Links of Randomness

♦  My wife has a new blog about quilting.

♦  This Nature article describes the subfield of quantum gravity which I've been working in—I know most of these people.  This is a lot more accurate of a description than your average pop journalism article, so check it out.  (Of course many of the ideas here are speculative and may turn out to be false.)

♦  What if you discovered one day that everyone except you has a magical superpower, and you never knew about it?  A true story, flippantly described yet also deeply moving.

(If you're curious to find out more, here's another decent article on the same subject.  [Warning: includes gratuitous disturbing art involving surreal faces])

♦  This is the best article about lichen I have ever read.  Well, maybe that isn't all that competitive an award, but it's still a pretty good article.  The runner-up lichen article is also pretty good.

♦  You already know that dolphins are really smart animals—but that doesn't mean you won't enjoy reading more about it.

♦  One way to go to college for free.  But not suitable for dolphins...

♦  Once upon a time, people thought that Jews were naturally the best at basketball, because of their short stature and scheming minds!

♦  How we know that the robots didn't take our jobs.

The scary chart (the one that shows how, as a result of poor structuring of government programs, poor people can actually be worse off as a result of getting a job or a pay raise) is from this article.  Honestly, how hard is it to phase out programs gradually with income so as to avoid truly stupid incentives?

♦  This critique of our current primary nomination process, may have changed the way I think about politics.  This article bothers me because, on principle, I dislike pretending to have a democracy when actually the important things are settled in the cliched "smoke-filled rooms" (I've disliked the Democratic superdelegates since I first heard of them) but it seems obvious in this election that that those methods have worked better.

Also one could question whether "democracy" should really mean majority (or plurality!) rule when we are talking about the plurality of a minority (those who vote in a given party).  The main way that the party establishment would like to modify raw democracy is to make the candidates more electable, which means in a way they represent the rest of the nation and make the results more democratic.

♦  How do you warn people thousands of years later about sites where radioactive waste is stored?

♦  Or for a more short term prediction about future developments: some predictions in the year 1900 about the year 2000.  About as accurate as these things ever are, i.e. not terribly but a few of them score some palpable hits.   (Here's a plaintext version if you find the first one hard to read.)

♦  "A Mathematician's Apology" by G. H. Hardy.  Still contains a lot of truth today, although when he lumps Quantum Mechanics and Relativity in with pure mathematics, and says that at least these things can never have any use in war... well, I think we have to count that as another failed prediction.

♦  Interesting article in the NY Times about a two sets of identical twins (2 x 2 = 4) where one twin from each set was swapped at birth, and what happened after they found out.

♦  An interesting series by St. Jason of Triablogue on some of the less well-known evidence in favor of the traditional authorship of the Four Gospels.

♦  Does wishing to believe in religion put one in a better or worse position, for learning whether it is true?  An interesting fictional dialogue [google books] on the subject by a Catholic author.  I read this on the strength of a quotation excepted at Siris.

♦  An actual dialogue about religion at First Things, with a Catholic and Muslim, both authors, about religion and their friendship with each other.

♦  Nobody Expects the Spanish Inquisition … to Be Explained Fairly, a review of a book addressing anti-Catholic history.

♦  Speaking of agencies that use the methodology of inquisition (the accused must prove themselves to be innocent)... please don't call Child Protective Services on parents for trivial issues unless you hate both them and their offspring.  Followup posts here, here, and [added later] hereSpoiler: happy ending.

Posted in Links | 13 Comments

Is Divine Simplicity compatible with other Doctrines?

Introductory Warning: please note that when theologians call God "simple" that's a technical term.  It doesn't mean he is easy to understand.  It means that he is not composed of any parts of any kind.  (Depending on who you talk to, this could also mean stronger statements which are believed to follow from that, e.g. that all of God's attributes are really different ways of talking about the exact some thing, and that they only appear different to us from our limited earthly perspective.)

A certain St. Matthew J. Thériault, whom I met at Ratio Christi, sent me the following questions about whether divine simplicity is compatible with other Christian doctrines such as God's Omniscience, the Trinity, and the Incarnation:

Attached [click on this for a word file---AW] is the abstract to a presentation I intend to deliver at Ratio regarding the doctrine of Divine Simplicity. Regarding the Trinitarian objection there's no relation to physics, and you already personally addressed the Incarnation objection when you last visited Ratio (though I'd be interested if you've given any more thought to the matter of the simultaneity of the ascended Christ and the Church on earth). However, I imagine you'd be able to offer immense insight regarding my objection to Omniscience, which is partially informed by a short article linked within. Thank you in advance for any feedback you might be able to offer.

