THROUGH THE LOOKING-GLASS

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Lewis Carroll’s second novel about his young heroine, Alice (you know Alice…of Wonderland fame), is titled Through the Looking-Glass and What Alice Found There. The title is usually shortened to just “Through the Looking-Glass” but it is “what Alice found there” that is the really important part of this story.

You see, from our side of a looking-glass, the world on the other side looks just like our side with one pesky little difference: everything is reversed. The right hand becomes left and the left hand becomes right.

If you reach out your right hand to shake hands, the man in the mirror will reach out his left. If you write a note from left to right, the man on the other side of the glass will write the very same note; but his note will be written from right to left.

Otherwise though, everything is exactly the same on both sides of the glass…or is it?

In Carroll’s novel, Alice notices that she can’t see the fireplace in the room on the other side of the looking-glass. Is there a fire in it? There may be smoke “but that may only be pretense, just to make it look as if they had a fire”.

Of course, on our side of the looking-glass there is a big wide world; on the other side, Alice can only see most of one room and a small part of another. What else is there on that other side? Nothing else? Exactly what’s on our side (only reversed)? Or something altogether different?

Alice wonders: “’You know, it may be quite different beyond…How nice it would be if we could only get through into Looking-glass house’…In another moment, Alice was through the glass…and she was quite pleased to find that there was a real one (fire), blazing away as brightly as the one she had left behind(but) all the rest was as different as possible.”

It turns out that what you can see from our side of the glass is “quite common and uninteresting” but all the rest, what you can’t see from our side, is very, very different. Alice had been right to be suspicious of the mirror after all.

The portion of Looking-glass world that can be seen from our side of the mirror is “entangled” with our side (by our observation?). Handedness (technically known as ‘parity’) is reversed but everything else appears to be the same. If something goes right to left on our side, we know absolutely that it will go left to right on the other side and vice-versa. If we measure parity on one side (either side) of the mirror, we determine parity on the other side. Beyond this entangled region, however, the two worlds are apparently free to evolve independently…and in this story at least, they do!

So what’s so different? Different creatures, different events, to be sure! But most importantly, the fundamental structures of the two worlds are radically different: space and time just don’t work the same way on the other side of the looking-glass!

For example, Alice tries to walk to the top of a near-by hill and finds “a path that leads straight to it”. Except it doesn’t! Every time she tries, she ends up right back where she started, “wandering up and down, and trying turn after turn but always coming back” to where she began.

But Alice is a very clever girl, so she decides to try a new plan: ”…walking in the opposite direction. It succeeded beautifully. She had not been walking a minute before she found herself…in sight of the hill she had been so long aiming at.”

Toward the end of the story, the same thing happens again. Alice decides, “’I should like to buy an egg, please.’” But when she goes to collect her purchase, she notices that “‘the egg seems to get further away the more I walk towards it.’”

We are used to ‘orientable’ space. You move in one direction and you get where you’re going. On the other side of the looking-glass, however, space is ‘non-orientable’. You travel in loops (Mobius strips) and always return to where you began. If you want to get somewhere else, you have to reverse your orientation (“walking in the opposite direction”). Then you may loop ‘back’ to where you did not begin…which is where you wanted to get to in the first place.

Early on, Alice meets the Red Queen (yes, from a chess set) and together “they were running hand in hand, and the Queen went so fast that it was all she (Alice) could do to keep up with her…(But) the most curious part of the thing was, that the trees and the other things round them never changed their places at all: however fast they went, they never seemed to pass anything. ‘I wonder if all things move along with us?’ thought poor puzzled Alice.”

“Just as Alice was getting quite exhausted, they stopped…Alice looked round her in great surprise. ‘Why I do believe we’ve been under this tree the whole time. Everything’s just as it was!”

“’Of course it is’, said the Queen…’Well, in our country,’ said Alice, still panting a little, ‘you’d generally get to somewhere else – if you ran very fast for a long time as we’ve been doing.”

But the Queen had the last word, “’Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!’”

On our side of the mirror, all motion is limited by the speed of light. As long as that limit is respected, events unfold from the past into the future.

Particles that move faster than the maximum speed are called tachyons. Because tachyons move faster than the maximum speed, they travel backwards in time. Tachyons are really just ordinary particles moving at ordinary speeds, only backwards. Saying that something moves faster than ‘the cosmic speed limit’ is just another way of saying it’s travelling backwards in time. Where tachyons are involved, events unfold from the future toward past.

