This is the fifth post in a series discussing Professor Lawrence Krauss' work A Universe From Nothing. In the first post, I gave an introduction to the underlying cosmology, and a quick overview of the rest of his work. Since his discussion of the cosmology was broadly correct, and very well presented, I am (largely) skipping over those chapters and concentrating on those parts of the book which I find problematic.
In the second post, I had a look at chapter 4, which is where things start to go awry. The chapter was about particle physics (my own speciality), and Krauss' claim was the uncertainty principle allowed matter to spontaneously emerge from the vacuum, with nothing before it, albeit only for very short periods. I suggested that his arguments for this were poor, and he made a number of mistakes with regards to the physics.
In the third post, I had a look at his chapter 8, which discussed the anthropic principle, which Krauss uses to support his contention of a multiverse.
The previous post looked at chapter 9, which discussed Krauss' definition of nothing to be the quantum vacuum, and his belief that various philosophers who specifically rejected his proposition would have agreed with him.
Now we come to ask whether the quantum vacuum is unstable. Or, as Krauss phrases it Is Nothing Unstable?
One would have thought that the answer to that question is obviously `No.' Stability in physics refers to (for example) a particle's propensity to decay, or to a system to move from a local minima of its energy to a global minimum. For example, one might have a particle in a particular potential energy, which varies from one location to another. For example, the gravitational potential energy in Newtonian mechanics is proportional to the inverse of the distance from the source of the gravitational field. One can also arrange to have the potential energy oscillating, with various minima and maxima. In classical mechanics, what happens is that the particle rolls down to a local minima of the potential energy, and it can't escape from it unless it is pushed with sufficient force to propel it over the next maxima. That local minima need not be the smallest energy around. It's like a ball sitting at the bottom of a valley. The next valley might be deeper, and if they were connected the ball might roll down there, but it can't because there is a mountain in the way. But in quantum physics, there is a small chance that the particle will tunnel through the potential barrier to the next minimum of the potential energy. So what we have here is a system that seems to be stable, but then suddenly jumps to a very different configuration.
But in both of these cases, you need to have something there in the first place. In the case of the decaying particle, there is the initial particle. In the case of the quantum tunnelling, you have the system in one local minima of the energy, but also, more importantly, whatever generates the potential energy in the first place. So the obvious answer to the question `Is Nothing Unstable?' is no, it isn't.
But, of course, Krauss has redefined the word nothing to mean the quantum vacuum, which is most definitely something (as pretty much everyone except him understands the words nothing and something), so he might have a chance of pulling this one off. Let's see what he has to say.
Back in chapter 4, Krauss suggested that in quantum physics, the vacuum is a seething mass of particles and anti-particles, which can pop into and out of existence continually. As I suggested in my review of that chapter, that idea is incorrect. If the vacuum was like that, it would be observable in gravitation, and the observed effect is much tiny in comparison to the prediction. Neither is it required by the formalism of quantum field theory, which describes objects changing from one type of particle to another (a process of substantial change and generation and corruption of particles), but not particles emerging without a predecessor. But Krauss needs his model, and so he continues with the assumption that it is correct.
He offers two examples of particles appearing from nothing. The first is when you have two parallel plates carrying a large electromagnetic potential. Krauss claims that in this set up, we can see a electron positron pair pop into existence in the vacuum between the plates, and the electron will travel to the positively charged plate, the positron to the negatively charged plate, and this current can be measured. However, this set-up doesn't do what Krauss claims of it. The two charged plates will continually exchange photons. One of those photons decays into a electron-positron pair. This electron and positron then interact with further photons, some from the plates drawing them apart, and others from each other, drawing them together. On occasion, the photons from the plates will win, and we see the electron/positron pair drawn towards the plates. It is not necessary to invoke particles emerging from nothing to explain this phenomena. Indeed, calculating the amplitude for this current uses photon decay, not creation from nothing. The electron/positron don't emerge from nothing; they emerge from a photon.
