Simon Saunders on the Nature of Time, Many-Worlds, and Bell's Inequalities

Simon Saunders

Ah, yes — the easy questions first. Richard Feynman said something about time: that time is how long you have to wait, and you can't really say anything more than that without getting into trouble. It goes back to a philosopher — I forget the name — who said something like: if nobody asks me what time is, I know perfectly well; but as soon as I am asked, I become deeply puzzled. So yes, I think time probably remains the central concept in physics that is least understood. At some fairly fundamental level we do not altogether understand time, but we have very adequate representations of it in physics, and we know how to deal with it in our ordinary lives — but it is extraordinary.

Think of it in terms of your own life. Whatever you've lived — perhaps only twenty or twenty-five years — there are many, many hours packed into those years. Consider yourself in any one of those hours: there's time to think, to comment, to reflect, to look around. That is you being you, living in an ordinary way over a period of, say, an hour.

But then there are thousands of such persons on that basis, because there have been thousands of hours where you have lived through those hours. There are thousands of "larger than moments" — I don't want to make it just something instantaneous — thousands of times of your life, each considering itself at the time to be all that there is. As I speak now, I'm not considering that there's more to me in the future. I'm perfectly adequate as a person with my past. But there are thousands of persons with my past going back in time, and I'm one of those thousands. If we compress it — not an hour but a minute, not a minute but a second — you can have as large a number as you like of previous moments. So what is the status of all of those moments? How do we deal with that reality, that multiplicity? Because each seems like a reality. You ask me what is real — I can't do better than point to stuff around us and say, well, look, this is what's real. They're all realities, and they all somehow exist. Well, they don't all exist at the same time — they exist with a sequential and, more than that, a causal structure. We build and build until we've got quite an elaborate, fairly integrated understanding of that multiplicity. And yet there remains something really problematic about it.

It seems we ought to take all of those momentary "me"s as equally real. They're separated by intervals of time, not all simultaneous with one another, but it seems we must take them to be all somehow real. This follows in particular from special relativity, and to some extent general relativity, though GR brings in extra considerations. In special relativity we learn that there's no such thing as a global present. If you think reality is a present, there's no such thing as a global three-dimensional reality — I mean, there's a momentary presence centred on me. I can have a momentary reality centred on me, a three-dimensional reality centred on me, but there's no intersubjective, public, three-dimensional spatial reality, because if there were such a thing, that would be a privileged frame of reference — and it is of the essence of relativity theory that there is no such thing. So it seems that the only public reality is the one that takes in all of the presents, all of those different presents.

So this is the block universe picture. You have the histories of people, the histories of objects, laid out — static, unchanging. Think of it as some extraordinary block of glass in which all of the myriad threads weaving through it are particles, objects, people. That would seem to be the reality. And that reality just doesn't look or feel like time at all — it looks and feels like space. Time has been specialised; it's been turned into a dimension similar to space. There are structural differences — the light-cone structure, absolutely inherent to Minkowski spacetime, the spacetime of special relativity — and with that one partitions straight lines, and curves too, momentarily, as timelike or spacelike, and if timelike, whether present-timelike or past-timelike. So you have that light-cone structure in the Minkowski spacetime, this block universe — but still it doesn't feel like time. It doesn't feel as though time is passing. If I ask where in that block universe time is passing —

What I can do is look at a segment of my worldline — a little piece of it, the piece that is "now," if you want. And in that little piece, you see me talking, discussing this crazy subject of what time is — my lips are moving, sound waves being emitted, my head changing, and so on. But all of that is, as it were, a static configuration. So in what sense does that capture my sense of the real, of passage? Stephen Hawking put it slightly differently: what breathes fire into the equations? We've got this abstract representational thing — what makes it come alive? Surely what makes it come alive is time actually passing — that this representation of spacetime fails to capture the felt experience of passage.

This is a very ongoing debate, and I think a fundamental one, and it connects fairly closely to the mind–body problem. One aspect that's little attended to is that awareness is local in time — and not even instantaneously local, more like a specious present of fifty milliseconds or whatever. That's the localisation of awareness. If you look at other animals, at insects, as you go down in scale the timescale of their awareness is much shorter. You've only got to look at a small dog, or a cat, and the way it moves — it's extraordinary. Birds even more pronounced. They are clearly living at an accelerated rate compared to us.

