The professor was heating up, his eyes flashing with proselytizing fire. By now, Helias was propping his chin up on his knees.
“The plusvariant formalism and the Copernican revolution thus have the same meaning in the epistemology of physics. Just as the solar system can be simply and symmetrically described if it is ‘seen’ from its center of mass—the Sun, approximately—which can be considered stationary in space relative to the system, so the physical world can be viewed and described more clearly and symmetrically from the standpoint of the photon, which, relative to us, is stationary in time. It’s a sort of bank standing high and dry between two canals that flow in opposite directions. A bank you have to cross if you want to reverse your direction in time.”
The professor lapsed into silence, and stood contemplating the tips of his shoes. Helias, fascinated, hardly dared draw breath.
“Are you familiar with the CPT theorem in relativistic quantum mechanics, young… Dr. Kadler?”
“Y…yes…. I believe I’ve understood….”
“Excellent. Tell me….”
“If I recall correctly, a CPT transformation…. No, let’s put it this way: the laws of physics are invariant for CPT transformations. In other words, if we have a system or an event and we apply a charge conjugation to it, and also a parity operator and a time reversal, we obtain an event described by the same physical laws.”
“Yes, you’ve expressed it poorly, but the concept is more or less that. And so?”
“And so I’d say that CPT symmetry is closely related to what you’ve said so far. If I’m not mistaken, it would mean that the description of an event in which a particle goes backward in time is equivalent to one where an antiparticle goes forward in time. But I don’t remember what parity has to do with it. And what’s the connection between charge conjugation and antichronous transformations?”
“Well, without going into the details, the CPT operation corresponds to the antichronous transformation called total inversion ‘’ combined with the reinterpretation principle. At this point, then, we should be convinced that the two descriptions, the particle going backward in time and the antiparticle going forward in time, are entirely equivalent, at least as far as special relativity and quantum physics are concerned. But what happens if, instead of considering single particles, we consider a macroscopic system of particles, one that’s also subject to other laws, like those of thermodynamics?”
By now, Helias was gnawing on his cuticles. Something he had never done in his whole life.
“Right, so if I’ve understood correctly, the two situations, the antiparticle going forward in time, and the particle heading toward our past, are indistinguishable from the standpoint of special relativity and quantum mechanics, precisely because at that microscopic level it’s not possible to identify a ‘preferential’ time direction, an arrow of time, as on the contrary is the case for macrosystems that have precise thermodynamic evolution, the only one possible, marked by increasing entropy.”
“Exactly. That’s the concept, though you’ve put it a bit picturesquely.”
“There’s another thing I’ve not quite grasped: why total inversion? Why invert all three spatial axes? For a parity transformation, which in practice provides a mirror image of the phenomenon, isn’t it enough to invert just one axis?”
“To a certain extent you’re right. You can invert all three or only one, but not two: the transformation would be improper, it would have a negative determinant. In the world of elementary particles, it doesn’t make any difference whether you have one or three: there’s nothing that distinguishes between ‘up’ and ‘down’. But if we take a system with non-negligible mass and we ‘switch on’ a gravitational field, then it makes sense to invert the other two axes as well.”
“So what you’re telling me is that if there were a macroscopic system going backwards in time relative to us, we would observe it not only with a reversed time evolution—as is to be expected, obviously—but also as a mirror image and upside down. Let’s take an example. Say we have a pot, a plain old-fashioned pot, the sort of thing you’d put flowers in, made of earthenware or glass, we put it on a table, on the left edge of the table, and we somehow send the whole setup back in time. What we observe then is an overturned table, with the pot ‘glued’ to it, upside down, on the right edge of the table which is now on our left. Or no, actually, seen from behind. Unless we walk around it. Is that right?”
“Yes. Except for the fact that if we, now, send it back in time, we would stop seeing it, whereas we would have seen it in our past. Go on.”
“Of course, you’re right. So let’s say that our table-pot system (and while we’re at it, we’ll also add a floor) comes from our future. Suppose that at a certain point, still in our future, say, in a few moments, an accident of some kind makes the pot fall off the table, and it shatters on the floor. And so, in a few moments, we will see all those broken bits collect together on the floor above the table and reassemble into a perfectly sound pot that would calmly rise off the floor and sit on, or rather under, the tabletop. Is that right?”
