Bohmian Relativity: Quantum matter may create time
What is time? Quantum theory may have the answer.
A fundamental question in physics is what is time? Albert Einstein believed it was a dimension like space. You could traverse through it like a sailor on a sea, changing course by accelerating or decelerating, or by manipulating gravity using matter to alter the curvature of space and time. Virtually every paper on Einstein’s theory of gravity, General Relativity, begins with something like assume a four dimension manifold M and a metric g. Implicit in the four dimensions of the manifold is that time is somewhere in there. The metric just describes distances on that.
It is as if I unfolded a map and said, here is a map of the Mediterranean, and I pointed to the compass rose and the scale factor. The manifold is like the sea and the metric is like the compass and the scale. Your place in space and time is some coordinate on the map, say, Malta. Your destination is somewhere else, say, Alexandria in Egypt. It hardly matters if you, say, go south and then east or east and then south. There is nothing special about those directions on my map. You might encounter different land features, but so what?
That is time to Einstein.
In almost all physical law, indeed, time is reversible. That means that the law works the same backwards and forwards in time. All trajectories are reversible. There is an exception: the 2nd law of thermodynamics:
The total entropy of a closed system cannot decrease over time.
The universe is a closed system. If you reversed time, the entropy would decrease, so the law is not time reversible.
This difference is called the “arrow” of time because it points time in one direction. It does not, however, quite contradict Einstein. It just adds a qualifier that, on the river of time, the current flows one way and you cannot ever go fast enough to get back upstream. What is that speed? Faster than light.
It seems like Einstein’s view of time is safe.
Another view of time, however, that is more fundamental is that time isn’t like another dimension. It isn’t like traveling down a river. It is, in fact, more like standing in one spot, say, at the bottom of a waterfall. Time falls on you and flows away never to be seen again. Things change. But you never moved.
In order to make this idea work with Einstein’s General Relativity, however, we need to make sense out of this four dimensional manifold, M. The way to do that is called a “foliation” which means that we slice the manifold up like a loaf of bread. Each slice is a moment in time.
One question is whether this foliation is “local” or “global”. That is, does the universe as a whole have a single foliation, defining the current present moment, or does every observer in the universe have its own foliation?
For most of the universe, you could define a global foliation, but, if you start to include the interiors of black holes, you run into problems. Time works differently inside a black hole. It doesn’t run into the same future as ours. Time runs into the singularity at the center while our time acts like a spatial dimension. It is fundamentally opposed to our view of time from outside the black hole.
Italian theoretical physicist Carlo Rovelli takes the point of view of defining time as local but that whether the river of time has a current or not and what that current is matters. It is not just some dimension we can travel in like on some barren desert plain. There is something pushing us along. He proposes that this something are tiny fluctuations in the gravitational field which, taken altogether, create a flow of time. This flow carries us along with it. Still, his view is more like the universe as a vast ocean with many different possible flows and currents, whirlpools and maybe even dead zones where there is no flow. It all depends on the statistical nature of the gravitational field.
Rovelli does all this without breaking one of the cardinal features of gravity, that there is no “preferred” direction or dimension. Time isn’t special. It’s just that all the statistical fluctuations in the gravitational field naturally create a flow that feels to us like time, in the same way that a river basin creates a flow, but that basin isn’t special. It is just a feature of the land. It can shift and move. It can be somewhere else. There is no external god telling it where to be or which way to go.
Still, Rovelli sticks with the four dimensional picture of the universe. It is a vast ocean, to be sure, with many currents. We can still sail it but we best catch the right winds to take us where we want to go. It does matter if we go south and east versus east then south because the currents and winds will take us one way but not the other.
German physicist Detlef Dürr takes a more radical approach through Bohmian mechanics. Unlike Rovelli, who starts with classical general relativity to create his theory of time and then adds quantum aspects to it later, Dürr takes an immediate quantum point of view with Bohmian mechanics.
David Bohm developed his mechanics in the 1950s as an alternative to standard quantum theory. John Bell later expanded it to include particle creation and annihilation, but integrating it with relativity, either special or general, was troublesome. Bohm fundamentally built time into his theory much like Isaac Newton did with his mechanics and Erwin Schroedinger did with his.
Scientifically, building time into a theory isn’t a problem as long as it makes good predictions, but philosophically it is a step backwards. Einstein’s great achievement was in removing the idea of a preferred direction from space and time. To add that idea back in would mean to have to explain why it is there. Ockham’s razor says that a theory should not add unnecessary explanations.
Dürr set out to show that you could remove this problem from Bohm’s theory and, in doing so, show that time could be both universal and the product of matter and forces itself.
Bohmian mechanics explains quantum theory as the evolution of two things: the quantum wavefunction that we all know from Schroedinger, and another function called the guiding function, Q, which holds the hidden locations of all particles in the system.
Theoretically, this system could be the universe as a whole. There should be no problem with doing that. This ultimately added into quantum mechanics something that Einstein wanted, which was the idea that particles had hidden states that we couldn’t measure. For the last 100 years, most physicists have argued that particles do not have hidden states. They have no state at all until measured. Bohm showed that they could provided that all particles could communicate with one another instantaneously (nonlocally) through the Q function.
The problem with instantaneous communication is that you have to say when this communication occurs. You need the entire system, spread across space, to have a single present moment. If the entire universe is your system, then you need to have a single present moment for the universe. In other words, you need foliation.
Also, since the quantum system is evolving from moment to moment across 3D hypersurfaces, you don’t really need the past or future to be involved except in determining the next step in the evolution. Unlike with Einstein’s theory, you can assume that the past and the future are not even there.
