Time is real, so I’m not a fan of timelessness. However, while you have a conservative flow in your model, you can still work your way back to histories and thus to the reality of time within a history.
Consider how Bohmian mechanics does it. You have the Schrodinger evolution of a wavefunction, with a conserved flow of probability density. If you chart a trajectory through configuration space according to the gradient of the phase of the wavefunction, you end up with a timelike foliation of (configuration space x time) with nonintersecting trajectories. Add the usual probability measure, and voila, you have a multiverse theory of self-contained worlds which neither split nor merge, and in which the Born rule applies.
In principle you can do the same thing with one of those timeless-looking “wavefunctions of the universe” which show up in quantum cosmology. Here, instead of H psi = i.hbar dpsi/dt, you just have H psi = 0 (where the Hamiltonian is general relativity coupled to other fields). So instead of an evolving wavefunction on a “configuration space of the universe”, you just have a static wavefunction. But you can still take the gradient of psi’s phase, everywhere in that configuration space, and so you can figure out Bohmian trajectories that divide up the universal configuration space into disjoint self-contained histories.
In practice, things are more complicated. In general relativity, you distinguish between coordinate time and physical time (proper time). The proper time which elapses along a specific timelike curve is an invariant, an objective quantity. But it is calculated from a metric, the exact form of which depends on the coordinate system. You can rescale coordinate time, according to some diffeomorphism, but then you adjust the metric accordingly, so that distances, angles, and durations remain the same. If you actually try to follow the program of Bohmian quantum gravity that I outlined, it’s hard to define the wavefunction of the universe without reifying a particular coordinate time, a step which is just like having a preferred frame in special relativity. I suspect that the answer lies in string theory’s holographic principle, which says that a quantum theory containing gravity is equivalent to another quantum theory that doesn’t contain gravity, and which is defined on the boundary of the space inhabited by the first theory. In terms of this second theory, the space away from the boundary is emergent, it’s made of composite degrees of freedom from the boundary theory. In the real world, it’s going to be time which is “emergent”, from the “renormalization group flow” of a Euclidean field theory defined at “past infinity”. In fact, excuse me while I run away and study the Bohmian trajectories for such a theory…
Anyway, bringing this back to Judea Pearl: As soon as mathematics represents a history as a “trajectory” in a state space, it is already becoming a little “timeless” in a formal sense. Consider something as simple as a time series. You can plot it on a graph and now it’s a shape rather than a process. You can specify its properties in a timeless geometrical fashion, even though one of the directions on the graph represents time. In talking about flows on state spaces, I don’t think you’re doing more than this. So what you’re doing is harmless, from the perspective of a time-realist like myself, but it also doesn’t really embody the full revolution of Julian Barbour’s ontological timelessness, which necessarily involves both general relativity and quantum mechanics. General relativity makes proper time a physical variable, and quantum mechanics matters by way of many worlds: Barbour’s multiverse is one of “many moments” (he calls them time capsules). In order to interpret an unevolving wavefunction of the universe, rather than divide it up into trajectories, he completely pulverizes it into moments, one moment for each point in configuration space.
If you want to imitate Barbour’s timelessness, then the crucial step is the ontological one of denying that the moments have a unique past or future. But if you have a conservative causal flow, you can always string the moments together into specific histories, like the Bohmian trajectories. Technical difficulties for the definition of trajectories only enter for relativistic systems, because you want to avoid reifying a particular coordinate time. But for nonrelativistic systems, it looks like formal timelessness in a causal model (in the sense you describe) is just a change of perspective that’s always available and can always be reversed.
The point of timelessness is not to say that time is unreal, merely that it is superfluous.
It’s difficult for me to follow your comment. While I’m familiar with the theories you discuss (with the exception of string theory and quantum cosmology), I don’t see how some of them are linked to this. I’m not trying to do anything so great as unify quantum mechanics and general relativity.
As soon as mathematics represents a history as a “trajectory” in a state space, it is already becoming a little “timeless” in a formal sense.
Yes.
Consider something as simple as a time series. You can plot it on a graph and now it’s a shape rather than a process. You can specify its properties in a timeless geometrical fashion, even though one of the directions on the graph represents time. In talking about flows on state spaces, I don’t think you’re doing more than this.
Time is no longer “one of the directions on the graph”. If you fix a trajectory, then it comes with it’s own time, but the more interesting object is the flow, which does not have any sense of time.
We agree that whatever I’m doing is mostly harmless.
So what you’re doing is harmless, from the perspective of a time-realist like myself, but it also doesn’t really embody the full revolution of Julian Barbour’s ontological timelessness, which necessarily involves both general relativity and quantum mechanics.