My reply email spiraled out of control, and having gotten permission to share it on my blog, I will now do so:

Dear Matthew,

Although as you say the simplicity of God is easier supported by philosophical than by scriptural arguments, nevertheless there is a little bit of Scriptural support for the doctrine of simplicity.  Traditionally, the doctrine was held to be taught in the Shema: "Hear O Israel, YHWH your God is one YHWH" (Deut 6:4).  Jews interpreted this to mean not just that there are no other gods, but also that this God has some kind of absolute unity of being.  Traditional Christianity, rather than deny this interpretation to make room for the Trinity, has tended to affirm it and then to assert that the Trinity does not contradict it, because the sense in which God is three is different from the sense in which God is one (the persons of the Trinity are not parts, or additional deities).

One could also gather some indirect support for divine simplicity from the tendency of Scripture to sometimes refer to God as "I am [attribute]" or "God is [attribute]".  Also, obviously a lot of the philosophical arguments depend on God's self-existence which is taught more explicitly by "I am who I am" in Ex 3, John 8, among other passages.

Nevertheless, I think one needs to proceed cautiously.  While it would be heretical and unreasonable to say that God is actually divided into parts, it seems to me that some theologians have subscribed to extensions of the doctrine of simplicity which go beyond what can necessarily be deduced from it.  As you know the very strongest formulations of divine simplicity can lead to a number of philosophical paradoxes which are difficult to resolve.

We must always remember that (whether we are talking about Natural Theology or Scripture), "for now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known", and also that "no man shall see my face and live".  Mortal human beings are not capable of seeing the divine nature directly, but only deducing its existence through either revelation or more remote lines.  God may be known to exist from philosophy but he is also the Invisible and Incomprehensible Glory, the numinous Sanctus, that haunts all our experiences but can never quite be contained in any of them.

Therefore we should not be surprised if different attributes of God seem to be in some degree of tension with each other, in fact it would be more surprising if we could fully understand how all of the various attributes can be consistently combined.  Christianity has never shied away from paradox, as can be seen from the doctrines of the Trinity and the Incarnation.  A paradox is not a logical contradiction per se, it is only a sign that our own understanding is limited, yet we can make progress by approaching the same thing from two or more viewpoints which appear to give conflicting information (just like our two eyes, by seeing slightly differing data, can produce a 3d stereographic perspective).

As St. Lewis said about a different theological quandary: "Heaven will solve our problems, but not, I think, by showing us subtle reconciliations between all our apparently contradictory notions. The notions will all be knocked from under our feet. We shall see that there never really was any problem."

What I have just said about paradox is really my primary answer to all of your questions.  It would be grossly misleading if I gave the impression that I could 'resolve' paradoxes in the divine nature in the sense of providing a clear logical schema in which the nature of divinity could be fully grasped with the mind; any such scheme would necessarily be misleading and even impious.  But nevertheless, as a secondary matter, I think I can say some specific things to resolve some of the specific difficulties you list:

A. Omniscience.  This is only an issue if you assume that God's knowledge is, like ours, representational, that is, that it proceeds by means of making something like an image or duplicate of the object known, in some other physical system (in our case, the brain).  That viewpoint seems excessively anthropomorphic, and I have already argued against it here:

Fundamental Reality IV: Does God Need a Brain?

To put it into bite sized arguments, God's knowledge cannot be representational because:

1. The Redundancy Objection: If God is omniscient then his thoughts about the universe would necessarily be an exact copy of the universe not differing in any details.  But that is silly because if two things are identical in every respect, they may as well be identified.  Furthermore it would imply that God's knowledge is limited to the form of the thing (the structural attributes which are the same between the image and the reality) rather than the essence of the thing (which would not be shared between the universe and the image in its mind.  But that would be a limitation on God's intellect.  So instead we must assert that there is no division between God's knowledge of a thing, and the thing itself.

2. Infinite Regress Objection: Even human knowledge is not purely representational, because that would threaten an infinite regress.  For example, suppose we look at a lamp and form a mental image of the lamp somewhere in our mind/brain.  We are then aware of the lamp outside ourselves by means of the lamp inside.  But how is it then that we know the lamp inside?  By means of a second image of the lamp inside of us?  That would threaten an infinite regress.  Instead we must somehow have the power to directly perceive, without any intermediary representations, some things that reside in our own brains.  But God would have the power to directly perceive anything, without any limitations.

3. And of course because it contradicts Simplicity, as you point out.  But that just means we have the wrong model of how God's knowledge should work.

I think the article you link to ["Information Storage and the Omniscience of God", by Hollis R. Johnson & David H. Bailey] is completely off-base when it proposes that God's knowledge must be understood as if God were a giant computer.  They should have realized that it was this ridiculous idea of their own which they were refuting, and not anything in the Bible or theology as traditionally understood.  They say that:

"Some defenders of the traditional doctrine of God’s omniscience may respond to this argument by simply declaring that God is omnipotent and thus omniscient, in the sense of residing and operating completely outside the confines of the Universe and the natural laws that govern our Universe. In short, they may assert omnipotence and omniscience by fiat: God can store knowledge, even an infinite amount of knowledge, without any plausible physical storage mechanism or medium. This is because God’s ways are not ours, and our finite mortal minds cannot possibly hope to comprehend the means employed by this supreme Being. Against such reasoning there is no counter argument."

