Alice in Quantum Land

A Quantum's Eye View

“Good morning, Alice,” a voice said. Or at least it seemed like a voice.

Alice rubbed her eyes. She'd fallen asleep in the orchard, and had been dreaming of tea parties with singing cakes and dancing oysters. She rubbed her eyes and looked around, but there was no one in sight. Had the voice been in the dream, she wondered? Or maybe she was still dreaming. She'd caught herself enough times in that trap, thinking she had woken up, only to discover she was still dreaming. It always annoyed her.

“Good morning, Alice.” There it was again. But where was it coming from? Alice had become used to voices that came from strange and unexpected places, or were disconnected from the people or things who were speaking, but not voices that came from nowhere.

“Good morning,” replied Alice cautiously but politely, not wanting to upset whoever, or whatever, this might be. “Who are you? Or more to the point, where are you?”

“I'm a quantum,” the voice continued. “You've been hearing a lot about quantum physics and all the strange conclusions that it leads to in your world, so I thought it was time you heard from me, and got a picture of how the world looks from a quantum's point of view.

“As to where I am, I am everywhere and nowhere. Always and nowhen.”

Alice knew better than to let her mind be worried by paradox. Just about everything she had heard so far was paradoxical in some way or other, and trying to understand paradoxes was bound to lead to even greater confusion.

“Let me introduce myself,” it continued, “and all the other zillions of quanta in the universe, for in many ways we're all exactly the same.

“Each of us is the smallest possible packet of energy in the universe. Any transfer of energy, whether it be from one electron to another in an atom, or from the sun to your skin, involves a whole number of us quanta. There may be 1, 2, 5, 117, or 19,387,463,728 of us, but never half a quantum or three-and-a-quarter quanta. That would be like you having a conversation with half a person, or three-and-a-quarter people.”

Alice wondered whether she could imagine having a conversation with three and a quarter people. Three-and-a-quarter bodies, perhaps -- she'd met stranger situations than that -- but three-and-a-quarter people, she was not so sure. But before she had a chance to try imagining a fraction of a person, the voice from nowhere was back.

“In your world you also call us photons -- the smallest unit of light.

“Now when I speak of light, I am talking not just of the visible light you see with your eyes; I mean the whole spectrum of electromagnetic radiation of which visible light is just one tiny range of frequencies. At higher frequencies are ultraviolet light, X-rays and, beyond them, gamma rays. At lower frequencies you find heat waves, and at the lowest frequencies of all, radio waves. All of them are just different frequencies of light. And they are all composed of photons, each one a single quantum.”

“Then why did you say you were all the same?” asked Alice. “Light has many different colors; heat I can feel on my skin; and I've been told to keep well clear of gamma rays. They all seem very different to me.”

“That is because the energies we carry vary enormously. The higher the frequency, the higher the energy. A gamma-ray photon, for example, packs billions of times more energy than a radio-wave photon. This is why gamma rays, X-rays, and even ultraviolet rays to some extent, can be so dangerous to you. When these photons hit your body, the energy released can blow apart the molecules in a cell. When heat radiation is absorbed by your skin, the energy released is much, much less, and all it does is warm you up a little.

“However, although our energies vary enormously, there is one thing about us that is always the same. We all, each and every one of us, possess exactly the same amount of action.”

“What,” Alice was about to say, “is action?” But before she had even finished thinking “What,” the quantum said, “I thought you might ask that.

“You're familiar with the terms mass, velocity, momentum and energy, I presume?”

“Yes,” thought Alice. She remembered learning about them at school.

“And you learned how they relate to each other. An object's momentum, for example, is its mass multiplied by its velocity. And work is energy multiplied by distance. Action is just another one of these qualities, but it is not one you normally hear about at school.”

“The amount of 'action' in any action is defined as the object's momentum multiplied by the distance it travels. Or it can also be expressed as the object's energy multiplied by the time it is traveling.”

“Imagine someone throwing a ball.” Suddenly, out of nowhere, the White Rabbit appeared, running around the orchard throwing large orange tennis balls into the air. “Some imagination!” thought Alice.