Krauss' second example is Hawking radiation around a black hole. This is a similar set-up, except instead of two charged plates and a strong electromagnetic field, we have a black hole and a strong gravitational field. Hawking radiation hasn't yet (to my knowledge) been observed. The theory is based on quantum theory in curved space time rather than a full theory of quantum gravity. In particular, it involves the thermodynamics of the black hole, and the equilibrium condition that if it is to remain in equilibrium it must emit as much as it absorbs. Thus Hawking radiation would only occur if the black hole exists within a thermal background. That background is provided by the cosmic microwave background (CMB). So, Krauss' example occurs when a photon from the CMB decays into a particle/anti-particle pair. One of those is sucked into the black hole, while the other escapes. Again, no particle emerges from nothing or empty space. It is all photon decay.
Krauss concludes his discussion of these examples with the statement
Nevertheless, all these phenomena imply that, under the right conditions, not only can nothing become something, it is required to.
Not only has he got his physics wrong, he is brazen about it. Conditions only exist in the presence of something. Nothing, by definition, can't depend on external conditions, because to be affected with whatever those conditions are it would have to interact with them. And only something can interact. So what Krauss is saying is that if there is something present, then nothing is sometimes required to become something. But if there is something present interacting with it, then you haven't got nothing.
The next topic that Krauss discusses is the imbalance between matter and anti-matter in the universe. This is observed: our galaxy is obviously made of matter. But there is enough inter-galactic matter and interaction between galaxies in the early universe to convince us that the same must be true of all the other galaxies as well (of course, if the universe was predominately anti-matter, we wouldn't notice: we would just swap the labels around, and call positrons matter and electrons anti-matter). It is natural to think that there ought to be equal balance between the matter and anti-matter in the universe. After all, a photon decays into an electron and positron, or quark and anti-quark, and the same is true for the other forces. So the observed imbalance between matter and anti-matter is something of a conundrum.
In particle physics, as well as the continuous symmetries such as rotational, translational or Lorentz symmetry, people also consider how the action also transforms under various discrete symmetries: T, or time reversal symmetry (changing the future into the past in the equations), P, or parity symmetry (similar, but with the spatial dimensions), and C, or Charge Conjugation symmetry (swapping particles and anti-particles). These are all individually symmetries of quantum electrodynamics and quantum chromodynamics (the theory of the strong interaction). However, the weak nuclear force violates charge conjugation and parity. It does, however, maintain the combination between the two of them, known as CP. This is the more technical statement of why we believe that matter and anti-matter ought to be in balance. If the universe (together with its initial conditions) were governed by CP symmetry, then that would imply that there ought to be a complete balance between matter and anti-matter. Which there isn't. The argument is a specific application of the general principle that if a symmetry is observed in the action, then it should also correspond to something in nature. (The principle breaks down if the symmetry is broken by the vacuum configuration, called spontaneous symmetry breaking, but that doesn't apply here.)
In the standard model, there is a small violation of CP symmetry. This occurs when we put together the theories of the electro-weak and strong nuclear forces. There are (we believe) three families of fermionic matter. You have the electron, neutrino, and up and down quarks. Then, for some reason, that pattern is repeated again at a higher mass. And then repeated once more at an even higher mass. So we have twelve fundamental fermionic particles in total. These particles are described by quantum states, and like all quantum states one can create superpositions of them. So we can describe a quantum state which is partly an electron and partly a muon, from the next family. To properly distinguish the particles, we need a preferred basis, and this basis is provided from the eigenstates of the operators representing the photon, W, Z and Higgs Bosons in the case of the electro-weak theory, and the gluons in the case of the strong nuclear force. But there is no reason to suppose that the mass eigenstates in the electro-weak theory should precisely coincide with those of the strong nuclear force. And they don't. What this means is that when we discuss up quarks in the nuclear force calculations and up quarks in electro-weak calculations, we are not actually referring to quite the same object. The nuclear force down-quark is a superposition of electro-weak down, strange and bottom quarks. The difference is small, but measurable, and this slight difference between the strong and electro-weak theories violates CP symmetry, and leads us to expect a slight imbalance between the matter and anti-matter in the universe. (As far as we know, CPT symmetry is not violated anywhere.)
