Quantum Mechanics and the Broken Mirror Symmetry (Script) (Patreon)

The foundations of quantum theory rests on its symmetries. For example, it should be impossible to distinguish our universe from one that is that is the perfect mirror opposite in charge, handedness, and the direction of time. But one by one these symmetries were found to be broken, threatening to break all of physics along with them.

In his famous lectures on physics, Richard Feynman talks about what it means to expect the universe to be identical in the mirror. For it to be parity-symmetric. He invites us to imagine a clock in a mirror reflection – numbers are backwards, components are all flipped left to right, and it ticks counterclockwise. And then we imagine building that mirror clock in reality. Everything is constructed as though reflected. Numbers get painted backwards. Every screw with right-handed thread or right-spiraling coil is replaced with a left-handed version. Our intuition tells us that the mirror clock should tick in exactly the same way, except counter-clockwise. 

Our intuition would be wrong. The laws of physics, and so the laws of clocks, are NOT symmetric to this sort of parity transformation. As we saw in our recent episode, the experiment that first proved this found that cobalt-60 nuclei decay by spitting an electron out in the opposite direction to their nuclear spin axis. But in a mirror-reflected universe the same decay should be in the opposite direction – WITH that spin axis –  which is fundamentally different physical behavior. 

So back to Feynman’s clock, which we’ll use to . He proposed a clock whose ticks are governed by the decay of Cobalt-60. Imagine an array of cobalt-60 atoms in a magnetic field. The cobalt nuclei have angular momenta that align with the magnetic field – let’s say upwards so the decay electrons travel down. A detector is placed to intercept those electrons the clock ticks with every captured electron. In our reflected clock, we need to replace the cobalt atoms with their parity-inverted counterparts, but now the decay electrons travel upwards, with the nuclear spin, and away from the detector. Such a clock wouldn’t tick at all. Taken to its literal extreme, a perfectly constructed mirror-reflected clock behaves differently.

The violation of parity symmetry poses a threat to an even deeper symmetry - CPT symmetry - the combined flipping of charge, parity, and time. And this is a symmetry lies at the foundations of quantum field theory - physics MUST work the same if we flip all three of these properties. If not, physics as we know it goes out the window, which seems like a big deal. 

To save physics, Richard Feynman proposes that we build a copy of our clock out of antimatter. Uh, sure Feynman, why not. Electrons become positrons, quarks become antiquarks and vice versa, sending protons and neutrons to their anti-versions in the nuclei of our anti-cobalt-60 and other anti-atoms. Sending matter to antimatter is the C part of CPT. Charge conjugation  All charges switch sign, electric charge, quark colour charge, weak hypercharge, etc. That’s what a switch to antimatter means

How does this work in our antimatter clock? Well, antimatter atoms have negatively charged nuclei, which means their nuclear magnetic fields point in the opposite direction to regular matter, relative to their angular momentum. The magnetic field in our clock will align antimatter nuclei in the opposite direction to matter nuclei. So in our mirror-reflected antimatter clock, the direction of the decay electrons are flipped once due to the mirror reflection and once due to the switch to antimatter. That leaves the electrons traveling in the original direction, down, and the clock ticks as normal.

So even though the universe isn’t parity symmetric, but perhaps it IS symmetric under a charge-partity - a CP transformation. Send right to left and send matter to antimatter. At first glance this CP symmetry appears to hold not just in imaginary clocks but also in the particles of the standard model. The great parity-violating process is the weak interaction, which only affects left-chiral fermions. Right-chiral fermions don’t feel the weak force at all. But the opposite is true of antimatter – right-chiral antifermions feel the weak force while left-chiral antifermions don’t. So a charge-parity flip leaves you in the same situation regarding the weak interaction.

If CP is what we call a “good symmetry”, you shouldn’t be able to do any experiment to tell whether you’re in this universe or a CP-transformed universe. All physics should work the same. Theoretically. But don’t get too comfortable. We haven’t looked at what the experimentalists have to say about this. You may recall from the parity episode that the first hint of parity violation was the so-called tau-theta problem, which turned out to result from the fact that the positively charged kaon particle decayed in ways that violated parity conservation. It turns out these kaon things are great at catching the universe doing weird stuff.

In 1964 James Cronin and Val Fitch looked at the outcomes of the decay of neutral kaons. These things are extra weird. A neutral kaon is a quantum mix of its own particle and antiparticle. There are two ways to do this mixing, yielding two types of neutral kaon. One type – let’s call it K-S – is short lived and has what we call an even CP state – which just means it doesn’t change under a combined charge-parity transformation. The other type – K-L – is long lived and has an odd CP state – its wavefunction gets a multiplied by -1 on a CP transformation. And that means … well, it’s different to the KS state. If CP symmetry is conserved, KS and KL should never transform into each other because they have different CP symmetries.

Cronin and Fitch tested this by sending a bunch of both types of neutral kaons down a tube with a detector at the far end. The K-S particles should never have made the journey given their short lifetimes. And yet a small but significant number of decay products from K-S particles were found at that far end. The only explanation is that K_L particles oscillated into K-S’s, violating charge-parity conservation.