Simon Saunders

Well, maybe that's a special case. But there's something rather extraordinary about the localisation of awareness. Presumably it can't get down to extremely small timescales — but why is it local at all? You could say, well, because dynamics is local — physics is built on locality, fields are local. But awareness is underpinned by all of this phenomenology, and it could have been underpinned by phenomenology spread out over times enormously longer than ours. To really capture an adequate representation of spacetime, it would help if we could see the thought of a creature on an enormously slower timescale as something spread out over hours or days. I struggle with it.

So these are the issues that arise with time, and they're especially interesting vis-à-vis other topics in physics, particularly the many-worlds interpretation of quantum mechanics — because there is something absent there too, and it comes in with probability. The nature of probability, and that absence of something felt that's missing in the theoretical representation of probability, is rather parallel to the sense that something is missing in the representation of time — forget about quantum mechanics altogether. It may be that there's something more fundamental going on, which is what's really involved in buying into a "view from nowhere," as it's sometimes called, or a god's-eye view — a timeless view, a view that is not in time. There are lots of debates in theology, aren't there, about whether God exists in time or is somehow outside of time, and I think those debates are probably extremely interesting if you drill down into them. What quantum mechanics is adding, I think, is that actuality is now missing too. Just as the god's-eye view is not temporal — it takes in all of time — the god's-eye view with quantum mechanics doesn't take in the actuality, the particularity, that we think is also a part of reality.

Simon Saunders

Yes — a perfect summary of the problem situation, because we are in this problem situation. We are all of us perfectly at home; there's nothing alien about our world. And yet when we try, through intellectual discovery and analysis, to construct a more accurate representation of the world, we find that it has to nullify the personal perspective — that to get the thing off the ground we've got to remove our personal perspective from the picture. In a way that's fine. It's a bit like drawing a diagram — you can draw your garden without saying where you are in it. It would be even lovelier to have, as in Harry Potter, a map showing where everybody is as they move about. But you have a map of your garden, and you don't have to put yourself in it, and that seems fine.

But now think about a period of time and envisage it without putting yourself in it. Then you've got a problem. The point about putting yourself in your garden is that you put yourself in it as something spatially local. The point about putting yourself in a duration of time is that you put yourself in it as something temporally local — something with a specious present, a particular fifty milliseconds which is "now," which has that immediacy and urgency. Well, the place I'm standing in has that immediacy and urgency too. The difficulty is: I can represent the three-dimensional space without myself being situated in it, but I struggle to understand a duration of time without me being situated in it. I can only see it as something frozen, lacking temporal characteristics. If I imagine putting a spot on that four-dimensional structure — we can represent it as one temporal and one spatial dimension, so two dimensions — you can't just label a particular point on the time axis and say "that's now," any more than you could label a particular point in the garden and say "that's where I am." I can put many other dots there too, which would be myself at different times. And the problem is that it seems arbitrary — there's something inadequate about the spatial representation of time to capture its sense of flow.

This is one of the philosophical debates that has persisted for a long time, but it became particularly urgent with relativistic physics. That's the point: it's no longer even an option to declare a privileged moment of time for the whole universe. There's no reasonable moment of time you could pick out, because it would be a preferred frame of reference — this is all within a special-relativistic context. When you go to general relativity, you also go to the real structure of the universe, and that includes a microwave background. The microwave background does very nearly pick out a unique global time: at each point you seek a velocity at which the microwave background is isotropic, and you can knit together neighbouring points, each moving with the velocity at which the microwave background is isotropic, and that picks out a foliation — a slicing of spacetime. The trouble is it only does so up to certain approximations — you cannot get to a precise, instantaneous time this way, or you can make certain conventional assumptions. So when you go to GR and the actual structure of the physical universe, the situation is different — there is even a way of theoretically changing how the "time works" here. But in special relativity, the theoretical fact is that there's no privileged frame of reference vis-à-vis dynamics: dynamics, which is all about change, does not acknowledge a preferred frame of reference — it has a symmetry with respect to changes in frame. That symmetry is what destroys the option of having a global three-dimensional space. Newton and people like him thought they had a global three-dimensional, absolute space, and it is this that was lost in the shift to relativity.

Simon Saunders

Excellent. I think one fundamental aspect has got to be that we can imagine a space that endures, and that we can visit every part of it as we wish. That's not really an instantaneous three-dimensional space — it's a space that endures, so it's already got additional structure in there; it involves space as well as time, this endurance of the parts of the space. But granted that endurance, just take the surface of the earth as it rotates around the sun: we have continuity, we can track objects on the surface, and as that's happening we can visit whichever parts of the earth we wish. And that is lacking in time.