However, the standard model CP violation is not enough to explain the observed matter/anti-matter imbalance. It is assumed that there is some additional beyond the standard model physics which will account for the missing CP violation. And this is quite plausible and possible. But it remains one of the biggest unsolved problems in particle physics.
Krauss states that if the universe began sensibly, with an equal amount of matter and anti-matter, then we wouldn't be around to ask the wherefores of it, since all the matter and anti-matter would annihilate each other, and all there would be would be a vast sea of photons. Nothing is obviously symmetric between matter and anti-matter, so for a universe from nothing, we might expect this to be case. One would think that this is a problem for Krauss' thesis. However, CP violation allows one to begin with equal amounts of matter and anti-matter, and end up with a matter dominated universe. All we need is a small imbalance between matter and anti-matter to arise, and then all the anti-matter will annihilate with some matter, leaving just the excess matter and a fistful of photons (the microwave background). Krauss relates the origin of this asymmetry as the moment of creation. This can't be the case: CP violation is an ongoing process, observed to this day, and therefore can't be considered to have happened in a moment. The asymmetry needs to correspond to approximately one part in a billion more matter than anti-matter. Known CP violations don't allow for this, but Krauss isn't overtly worried by this because the standard model is almost certainly incomplete (and here I agree with him).
The next question is why a matter dominated universe rather than an anti-matter generated universe? If we start with an equal amount of matter and anti-matter, then it could go either way. Imagine that you manage to balance a pencil on its point. It might be stable, but the smallest perturbation will cause it to fall down. But we can't predict which way it will fall down. Presumably, Krauss muses, even if the laws of physics are fixed, the ultimate direction of the asymmetry between matter and anti-matter was driven by some random initial condition. (Recall that randomness isn't a thing and therefore can't be a cause; when we describe something as random we are either being lazy or saying that we don't know the causes.)
This idea that an initially symmetric universe could collapse into one which is either matter or anti-matter dominated is Krauss' motivation for the title of the chapter: nothing is unstable. He assumes that the universe starts in a featureless state, where no matter existed: the universe is a vacuum. A second possible state is one where matter exists, with slightly less symmetry but lower energy. There is a quantum tunnelling to this second state, and the energy released emerges in the form of particles. Then, due to Bayron number and CP-violating processes, we get the matter/anti-matter imbalance.
This doesn't seem very plausible to me. A featureless state would have energy zero, so it can't tunnel into a lower energy state. Indeed, (the quantum mechanical definition of) energy only makes sense in the context of matter: it is a quantum number of a particular particle. It doesn't make sense to discuss the (quantum mechanical) energy of nothing or empty space, because nothing is not a particle in an eigenstate of the Hamiltonian and therefore can't have an energy associated with it. So this means that to have the initial symmetric system in a higher energy state than the one where the symmetry is broken, Krauss needs something there to carry the energy. Thus he is not starting from a state of nothingness.
Plus, of course, the mechanisms of CP violation involve particle interactions, such as kaon decay. To get from that to Bayrogenesis, you also need to make use of electroweak anomalies. Both of these interactions assume that you have some starting state; Baroygenesis can't start from nothing.
Mechanisms for CP violation and Baryogenesis are certainly known. They can explain matter/anti-matter asymmetry. But they don't allow one to conclude that nothing is unstable because all known mechanisms for this assume that you have to start with something. The best that they can say is that certain somethings are unstable. But then we already knew that.
Not much is known about the quantum theory of gravity, or how to quantise general relativity. I personally believe that we should be focussing more attention on quantum field theory in curved space time. The more conventional approach, however, is to introduce a new quantum particle, the graviton, as the conveyor of the gravitational force, just as the photon carries the electromagnetic force. The naive way of doing this is not mathematically self-consistent (not renormalisable), but there are various more sophisticated models which might work.
The naive way of performing quantum gravity is to expand the space time metric as the Minkowski metric plus a perturbation. This perturbation is then quantised and called the graviton, the mediator of gravity. The process of quantisation involves upgrading the graviton to a creation operator, which (when placed into a time evolution operator) will be used in the description of how gravitons are emitted from other matter, or decay into other matter.