On the other side of the looking-glass, things are reversed. There is a maximum (but more concrete and intuitive) speed as well, “all the running you can do”. There, ordinary events, as we shall soon see, regularly move from future toward the past. If you want to reverse that order and move from past to future “you must run at least twice as fast…”

Interestingly though, both sides of the glass have one thing in common. If you move exactly at the maximum speed (light speed or “all the running you can do”), you simply stand still. On our side of the looking-glass, a photon traveling at the speed of light is stationary (from the point of view of the photon of course). It’s just the same on the other side: “it takes all the running you can do to keep in the same place” (from the point of view of Alice of course).

Later, Alice encounters the White Queen and her majesty is looking quite disheveled. Alice does her best to make the Queen presentable and then offers, “But really you should have a lady’s maid!”

“’I’m sure I’ll take you with pleasure!’ the Queen said. “Two pence a week and jam every other day.’” Alice protests that she does not care for jam, “Well, I don’t want any to-day at any rate.”

“’You couldn’t have it if you did want it,’ the Queen said. ‘The rule is, jam tomorrow and jam yesterday – but never jam today.’ ‘It must come sometimes to jam to-day,’ Alice objected. ‘No it ca’n’t,’ said the Queen. ‘It’s jam every other day: to-day isn’t any other day, you know.’”

“’I don’t understand you,’ said Alice. ‘It’s dreadfully confusing!’”

“’That’s the effect of living backwards,” the Queen said kindly…‘But there’s one great advantage in it, that one’s memory works both ways.’”

“’I’m sure mine only works one way,’ Alice remarked. ‘I can’t remember things before they happen.’ ‘It’s a poor sort of memory that works only backwards,’ the Queen remarked.”

“‘What sorts of things do you remember best?’ Alice ventured to ask. ‘Oh, things that happened the week after next.’ the Queen replied.”

The White Queen describes experience in Looking-glass world as “living backwards”; but she also says that Alice’s “poor sort of memory…works only backwards”. So we must conclude that all memory works in a direction opposite to the direction of time. When time is moving from past to future, we remember the past; but when time is moving from future to past, we remember the future.

Causality (cause and effect) runs both ways as well. The Queen gives an example: “’There’s the King’s Messenger. He’s in prison now, being punished: and the trial doesn’t even begin till next Wednesday: and of course the crime comes last of all.’”

“’Suppose he never commits the crime?’ said Alice. ‘That would be all the better, wouldn’t it?’ the Queen said.”

(I used to know parents on our side of the looking-glass who subscribed to this same philosophy. If an opportunity for misbehavior was about to present itself, these parents would punish their children in advance in the hope that the misbehavior might be avoided. Fortunately, this parenting strategy is not in vogue today.)

“Alice was just beginning to say ‘There’s a mistake somewhere –‘when the Queen began screaming.” Alice rushed to comfort her. “’What is the matter?…Have you pricked your finger?’ ‘I haven’t pricked it yet,’ the Queen said, ‘but I soon shall.’” And sure enough, a moment later, “the Queen had pricked her finger,” clutching at a brooch she used to fasten her shawl.

Later, “Alice had seated herself on the bank of a little brook, with the great dish on her knees, and was sawing away diligently with the knife…’I’ve cut several slices already but they always join on again!’”

“’You don’t know how to manage Looking-glass cakes’, the Unicorn remarked (oh yes, there are unicorns beyond the looking-glass as well; did I forget to mention that? And they talk!). ‘Hand it round first, and cut it afterwards.’ This sounded nonsense, but Alice very obediently got up, and carried the dish round, and the cake divided itself into three pieces as she did so. ‘Now cut it up,’ said the Lion…” In Looking-glass world, future actions (“cut it afterwards”) can cause past events (“the cake divided itself into three pieces”).

So yes, time, memory and causality do all run both ways on the other side of the looking-glass; but something even stranger is afoot here.

Remember back to the matter of the jam? Alice explained to the Queen that she did not like jam, “Well, I don’t want any to-day at any rate.”

And remember what the Queen said? “’You couldn’t have it if you did want it…the rule is, jam tomorrow and jam yesterday – but never jam today.”

Now, further on in our story, Alice finds herself in a shop and “the shop seemed to be full of all manner of curious things – but the oddest part of it all was that, whenever she looked hard at any shelf…that particular shelf was always quite empty”.

