In the Beginning was the ontologically real Possible and it was without Word and all was Void. But it was full of an interacting, seething broth of ever becoming, enabling, blocking ontologically real possibilities. On average, the number or total likelihood of these propagating and interacting possibilities stayed roughly constant.
We can, in principle, try to test that the possibilities stayed roughly constant using random sets of non-linear partial differential equations as a zeroth order mathematical model of this Possible. Amplitudes of the equations or even newly interacting nonlinear partial differential equations, on average, neither grew nor died out. Any ontologically real possibility was as likely to be more or less blocked as it was likely to be enabled.
Thus, the total "amount" or number of possibility stayed low. On average, not much changed in The Possible. (At least I can hope so.)
But one non-day, a set of Quantum Mechanical possibilities came forth from the Possible, that is, from Actual nothingness, and there was a sudden burst of a vast number of Possibilities due to the linearity of the Schrodinger equation describing their ontologically real behavior. And magically, the squared amplitudes summed to 1.0, so there was something conserved in the vast sea of possibilities. The proliferative advantages of the Schrodinger equation vastly outpaced all other possibilities in the Possible. (Since the electroweak and strong forces have been unified, my creation myth actually needs those partial differential field equations to generate all the burst of possibilities.)
And lo, particles forming a group came forth and even replicated, preserving exactly themselves as a Group identity, and were describable by the same Quantum Mechanics. The particles proliferated and persisted. Later they would stop proliferating.
Suddenly there was a vast set of ontologically real possibilities, the different excited states of all the particles as they transformed as a group into one another. Or it all started just with photons, quarks and gluons, and later the whole particle group formed.
If we are allowed an energy state to each of these modes or the spectrum of the eigenfunctions of the partial differential equations, there was an explosion of a vast amount of energy, where the energy of a photon is proportional to its frequency.
You may object, "Where did the energy come from?" But that sudden emergence of energy is already postulated from nowhere in the Big Bang which is obviously still magical, so I don't see why I cannot magically say the diverse modes of the Schrodinger equation and photons and other particles it described did not have the energies they do. (I hope the LHC finds the needed HIggs particle to give mass and energy to the particles.) And in any case a universe with gravity and spacetime and the Standard Model does not, my physicist friends tell me, conserve energy.
Do I believe my creation myth? No. But myths can become a shared framework that later can become science.
I am about to try to state a creation myth. I do not believe it. But I do not not believe it either. There will be just enough sense in what I will say, that the right skeptical response seems to be, "Well......maybe."
Paul Dirac, famous physicist, famously told one young student that his theory, "Was not even wrong!". It is a wonderful line and, I hope a true tale. But in some defense I say back, "Any idea at all cannot be worse than no idea."
I think, however, we are safe. I do not envision armies of the converted rising to defend my speculations below.
It all has to do with some odd characteristics of "Possibilities". I am going to try to build a prototheory in which a sudden explosion of possibilities underlies the creation of the universe out of nothing but "The Possible."
I begin with the fact that Quantum Mechanics has, as one consistent interpretation of the famous Schrodinger equation, as I have discussed in previous blogs, that what is "waving" are ontologically real possibilities.
A first clue that we might want to take an ontologically real "possible" seriously arises from the 19th Century American philosopher, Charles Sanders Pierce. Pierce noted that there were three categories of "the the Modal: Actual, Possible, and Probable. It all is related to "The Law of the Excluded Middle" in logic:
Consider the statement, ("Princess Diana died in a car crash." AND "Princess Diana did not die in a car crash.") This can be schematized as (A AND NOT A). Taken together, "A AND NOT A" is a contradiction, forbidden by the Law of the Excluded Middle.
Now flip a coin 10,000 times and conisder the statement, ("The probability of 4723 heads is 0.214 AND the probability of 4723 heads is NOT 0.214"). Again this is a logical contradiction forbidden by the Law of the Excluded Middle.