So there's the obvious point: time is one-dimensional, space is three-dimensional. You're right that in Minkowski spacetime there's a sort of fusion of the two, but this aspect remains — if you consider the analogue of an object moving through time, it is not possible to visit any point along that trajectory, whereas if you had a three-dimensional space that endures, you can visit each part of it as it endures. It's a more complex idea. What we've just discovered is that as soon as you speak of space and time in terms of points, events, instantaneous three-dimensional shapes, things are reasonably clear. As soon as you introduce things that endure, you've got a problem, because you have to say what it means for an object to endure, to persist, not to vanish in a puff of smoke. The answer: resolve it into a bunch of timelike lines representing all the pebbles, if you like, of a beach. Give each pebble a timelike line — a line connecting together events in spacetime but always timelike, in the vertical part of the light cone — and look at the neighbouring pebble, and the neighbouring pebble, and what you've got is a congruence of timelike lines all roughly parallel to one another. They don't move about too much; the structure is preserved. When you've got all of that physically in place, you have a notion of a three-dimensional space that endures — which is everybody's idea, whether of my garden or your home: it doesn't suddenly disappear on you.

So what's involved, thinking about it in terms of four-dimensional events, spacetime points, is that you've got this congruence of timelike lines, roughly parallel, with characteristics that are relatively frozen — they don't change too much from one day to the next, aside from things like the change in lighting. When I revisit and visit every part of it, I'm always going timelike, but I can move timelike and check out different parts of the big congruence of lines. This needs a graph, really, to depict it properly.

Simon Saunders

No — what you've just done is point to two contrasting systems of coordinates. One system is just spatiotemporal points, external to the earth as it orbits the sun and everything else that's happening. You can refer the question to the level of a spacetime point, and then the answer is no — you never revisit the same spacetime point. It's like you never step into the same river twice. So that's not what's meant. What's meant instead is a different system of coordinates: a set of rigid bodies. Think of it as a mechanical system — electric, or mechano, or something — made out of approximately rigid bodies. We don't need perfect rigidity — there are difficulties making sense of perfect rigidity in relativistic terms — approximate rigidity will do just fine.

The point is that the coordinate system thus defined persists over time, and we can think of events as labelled by position with respect to the special parts of that coordinate system. As we move around, at later and later times, we can visit many different points of that mechanical system of rods — we can revisit all of those points, and in that sense revisit "spatial points," but not spacetime points — that's the difference. It's a hard thing to convey with precision and clarity — part of why teaching special relativity isn't easy. It becomes easier in terms of visual diagrams and equations, but translating and expressing these ideas in ordinary words is more difficult. That's one of the things that makes philosophy of physics so much fun, and I think makes it important to do, because physicists can get away too much with merely relying on the equations.

Simon Saunders

It's something I've often found in discussions with physicists. As a lowly philosopher of physics, I can admit my ignorance of physics without damage — I can say, oh, please explain, there's something here I really don't know. And the physicist present will enjoy explaining to someone who's admitting their ignorance. But then what happens is that another physicist who's present finds they don't agree with the explanation being given — and because I know some of the game, I can point out, well, that's not quite right, is it? — and they find themselves having to discuss things they mostly didn't need to discuss.

Simon Saunders

I think it would be two people talking about stochastic quantum mechanics, the dynamical collapse theories, having very different views of how localisation works in that context. When pressed, with me saying I really don't understand, please explain, one would attempt to explain it — but then the other person present took a very different line. I'd have to give names, wouldn't I, not to.

Simon Saunders

Yes, absolutely — both of these were physicists, not philosophers of physics. They were talking about certain kinds of diffusion equations and how they relate to a conceptual question I was interested in. I could express ignorance as to how the mechanism worked — something we do all the time in ordinary conversation, pretending to be ignorant when actually we know quite well what's going on, but asking "can you explain that?" — and then they find themselves not in their comfort zone. Any conversation about the measurement problem in quantum mechanics has this result among ordinary physicists — they're not really prepared to discuss it in a very serious way, and typically haven't thought very hard about it. So that's an example of how it's possible to get away without talking about something very fundamental — not an example where I can say "tell me about quantum mechanics, I don't know anything about it," because they know me well enough that that's why I'm there. But it's an example of how physicists will not engage in a conversation, whereas as soon as somebody genuinely, without guile, says "please explain how this works," they will interestingly differ from one another.