In Looking-glass world, there is no now (“jam to-day”), only past (yesterday) and future (tomorrow): It never comes to “jam to-day”. And there is no here, only there: “that particular shelf was always quite empty”. Here-and-now, the present, simply does not exist.

But this raises a problem. How can there be a world where everything is then, never now, and there, never here? There cannot be such a world! At some point, something actually has to ‘be’; if nothing ever is than nothing ever was and nothing ever will be.

So where does that leave us? It turns out that Looking-glass world is totally dependent on our world after all. The world on our side of the looking-glass supplies the missing but desperately needed here-and-now, aka ‘the present’.

In fact, just as there is no here and no now in Looking-glass world, there is only here and only now in our world. Appearances and assumptions notwithstanding, the past and the future exist on our side of the mirror only in so far as they are immanent in the present as cause or goal (causality/teleology). Likewise ‘there’ exists only in so far as it is somehow felt ‘here’.

But just as there cannot be a world of there-and-then without here-and-now, so too there can be no non-trivial world of here-and-now without there-and-then. Otherwise every event would come to be ex nihilo and that is certainly not the way our world seems to work.

So where does this leave us? It turns out that our world is just as dependent on Looking-glass world as Looking-glass world is dependent on our world. We supply the here-and-now; they supply the there-and-then. Neither world can survive without the other.

Now at this point you might shrug your shoulders and say, “So what? Of course, we need a here and a there and a now and a then. What’s the problem?”

The problem is that neither here and there nor now and then can co-exist. The world of there-and-then obeys the laws of classical physics, including General Relativity. The world of here-and-now does not obey those laws; instead it obeys the ‘laws’ of Thermodynamics and of Quantum Field Theory.

On the other side of the looking-glass, time is indefinitely divisible and past and future are interchangeable (time can flow either way). On our side of the mirror, ‘the arrow of entropy’ defines the sequence of events, and quantum de-coherence, the collapse of the wave function, shatters the classical symmetry. Entropy and presence polarize time. On our side of the looking-glass, time, memory and causality only flow one way.

The fact is, the world we experience is both classical and quantum mechanical. But try as they might, scientists have been unable to come up with an overarching ontology that will reconcile the two. If we are to believe Lewis Carroll, they never will!

Through the Looking-Glass proposes an entirely different solution. The classical world of there-and-then co-exists symbiotically with the quantum world of here-and-now. But to maintain logical and physical consistency, the two worlds must be separated by an ontological membrane. The looking-glass is Carroll’s metaphor for that membrane.

The two sides of the looking-glass (membrane) are what’s called ‘complementary’. The two worlds cannot coexist yet neither world could exist without the other. Space needs to be both orientable and non-orientable; but the same space can’t be both. Time needs to be both polar and non-polar; but the same time can’t be both.

Membranes play an important role throughout modern physics. One sort of membrane is the event horizon of a black hole. Recent theories concerning black holes suggest that there may be some sort of connection between particles on our side of the horizon and particles on the other side. Perhaps the particles are ‘entangled’ (as we discussed above) or perhaps there are worm holes connecting every particle beyond the horizon with a sister particle on our side of the horizon.

Or perhaps there is nothing at all on the other side the event horizon. Perhaps the event horizon is the boundary of spacetime. Even so, the reality of our world may be somehow dependent on the ‘imaginary’ world inside the black hole. That imaginary world may be encoded in the membrane (event horizon) much as the world beyond the actual mirror in your living room is empty but ‘encoded’ in the mirror itself.

While in Looking-glass world, Alice encounters the legendary Humpty-Dumpty. Carroll did not invent HD; he seems to date back at least 100 years earlier; but Carroll exploited Humpty to further his message. (Who hasn’t exploited that good egg?)

HD and Alice engage in banter until out of genuine concern for Humpty’s wellbeing, Alice offers, “Don’t you think you’d be safer down on the ground?…that wall is so very narrow!”

In response, “Humpty Dumpty growled out. ‘Of course I don’t think so. Why, if I ever did fall off – which there’s no chance of – but if I did…the King has promised me – with his very own mouth…’”

Here Alice interrupted, “’To send all his horses and all his men…’”

A little while later, just as Alice was leaving, “…a heavy crash shook the forest from end to end. The next moment, soldiers came running through the wood, at first in twos and threes, then ten or twenty together, and at last in such crowds that they seemed to fill the whole forest.”