String theory supposes that our universe is embedded in a higher (10 or 11 dimensional) space, and that the fundamental objects are one dimensional strings. The particles we know about are vibrational modes of those strings. In addition, there are higher dimensional surfaces called branes, and massless spin 2 particles emerge which are associated with the graviton. These are in turn quantised.
Loop quantum gravity sticks with our four dimensional universe. It treats the affine connection (which is the object used to transform the differential operator so that it can be expressed in a different reference frame) as the thing to quantise. So, again, it is upgraded into a creation or annihilation operator
I list these to make a point: in none of these approaches, nor any others that I know of (although I admit that I am not an expert on quantum gravity) is space time itself replaced by a creation or annihilation operator. I am not sure even what doing so would imply. One's location in space is a property of matter; as such it is something which subsits in a material substance, not a substance itself. Key to quantum physics are the energy and momentum operators, related to the differential operators in time and space. For a quantum particle to have energy, it presupposes that you can differentiate the field representing that particle with respect to time. Yet if time itself were a quantum field (capable of undergoing creation and annihilation), then you couldn't differentiate it with respect to itself. That quantum field would not be able to carry energy, leaving us with something entirely different from the sort of quantum field theory that describes matter. But if it is very different, the arguments from analogy which Krauss uses (since he doesn't have a theory of quantum gravity, he can only use analogy) break down. An analogy relies on similarity, not difference. And if something other than space and time are the quantum field, then how can you use an analogy with quantum field theory to say that space and time emerge from nothing?
Krauss, however, disagrees,
General relativity, as a theory of gravity, is, at its heart a theory of space and time. As I discussed in the very beginning of this book, this means that it was the first theory that could address the dynamics not merely of objects moving through space, but also how space itself evolves.
Having a quantum theory of gravity would therefore mean that the rules of quantum mechanics would apply to the properties of space and not just to the properties of objects existing in space, as in conventional quantum mechanics.
I broadly agree with the first of these paragraphs. But the second one seems to be way off. Firstly, quantum field theory isn't fundamentally concerned with the properties of objects, but how those objects come in and out of existence during interactions with other objects. If we transpose this to space, Krauss' quantum theory of gravity would be mainly about a description of how space itself would come into and out of existence. And this is certainly how he wants to bring the argument forward in subsequent paragraphs.
In quantum field theory, there are two types of operators. Those (inherited from quantum mechanics) which are used to describe properties of matter, the location operator x, the momentum operator d/dx, the spin operator and so on. Secondly, there are the creation and annihilation operators for the fermions, photons and so on. Things like location, momentum and spin are related to the indices on the creation operators. This distinction is of particular philosophical importance; it matches the distinction between substances (which exist in themselves) and properties (which subsist in substances). To create space, one would have to convert x into a creation operator. I know of no theory of quantum gravity that does this.
Krauss next discusses Feynman's path integral approach to quantum physics. Here, one sums over all possible trajectories or paths for a particle, even those which are classically forbidden. I'm not quite sure what Krauss means by "classically forbidden." The natural interpretation is that he is referring to those paths which vary from the path designated by the principle of least action. Another might be that it includes "off-shell" paths. In classical special relativity, the energy and momentum of a particle are related by the dispersion relation E2 = m2 c4 + p2 c2. In field theory, it is a bit more complicated: one integrates over all energy and momentum, although the propagator has poles (or divergences) at the energies implied by the dispersion relation, meaning that those momenta dominate.
Feynman's approach has the advantage that the central feature in calculating the weights in the sum, the action, is invariant under coordinate transformations. What this means is that one can use the same basic approach in any reference frame; we don't have to specify how to label space time points before performing the calculation. This formulation is no doubt even more useful in general relativity, where we consider even more possible coordinate transformations and ways to label space time. In a quantum theory of gravity, we would presumably have to sum over, as well as all the possible particle trajectories, but also all the ways in which space time can be curved. So far, this seems reasonable to me.
But now Krauss starts move beyond reasonability. Using the analogy that particles can pop into existence in quantum field theory out of nothing (which I have repeatedly stated is untrue), doesn't this imply that in a quantum theory of gravity, space itself can pop into and out of existence from nothing? No, in a quantum theory of gravity, the graviton, or the particle representing the affine connection, would come into and out of existence (and not from nothing). Krauss asserts that unless one can come up with a good reason for excluding this possibility, then, since everything that can happen must eventually happen, we must expect that small regions of space must pop into and out of existence.