Somebody else who made the same point was Steven Weinberg, in one of his last essays on the foundations of quantum theory — in the New York Times Book Review, I think around 2017. He said he wasn't as happy as he once was about the foundations of quantum mechanics; the trouble is that the experts don't seem to agree, and that's a bad sign. There is this failure to agree, but very often one doesn't want a conflict — if it's a matter that affects the equations, it becomes a real dispute, something that can't be amicably nodded through. But if you can keep it at the level of conversation, one can pretend it doesn't really matter.

Simon Saunders

Well, let me give an honest answer to what I think time is. I think time is precisely this geometrical structure — and not just any old geometrical structure, but one threaded together by certain dynamics, represented in this four-dimensional picture. I think that is the correct characterisation of reality. Its seeming lack of "where is the fire in the equations," or where is the passage taking you from one moment to the next, is precisely a price you've paid for a perspective, a view from nowhere, a god's-eye view — sub specie aeternitatis, as Spinoza put it. It's very alien to ordinary sensibility; it's very difficult to inhabit that view. There's a tremendous challenge in really trying to grasp what spacetime is, if you're serious about it — and equally, in trying to understand cosmology, the universe as we understand it. I think it's character-affirming, but life-changing, to properly engage with the enormity of the universe. The only way I can put it is in terms of the experience of seeing the night sky — a real starscape, on a properly dark night, which is so rare, is one of the most extraordinary experiences we can have. I hope you've had such an experience, because it's profound. We can't live with that all the time — we go about our daily lives and forget about it — but it's there, and this is the reality we live in. Something similar happens in the foundations of quantum mechanics.

Simon Saunders

A great set of questions — simple ones. I very much took that direction in my own early career. I thought, if we're going to be serious in doing philosophy of quantum physics, we ought to be looking at relativistic quantum field theory, and I spent a lot of time studying it — it's enormously rich and extraordinary. But I think the most important lesson to come away from it is that physics is scale-relative, and the physics adequate at one scale will not be adequate at another — and these different scales are not in tension with one another. In particular, what's of importance to conceptual questions about probability, and the measurement problem, can all be articulated at low energy scales. That doesn't mean you get rid of relativistic stuff altogether — you've always got radiation, photons, playing an important role in the foundations of quantum mechanics — but it does mean you can be fairly sure that if you've got an analysis that works in non-relativistic quantum mechanics, and it's not too dependent on structures we don't also find in relativistic quantum theory, then we can make do with the language of non-relativistic quantum mechanics reasonably well.

What's also been found is that you can express or probe many profound-looking concepts in quantum gravity using elementary concepts of quantum theory, like entanglement. There may be great progress to be made not through looking at the standard model, high-energy phenomenology, or structures relevant in high-energy regimes — we don't learn about quantum gravity so well like that. We better learn about it by applying quantum concepts directly to recover something like spatiotemporal concepts. I find these questions absolutely fascinating, but they give further emphasis that the answer isn't to go immediately to relativistic quantum field theory — unless, of course, you think people are making arguments in the non-relativistic regime that cannot be recovered in relativistic quantum theory.

A good example is the localisation of particles. In what sense can particles be localised within regions? Can quantum states be localised within a region? Answer: yes. Pose the same question in the relativistic case — can particles be localised? The concept becomes problematic. We don't mean localised in spacetime — that would mean they came into existence and went out of existence — we mean spatially localised. Can they, in relativity, be spatially localised? Not in a covariant way — not in a way that respects the symmetries of relativity theory. There's something problematic about the localisation concept if you ground too much foundational work in quantum theory on position — there is no covariant position operator in relativistic quantum field theory. As long as you're careful not to rely too heavily on the notion of a position operator in foundational questions, then that potential problem is dealt with.