“Very good. You’ve just given a description of the principle of general CPT invariance, called CST invariance, whereby all the laws of physics, and not just those of quantum mechanics, are invariant for joint CST, transformations where T is the time reversal, and S the space inversion, which, together, correspond to the Lorentz total inversion ‘−1’, and where C is no longer only the charge conjugation, but the scalar quantities, like rest mass and proper time, also change sign. Consequently, our anti-pot, repelled by the gravitational field, sits happily below its anti-table, perfectly comfortable. And up to here the description ‘antimatter going forward in time’ would still be valid and indistinguishable from matter going backward in time. The same is true if our anti-pot were unbreakable and its impact against the floor were perfectly elastic: we would have an anti-pot that, after infinite equal rebounds, would come to rest exactly on the table, transferring momentum and kinetic energy to the elastic body that made it fall. Strange, but mechanically possible. But if the impact is not elastic, and the pot breaks or at least is distorted, or in other words if energy degrades and entropy increases, changing sign for ‘dτ’ would indeed save the laws of thermodynamics and everything would be formally okay, the formulation of the law is not changed, but can we still believe it? Do we still believe that we are dealing with antimatter that travels with us in time? Would an anti-floor really have the power to challenge all the laws of probability, giving each of the anti-pot’s fragments the right momentum to go reassemble in the exact position where the anti-molecules would stick to each other, and finally give a good kick to the whole pot so that it comes to rest on the anti-table? No living anti-being could do that, let alone mere chance. Another arrow of time is the only, obvious solution.”
The professor was visibly tired from the discussion. Helias even more so.
“An ice cream, my boy?”
Helias ate half of his ice cream in silence. The professor had already finished his. He had devoured it, practically attacking it.
“When did you discover all this? I mean, when were you able to ‘produce’ enough antimatter to allow you to observe its thermodynamic behavior?”
“A long time ago. And thanks above all to the plusvariant formalism, which enables us to find the most direct way of exchanging energy with the photon state. In addition, we already knew that it was a question of reversing the time arrow of something that already existed, and not of producing something new, and this was a major advantage.”
The professor remained in silence for a while, bowl in hand. He seemed to be weighing whether to have another helping of ice cream or not. His conscience quickly got the better of him, and he put the bowl back on the table, with an air of regret. He stood musing for a little longer, looking down. Then he raised his face to Helias, eyes sparkling in amusement.
“You gave a very nice example, with the breaking pot. Now I’m going to give
you one that’s even more challenging. Let’s say we have an anti-man. Rather, it’s you and me, in the same room. Each of us is going toward his own future, but in opposite directions. Each of us is aging, but sees the other getting younger, head where the other’s feet are, and heart on the right hand side. The room, let’s say, is on my side, together with the air it contains. You have your armchair and the air you need to breathe. The whole business, obviously, is surrounded by magnetic fields that isolate the two systems, otherwise they’d wipe each other out. You’ve arrived a short time ago, where you came from and how you got here doesn’t matter. In reality, when you appeared you were going away. I know now that you’ll be leaving in a few minutes, because I saw you do it a few minutes ago. But I don’t yet know anything about our meeting, or how long it will last. You, however, know everything, because it belongs to your past. As my time passes, I will come to know about the meeting, while you will lose that knowledge. We look at each other, but I don’t know if we can communicate verbally. Let’s suppose we can: the magnetic fields can transfer the sound waves. At a certain point, my eardrums begin to vibrate: they’re the last things you said to me, the words are backwards, incomprehensible. Perhaps they reach me even before you pronounced them, it depends on how far I am from the magnetic field, beyond which the sound waves travel backwards, until they make your vocal cords vibrate. I don’t know how long you’ve been talking to me, but I record it all, so that I can look at it and listen to it at my convenience, backwards. You talk for a long time. You’ve probably repeated the same things, because otherwise I would have missed the last parts, before starting to record. At a certain point you stop talking, that’s the moment when you had started. I stop recording, very noticeably, so that in this moment you will know I recorded you, or rather, that I will record you: it’s the signal for you to begin talking. You do the same thing, you turn on the camera, or rather, you’ve just turned it off: and that’s the signal for me, it means that I will speak to you, rather, I’ve spoken to you. To make sure, you show me a sheet of paper where you’ve written, ‘Now you talk, and then repeat what you said’. Then I begin to talk, I introduce myself and I tell you a few things, twice. A bit before I finish, you turn off/on the camera, so that I know I am about to finish, and in fact that’s the way it is, because you, like me earlier, have turned on the camera after you heard me finish talking. And so, after waving to me, you leave relative to my timeframe, whereas for your own, you’re just arriving. Right. I’ll let you think about this for a while. When I come back, I’d like to hear your comments about this business.”
The professor, taking the two bowls, left the room. Helias knew what he was going to do. And he took advantage of the time to go to the lavatory.
The professor was still licking his lips when he returned. Helias had curled up again in the armchair.
“First, I’d say that we would have communicated more efficiently in writing, with messages prepared and left at appropriate times. This, though, is only a technical detail that doesn’t change the substance of what you described to me. But before I explain my doubts to you, tell me if things like this actually happen.”
“No. At least to date, and as far as I know. As you’ve been informed, we’ve forbidden ourselves from interfering with the past. So if this prohibition continues in force, we cannot receive information from the future. It was just a little intellectual exercise, invented on the spur of the moment. There are people who study these things much more seriously and in greater depth. But we can learn something even from this little example. Let’s hear what you have to say about it.”