On our side of the Looking-glass, the story of Humpty Dumpty is generally used to illustrate the proposition that it is impossible ever to restore things to the way they once were. Specifically, the story is often cited as an example of thermodynamic entropy but it could also be used to illustrate the boundary between quantum coherence (HD before the fall) and decoherence (HD afterwards).

What did Carroll have in mind? Was he was being ironic, making fun of HD’s naïve confidence and the King’s glib assurances? Upon reflection, the King did make good on his famous promise. Perhaps beyond the looking-glass, Humpty Dumpty can be made right again. Entropy and quantum decoherence apply only on our side of the looking-glass after all.

This brings us back to Alice’s fundamental question: what exactly is the relationship between the two sides of the glass? Early on, Alice is introduced to the idea that the two worlds are somehow connected by dreaming. While visiting Tweedledum and Tweedledee, Alice encounters the Red King. He is sleeping with “a tall red night-cap on”.

“’He’s dreaming now,’ said Tweedledee: ‘and what do you think he’s dreaming about?’”

“Alice said, ‘Nobody can guess that.’”

“Why about you!’ Tweedledee exclaimed…’And if he left off dreaming about you, where do you suppose you’d be?’”

“’Where I am now, of course,’ said Alice.”

“’You’d be nowhere. Why you’re only a sort of thing in his dream!’”

“’If the King were to wake,’ added Tweedledum, ‘you’d go out – bang! – just like a candle!’”

In our orientable space, there is an absolute ontological barrier that prevents us from knowing the contents of another’s mind (“nobody can guess that”). But in the non-orientable space of Looking-glass world, the contents of another’s mind (e.g. dreams) are on full display for all to see.

On our side of the looking-glass, inside and outside are absolutes. But in Looking-glass world there is only one ‘side’ (think Klein Bottle).

At the end of the story, when Alice is safely on her own side of the glass again, she ponders her experience:

“’Now, Kitty, let’s consider, who it was that dreamed it all…it must have been either me or the Red King. He was part of my dream, of course – but then I was part of his dream too!’”

Carroll ends the story with these final words: “Which do you think it was?”

At first glance, this ending suggests that Carroll is expecting the reader to say, “Alice, of course; Alice dreamt it all. Chess pieces don’t dream.” But is this answer really consistent with the rest of Carroll’s tale?

It seems a better answer would be both. Alice dreams the Red King as the Red King dreams Alice. Alice is dependent on the Red King, as Tweedledee professes, but the Red King is just as dependent on Alice.

We find such a concept difficult to express in our language, comprised as it is of active and passive verbs and subjective and objective nouns. Many ancient languages incorporated a ‘middle voice’ that eliminated the subject/object dichotomy; these languages were much better suited to express the symbiotic relationships that lie at the heart of modern ontology.

Lewis Carroll (nee Charles Dodgson, 1832 – 1898) did not intend Through the Looking-Glass to present a comprehensive, coherent and consistent cosmology. In searching his pages for deeper meaning, I have taken the liberty of introducing concepts that were entirely unknown in Carroll’s time. He may have intuited some pretty deep ideas but he lacked the tools to translate those intuitions into actual theory.

Carroll was a distinguished mathematician, but in what might be considered the Dark Ages of modern physics. In Carroll’s time, physicists believed that they knew pretty much everything there was to know about how the world works; in fact they knew almost nothing. And the world was to discover this rather quickly over the next 30 years.

Carroll knew almost nothing of Atomic Structure, much less Relativity or Quantum Mechanics. But he may have sensed that something was missing from the pseudo-complete physics of his day. Though the Looking-Glass hints at things that decades later became accepted science…and at other things that are still under investigation today.

My admittedly sympathetic reading suggests that Carroll may have used the fantasy genre to challenge the accepted cannon, and point at possible holes in its fabric, without confronting it head-on in what would have been a futile intellectual exercise.

Perhaps Through the Looking-Glass should be read as Carroll’s note in a bottle to future generations. If so, it might translate something like this:

“There is something wrong with today’s accepted Theory of Everything. I don’t know exactly what it is but I have few roughly worked out ideas. Anyway, right now we lack the mathematical and experimental tools needed to correct it. But let Alice’s adventures be a cautionary tale for you. She thought she knew how the world worked…but she found out otherwise. Pray that you will be as open to new experiences and their lessons as Alice was; because I can assure you, they’re coming!”

And come they did!