“If he were to throw the balls twice as fast, would there be more or less action?”

“More, of course.”

“Twice as much?”

“I'd think so.”

“And if the balls were much heavier, like croquet balls, would there be more or less action to his action?”


“And if he ran around for twice as long, how much action do you think there would be?”

“Twice as much, I suppose.”

“So the concept of action isn't really that strange, is it?”

“No,” replied Alice, wondering why she had never thought about action in this way before. And why hadn't she heard about it at school? Maybe it hadn't been important?

“Oh, it's very important,” said the voice from nowhere. “Your mathematicians have discovered that whatever happens in the universe happens in such a way that the total amount of action is always the lowest possible. It's what they call 'The Principle of Least Action.' And your scientists use it all the time to predict how things will happen. Those balls the White Rabbit is throwing trace out a curve in the air, yes? Well that curve happens to be the one that involves the least amount of action. Any other curve you could imagine would require more action.”

“A sort of cosmic efficiency principle,” thought Alice.

“Yes. And it apples to everything. Even light. When you see a reflection in the looking glass, the light comes back to you at the precise angle that involves the least amount of action.”

“Hmm, I'm beginning to see why action is so important.”

“Yes, it's absolutely fundamental. And, as I was saying, every single quantum in the universe, every photon, whatever its frequency and energy, is an identical unit of action. The amount is exceedingly small -- after all, we're very, very, very tiny. In your units of measurement, each of us is about 0.00000000000000000000000000663 erg-seconds. And before you even think of asking what an erg is, it is a unit of energy, a very small one. To lift a one-pound croquet ball a distance of one foot takes about 13.5 million ergs. If you took one second to lift the ball, your action would have involved about 13.5 million erg-seconds. Now each of us quanta is a tiny, tiny, tiny fraction of an erg-second -- point zero zero zero zero zero zero zero zero . . . ”

“Stop, please. I get the picture. You are a very, very, very, tiny unit of action.”

“Yes, the smallest possible action in the universe. It's called Planck's constant, after Max Planck, who first discovered us. Each one of us, each and every one of us, is exactly this amount of action.”

Alice thought about this for a while. “Light is action,” she mused. “I'd never thought of it like that before. But I suppose it sort of makes sense. After all, light never stops moving. It can travel right across the universe, and at great speed. Light never rests, it never slows. Yes action seems kind of appropriate.”

“Not so fast,” the quantum interrupted. “That may be how you see light, but we see ourselves very differently. As far as we are concerned, we don't ever experience ourselves traveling anywhere. We never move at all.”

“Now, that's ridiculous!” cried Alice. “I'm used to paradoxes in this quantum world of yours, but how can you say you never travel anywhere when you so clearly do? If you never go anywhere, how come light gets to us from the sun, and how come light has speed?”

“Hold your horses, my dear, and I'll try to explain. But first I'll need to take you on a little excursion into the theories of another of your great scientists, Albert Einstein.

“Like many other scientists of his time, Einstein was puzzled by the fact that light always seemed to travel at the same speed, no matter how fast you might be moving. At first this seemed nonsense. If you were to walk along at 3 mph, and the White Rabbit ran by at 7 mph, simple arithmetic tells you he'd be going 4 mph faster than you. If you speeded up and ran along at 7 mph you'd be able to keep up with him, and there would be no difference in speed. But light didn't seem to behave like this at all. Experiments showed that however fast you go, you can never catch up with light; it always passes by at 186,000 miles per second. Even if you were to travel at 185,990 miles per second, light would still whiz by 186,000 miles per second faster.”

“Faster! Faster!” the Red Queen's voice echoed through her mind, along with images of chessboards and talking lilies. Alice remembered what it was like to never get anywhere however fast you ran. “Was the Red Queen a friend of yours?”

“No, but maybe young Albert had read about your adventures with her.