Now consider the two slit experiment at the statement, (It is possible that the photon went through the left slit AND it is possible that the photon did not go through the left slit."). This statement is not only not a contradiction, but on the Copenhagen interpretation of Quantum Mechanics, the statement is true!
As Pierce pointed out, Actuals and Probabilities obey the law of the Excluded Middle.
Possibilities do not obey the Law of the Excluded Middle.
This fact alone is an important clue that an interpretation of quantum mechanics in terms of an ontologically real "possible" is a legitimate interpretation, and can be taken as one line of evidence for an ontologically real "possible."
Pause to take this in. Empedocles said that only Actuals exist in the world. Aristotle seemed to say both Actuals and Potentia were ontologically real. The early 20th Century philosopher and mathematician, Alfred North Whitehead, believed that both Actuals and Possibles were ontologically real. He held that Actuals give rise to Possibles which give rise to Actuals which give rise to Possibles."
In an earlier post, I pointed out that with the interpretation of quantum mechanics in which quantum possibilities, like Feynman's sum over all possible histories, are ontologically real, a very important feature arises: Constructive and destructive interference of the Schrodinger wave to give the famous interference light and dark band pattern in the two slit experiment. If the "possibility waves" are ontologically real, then interference must be interpreted as ontologically real interactions among ontologically real possibilities, just as real possibilities "out in the world." As real possibilities they can partially or completely block one another in destructive interference, when Possibility peaks coincide in space with Possibility troughs, yielding a dark band on the photodetector screen. Conversely, these ontologically real possibilities can augment one another when wave peaks coincide or wave troughs coincide.
Then, on this interpretation of quantum possibilities, in a sum over the history of all possible pathways photons can take to the light detector via the two slits, those ontologically real possibilities "out there" can interact with one another.
It is, in fact, a mind bending idea. But let's hold onto it.
This interpretation suggests that the Possible is real.
But first, let's back track to Newton. His laws, say for billiard balls on a table, are a predicted succession of only Actuals. The Actual positions and momenta of the balls exactly determine, via integration of his laws of motion, the next Actual position of the balls. There are no possibles, except perhaps in the weak sense of the forward and backward time trajectory of the deterministic Newtonian system.
With Einstein's wonderful General Relativity and the four dimensional block universe of space-time, there are only world lines of events that weave through the block universe. In this block universe, time itself disappears. This disappearance is called the problem of time in General Relativity. But even more strongly, there are no Possibles at all. All is purely Actual.
It seems deeply interesting to me that Einstein's General Relativity, widely regarded as the highest culmination of classical physics, deals only with Actuals. Yet Einstein received his Nobel Prize for the photoelectric effect, a major step towards modern Quantum Mechanics which can deal with possibles.
It is commonly realized that fitting General Relativity together with Quantum Mechanics is very hard because GR is a strongly non-linear theory and Quantum Mechanics is a linear theory. I'll return to this below, for the linearity of Quantum Mechanics is the heart of this blog.
But beyond non-linearity and linearity, there can be a metaphysical difference between General Relativity, with no Possibilities, and Quantum Mechanics which at least has a consistent interpretation in terms of Ontologically real Possibilities. The two cannot match one another, if this claim is true, and these fundamental ontological differences may be a deep reason for difficulties finding a theory of quantum gravity.
So: I begin with the assumption of an ontologically real Possible. In my creation myth, The POSSIBLE WAS, before a single universe emerged.
The next thing I want us to do is to start without any laws at all. I want the laws of the universe to emerge. Indeed, I want the laws of the universe to emerge by a kind of abiotic selective advantage, out of the Possible, all on their own, and naturally.
Next, let's notice what might be two huge clues:
First, consider all the known subatomic particles in the Standard Model. It is a famous fact that these form a mathematical group. That is, they reflect the symmetries of an underlying mathematical structure with the property that each particle can stay the same, can convert to some other particles, can revert back to the initial particle and most importantly, this entire process gives rise to exactly the same set of particles.