That will have to wait for someone else. I haven’t read Barbour, and it sounds horrifically difficult.
But for nonrelativistic systems, it looks like formal timelessness in a causal model (in the sense you describe) is just a change of perspective that’s always available and can always be reversed.
Probably. But such a thing could still be worthwhile.
Time is real, so I’m not a fan of timelessness. However, while you have a conservative flow in your model, you can still work your way back to histories and thus to the reality of time within a history.
Consider how Bohmian mechanics does it. You have the Schrodinger evolution of a wavefunction, with a conserved flow of probability density. If you chart a trajectory through configuration space according to the gradient of the phase of the wavefunction, you end up with a timelike foliation of (configuration space x time) with nonintersecting trajectories. Add the usual probability measure, and voila, you have a multiverse theory of self-contained worlds which neither split nor merge, and in which the Born rule applies.
In principle you can do the same thing with one of those timeless-looking “wavefunctions of the universe” which show up in quantum cosmology. Here, instead of H psi = i.hbar dpsi/dt, you just have H psi = 0 (where the Hamiltonian is general relativity coupled to other fields). So instead of an evolving wavefunction on a “configuration space of the universe”, you just have a static wavefunction. But you can still take the gradient of psi’s phase, everywhere in that configuration space, and so you can figure out Bohmian trajectories that divide up the universal configuration space into disjoint self-contained histories.
In practice, things are more complicated. In general relativity, you distinguish between coordinate time and physical time (proper time). The proper time which elapses along a specific timelike curve is an invariant, an objective quantity. But it is calculated from a metric, the exact form of which depends on the coordinate system. You can rescale coordinate time, according to some diffeomorphism, but then you adjust the metric accordingly, so that distances, angles, and durations remain the same. If you actually try to follow the program of Bohmian quantum gravity that I outlined, it’s hard to define the wavefunction of the universe without reifying a particular coordinate time, a step which is just like having a preferred frame in special relativity. I suspect that the answer lies in string theory’s holographic principle, which says that a quantum theory containing gravity is equivalent to another quantum theory that doesn’t contain gravity, and which is defined on the boundary of the space inhabited by the first theory. In terms of this second theory, the space away from the boundary is emergent, it’s made of composite degrees of freedom from the boundary theory. In the real world, it’s going to be time which is “emergent”, from the “renormalization group flow” of a Euclidean field theory defined at “past infinity”. In fact, excuse me while I run away and study the Bohmian trajectories for such a theory…
Anyway, bringing this back to Judea Pearl: As soon as mathematics represents a history as a “trajectory” in a state space, it is already becoming a little “timeless” in a formal sense. Consider something as simple as a time series. You can plot it on a graph and now it’s a shape rather than a process. You can specify its properties in a timeless geometrical fashion, even though one of the directions on the graph represents time. In talking about flows on state spaces, I don’t think you’re doing more than this. So what you’re doing is harmless, from the perspective of a time-realist like myself, but it also doesn’t really embody the full revolution of Julian Barbour’s ontological timelessness, which necessarily involves both general relativity and quantum mechanics. General relativity makes proper time a physical variable, and quantum mechanics matters by way of many worlds: Barbour’s multiverse is one of “many moments” (he calls them time capsules). In order to interpret an unevolving wavefunction of the universe, rather than divide it up into trajectories, he completely pulverizes it into moments, one moment for each point in configuration space.
If you want to imitate Barbour’s timelessness, then the crucial step is the ontological one of denying that the moments have a unique past or future. But if you have a conservative causal flow, you can always string the moments together into specific histories, like the Bohmian trajectories. Technical difficulties for the definition of trajectories only enter for relativistic systems, because you want to avoid reifying a particular coordinate time. But for nonrelativistic systems, it looks like formal timelessness in a causal model (in the sense you describe) is just a change of perspective that’s always available and can always be reversed.
The point of timelessness is not to say that time is unreal, merely that it is superfluous.
It’s difficult for me to follow your comment. While I’m familiar with the theories you discuss (with the exception of string theory and quantum cosmology), I don’t see how some of them are linked to this. I’m not trying to do anything so great as unify quantum mechanics and general relativity.
Yes.
Time is no longer “one of the directions on the graph”. If you fix a trajectory, then it comes with it’s own time, but the more interesting object is the flow, which does not have any sense of time.
We agree that whatever I’m doing is mostly harmless.
That will have to wait for someone else. I haven’t read Barbour, and it sounds horrifically difficult.
Probably. But such a thing could still be worthwhile.