Well that sucks. Our mirror-reflected anti-matter clock doesn’t work right after all. And that’s not the worst of it. The violation of CP symmetry has a much more dire consequence than a broken imaginary clock – although it does have a lot to do with time. It also suggests that time reversal symmetry is broken. And to understand this we do need to reverse time a bit – back to the 1950s.

This was the decade of the foundation of quantum field theory, and as QFT emerged it became clear that there IS a certain symmetry that was not just intuitively expected, but also theoretically required. Starting Julian Schwinger’s spin-statistics theorem in 1951, it became increasingly clear that quantum field theory demands symmetry under the combined action of charge conjugation, parity inversion, and time reversal. The very axiomatic foundations of QFT state that an antimatter, mirror-reflected, time-reversed version of our universe should have exactly the same laws of physics. Quantum field theory should be CPT invariant. And we know that QFT is right – at least as far as it goes. It’s just about the most right theory we’ve ever come up with.

I’ll get back to what this new time reversal symmetry means. For now, let’s just accept that the laws of physics must work the same under a simultaneous flipping of charge, parity, and the direction of time. That was certainly the view of physicists in the late 50s. So then come the experiments that show violations first of P and then CP symmetry. Big deal? We still have T! CPT can still be conserved. Ah, but here’s the issue: if CP symmetry is violated and CPT symmetry holds, then T-symmetry MUST be violated. 

Why? Because that time reversal operation needs to bring us from a broken CP-reflected universe into a fixed CPT universe. That means a T-transformation from our working CPT universe sends us to a broken CP universe. Ergo a time reversal transformation changes the way the universe behaves. Time symmetry is out the window. Theoretically.

That sounds bad. Isn’t physics supposed to work the same whether we go forwards or backwards in time? As we talked about here, don’t we absolutely require time-reversal symmetry to in order to conserve quantum information, which itself is required for all quantum mechanics to make sense? To get this we’re going to need to talk about what we mean by reversing time.

The most obvious interpretation of time reversal is literally reversing the arrow of time and causing the universe to travel backwards in time. That is NOT what we mean by the T in CPT. Which I’ll explain. But first let’s think about this simple type of T transformation as a literal rewind. Rewind the universe and you get back to where you started, pretty much by definition. So presumably quantum information is conserved in this type of time reversal. Mathematically, the particles in a rewinding universe actually look like they underwent a charge-parity inversion. Matter that was going forward in time looks like parity-flipped antimatter going backwards in time. This interpretation of antimatter as time-reversed matter was first proposed by Ernst Stueckelberg in 1941, but is now largely associated with Richard Feynman. It’s essential to his path-integral approach to quantum mechanics and to Feynman diagrams. Maybe that’s why he was into building antimatter clocks.

So, yeah, the universe is not symmetric under this simple version of T-reversal. It’s the precise inverse of a CP transformation: the two undo each other. Do a CP transformation and then a simple T transformation and get you back to where you started. If CP is violated then this simple time reversal is also. We see the violation of simple-T symmetry in the imbalance between matter and antimatter. Then there’s that whole entropy business, although its connection to quantum mechanics is still not well understood.

But like I said, this simple interpretation of T as rewinding the universe is not what is usually mean by the T in CPT. The T in CPT is very specific is more accurately thought of as flipping the direction of evolution of a physical system. An explosion becomes an implosion and particle decay becomes particle creation. You’re not rewinding time and you’re not converting matter to antimatter. You’re just reversing all momentum and spin. Essentially you’re taking all particles in the universe and pointing them back in the direction they came from. If the T in CPT is conserved then after reversing all particle motion, those particles should perfectly retrace their steps and perfectly reverse all reactions in their histories. As this motion-reversed universe evolves forward in time it should end up back in its starting configuration. On the other hand, broken T-symmetry says that if you do this reversal, the future won’t mirror the past.

One prediction of T-symmetry is that processes should take the same amount of time forwards versus backwards. For example, a quantum transition between one particle type and another should take the same time in either direction. And that gives us a test. In 2012 physicists from the BaBAR collaboration at the Stanford Linear Accelerator Centre tested the speed with which B mesons transition between two types. In a T-symmetric universe the speeds should be the same in either direction. They weren’t. Reversing the direction of the interaction changed something fundamental about the physics, indicating a violation of T-symmetry. But remember, killing T-symmetry is a good thing because it saves CPT symmetry.

The last 70 years has been a true rollercoaster ride for the symmetries of nature. As experiments found broken symmetries – first parity, then charge-parity, the hallowed CPT symmetry looked in danger unless we gave up on the symmetry of time itself. But now, with time-reversal symmetry also proved broken, the CPT theorem looks safe. Feynman’s mirror-reflected antimatter clock will work just fine. It just needs to tell time backwards. A perfect mirror reflection reflects all three: charge, space, time.