The reason there is no counterargument against this position is that it is obviously correct.  The idea that God is outside the confines of the physical universe and therefore does not store his knowledge on some physical medium such as a film reel, is not some arbitrary stipulation made to avoid falsifying Theism.  It is part of the definition of Classical Theism that God is outside the universe, for goodness sakes.  To explain God's knowledge by means of ordinary causal mechanisms, far from being a "scientifically tenable theology" would simply be the denial of classical theology, which holds that God as creator is not subject to the limitations and natural laws which govern creation.  If we found the giant film reel it would refute, not confirm, Classical Theism.  One may as well say that any theism compatible with modern biology would need an evolutionary explanation of why God the Father has a beard!

[Not in my original email: I looked it up, and the authors are actually Mormons.  That explains a lot.  Mormons are polytheists who believe that God the Father is merely an exalted human being, one of many deities who worked his way up in some sort of cosmic pyramid scheme, and that he has a physical body.  So according to them God isn't omniscient and probably does have a literal beard, and a brain with finite information storage capacity.  This is, of course, completely different from the classical Jewish-Christian-Islamic concept of God as the absolutely powerful and wise being who is the source of all existence.   Such a being, if he happened to exist, would not be a God at all in the traditional Classical Theist sense.  Why should I worship a being simply because he happens to be a finite amount more powerful, wiser, or moral than I am?  There are already human beings who are better than I am, that doesn't mean they are worthy of my worship!]

B. Trinity.  Here I think I need to quibble with some of your language.

While it is true that the Second and Third Persons in some way originate from the First, I do not think it is orthodox to say that only the Father is the First Cause, apart from the other two persons.  That would seem to gloss over the crucial distinction between "making" and "begetting".  The Father is not a separate metaphysical entity from the Son and the Spirit, indeed the Father has no separate existence apart from his act of begetting the Son and breathing forth the Spirit, since these acts were by metaphysical necessity; it could not have been otherwise.  (This does not mean it was involuntary, for God is spirit and his acts are therefore by will and love, not physical compulsion.)  The persons are so united that you can't have any of them without having the other two as well.  For this reason, I would say that the Triune God taken as a whole is the First Cause, rather than the Father alone.

John 5:26 says that "For as the Father has life in himself, so he has granted the Son also to have life in himself", that is even though it is a gift, the nature of the gift is that the Son has life intrinsically, according to his identical divine nature, rather than derivatively and externally through grace.  Otherwise Christ could not say, in the divine sense, "I am", but rather should have said "I was made to be".

As you recognize the persons of the Trinity are not parts (since they are indivisible), but they are real distinctions in the divine nature.  It therefore seems inevitable that the analysis of the "relations" which define the persons (things like Paternal, Begotten, and Proceeding) must necessarily differ from the divine "attributes" such as power, wisdom, or love which are common to all persons because they belong to the single divine nature.  To that extent I agree with you.

But if this admission seems to contradict some specific analytic formulation of "Property Simplicity", why not simply acknowledge an accidental misstep, coming from an over-strong formulation of Simplicity, and retreat to a slightly weaker version of the doctrine?  For example, one might tentatively say that if God is One, then any of his properties must either be identical to himself OR ELSE to one of the persons, and then say that the latter possibility does not contradict simplicity because each person of the Trinity contains the other two within by "perichoresis".

You say that "Begotten" is "unidentifiable with and alien to" the First Person, but this seems to be stating it too strongly, since even though the Father is not Begotten he does have the reciprocal property of "Begetting", which is the exact same thing viewed from the other side.  The one implies the other.  Apparently God can have real distinctions within himself, but only involving relational terms, of the kind we are discussing.

Thus, since the doctrine of the Trinity is clearer in Scripture than the doctrine of Simplicity, we should adjust the latter to make room for the former, but without of course abandoning the doctrine of Simplicity altogether!  Implicit in this is the idea I sketched in my introduction, that God is mysterious, and that the philosophical "proofs" of his attributes, while perhaps compelling, do not amount to strict logical implications.  And therefore that there is room for "adjustments" in our very provisional understandings when we run into trouble!

C. Incarnation.  As you note, I was asked about this during the Ratio Christi meeting, but for clarity I'll repeat myself a little.  According to the Chalcedonian understanding, the Incarnation involves the union of a complete divine nature with a complete human nature into a single person, Christ.  Properties like Simplicity would apply only to the divine nature, and therefore it would not contradict Simplicity to note that Christ's physical body had parts and could change etc.