Why should this be true? One can imagine particles giving rise to jets of ever new particles forever. Why should the particles form a self recreating group?
I will suggest that any such group has an enormous abiotic selective advantage in an early universe or pre-universe. compared to particles that jetted off in streams of ever new particles. In biological terms, the group of particles is a "self maintaining set" in the minimal sense that, once formed, the set recreates only itself. Let the jets of particles jet away, the self maintaining set "gets to exist", even as quantum objects. More if the particles in the group can ever multiply, so particle number in the pre-universe or early universe was not conserved, a group becomes the abiotic analogue of a "replicator". It produces more of exactly itself in The Possible.
Second, Quantum Mechanics has two magical properties. It is a linear wave equation and the square of the amplitudes of the all the waves, representing all the possibilities, add exactly to 1.0. The latter property means that a global property of the amplitudes is exactly conserved. Each property confers what I am again going to call enormous "abiotic selective advantage" on such a set of Possibilities.
The first linear property of the Schrodinger wave equation, say of an electron in a box, or potential well, has as mathematical solutions what are called eigenfunctions, showing the space-time pattern of amplitudes for the position of the electron in the potential well. But in fact, mathematically, there are for a linear theory two further magical properties. There are an essential infinity of eigenfunction solutions to the Schrodinger equation for the electrons in the potential well.
More strikingly, since the theory is linear, all the infinite possible sums and differences of any pair of eigenfunction solutions, are also possible solutions of the Schrodinger wave equation.
Thus there are vastly, indeed, infinitely, many possible solutions to the Shrodinger partial differential equation. The possibilities of the Schrodinger equation can diversify wildly, yet their squared amplitudes sum to 1.0 so a feature of their total amplitudes is exactly conserved. In The Possible, solutions, or possibilities, derived from the Schrodinger equation can explode yet, in total, via the sum to 1.0 of their squared amplitudes, the ontologically real possibility amplitudes do not disappear. In a pre-universe, such possibilities have enormous abiotic selective advantage compared to possibilities for which these properties do not hold.
Of course there is no normal biological selection and competition, but if we can think about the total number of possibilities in the Possible, the Schrodinger equation real possibilities would do very well compared to possibilities that remain few in number.
These abiotic selective advantages will be the basis for my hoped for natural emergence of both the group property of our particles and something like the linear Schrodinger equation linking the behaviors of those particles.
For those of us not familiar with eigenfunctions, we can be helped by a familiar guitar string. It's ends are fixed. It can vibrate in its harmonic mode, or any octive above that to infinite frequencies in classical physics. These patterns of vibrations are the eigenfunctions of the equations for the guitar string. Just as in quantum theory, the sums and differences of these different string vibrations correspond to different proportions of the diverse harmonics of the base tone.
In short, a first magical property of the linear Schrodinger equation is that it yields an infinite spray of Possibilities. We'll see below that this does not seem to be true of most possibilities and that fact is central to my creation myth.
And again, the other amazing property of the Schrodinger equation is that the square of the absolute values of the amplitudes of the ontologically real possibility waves, sum exactly to 1.0. So as Max Born first pointed out, these squared amplitudes can be interpreted as the probability (probability, not mere possibility) that if the electron is measured in the potential well, the probability of its location being detected in such and such a spot and moment is as given by the squared amplitude for this possibility.
Thus, for my creation myth, we have two remarkable features of Quantum Mechanics and the Standard Model which unites the strong, weak and electromagnetic forces, but not gravity. The particles form a self maintaining group. The probabilities of the Schrodinger equation, by always summing to 1.0, maintain themselves. Both exactly.
Why?
Let's turn to what we know in real life about possibilities in biological evolution and our practical life as, I hope, free willed agents.