This argument has numerous flaws. To start with, he confuses two different types of possibility. Firstly, there are possible expressions of the laws of physics. Secondly, there are possible events given the truth of a particular expression of the laws of physics. The first of these represents our uncertainty -- one expression of the laws of physics is true, but we don't know which one. The second of these refers to the frequency at which events can occur. It is true that if the laws of physics were such that space could come into existence from nothing, then it would happen and, given an infinite time, it must happen. But it is not true that because such an expression of the laws of physics that might allow for this to occur is possible (in that we currently know no reason opposing it), then it must happen. It must happen only if there are an infinite number of universes each sampling each possible laws of physics. But we are not interested in what happened in some purported alternative universe completely disconnected from our own; we are interested in what is happening in this universe. And when there is only a sample of one object, one can't get from a premise that it's possible, to a conclusion that it must happen.
And I would be very surprised if it were possible. All we have got from Krauss is a suggestion which contradicts every model of quantum gravity I know of, no proposal for an alternative model which might explain it, and a complete misunderstanding of quantum field theory -- particles do not pop into existence from nothing. Krauss' speculations are of no more use than an idle daydream.
Krauss' next problem is that just as his virtual particles of quantum field theory pop into and out of existence for short time periods, the same will be true for his virtual areas of space. These areas of space would invariably immediately collapse -- not much use for explaining the origins of our universe. But he has a proposed solution.
He starts by asserting that a zero energy virtual photon is not subject to the uncertainty principle and its restrictions on how long the particle can pop into and out of existence: it can last forever. Of course, a zero energy photon (if any such object exists) wouldn't interact with anything, since every interaction involves the transferral of either energy or momentum, and most usually both. This is also a misuse of the uncertainty principle (which is not about the length of time that particles can exist, but the spread of the wavefunctions of existent particles); although in this case his conclusion is correct: a zero energy particle would endure forever.
His analogy is that a zero energy universe might emerge without collapsing immediately. Such a universe would be closed (i.e. eventually it will collapse in on itself). This is inconsistent with our flat universe, but if it suddenly shifts to a period of inflation, then that is enough to make anything flat. So closed universes popping in and out of existence plus inflation might give us the universe we observe.
There are, of course, various problems with this. Firstly, and most obviously, he requires that the universe would be both closed (zero total energy) and flat (non-zero total energy). Secondly, his argument relies on his interpretation of the uncertainty principle. The uncertainty principle is applicable to single particle wavefunctions. The universe is not a single particle. Thirdly, given that space and time are interconnected, it is not just space that he requires to pop into existence from nothing, but time. But the uncertainty principle is derived from the relationship between the energy and temporal wavefunctions. It is only applicable to particles in time; but not for time itself, as Krauss demands (if his universes are to obey Lorentz symmetry). Finally, he once again confuses energy as it is used in cosmology (the stress energy tensor) with energy as it is used in quantum physics (eigenvalue of the Hamiltonian operator). A closed universe is one with a zero cosmological energy. But he jumps from that to discussing quantum physics, i.e. he has transposed to a zero quantum mechanical energy. But energy is always positive in quantum physics. The only way for an universe to have total quantum mechanical energy of zero is if everything in it has energy of zero, which is clearly not the case.
So, by misusing of quantum field theory, inventing a new theory of quantum gravity, and despite claiming that the uncertainty principle allows something of zero energy to last for ever, he proposes that a zero-energy closed universe can pop into existence for short times. However, they would invariably collapse within a Planck time of emerging (what does he use as a scale for measuring time here, since time would come into existence with the universe). But add in inflation (where did the inflation, the hypothetical particle that drives inflation come from?) and he gets a universe which lasts a long time and has the appearance of being infinitely big and flat. So having spent the first part of the book laying out his case (and with good physics) that the universe is flat, he now demands that it isn't.