After the Resurrection and Ascension, Christ continues to have a human nature, but now his body and soul are glorified, possessing additional abilities and attributes.  This glorified Resurrection body transcends our current earthly state (although we too will be glorified when our bodies are raised from the dead.)  I would love to know more about this but our data from the Gospels and Acts is limited.  What we do know is that Christ's body was capable of being recognized (though not always immediately) and touched, that he could speak and eat, and that he was capable of teleporting instantaneously.  After which he ascended into "heaven" (i.e. somewhere else outside of spacetime as we know it, in which God's will is more fully done as in the Lord's Prayer, as in the angelic world), which he conceptualized as a return to the Father from whom he came, triggering the pouring out of the Holy Spirit on the Church.

Obviously this cannot be understood in an excessively anthropomorphic way.  While Christ has a body by virtue of the Incarnation, God the Father does not, and therefore "sitting at the right hand of the Father" cannot be taken literally, to mean the proximity of two physical bodies at a common time.  Instead it is an Aramaic way of saying that Christ is placed in a position of full welcome and authority, that a formerly crucified and rejected man is now being given the governance of all Creation, with rebel angels and authorities now fully subject to him.  (Try searching the Psalms for "right hand".)  Christ's body continues to have an objective and real existence---and our earthly imaginations cannot conceptualize this except by imagining him residing in something like a "place"---but the nature of that "place" is not one that we can understand, until we ourselves follow him there (John 14:2-3).  There is no reason to think that the visible universe as we know it is anything other than a small portion of God's creation:

"There may be Natures piled upon Natures, each supernatural to the one beneath it, before we come to the abyss of pure spirit; and to be in that abyss, at the right hand of the Father, may not mean being absent from any of these Natures – may mean a yet more dynamic presence on all levels."--St. Lewis, "Miracles" (essay in God in the Dock).

How "time" works in this "place" is not something which I think we are in a good position to know.  I agree that Einstein's theories suggest that time is a feature of our own material universe, so that a completely disconnected universe would probably have a different timestream, if any.  However, if there are interactions between two such universes, then there would presumably still be causal relations between them, and hence (I suppose), some partial notion of prior/posterior events.

This agnosticism about the details might seem a little depressing, but I am afraid it may be the best we can do right now, fun as it may be to speculate on the details of the "control room" from which Christ currently reigns!

I am not sure why you think that there needs to be any "simultaneity" between Christ's body in heaven and the "physical universe in between his Ascension and Return", any more than there needs to be simultaneity between the eternal God and us, in order for God to answer our prayers.  Christ is present in the Church in a number of ways; as the Head who gives the Body life, through the presence of the Spirit, sacramentally in the Eucharist, and so on; but none of these ways seem to involve or require any one-to-one map between individual moments of Christ's existence in heaven and our individual existence on earth.  Even Christ's everlasting intercession for us comes not through continual labor, "offer[ing] sacrifices day after day", but rather by presenting his wounds to the Father once for all, as an eternal atonement for the sins of the whole world: past, present, and future.  We may be sure that he knows and cares for all of our needs, which suffices for practical spirituality, without getting into the mechanics of exactly how his glorified human nature shares in the universal knowledge of his divine nature.  Psalm 131.

All right; this has become quite a treatise.  Hope it helps!


PS Do you mind if I post this exchange on my blog?  This could involve as much or little of your identifying information and words as you wanted.

Posted in Theology | 33 Comments

In the Red Light District

I'm in the middle of a six week trip in Europe; currently I'm attending the Amsterdam String Workshop.

I'm reminded of something that happened to me a year-and-a-half ago December when I visited the String Theory group in Amsterdam.  I didn't realize until I starting doing touring on Sunday that my hotel was close to the main "red light" district, where the alleyways are full of semi-naked women in booths selling their bodies to the tourists.  The main red light district is right in the middle of the oldest part of town, well worth seeing for the architecture, if you can ignore the vice peddling (which is easier during the daytime).

I was absolutely shocked in the red light district—but not by the prostitutes or the drug use, which I had expected.  (Although these things are bad and degrading, don't do them.)  There is a beautiful old Dutch Reformed church there, dating from the 1300's, which I wanted to see.  I went in to see the church, but whoever was in charge had allowed an artist to set up a crass avant garde multi-media work of art in the interior, with disturbing images of unwholesome faces projected on the blank walls speaking nonsense phrases, and even representations of bright neon casino scratch pads, glowing on the floor!  I felt it was an extremely disrespectful, if not diabolical, use of a space dedicated to our Father in heaven, and in which faithful Christians were buried.

There were a small number of middle aged couples roaming around looking a bit perplexed.  I was outraged.  I said to myself "How DARE they do this to my Father's house!" and I couldn't stay there any longer because I could not contain my rage.  (I said something about it to the poor lady handing out tickets at the entrance.  I tried to make it clear to her that my anger was not directed at her, but I had to say it to somebody.)