Other potential problems arise with things like the number of particles. You can have interpretations of quantum theory that drill down into the actual number of particles involved — imagine one degree of freedom per particle, n particles, n degrees of freedom, and that's your physical arena. You conclude you've got a 3n-dimensional configuration space, and a quantum state assigns a complex number to every point of that 3n-dimensional space — that's what the quantum state is doing for those n particles. You can ground an interpretation of quantum theory on that structure, recovering objects in three-dimensional space from a complex field in a much higher-dimensional space. That's a whole programme you can engage with, and perhaps carry through successfully — but it's not workable if you've got a changing particle number. And what then is the arena where particle number is changing? We know how to do it in quantum field theory — it's called Fock space. But the people playing this kind of game aren't really interested in Hilbert space structure, and Fock space is a Hilbert space structure — they're interested in something more attuned to philosophy: we all understand what it is to have a scalar field in three-dimensional space changing over time, so we're happy with that, and there are lots of interesting philosophical questions to raise. So, looking at quantum mechanics, can we not construe it as a complex field in a much higher-dimensional space, and use that to understand all the peculiarities of quantum mechanics? Fine — but then how do you carry that through when particle number is changing over time?

It's the same with pilot-wave theory, or de Broglie–Bohm theory. You have a reasonably adequate non-relativistic quantum mechanics in terms of local hidden variables that give you the same probability distributions as non-relativistic quantum mechanics — a great achievement. Can it extend to the relativistic domain? Work in progress — and it's been work in progress for a very long time. If you look at what's going on in pilot-wave theory, the guidance equations give you the point of configuration space wandering around — the motions of all the hidden-variable particles — as just the integral curve of the probability flux, the probability current. That's got a very simple mathematical expression: the probability current is an object you can build out of the quantum state, like a vector field, and each integral curve of that vector field is a de Broglie–Bohm trajectory. A very natural structure. Try to do the same in a relativistic quantum theory, and it completely fails — one of the major reasons being the lack of permanency of particle number, and the associated negative-energy difficulty present in relativistic wave equations.

I'm going a little theoretical just to indicate the examples where using non-relativistic theory outside a proper relativistic framework can be damaging — you can be misled, make wrong choices, if you rely on structures that either don't work at all in the relativistic case, or work in a way dramatically opposed to what you want — which is roughly what happens when you try to do with pilot-wave theory in the relativistic wave equations what you do with the Schrödinger equation. I spent years on relativistic quantum field theory, and at the end of it all I thought: no, it's just not really of substance in engaging with the conceptual questions I'm interested in — which can indeed nearly all be expressed at the non-relativistic level, if you allow photons. So that's my answer for why I frame things in terms of quantum mechanics — though another answer is that "quantum mechanics" is just shorthand for quantum theory, and means to include field theory.

Simon Saunders

Particle localisation — I think that's a very distinctive question, and one I still think is unresolved.

Simon Saunders

What does one do without a covariant position operator? What does it mean not to be able to express spatial localisation without privileging a frame of reference? There's a way of doing it — the so-called Newton–Wigner representation — which breaks the Lorentzian symmetries. It's a fascinating field, though a bit mathematical.

Simon Saunders

Well — one example of what happens with the Newton–Wigner representation: the vacuum ceases to have the Reeh–Schlieder property. The Reeh–Schlieder property itself, in the Minkowski-space vacuum, the covariant or invariant vacuum, is rather extraordinary — it seems that by local operations you can approximate any state in the whole space of states. It's led to a great deal of bafflement about what's quite the right thing to say about it, and it's connected, interestingly, with Newton–Wigner localisation — as is the way complex fields are represented in the covariant wave equations, like the Klein–Gordon equation, the Dirac equation, the Yang–Mills equations, equations for QCD and QED.

The complex numbers that occur in those equations are, as it were, fixed numbers, which don't change in the dynamical evolution of the fields — not the complex numbers used in a Hilbert space representation in terms of particle number, and arguably not the complex numbers of any Hilbert space representation at all. What's used in the Hilbert space representation is a decomposition of these covariant fields into positive- and negative-frequency parts — a decomposition that works so that each part can function to have positive energy and contribute to total particle number. The way that representation works is a non-local operation with respect to the complex unit that occurred at the covariant field level — the complex numbers in the Hilbert space representation are non-locally related to the complex numbers used in the covariant fields. That's the reason there cannot be a position operator in relativistic quantum field theory. If you do a Newton–Wigner representation, these things essentially go away — but fields constructed locally using the complex numbers of the Newton–Wigner representation would then be non-local fields, understood in the standard Lorentz-covariant representation.