If inflation doesn't cut it, then Krauss has a couple of alternative proposals. The first is a proposal by Vilenkin, the second the Hartle-Hawking no boundary universe. These two approaches are similar. I'll just discuss the earlier paper of Vilenkin. Vilenkin's paper starts from the standard first approximation model for the universe. The Robertson-Friedmann-Walker metric describes a perfectly homogeneous and isotropic universe, i.e. one which is exactly the same in every place and direction. Obviously the universe isn't like this, but this is actually a good approximation, and usually used as the starting point of cosmological calculations. One key part of the solution is the scale factor, denoted as a(t), related to the distance between two objects over time. Vilenkin noticed that a solution to this universe led to a behaviour of a(t) which resembled that of a classical particle approaching a potential barrier. This solution did not permit values of a smaller than the inverse of Hubble's constant, as long as time is treated as a real number.
One trick used by physicists is to rotate time to the imaginary axis, replacing t with it, where i2 = -1. You might wonder what the point of doing this is. What it does is convert the hyperbolic space time of special relativity to a Euclidean space time, which tends to be an easier place to perform certain calculations. So we rotate to Euclidean space, do the calculations, and then analytically continue them back to Minkowski space. What Valenkin noticed was that in this rotated time, the scale factor a also resembled a classical particle that hit a potential barrier, only this time only values smaller than the inverse of Hubble's constant were permitted. Instead of a universe where space expanded in time, this universe would describe a system where space and time are described by a four dimensional sphere (recall that time is now equivalent to just another space dimension).
Valenkin's proposal, as far as I can make out, is that we treat a as a quantum mechanical wave function. There are two stable states, separated by a potential barrier. But quantum mechanical particles can tunnel through potential barriers. Thus if we started with this static, four space-dimensional spherical universe, there would be a small chance that it would suddenly switch to the three space and one time dimensional expanding universe that we are all familiar with. And this, Valenkin claimed, signified the emergence of the universe from nothing.
Of course, there are issues with this approach. Firstly, there is little motivation to treat a, one part of the metric, as a quantum mechanical particle. There are further problems when we try to upgrade it to a quantum field theory, particularly concerning the renormalisability (or lack of infinite cross sections). There are further philosophical problems with switching to an imaginary time coordinate: fine as a mathematical trick to ease computation, much harder to accept as a feature of the universe. Time is, after all, an observable coordinate of the universe, flowing in one direction. We would need a tunnelling of time itself from imaginary to real, but time is not a quantum mechanical particle and therefore cannot exhibit quantum mechanical tunnelling. All observables are represented by real numbers. Time is something we observable. And then there is the major problem that the starting spherical universe in Euclidean space time is not nothing. It is another universe with a different geometry. So instead of a universe from nothing, we have a universe from another universe.
So this picture manifestly fails to generate space time from nothing. Even if we grant its assumptions, it doesn't begin with nothing.
Krauss asserts that this mechanism will allow universes to be generated from nothing, which he now defines as an absence of space and time. But Valekin's spherical Euclidean universe which is the starting point of his tunnelling event is not the absence of space and time; space and time are still there, just with a different geometry.
Krauss admits that this proposal does not prove his thesis. But he thinks that it is a possibility. The problem is that there are too many assumptions, too much bad philosophy, and too much bad physics for me to take his belief that nothing is unstable seriously.
I find Krauss' argument in this chapter to be really poor. He continues to compound his error of redefining nothing, but goes further than this. He builds on his errors in the interpretation of particle creation and the uncertainty principle from chapter four. He relies on applying an analogy from his interpretation of quantum physics to quantum gravity, but his argument only makes sense if space and time are quantum fields themselves, and no prototype quantum gravity theory I know of asserts that. He then adopts the speculative theories of Valekin, which also rely on an analogy and the idea that some specific parameter should be quantised. But that parameter is not quantised in any quantum gravity approach I know of, and even if it was, it would not show that nothing is unstable, but that something which is most definitely a something is unstable. And he tries to tie all this together with various arguments from analogy and other logical fallacies. In short, the arguments of this chapter are pretty much worthless.
Next time, Krauss tries to get the laws of physics from nothing.
All fields are optional
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