“After a lot of thought he decided to accept that you could never catch up with light, however fast you went. It is just the way the universe works, however non-sensical it might seem. This led him to his famous 'Special Theory of Relativity,' and to some conclusions that at first seemed even greater nonsense.

“His equations predicted that the faster something went the more slowly its clocks would run. The precise relationship between speed and time is not a straightforward one, and I won't bother you with the detailed mathematics, but the result is that if you were to travel past someone at 87 percent the speed of light, they would observe your clocks to be running at half the speed of theirs. This slowing applies not just to clocks, but to all physical processes, all chemical processes, and all biological processes. Your whole world would run at half the rate of theirs.”

“Sounds more like the looking glass world than my world.”

“Well, it turns out that your world really is a bit like the looking glass world. Scientists have flown clocks around the world on jets and found that they do indeed run slow -- by a factor of about one in a trillion -- not enough to worry anyone, but enough to prove that Einstein's theory is correct.

“And it's not just time that shrinks. Space is also changed. Lengths measured in the direction of motion become shorter, and in exactly the same proportion as time slows. If you were to travel a measured mile at 87 percent the speed of light, you'd measure the distance to be only half a mile.”

“You mean space and time really aren't fixed after all?”

“Right, they're not as absolute as people had thought. How much space and how much time you observe is relative to your speed. That's why Einstein called it 'relativity'.

“But he also discovered that not everything about time and space was relative. People moving at different speeds may disagree on how much space and how much time they observe, but they all agree on the total amount of space and time.”

Alice thought it must be a bit like cutting a string in two. Cutting it in different places would give pieces of differing lengths, but the total length of string would always be the same.

“Exactly. Or rather, not exactly. Space and time don't add up by simple arithmetic. In fact, you get the total by doing a subtraction.”

“Doing addition by subtraction! Now that's the sort of arithmetic the Red Queen would like.”

“But it isn't simple subtraction,” the quantum continued, “the mathematical formula for combining space and time is more complicated than that. It's something like 'the square root of space squared minus time squared.'”

“I think I'll skip that. I'm confused enough as it is. But what's all this got to do with light, and you saying that light never travels anywhere?”

“Well, the equations of relativity predict that at the speed of light, length shrinks to nothing, and time slows to a complete standstill.”

“You mean space and time just disappear? That is bizarre.”

“Yes, and it's quite troublesome to your physicists. Their equations of motion get littered with zeros and infinities, and it's very hard for them to make much use of them. So they usually ignore this extreme case, consoling themselves with the thought that because nothing can ever actually travel at the speed of light, they don't have to worry about these bizarre effects.”

“Why do you say things can't travel at the speed of light?” Alice asked, sensing a possible contradiction.

“Ah, that's because not only space and time change with speed, but so also does mass. Whereas space and time decrease with speed, mass does the opposite. The faster you go, the heavier you become. If you reached the speed of light, your mass would become infinite.”

Alice tried to imagine having an infinite mass. Being very, very heavy she could just about handle. But infinitely heavy? She couldn't even imagine infinity, let alone an infinite amount of anything.

“Don't worry. You'll never go that fast. To move an object of infinite mass would take an infinite amount of energy. A lot of energy might get you close to the speed of light, but there simply is not enough energy in the whole universe to accelerate you all the way up to light speed. That's why it's impossible for anything to ever travel at the speed of light.”

“But some things do travel at the speed of light,” interjected Alice, pleased that she had caught the quantum contradicting itself. “You, for example, travel at the speed of light.”

“Of course. To say that light couldn't travel at the speed of light would be pretty ridiculous, wouldn't it? But light is not really a 'thing' as you think of things. Photons have no mass at all. Each of us weighs absolutely nothing -- no matter how fast we go. Even at the speed of light, we still weigh absolutely nothing.”

“So you aren't subject to the same cosmic speed limit as we are.”


“And so you always travel at the speed of light.” Alice proudly concluded.

“On the contrary. We never travel at any speed.”


“No, that's just how it appears to you in your world. On our side of the quantum looking glass, things look very different.