As I was wandering around in a daze, I noticed that there was another church in the district, a Roman Catholic church, which was free for anyone to enter.  (The first church had had a 10 euro entrance fee, which is also wrong—what if one of the prostitutes felt a sudden urge to go into a church and pray?—but one quickly becomes desensitized to fees for entering famous churches in Europe).  It was full of tourists but pious ones, and I felt such relief to know that, despite the theological differences, there was some place in the area dedicated to God which was still held sacred, and where the people had natural feelings.  I sat down in an empty pew and wept.

Posted in Ethics, History | 9 Comments

Open and Closed

A reader asks:

This seems to be as good a place as any to ask a question about closed universes.

See, in a lot of popular science books, they teach you that an "open" universe is one where space is infinite, saddle-shaped, and keeps expanding forever; a "flat" universe is infinite, plane-shaped, and the rate of expansion eventually peters out to zero; and a "closed" universe is finite, sphere-shaped, and eventually contracts in a big crunch. They then talk about the cosmological constant and "dark energy," which make our universe expand at an accelerating rate, something that doesn't fit the taxonomy of possibilities for the universe's topology, and which they do not relate back to that taxonomy in any way.

Can a universe with lots of dark energy be a closed universe? Will a closed universe with dark energy keep on expanding and accelerating, or will it eventually collapse in a big crunch like a "normal" closed universe? Is the three-type Taxonomy only relevant given certain energy conditions? (Strong/weak/null)

Oh, and I almost forgot:

are there any good reasons to think that the universe is closed in the first place, other than Kalam-esqe arguments against actual infinities?

It sounds like these books were just adding the new material about the cosmological constant to the old discussions without doing the hard work of going back and revising it so that it makes sense.

The Bad Old Days

In the old days (pre circa 1998) people didn't know about the acceleration of the universe, and they thought that the universe just consisted of ordinary radiation and matter (where for these purposes, dark matter is a form of matter).  In the old days, the model of closed, flat, and open works exactly as you say: a closed universe (spherical geometry) will recollapse, and open one (hyperbolic geometry) will trend to a constant rate of expansion (in terms of distance / time) and a flat one is right on the edge and will expand forever at a slower and slower rate (but still getting arbitrarily large).

Given the rate of expansion, it takes a certain amount of energy density to get a flat universe.  Too much, and you get a sphere, too little and you get hyperbolic space.  (The expansion or contraction of the universe makes it hyperbolic in the absence of matter.)  These are the 3 kinds of geometries which are homogeneous (the same everywhere) and isotropic (the same in every direction).  On average, the observable universe seems to be homogenous and isotropic, so it's got to be one of these three (a.k.a. an "FRW cosmology").

However, this was confusing for several reasons.  One is that the cosmological data kept suggesting that there wasn't enough energy in matter to get anywhere close to a flat universe, yet other data seemed more consistent with a flat universe.  A flat universe is also a natural consequence of inflation since it stretches out the pre-existing geometry to exponentially large distance scales.  Also, the universe seemed like it wasn't quite old enough to explain all the structures in it.

Concordance Cosmology

Now we know that there is an additional form of energy which is confusingly called "dark energy" (but I dislike this name, because it makes people think it has something to do with "dark matter".)  Most likely it is just a cosmological constant, a constant energy density in all of space.

Now it turns out that for purposes of determining the spatial geometry, a positive cosmological constant counts positively (so it helps to close the universe).  But when you calculate its effect on the expansion of the universe, it counts negatively, as repulsive gravity.

This may seem like odd behavior because energy and mass are equivalent, and we all know that mass causes gravitational attraction, not repulsion.  But in turns out that in General Relativity, both energy density (associated with time) and pressure (associated with space) lead to attractive gravity.  Negative pressure is called tension, and tension therefore causes antigravity.

In ordinary matter travelling at low speeds, the amount of pressure/tension is typically very small compared to the energy density.  Radiation which travels near the speed of light has a lot of pressure, but that only makes gravity stronger.

On the other hand, a positive cosmological constant has tension equal to its energy density.  Something has tension if, when you stretch it out, it's energy increases.  But the energy of the cosmological constant is proportional to the volume, so when the volume increases the energy increases proportionally.  Hence the tension in each spatial direction is equal to the energy.  Since there are 3 dimensions of space and only 1 of time, the antigravity due to the tension is 3 times larger than the gravity due to the energy density.  Hence the antigravity wins!  So paradoxically, the gravitational effects of this tension just make the universe want to grow faster!  Unlike the usual effects of tension, which cause things to shrink in on themselves.

On the other hand, if the cosmological constant were negative (it isn't, but suppose) its effects would be reversed: it would make the spatial geometry more hyperbolic, but would decelerate the expansion.