This is a whole dimension of analysis. The major question that stares a philosopher of physics in the face is: you can do statistical mechanics, non-relativistic quantum mechanics, thermodynamics, GR, really without too much difficulty — after one or two years of graduate work it's not so hard. But to do anything serious in relativistic quantum field theory is much more — relativistic is like falling off a cliff. There's a mathematical depth and profundity to relativistic quantum field theory that we still haven't understood properly. It's amazing that there demonstrably does not exist a non-trivial relativistic quantum field theory satisfying the Wightman axioms — that these axioms are so constraining they seem to have an effectively unique solution, as free field theory. Relativity and quantum mechanics go together beautifully — the Dirac equation is, I think, the greatest work of art in mathematical physics. I don't think I'm exaggerating: it has a cathedral quality of beauty and elegance.

Simon Saunders

It's difficult to characterise what is beauty, but I think one feature of it is the way your understanding of the equation can be greater or less profound. If you understand the equation in terms of various symmetries, it acquires an elegance that some random partial differential equation doesn't remotely convey — it's already, in the symbolism and notation, a unification of a whole system of partial differential equations. The theory of geodesics is exactly the same.

Simon Saunders

They are related — because I think that the multiplicity of moments in time, we've kind of got our heads around: there's a systematic way of understanding it that makes sense. And we're not there yet with the many worlds.

Simon Saunders

We have yet to have a way of articulating this multiplicity with the same familiarity, confidence, and competence that we have in dealing with linear time. Here's where it exists very much in our ordinary lives: in terms of ordinary notions of probability. If you think about all the contingencies that happen in our lives, that's what we're managing — all of those contingencies carry with them all of the contingencies that did not take place. The very notion of contingency is bound up with so much else that is not the case. We live with that, and it's deeply shocking to us. I find it almost insane that, from the beginning of the universe, this utterance of mine would be deductively, deterministically arrived at, along with all of the minutiae of our present existence with all its craziness and insanity built in from the beginning of the world — that all of that was somehow necessary. No, I cannot believe it. It is not necessary. This is one of a multiplicity of possibilities — and that multiplicity is vast and mind-boggling. Mostly we try not to think about it, but we do have to when making choices relating to events in the future, among the possibilities. The choices we have to make are really about how seriously we take these various possibilities — there's a possibility you'll be run over by a bus the moment you step out of your front door; there's a possibility you'll win a fantastic lottery. What weight do you give them in how you live your daily life? The answer is that we're all extremely adept at correctly estimating risk — we're mostly really good at avoiding it. As long as there aren't wars or crazy things happening, year after year we'll manage to negotiate all of the potentially disastrous things that can happen in ordinary life. That goes with having a sort of primitive theory of everything — a kind of Aristotelian common-sense reasoning about the world, with notions of causation, propensity, disposition. Not quite the notion of probability, but we can fairly easily put it in there — and once it's in there, it starts to make clear what's going on. We have to negotiate amongst probabilities all the time, and that negotiation, from an Everettian point of view, is exactly an understanding of branching structure. Branching structure, from an Everettian point of view, is the source of the many worlds — by virtue of the unitary evolution of the quantum state having this branching structure, and with nothing else added, we take it seriously not just as representing possibilities but as representing actuality — of which the actuality we see, the branch we are located in, is just one.

Simon Saunders

That's a very interesting question. I think the right answer comes back to the issue of whether consciousness is localised. The point is that you and I, as we talk, are able to exchange signals at a rate of a few hundred, probably several thousand, bytes a minute — that's fairly engaged, and fairly integral to the conversation: there's time for my thoughts to have oral impact, for you to hear, to think, to respond. That cooperative effort, this conversation, is taking place over several thousand miles, but it might as well be understood as a common observer. As long as we're not bringing in localised quantum systems that we're looking at, we can coherently treat ourselves as a sort of extended observer, and as such speak of inhabiting one Everettian branch, or a common Everettian branch. I think the quicker and sharper reply must be: you're right, I am just in my Everettian branch, you are in your Everettian branch, but we are correlated — and if you look at the branching structure that unfolds, mostly we are tightly correlated with one another, and in that sense we share a common exchange of ideas. But if we were to start doing quantum experiments, you in the US, me in the UK, we might find, from a branching-structure point of view, some very interesting aspects that underlie the appearance — at least — of Bell non-locality.