3.7 billion years ago, life emerged on the earth or jumped here via space. In any case, over these eons, species have come to be, created opportunities, or niches, for other species to come into existence and make a living, have gone extinct, and a rolling wave of becoming. Each species creates possibilities, adaptive opportunities, for other species, which in turn create opportunities and also block other opportunities for other species to come into existence. Thus, I have written about the non-prestatable emergence by Darwinian preadaption of the swim bladder from the lungs of lung fish. But once there were swim bladders, we can imagine bugs that could only live in swim bladders arising in evolution. It is not fanciful. A very small bacterium lives only in the lungs of sheep. So the coming into unforeseeable existence of some species and organs creates the possibilities for other species to come into existence. And presumably the same process blocks the coming into existence of still other species. Given the wolf, a similar predator cannot easily come to occupy its niche.
In evolution, selective opportunities - or selective possibilities - arise, are selected, enable some other further possibilities, swim bladder to swim bladder bug, but block other possibilities, wolf blocking evolution of near wolves from alternative founder species.
Now lets try our real life. You go to your lawyer about founding a business, business plan all worked out. You start talking. You say, "But of course the plan is reasonable. I've assessed the risks as I must. They check out. But of course, if X, which is quite unlikely to happen, did happen, that would ruin or at least lower the likelihood of this part of my plan. On the other hand, if Y then occurred, now possible because X occurred, it would tend to wipe X out, so my plan would be safe. On the other hand...."
Your lawyer, looking at her watch, says, "Enough, we can go down these alleys of ever more remote possibilities until Doomsday. We don't know, let's cut back to the short term and get real."
We all know this experience. In short, in our real lives, opportunities seem to have likelihoods and to enable or block one another. More we are aware that as we extend to possibilities further in the future that tend to be enabled by or blocked by earlier opportunities in our planning imagination, they become ever more unlikely, precisely because the become ever more "contingent". X only happens if Y does not happen, but could, and Z occurs first to make X more possible, and...
Both in human life and planning and in the evolution of the biosphere, possibilities, like the quantum wave constructive and destructive interferences, enable or block one another.
But something critical is different about these possibilities compared to Quantum Mechanics, which thanks to its linearity gives rise to an infinite set of eigen-function solutions of possible behaviors for the Schrodinger equation for the electron in the potential well, and also the infinite set of the the sums and differences of these solutions are further possibilities, and more magically, the square of the amplitudes of all these possibilities sum to exactly 1.0 and is conserved.
No, in normal common variety possibilities, they do not give rise to an infinite, or at least vast spray of new possibilities, nor do they have a known measure, which, summed over all the possibilities and their likelihoods, let's call those likelihoods "amplitudes", can be squared or some other simple constant mathematical operation acting simultaneously on all the likelihoods to sum to exactly 1.0.
In short, our familiar possibilities are not like quantum possibilities at all.
A brief comment about "bifurcation theory" will be useful. Consider a bowl with a dip in its bottom, honey on the side and a marble rolling down the bowl to the bottom of the dip. This bowl is a "potential well." It has a single minimum, the dip in the bowl. Now imagine an outside control "parameter" P, for piston, that can move the dip upward into the bowl. At some point the dip disappears and a little bump appears in the bottom of the bowl. Now the bowl suddenly loses its single minimum, and the marble will roll into the "mexican hat" well around the bump. If, for simplicity, we tip the bowl a bit so the Mexican hat has a lowest point in its "well", the marble will roll there. OK, we have just seen a "bifurcation", one solution of the ball's behavior disappeared, the dip in the bowl, and a new behavior, a steady state of the marble on the lower side of the Mexican hat well, appeared. If the piston goes down again so the dip reappears, the marble will again roll into the dip.
The point about this is that as the piston moves, say very slowly, old possibilities disappear, (the dip minimum), and new ones appear, (the low point on the Mexican hat well), then the new ones can change further, in our case, disappearing and the dip reappearing.
These bifurcation appearances and disappearances are well known in ordinary and partial differential equations, and may bear a similarity to the possibilities in our life appearing and disappearing. Note there is no infinity of possibilities here. I will try to use this below.