“Well, I’m certainly quite perplexed. All of this clearly runs counter to every notion of causality and free will. Not that this frightens me, mind you, but for most scholars on Earth this would all be repugnant, to say the least. Take, for example, the message, ‘Now you talk’, and so forth, that I show you at a certain point. But why do I show it to you? I already know that you’ve talked, or that you will talk, I’ve even recorded you. For me, it’s already happened, by now it’s part of an indelible past, I’ve also got the proof. On the one hand, what need do I have to urge you to do it if I already know that you’ll do it anyway? On the other hand, if I don’t do it, you might also not decide to talk, even if that seems impossible. In other words, you have no choice, you will definitely do it, regardless of what I do, and I can always decide not to show you the message. And this is just an isolated example. In actual fact, the whole thing seems to be like an unalterable script that both of us have to stick to, without being able to do otherwise: what’s become of our free will? So it doesn’t exist? Is everything already in the script and the only thing we can do is to act our part and speak our lines? So it would seem, since the two of us can’t do anything except what the other one has already seen us do.”
“Good. You’ve hit the crux of the problem. Except that you’re not able to shake loose a ‘single-time’ outlook that sees the future as something that is yet to come and can still be changed. Let me explain that better. In our example, I, and I alone, have chosen to speak, I didn’t stick to a script that was already written. You induced me to speak: it was your future choice to induce me to speak in your past. Both of us made our choices, and the result is what happened. We just have to liberate ourselves of the idea that causality has only one time direction. In other words, causes lie not only in the past, but in the future as well. What you chose in the future—to show me the message—influenced what happened in your past, it induced me to speak. It’s still a matter of cause and effect, even though in one of the two time directions it is necessarily seen the other way round. It always happens when you’re dealing with time travel, of any kind. Take the famous ‘grandfather paradox’, where a time traveler has to choose whether to kill his own grandfather when the latter is still a little boy: were he to do so, he would never have been born and he would not even have been able to kill his grandfather. The solution is not that time travel doesn’t exist, but simply that the time traveler chose not to kill his grandfather: a future choice that affected his entire past, his very existence.”
They both remained in silence, while Helias pondered all those new and amazing things, trying to assimilate it all. It was the professor who spoke first.
“There’s another important physical problem hidden in our example, which you may not have noticed. We said that the two of us, you and I, speak to each other in some way. In other words, we assumed that the sound waves can somehow propagate from one system to the other. Let’s not worry for the moment about the details of crossing the magnetic barrier, but focus on ‘before’ and ‘after’. You speak, and the sound waves produced by your vocal cords propagate around you, moving away from you as your time moves on. Simultaneously, as we said, if I could visualize the phenomenon I would see those waves traveling backwards in your direction until they make your vocal cords vibrate. What happens when the signal ‘passes’ into my system? Well, it must necessarily obey the thermodynamics on my side, in accordance with my time arrow. And so it will propagate normally, forward in my time, and I will receive it a few instants after it passes across the barrier.”
“Yes, I had started to think about that, but then I got caught up in the other problems. What happens with the light signals or, in general, for electromagnetic waves?”
“Excellent. And that opens a whole new chapter, and a crucial one. As you may recall, Maxwell’s equations contemplate both retarded and advanced solutions for the propagation of electromagnetic waves. In other words, waves can be emitted both forward and backward in time, with positive energy and negative energy respectively. Accordingly, advanced radiation reaches us from the future, but we can’t see it because its energy is opposite in sign. Not only, but for us it would be radiation that takes the reverse path: from the absorber to the source. Remember the Wheeler-Feynman absorber theory and Lewis’s paradox? No? It doesn’t matter. In other words, the photon, as we know from its ‘dτ = 0’, does not distinguish b
etween past and future, it’s arrived even before it’s left, whatever the distance that we see it cover. It’s just us, who belong to space-time, who are obliged to separate its presence into space covered and time taken. Here is where the constancy of the speed of light arises: it’s space and time separating, with a dimensional ratio equal to c, equal for everyone. We detect the photon in our future because that’s where it has positive energy in our time direction. By contrast, somebody going in the opposite direction will see what for us is the invisible advanced radiation, because for him it will be the radiation with positive energy, coming from his past. The photon makes no distinction between the two arrows of time, it’s the middle ground, present for both, that acquires positive energy according to the direction of time. Just as our flowing in time gives us positive mass.”
“So we see so-called antimatter through its advanced radiation, which to us appears as retarded, and vice versa.”
“Exactly.”
Some minutes of silence before Helias spoke again.
“Kathia talked to me about ‘back-now’, going back in time through the light-years that separate us from the Earth. I think I can fit that into what we’ve said so far. At the same time, though, I have the feeling that I’m missing something. Maybe I’m a bit tired….”
“When I’m tired I use the Feynman diagrams, which can be read in both ways: matter and antimatter that go forward in time, or matter only, going forward and back in time, as Feynman himself did. Do you remember them?”
“Yes. More or less.”
“Now, though, don’t imagine we’re dealing with elementary particles, but with something macroscopic, like our craft full of passengers, for instance.”
Craft?… That’s what Kathia had called it too…. Where was Kathia now?… What was she doing?…
The professor picked up the tablet and drew some lines and symbols.
The Dark Arrow of Time Page 9