This sort of negotiating with contingency — in a sense you could say it's nonsense to suppose this comes uniquely with Everettian quantum mechanics; it would arise in pilot-wave theory or dynamical collapse theory too. In any of those you equally have to manage possibilities, and the possibilities you'd come up with would be much the same regardless of which approach to quantum foundations you take. And, sure, that's sort of right. But if you go to the theoretical point of view and really look at what the contingencies are in the real world, using unitary quantum mechanics, you're doing something that from their perspective is either just a mistake — as in a dynamical collapse theory — or, from a pilot-wave-theoretic point of view, a very incomplete description.

So, in a sense, the perspective that looks at this multiplicity, this network of multiplicities, and says this is what we ordinarily negotiate in our lives — when we really look at it from the perspective of physics, from the point of view of Everettian quantum theory — what "one-world" requirement is doing is rather like the block universe, where you put in a point and say that's where you are: you're picking out a single one of the branches, saying that and only that is unique, and all the others do not exist, and with each branching event, all are cold but one. The Everettian is taking seriously this space of possibilities as all actual — and if that's the case, there is no contingency to reality any longer, because all of the particularities exist. What seemed extraordinarily contingent was that this unique particular world, as it is now, this instant, should exist — that seemed too specific. The multiplicity in Everettian quantum theory returns us to something much more — not that all possible scenarios exist, most, not all — but that there's a probability distribution over these possibilities, and the really crazy ones have zero probability, or close enough to it.

Simon Saunders

Well, I think it's a difficult question, and it depends partly on how you actually analyse probability. A recent approach I've been developing analyses quantum probability in terms of frequentism, involving decompositions of the universal state into microstates of equal amplitude.

Simon Saunders

With such a decomposition, you examine any proposition, property, or projection operator, and you can ask: given the total quantum state, from an Everettian point of view, and given a decomposition that diagonalises that projection operator, what fraction of microstates give the answer yes to the property, what fraction give the answer no? You can do that with finite decompositions of the total state. If you look at projection operators corresponding to things like records of multiple experiments radically in disagreement with the Born rule — take a projection onto that — you can never give it a non-zero probability. Or rather, to give it a non-zero probability, the decomposition of the quantum state into microstates has to be so insanely detailed that you're looking at distinctions of amplitude comparable to the amplitude of the very low, Born-rule-violating branch state itself. It's a conception of probability on which you can never, on any finite analysis, on any finite expansion of the state, see the real extremes. You will not see the very low-amplitude branch. It doesn't mean it's not there — that's one of the interesting features of this whole framework. There's always going to be a Schrödinger-cat state for the projection operator, the property of interest.

I've rather jumped into this without introductory remarks, but I wanted to give it as an illustration of how the extremely low-amplitude scenarios may not have quite the consequence usually thought. Even if there did exist people staring in the face of the huge Born-rule-violating statistics — the experiments have been functioning properly, it hasn't been confected, it's not a put-up job — in the very next second, with enormously high probability, they will see things go on as normal, with Born-compliant probabilities. But then you can play the same game again and again. Perhaps another way of answering the question is a bit like the Boltzmann-brain scenario, which is also a serious problem — I'm not suggesting one should forget about it, but I'm saying this may be very related to Boltzmann brains, except in this perspective you will not even see the probability for the Boltzmann brain, because it cannot be captured in a finite analysis.

Simon Saunders

Yes — the theory comes in two or three forms. Initially I presented this as a theory only involving finite decompositions of the state, so you can approximate the Born quantity extremely closely, and the nature of the approximation is not that with very small probability it differs — no, it's giving you a number very, very close to the Born-rule quantity. It's completely categorical — it doesn't involve potentialities or propensities. It's just the number of microstates that fall within the projector, acting on them as value one. But one microstate — usually, almost always — will be indeterminate for that projector, a Schrödinger-cat state for it. So what you've really got is a set of microstates that give the answer yes to some question, a bunch that give the answer no, and then one, or perhaps more, in between the two — neither yes nor no, a superposition of the projector with one answer and the projector with the other.

Simon Saunders

Think of it this way: we're trying to give probabilities to properties. These could be properties attached to times — say, the property of displaying spin-up on a register of an experimental apparatus. It's just a physical property in that sense. So the question becomes: what is the probability of that property? Whether the property actually exists or not — is it hypothetical? — we don't have to engage with that at the moment; we're just assigning probabilities to different properties, or propositions. Properties are in one-to-one correspondence with propositions here. Propositions are often understood as yes/no questions, but equally can be assertions — if I assert something to be the case, you can turn it into a question. This is very much how properties, propositions, and yes/no questions have been handled in foundations of quantum mechanics — there's a whole logical, algebraic tradition.