“I said that at the speed of light distance and time shrink right down to zero. Well, that means that, from our point of view, we never experience ourselves traveling any distance whatsoever. In your world you see us traveling through space, but at the speeds we travel space has become so warped there is no distance between where we start and where we end up. And since our clocks have slowed to a standstill, we never take any time at all. We go nowhere in no time.”

“Makes the Red Queen seem positively sane.”

“The Red Queen was still living in the world of things, the world of space, time and matter. We quanta live in a very different world. We are not things. We have no mass, we never travel any distance, and we know no time. So, because we travel no distance in no time, the notion of speed is meaningless for us. In our frame of reference -- and what frame of reference could more appropriate for light than our own -- we have no need of speed.”

“But I thought Einstein said that the speed of light was the same for all observers. How can you say you have no speed?”

“What you think of as the speed of light is from our perspective something very different. You remember me saying that all observers always agree on the total amount of spacetime separating two events, even though they disagree on how much actual space and how much actual time they observe?”


“Well, when you calculate the total amount of spacetime between the two ends of a light beam the result is always exactly zero. This is because the total is arrived at by that complicated formula that involves 'space-squared minus time-squared'. For any photon, anywhere in the universe, the amount of space it appears to travel is exactly balanced by the amount of time it appears to take, and when you subtract the two, they cancel each other out, leaving a total of zero.

“This is something even we photons agree upon. Except that we don't experience a beginning and an end. We observe ourselves traveling zero distance in zero time. Subtract zero from zero and what do you get?”

“Zero, of course.”

“Exactly. However you look at it, the combined amount of spacetime for light is always zero.”

What felt like waves of significance passed through Alice. She felt that this was somehow very, very important, but she had no idea how or why.

“In your world,” it continued, “you observe a separation between the beginning and end of a light beam. The zero spacetime of light has manifested as some space, along with some time. Since the total must remain zero, the amount of space that appears is exactly balanced by the amount of time that appears.

“What you observe as the speed of light can be thought of as the ratio of manifestation of time and space. For every 186,000 miles of space, there appears 1 second of time. It is this ratio that is fixed. This is why the so-called 'speed' of light in your world is always the same.”

Alice didn't quite know what to think. She sat back and tried to imagine what it would be like to be light. She tried imagining space and time disappearing, but it didn't work. However hard she tried, space and time would not go away. Maybe that's just the way the mind thinks, she thought.

Then she tried the opposite, trying to imagine nothing, and to imagine that nothing being stretched out into space and time. But that was just as difficult.

She was just about to give up trying to understand any of this when suddenly the thought came to her that if light doesn't experience itself traveling anywhere, then what's all this stuff about light being both a wave and a particle? “It can't be either!” she exclaimed

“You're catching on fast. A wave that traveled nowhere would be ridiculous, wouldn't it? So would a particle that existed for no time at all. Waves and particles are concepts your scientists use to try to understand us in their world. They are both 'thing' words, but we quanta aren't things at all. Trying to make us seem like things is why your scientists find us so puzzling and paradoxical. They are seeking to understand us from their world, the world of time, space and matter. But that's the source of their problems. We have no mass; we don't inhabit space and time. We don't belong to the world of things. We belong to the world of light. If they could step into the world of light they would realize that there is no paradox at all.”

Alice lay back, and closed her eyes. There was the Red Queen again, with that incessant grin. Or was it the Cheshire cat's grin? And why was the Red Queen, or the Cheshire cat, or whoever she was, wiggling like that? Her edges grew fuzzy, her colors blurred and her crown began to turn into golden light. Before Alice knew it, the Red Queen had completely dissolved. And so had everything else. There was nothing in her mind but light. No thing. Nowhere that was anywhere else. No time that was not now. Just light.

“Is this the same light?” she wondered? “Is the light I see in my mind the same light as the light I see in the world?”

“What else? But I think we should leave that for another time,” she heard the voice from nowhere say. But at least she now knew where the voice was coming from. It was coming from everywhere.


A quamtum shows Alice the world from the point view of light.

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