We do not have yet any clear idea how to mathematize the likelihood - amplitudes - of normal possibilities which, in some sense, match how the biosphere evolves and the world in which we human live. But it is important to point out that those amplitudes, if we try to mathematize them, would very likely be non-linear and, ifcoupled, interact non-linearly. And they would probably have a modest number of solutions and have bifurcations.
Why nonlinearly? Well, we're just creating a creation myth. However, it is mathematically true that there are vastly many nonlinear partial differential equations. Linear partial differential equations like the Schrodinger wave equation are a "set of measure zero" in the space of all mathematical partial differential equations. So if one picked a pot full of partial differential equations as a zeroth order trial mathematical model of our familiar possibilities propagating in time and space, almost all would almost certainly be non-linear.
I am about to propose that such non-linear partial differential equations for possibles would be expected to give rise to blocking and enabling of one another of these possibles, more or less as we are familiar with in our everyday and evolutionary experiences of possibilities.
The next point to consider is mathematical, and a difficult area concerning partial differential equations. Some partial differential equations are known to have a set of solutions, eigenfunctions. This set is the spectrum of the partial differential equation. But some partial differential equations are known NOT to have solutions, hence do not have eigenfunctions and a spectrum of solutions. The relative density of arbitrary linear and arbitrary non-linear partial differential equations which have solutions, eigenfunctions and a spectrum of solutions is, I feel confident, not yet known.
More some nonlinear partial differential equations have only one or a few solutions and undergo bifurcations in which old solutions disappear, new ones appear, further new ones can appear, then some can disappear. This again is rather like possibilities in our ordinary life.
I am going to hope the mathematicians one day prove that arbitrary linear partial differential equations are far more likely to have solutions, eigenfunction spectra, than do arbitrary non-linear partial differential equations. One day, we may know. If non-linear partial differential equations often do not have solutions, or better, have only a few solutions and have bifurcations as is already known for some nonlinear partial differential equations, Then such nonlinear partial differential equations propagate no possibilities at all if they have no solutions, or only a few bifurcating solutions otherwise!
But a further point that seems likely, and may well be known, is that linear partial differential equations often will allow all possible infinitely different sums and differences of solutions to be further solutions. Such equations generate infinitely many possibilities.
By contrast, why should linear sums and differences of a set of solutions to some non-linear partial differential equation also be solutions? If not, for those, probably more rare, nonlinear partial differential equations that even have solutions, they cannot generate the infinite set of all possible sums and differences of solutions as further solutions.
Then if we can imagine "mathematizing" possibilities by arbitrary non-linear and linear partial differential equations, (in some unknown dimensions of the Possible before time and space), linear partial differential equations are special in that those that do have a spectrum of solutions also have all the possible sums and differences of those solutions as further solutions. So such linear partial differential equations are expected to generate vastly more possibilities than, I hope, non-linear partial differential equations.
Therefore, this may be a hint that a subset of linear partial differential equations that have vastly many possible solutions may have, as suggested above, the abiotic "proliferative advantage" in a "Possible" with a welter of vastly many arbitrary nonlinear and the many fewer linear partial differential equations in my attempt to even imagine mathematizing propagating possibilities.
Then just perhaps, the linear Schrodinger partial differential equation is "the winner" in this space of "The Possible." Its possibilities proliferate wildly and it "wins". If so, the start of quantum mechanics emerges on its own by a rough but natural abiotic natural selection.
But the Schrodinger equation operates on photons, electrons and other quantum particles and degrees of freedom in the Standard Model. But here I have noted another clue above: why do the particles of the Standard Model form a GROUP, all transforming into one another? Why don't particles generate jets of ever new particles?
Could this Group property, of obvious selective advantage in a soup of possible types of particles since it recreates itself, possibly emerge on its own in the Possible?
Just maybe.