So, once you include a cosmological constant, the rules change (as you guessed).  You can still have the same 3 types of spatial geometry (the words "open", "flat", and "closed" describe the spatial geometry, not the dynamics).  But with a positive cosmological constant, even a universe with closed topology can sometimes expand forever, if it gets big enough for the cosmological constant to take over.  (Matter thins out, while the CC doesn't, so when the universe is small the matter is more important, and when it gets larger the CC is more important.)  On the other hand, with a negative cosmological constant, even an open cosmology will always eventually recollapse when it gets big enough.

(The various energy conditions you mention place limits on the allowed energy density and/or tension/pressure, so not surprisingly these have certain implications for what a cosmology can do.  Note that a positive CC violates the strong energy condition—which allows for a bounce, at least in the case of a closed universe.  While a negative CC violates the weak energy condition, which requires that any FRW cosmology which is neither expanding nor contracting at some time, must be closed.  (OK, technically it also allows space to be flat, but only if the matter energy is exactly 0, which is unrealistic.))

Our universe seems to have a positive cosmological constant, which fixes all of the problems I mentioned above.  The cosmological constant seems to give us exactly the extra energy density we need to get a flat universe.  Yet it also causes the universe to be currently accelerating in its expansion (lengthening the projected time back to the Big Bang); this acceleration has been confirmed by surveys of supernovae in the past.  So everything seems to hang together consistently.

As far as we can tell from current observation, the universe is exactly flat (with experimental error of about 1-2% over scales comparable to the observable universe)  However, a flat geometry is right on the knife's edge between the spherical and hyperbolic cases, so actually this is perfectly compatible with the universe having a tiny positive or negative curvature, as long as the radius of curvature is big enough.

So really it could still be any of the three cases, or else something more irregular.  As I said, inflation blows up the size of the universe, so regardless of the initial geometry, the observable universe will look flat after enough inflation.  Outside the observable universe, for all we know, it could be some other shape, perhaps it isn't even symmetrical.

There is really no particularly good physics reason, apart from aesthetics and philosophical bias to think that the universe should be closed or open.  I personally don't think much of the "Kalam" argument that actual infinities are impossible, but I do find it distasteful that in an infinite homogeneous universe everything (including all possible histories of the Earth) would happen infinitely many times in different places.

Also on the speculative hypothesis that the universe originated from some kind of quantum fluctuation, or no-boundary condition, I think one expects it to be closed.  But this kind of thing is extremely speculative.

If I had to place a bet with a metaphysical bookie, my money would be on closed (but enormously large so that we could never tell).  But this is my own personal guess, not a conclusion of Science!

(Incidentally, even if the topology of space is flat or hyperbolic, it would still be possible for the universe to be finite in size and therefore closed, so long as it has nontrivial topology.  For example, space could be a really big "torus" where if you go far enough in one direction, you come back around on the other side, like in some video games.  Locally, such a universe couldn't be distinguished from the infinite case, but globally it would be different.  Astronomers have done measurements looking for nontrivial topology in the sky.  They haven't seen anything, but of course they wouldn't if it happened on a scale much bigger than the observable universe!)

On the other hand, if the universe really does have a positive cosmological constant than (regardless of its spatial geometry) the final outcome seems secure.  If we extrapolate the current laws of physics to the far future (assuming no changes or interventions), we get an exponentially growing universe.  The matter thins out and becomes unimportant, and you end up with a very tiny final temperature (corresponding to the analogue of Hawking temperature but for cosmological horizons instead of black hole event horizons).

Posted in Physics | 13 Comments

Quantum Mechanics III: Wavefunctions

[Fixed typo in Schrodinger's equation below—AW]

Previously I talked about interference, the chief weird thing about QM that makes it different from Classical Mechanics.  You have to think about complex-valued "amplitudes", from which you derive (real-valued) probabilities.  From this you can also derive the notion of a Hilbert Space of states.  We discussed the space of states for the polarization of a photon (a 2 state system), and how there are many different choices of "basis", representing different ways of identifying a mutually exclusive set of two possibilities.

Now let's consider a more complicated system: a single particle moving around in empty space.  There are infinitely many states, because space is a continuum.  Hence we need to use an infinite dimensional Hilbert space.  This is harder to visualize than a two-dimensional one, but it will still be true that, in any given basis, each state can be regarded as a quantum superposition of a bunch of possibilities.

There are many possible choices of basis, but two of them are particularly nice.  You can choose to either express the system as superposition of position states, or as a superposition of momentum states, but you can't specify both at the same time, because they are two different bases of the Hilbert Space!  This is the origin of the Heisenberg Uncertainty Principle.

Of course, since the position and momentum are continuous variables, the probability of having any particular exact value of position or momentum is always 0.  So we have to generalize the framework slightly and talk about amplitude densities.  (However, there are other choices of basis where you don't have to do that).

An amplitude density is an amplitude per unit square-root-of-volume.  I know these units sound a bit strange, but that way when you square it, you get a probability per unit volume, which is as things should be for purposes of doing measurements.  The amplitude density is more commonly called the wavefunction of the particle.  So the wavefunction can be written as a function of position: \Psi(x,\,y,\,z), or as a function of momentum: \Psi(p_x,\,p_y,\,p_z), but not both at the same time.  However, if you know one of them, you can calculate the other one by a Fourier transform (should you be lucky enough to know what that is).

