I don’t buy this abolition of time at all, but this question of how CP violation appears in Barbour’s scheme seems like a good test of one’s understanding.
The abolition of a time coordinate in quantum gravity is not Barbour’s invention. The usual Schrodinger equation is Hψ = -i/hbar dψ/dt, where the H operator represents total energy (typically, sum of a potential and a kinetic term). But in general relativity, the total energy persistently shows up as zero (gravitational potential energy cancelling out against mass-energy, I believe; I confess I’m just relating this secondhand). So H=0, and there’s no time evolution, just the stipulation that ψ has an H-eigenvalue of zero. Barbour’s contribution is to offer an interpretation of configuration space as a set of “time capsules”, static configurations containing static observers experiencing an illusion of time due to their memories. (I don’t know what meaning Barbour ascribed to the amplitudes, and perhaps he has problems comparable to those suffered by the usual, timeful many worlds interpretation.) So that’s it: H=0 gives you a standing wave in configuration space, and Barbour proposes a timeless variant on many worlds, with variables internal to the cosmic configuration, like cosmological radius, acting as clock variables, proxies for time.
Back to this CP violation problem. It is a theorem that quantum field theories have CPT symmetry, and it is an experimental result that CP symmetry is violated, i.e. that T symmetry is violated, in kaon-antikaon transitions. We actually have a quantum field theory implementing T-violation, the Standard Model, where it’s implemented in the quark mass matrix. The numbers in that matrix would be coefficients of interaction terms in the potential-energy part of H, connecting quark fields and weak-boson fields, I think. It is something like postulating that the amplitude for (quark disappears) (weak interaction happens) (antiquark disappears) is different from the amplitude for (antiquark disappears) (weak interaction happens) (quark appears). In a sense it is quite unmysterious, since there is no mathematical barrier to postulating such an asymmetry, though one would like to know a deeper reason for it; the CPT theorem only says that if you transform those two processes by CPT, the amplitudes for the corresponding processes had better have the same relation.
Now formally it is a straightforward thing to take a quantum field theory and couple it to general relativity. For example, you can just take the QFT’s old H, add a term for scalar curvature, and multiply the whole thing by “sqrt(determinant(metric))”. Set the new H equal to zero, and you now have what should be the equation for a Standard Model universe with Einstein gravity thrown in; and though you probably can’t solve that equation, you can still go ahead and follow Barbour’s procedure in the resulting configuration space.
It seems obvious(?) that the expression of CP violation in the timeless picture will have something to do with those quark/antiquark fields in the CP-violating terms, and I would point out that although my informal description of those terms might seem to make one the time reverse of the other, actually there is an algebraic difference. Algebraically, “X appears” means that you use a “creation operator”, while “X disappears” involves an “annihilation operator”. So understanding the implications of those operators for amplitude gradients in the timeless picture may be the key to figuring this out. Also, since other physical variables act as clocks in the time capsules—proxies for an actual time—kaons and antikaons ought to somehow have a different relationship to the clock variables. If I was seriously trying to figure this out, I’d be thinking at the intersection of those two approaches. (And I’d be doing it using the simplest T-violating QFT I could find, rather than with the full Standard Model.) This is an excellent question to think about, for anyone trying to understand Barbour’s interpretation in detail.
I don’t buy this abolition of time at all, but this question of how CP violation appears in Barbour’s scheme seems like a good test of one’s understanding.
The abolition of a time coordinate in quantum gravity is not Barbour’s invention. The usual Schrodinger equation is Hψ = -i/hbar dψ/dt, where the H operator represents total energy (typically, sum of a potential and a kinetic term). But in general relativity, the total energy persistently shows up as zero (gravitational potential energy cancelling out against mass-energy, I believe; I confess I’m just relating this secondhand). So H=0, and there’s no time evolution, just the stipulation that ψ has an H-eigenvalue of zero. Barbour’s contribution is to offer an interpretation of configuration space as a set of “time capsules”, static configurations containing static observers experiencing an illusion of time due to their memories. (I don’t know what meaning Barbour ascribed to the amplitudes, and perhaps he has problems comparable to those suffered by the usual, timeful many worlds interpretation.) So that’s it: H=0 gives you a standing wave in configuration space, and Barbour proposes a timeless variant on many worlds, with variables internal to the cosmic configuration, like cosmological radius, acting as clock variables, proxies for time.
Back to this CP violation problem. It is a theorem that quantum field theories have CPT symmetry, and it is an experimental result that CP symmetry is violated, i.e. that T symmetry is violated, in kaon-antikaon transitions. We actually have a quantum field theory implementing T-violation, the Standard Model, where it’s implemented in the quark mass matrix. The numbers in that matrix would be coefficients of interaction terms in the potential-energy part of H, connecting quark fields and weak-boson fields, I think. It is something like postulating that the amplitude for (quark disappears) (weak interaction happens) (antiquark disappears) is different from the amplitude for (antiquark disappears) (weak interaction happens) (quark appears). In a sense it is quite unmysterious, since there is no mathematical barrier to postulating such an asymmetry, though one would like to know a deeper reason for it; the CPT theorem only says that if you transform those two processes by CPT, the amplitudes for the corresponding processes had better have the same relation.
Now formally it is a straightforward thing to take a quantum field theory and couple it to general relativity. For example, you can just take the QFT’s old H, add a term for scalar curvature, and multiply the whole thing by “sqrt(determinant(metric))”. Set the new H equal to zero, and you now have what should be the equation for a Standard Model universe with Einstein gravity thrown in; and though you probably can’t solve that equation, you can still go ahead and follow Barbour’s procedure in the resulting configuration space.
It seems obvious(?) that the expression of CP violation in the timeless picture will have something to do with those quark/antiquark fields in the CP-violating terms, and I would point out that although my informal description of those terms might seem to make one the time reverse of the other, actually there is an algebraic difference. Algebraically, “X appears” means that you use a “creation operator”, while “X disappears” involves an “annihilation operator”. So understanding the implications of those operators for amplitude gradients in the timeless picture may be the key to figuring this out. Also, since other physical variables act as clocks in the time capsules—proxies for an actual time—kaons and antikaons ought to somehow have a different relationship to the clock variables. If I was seriously trying to figure this out, I’d be thinking at the intersection of those two approaches. (And I’d be doing it using the simplest T-violating QFT I could find, rather than with the full Standard Model.) This is an excellent question to think about, for anyone trying to understand Barbour’s interpretation in detail.