Thus, one more preamble then my creation myth. Some years ago Walter Fontana, then at the Santa Fe Institute, did a wonderful computer experiment. Lisp is a computer language. Lisp expressions can act on Lisp expressions to yield Lisp expressions. Fontana populated his computer with 60,000 random Lisp expressions. Random pairs of expressions bumped into one another, one was chosen at random to act on the other. Fontana iterated the process for a long time.
He also created "selective conditions". If the total number of lisp expressions in the computer pot became larger than 60,000, he randomly threw out some Lisp expressions down to 60,000. So he was selective for Lisp expressions that got themselves formed easily.
Here is what he found. First, he saw a very long sequence of ever new Lisp expressions, then began to see some of the same Lisp expressions. In due course, a Lisp expression arose that could copy any Lisp expression, including itself. This copier Lisp expression took over the computer pot and became the only expression. Note that the copier is a self maintaining Lisp expression.
Then Fontana tired of copiers and just disallowed them and reran his experiment. He got a wonderful results: He got a collectively autocatalytic set of Lisp expressions" that each made one another. This collectively autocatalytic set of lisp expressions is also an "identity operator" in the vast space of Lisp expression. A second wondrous property of Fontana's collectively autocatalytic sets is that they formed a mathematical algebra. It is not a group, for it lacks an identity operator in which a Lisp expression stays the same, and more importantly it does not have an inverse. That is expression A acting on B gives expression C. But B acting on C does not, in general, give A.
Fontana's algorithmic chemistry, or, in Santa Fe, Alchemy, demonstrates that random rules can evolve to form a self identity set, and in his case it can also reproduce.
I'm ready for my creation myth: In the Beginning was the ontologically real Possible and it was without Word and all was Void. But it was full of an interacting, seething broth of ever becoming, enabling, blocking ontologically real possibilities. On average, the number or total likelihood of these propagating and interacting possibilities stayed roughly constant.
We can, in principle, try to test that the possibilities stayed roughly constant using random sets of non-linear partial differential equations as a zeroth order mathematical model of this Possible. Amplitudes of the equations or even newly interacting nonlinear partial differential equations, on average, neither grew nor died out. Any ontologically real possibility was as likely to be more or less blocked as it was likely to be enabled.
Thus, the total "amount" or number of possibility stayed low. On average, not much changed in The Possible. (At least I can hope so.)
But one non-day, a set of Quantum Mechanical possibilities came forth from the Possible, that is, from Actual nothingness, and there was a sudden burst of a vast number of Possibilities due to the linearity of the Schrodinger equation describing their ontologically real behavior. And magically, the squared amplitudes summed to 1.0, so there was something conserved in the vast sea of possibilities. The proliferative advantages of the Schrodinger equation vastly outpaced all other possibilities in the Possible. (Since the electroweak and strong forces have been unified, my creation myth actually needs those partial differential field equations to generate all the burst of possibilities.)
And lo, particles forming a group came forth and even replicated, preserving exactly themselves as a Group identity, and were describable by the same Quantum Mechanics. The particles proliferated and persisted. Later they would stop proliferating.
Suddenly there was a vast set of ontologically real possibilities, the different excited states of all the particles as they transformed as a group into one another. Or it all started just with photons, quarks and gluons, and later the whole particle group formed.
If we are allowed an energy state to each of these modes or the spectrum of the eigenfunctions of the partial differential equations, there was an explosion of a vast amount of energy, where the energy of a photon is proportional to its frequency.
You may object, "Where did the energy come from?" But that sudden emergence of energy is already postulated from nowhere in the Big Bang which is obviously still magical, so I don't see why I cannot magically say the diverse modes of the Schrodinger equation and photons and other particles it described did not have the energies they do. (I hope the LHC finds the needed HIggs particle to give mass and energy to the particles.) And in any case a universe with gravity and spacetime and the Standard Model does not, my physicist friends tell me, conserve energy.
Do I believe my creation myth? No. But myths can become a shared framework that later can become science.
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