This statement is a lot stronger than Rovelli, who is content to define time as a local flow. While Dürr’s time could be only local, it makes more sense if you can expand it to be global, even inside black holes.
Dürr points out that you can recover “Lorentz” symmetry, the 4-D nature of equations compatible with relativity, by replacing all your equations with Lorentz symmetric ones. You can then say that the present moment is not discoverable by measurement. It is hidden in the quantum uncertainty.
He rightly points out, however, that this is not enough. Ockham’s razor still comes in. Where does the foliation come from? He suggests, like Rovelli, that it comes from matter itself, specifically, it comes from the quantum wavefunction.
He presents the possibility that matter itself may create an arrow of time in Bohmian mechanics. His solution is general to all quantum theories however. The idea is that the wavefunction of matter on average moves from the past into the future. Rather than that flow being because of time, he suggests it creates time.
He shows how you can extract a foliation from the wavefunction by applying it to some standard “pointing” structures from Quantum Field Theory (QFT) such as current, spin, and even the stress-energy tensor. Current and spin have magnitude and direction in four dimensional spacetime while stress-energy tensor has magnitude, direction, and orientation.
The basic premise here is that all matter flows in time and creates an average direction of time.
Dürr also attempts to formulate a more general Bohmian dynamics that is relativistic without a foliation. Instead, it has a “timelike” vector field that becomes a foliation under certain assumptions. This would be able to include spacetime that is not easily sliced, abandoning the idea that there is a single present moment. Unfortunately, without the foliation, the correspondence between Bohmian dynamics and standard quantum theory breaks down. Thus, Bohmian dynamics appears to require a single present moment or it is just plain wrong.
One of the drawbacks in Dürr’s work is that he relies on massive Fermionic matter (electrons, quarks, etc.) to define his foliation, ignoring the gravitational field itself, unlike Rovelli. This means that the gravitational field plays a backseat in defining the foliation of itself. It is as if the river basin had nothing to do with the flow of the river, but the river simply chose to move in a particular direction. Both need to be taken into account.
Others, such as Tumulka, have tried to incorporate general relativity into Bohmian mechanics with some limited success, defining space time as a set of gravitons which generate a gravitational field, evolving according to the universal wavefunction like anything else.
Tumulka suggests some possible rules for the foliation but doesn’t venture to develop a theory about it. He only suggests that it cannot be based on the particle configuration (which depends on the foliation) only on the wavefunction. He does suggest a foliation based on constant timelike distances from the Big Bang. This is the most natural foliation. In fact, it is by invoking this foliation that we know how old the universe is.
A number of Bohm-like theories attempt to do away with the time foliation entirely. There are multi-time wavefunctions, for example, that resemble Rovelli’s statistical general relativity, where there is no single present but many, many present moments, one per particle.
There are some additional tricks that have a concept of simultaneity that the guiding function needs without a foliation. For example, there is the synchronized trajectories approach where every particle communicates with every other particle according to its own local time. This is foliation-like, but is not necessarily a true foliation. It makes sense because each particle needs to “know” how to move at each point in its trajectory. Entangled particles, you imagine, would move in lockstep.
The unfortunate problem with trying to define foliations over the entire universe is what to do with areas of spacetime where local time doesn’t flow in the same direction as others such as inside black hole event horizons. One option would be to exclude these from the universe, but quantum mechanics suggests that the instantaneous communication that occurs between particles must also cross the boundaries with black holes as well.
Consider, for example, the case of two entangled photons, one that has fallen into a black hole and the other that has escaped. Surely they share a quantum wavefunction; therefore, how do these particles evolve together when the time they experience is so different? Is it simply according to the proper (local) time of each, like a synchronized trajectory? It is hard so say.
One can still use foliations even across black hole boundaries. One foliation that proves useful in these cases is known as “York” time. You can prove York time to be geometrically unique and it handles singularities like black holes well. Each “tick” of a York time clock is a 3D hypersurface that has constant mean curvature. From the perspective of cosmology in particular and astronomy in general, having some measure of time like York time is extremely valuable. Could it be that York time is more than a geometric convenience? Might it be the present moment we are looking for?
From the perspective of cosmology, Callender argues that, for the universe as a whole, something like York time is natural. After all, cosmology is deeply concerned with the evolution of the universe as a whole in a unique time, notwithstanding relativity. Moreover, Bohmian mechanics is uniquely adapted to quantum cosmology because it can support a quantum universe that changes with time, something that standard quantum theory has difficulty with, so dependent is it on the interaction between quantum and “classical” systems. A purely quantum system without Bohm’s modification does nothing.
If the entire universe is a quantum system, then it cannot interact with anything outside it unless you subscribe to many-worlds, which has its own set of problems. If you want a single universe, then you have to resolve the problem of what does it mean for a quantum universe to evolve? Bohm gives an answer that it is the interaction between particles and their wavefunctions from moment to moment. The universe itself decides on a cosmic level what the present moment is.
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Dürr, Detlef, et al. “Can Bohmian mechanics be made relativistic?.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470.2162 (2014): 20130699.
Tumulka, Roderich. “The ‘unromantic pictures’ of quantum theory.” Journal of Physics A: Mathematical and Theoretical 40.12 (2007): 3245.
McCoy, C. D., and Craig Callender. “Time in Cosmology.” (2017).
Callender, Craig, and Robert Weingard. “The Bohmian model of quantum cosmology.” PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association. Vol. 1994. №1. Philosophy of Science Association, 1994.