If you have multiple particles, you shouldn't think that each particle has a separate wavefunction.  Instead, you use a single wavefunction which depends on the positions (or momenta) of all of the particles.  For example, if there are two different particles which we'll call #1 and #2, then you'd write:

\Psi(\vec{r}_1, \vec{r}_2),

where I'm now using vector notation as a shorthand; but each vector still has x, y, and z components.  Hence the wavefunction for 2 particles actually is a function living in a 6 dimensional space!  (More generally, the wavefunction of N particles will live in 3N-dimensions, assuming that space is 3 dimensional.)

(Given these two particles, it might be that the wavefunction factorizes, so that

\Psi(\vec{r}_1, \vec{r}_2) = \Psi_1(\vec{r}_1)\Psi_2(\vec{r}_2).

That's what would happen if you independently prepare each particle in a state, and don't let them interact with each other.  But in general, there's lots of wavefunctions you could write down which do not factorize in this way.  This allows the particles to be correlated in strange ways not allowed by classical physics.  We call this phenomenon entanglement.)

Now the two particles might be either different type, or the same type.  One of the principles of particle physics is that apart from a limited number of attributes such as position/momentum, "spin", and a few other things, all particles of a given type are identical.  (E.g. all electrons have identical properties, and all photons also have identical properties.)

If the two particles are identical, then it shouldn't make any difference which particle we choose to label as "1" and which we choose to label as "2".  So there should be a symmetry of the wavefunction if we switch the two particles.  (Remember, in QM we have interference whenever two histories end up in the same place, so to get things right we have to obsess about exactly when two situations count as exactly the same, and when they don't.)  There are two different ways to implement this symmetry.  The obvious thing to do is to say that:

\Psi(\vec{r}_1, \vec{r}_2) = \Psi(\vec{r}_2, \vec{r}_1),

so that the amplitude is the same in both cases.  This sensible approach is taken by identical bosons, which includes particles such as photons, gluons, gravitons, mesons, He-4 nuclei, and so on.

Another, more perverse way to implement the symmetry is to insert a minus sign:

\Psi(\vec{r}_1, \vec{r}_2) = -\Psi(\vec{r}_2, \vec{r}_1).

This bizarre form of identicalness is used by fermions such as electrons, quarks, neutrinos, protons, neutrons, and He-3 nuclei.  (In general, something made out of an odd number of fermions is also a fermion, since if you switch two copies, you'll get an odd number of minus signs.)

So photons are strictly identical, while electrons are almost identical, but you get a minus sign if you switch them.  But remember, the overall phase of a QM system doesn't matter.  So you won't actually notice anything weird if you definitely switch two fermions.  The minus sign only matters in situations where they might-or-might-not have gotten switched, because then the interference between the two histories will be different.

A somewhat more straightforward implication is that no two identical fermions are ever in exactly the same position, because then the weird antisymmetry tells us that \Psi(x_1, x_1) = - \Psi(x_1, x_1) = 0.  This is the Pauli Exclusion Principle, which is the reason for the Periodic Table.  Since electrons can be either "spin up" or "spin down", you can only put 2 distinct electrons in each energy level of an atom.  Then they get full and you have to put them in higher energy shells.

Bosons, on the other hand, are gregarious and love to be in the same place.  Or rather, to speak less anthropomorphically, their probability to be in the same place is greater than you would expect from classical probability theory.  This is what makes lasers (a bunch of photons all in the same state) practically possible.

I've mentioned "spin" several times, but I haven't actually said what it is.  In QM, some particles also have an intrinsic angular momentum or polarization, which gives them a certain sort of directionality in space (even though they are point particles).  Unlike a lot of the cute terms used in particle physics such as "color" or "charm", the term "spin" really does refer to actual literal angular momentum.  But it works in a weird way.  The angular momentum along any axis is quantized, meaning it has to be either an integer, or an integer + 1/2 (times the Planck constant \hbar).  The maximum possible angular momentum along any axis is called the "spin" of the particle.

In Nature, there is a rule called spin-statistics which says that particles with integer spin are always bosons, and particles with half-integer spin are always fermions.  (You can prove this rule mathematically in QFT, but it requires Special Relativity and some additional physical assumptions.)

Every known fundamental fermion is spin 1/2, which means that along any given axis it is either spinning clockwise (a.k.a. "up" or +1/2) or counterclockwise (a.k.a.
"down" or -1/2).  You only get to specify the spin along one axis, say the vertical one.  This is not to say that an electron can't spin "left", "right", "in", or "out", but these states are quantum superpositions of the "up" and "down" states.  By rotational symmetry, we could pick a different basis (e.g. right / left) and instead think of up and down as superpositions of right and left.  The (2 complex dimensions = 4 real dimensions) space of possible electron spins is called a spinor.

A spinor needs to be rotated by 720º (2 full circles) to get back to its original state.  Yes, you read that right.  If you only rotate it by 360º (1 full circle) then it comes back to itself with an extra minus sign in the amplitude.  Just like when you switch two electrons.  They're just perverse that way.

Most of the fundamental bosons are spin 1, so their polarization is given by a vector, as in the case of the photon which we discussed last time.  Vectors get a minus sign when you rotate them by 180º, and return back to the way they were after 360º, just like you were taught in school.

However, the Higgs boson (which gives mass to most of the other fundamental particles) is a spin-0 or scalar field.  That means it doesn't change at all when you rotate it. On the other hand, the graviton is a spin-2 particle, which means it gets a minus sign when you rotate it 90º, and goes to itself under 180º.  It's polarization is described by a matrix, but let's not get into that here.

The bottom line, is that for anything more complicated than a scalar field, in addition to the position or momentum variables you also need to include the spin degrees of freedom.  So if we have one electron and one photon, the wavefunction will look like e.g.

\Psi(\vec{p}_e, s_e,\vec{p}_\gamma, s_\gamma),

where s represents the spin of the electron or photon along e.g. the z-axis.  This is still a 6 dimensional space since s_e can only take the values (+1/2, -1/2), and s_\gamma can only take the values (+1,0,-1).  (Incidentally, since the photon is massless, its spin is always required to be perpendicular to its momentum, so there are really only 2 polarization states, not 3.  But the explanation of this involves relativity and gauge symmetry and a bunch of other things from QFT...)

There is also a third kind of basis, distinct in general from both the position and the momentum basis, in which time evolution is particularly simple.  This is the basis where the energy of the system takes on a definite value.  In this basis, the only thing that changes is the phase of each energy state.  The phase changes with time at a speed proportional to the energy.

So one way to specify the dynamics of a QM system is simply to say what the formula for the energy H is, as a function of all the positions and momenta of all the particles in the problem.  (You think of this an operator, a gadget which acts on the wavefunction to get another wavefunction.  So if you are in the position basis, the "momentum operator" is given by \vec{p} \Psi = (i / \hbar) \vec{\nabla} \Psi, which is equivalent to switching from the position to the momentum basis, multiplying by p, and then switching back.)  Then you can figure out how the wavefunction changes with time by using the Schrodinger equation:

\frac{d\Psi}{dt} = -\frac{i}{\hbar} H \Psi.

Thus, if you know what the formula for the energy is, you can predict the dynamics of the wavefunction as time passes.  This is related to the Hamiltonian approach in Classical Mechanics.  As you take the \hbar \to 0 limit, you recover classical mechanics.

There is also a "path integral" picture due to Feynman, related to the Lagrangian or "Least Action" approach to physics mentioned at the same link, where you assign to each history an amplitude proportional to e^{iS/\hbar}, where S is the action.  This approach is actually more closely related to the picture I started with in part I!

In this sense, Quantum Mechanics is a fulfillment of Classical Mechanics, just as (in Christian doctrine) the New Covenant fulfills the Old Covenant.  The new model explains the relevance of some of its quirky, previously unexplained features of the old model, in terms of a more basic (yet also more mysterious) set of ideas.  Concepts such as action, energy, momentum, the associated conservation laws, and so on, all follow naturally from the interference of wavefunctions over space and time.

Incidentally, there's a very important flaw in what I've told you so far.  Generally speaking, in modern physics it's better to think of the universe as being made of fields, not particles!  This is the subject of Quantum Field Theory.  The idea is that we should really think of the universe as being made of some finite number of types of fields (e.g. the electron field, the photon/EM field, the quark field, etc.).  Consider a scalar field \Phi(t,x,y,z), which is basically a function of the spacetime points.  If we want to keep track of the amplitude for any possible configuration of the field, then we really need our wavefunction to be a function of all possible configurations of \Phi at one moment of time.  Morally speaking (i.e. I am about to make certain dreadful oversimplifications) this means that the state of the universe at one time is something more like:


in other words the wavefunction is a function of functions!  The "particles" are then quantized excitations associated with different modes of this field.  (The relationship between QFT and the QM of multiple particles should not be obvious from what I've said so far.)

QFT gets kind of complicated, but the advantages are that 1) it is easier to make it compatible with Special Relativity, and 2) it allows one to consider situations where particles are created and destroyed, e.g. an electron can emit or absorb a photon.  Since this happens all the time in the real world, that's kind of important!

But as long as you're dealing with situations where the particles are all going much slower than the speed of light, and none of them decay into other particles, you can use QM as described above.  (Perhaps I shouldn't have given the photon as an example, because it always travels at the speed of light and is never nonrelativistic.)

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