So on this picture, you've got a total number of microstates, n, and m of them say yes to the projector, and the residue — the remainder — say no, except for one. That means you've got a lower bound to the probability of that projection, given by the number of microstates that assigned it yes, and an upper bound, given by that bunch which assigned it no, and then the one in between — a grey zone, a no-man's-land, if you like. So rather than having probabilities as rational numbers, we're getting probabilities as small intervals of real numbers, an upper and lower bound. This is quite an interesting construction — I hadn't come across it before, but it existed before my own work: so-called imprecise probabilities, or interval probability — a whole recent branch in probability theory. The point about the grey zone, the interval — you've got a lower and upper bound — it's a bit like instead of a point probability being a point, it's a blip. It can be a bigger blip or a smaller blip, and the bigger the blip, the less informative the probability is, because it's bounded by zero and one. If you got a blip that took up the whole 0-to-1 interval, you'd have no information at all. The point about the very low-probability stuff, using the Born rule, is that it's always in the grey zone. The precision required to pick it out cannot be increased beyond that grey zone.

Simon Saunders

I do understand what you're saying. It's a very interesting set of questions. Perhaps just a comment on the last one: it surprised me, the sort of cacophony that emerged over the last twenty years — if I may call it that — of very different views. My own understanding of Everett was worked out in my mind and appeared to harmonise beautifully with the development of decoherent-histories theory. I thought this would be immediately obvious to anybody who studied these ideas, and I expected decoherent-history-based Everett interpretation to be widely pursued, investigated, examined as to its various conceptual challenges. But that didn't happen. What instead happened is that most people were very distrustful of decoherent histories altogether — quantum-histories formalism seemed somewhat alien to ordinary quantum theory — and instead people pursued their own imaginative reconstructions of Everett's ideas, many of them without regard to decoherence theory at all. I did find this odd. David Wallace and I were highly at one throughout this period, up until he left Oxford, in 2014 or so — there wasn't much difference between us. What differences there were concerned certain issues bound up with the nature of divergence and branching, and certain metaphysical questions I was probably more insistent on than was to his liking.

In the meantime there arose other very articulate proponents of pictures of many worlds involving ideas that, to my mind, are completely antithetical to anything in Everett or in the development that came out of Everett. For example, Sean Carroll's picture, that worlds are in one-to-one correspondence with the discrete spectra of the energy operator — I find that very different from anything I've gathered from Everett. There's another point of view some people have been interested in, a sort of hybrid of Everettian worlds that are nevertheless able to interact with one another, involving modifications to the quantum formalism. And yet further — and I do find this surprising — there's remained a tenacious view that Everett himself was not really committed to many worlds, and that there's a different reading of his work that should be an open inquiry in its own right. Well, I'm fine with that, opening the enquiry — but I do think it's fairly obvious that Everett grasped the many-worlds aspect to his work, even though it wasn't his favoured way of framing it. And rightly, I think, "many worlds" doesn't quite get at what was special about the Everett interpretation — which was multiplicity. In that sense, many worlds — but the multiplicity did not have to involve worlds. It could involve just trajectories of particles, a multiplicity of trajectories arising with the same degrees of freedom — sequences of localised quantum states, all in superposition. That is Everettian thinking, and you don't have to think of those things as worlds.

Simon Saunders

Well, they're very small worlds. No, I agree with you — of course they aggregate, have very large numbers of particles, superpositions of them doing very different things, and then, heavens above, aren't these like many worlds, and you can blow it up and look at the actual solar system. It's quite interesting to look at the actual solar system from an Everettian point of view. This is a little-known truth: Everett's most detailed model of how to interpret the evolving Schrödinger equation was in terms of something very close to a model of the solar system. He could have put it in terms of a model of the solar system — forgetting about chaotic orbits and moons of planets like Jupiter — just four or five very large masses in gravitational interaction, in a superposition: have them initially in localised states, then bring about a superposition of two of those localised states, involving some very serious collision between the particles involved, so it would be non-trivial. What would develop is a superposition of motions, each of which would be perfectly akin to a classical mechanical system, satisfying, to a very high degree of approximation, classical equations of motion. Everett more or less indicated this in the last section of his thesis — I think called "Supplementary Topics" — where he only sketched the idea of how you could recover classical motions from quantum states.

The key is that it's not the degree of freedom that gives you the handle on tracking an object over time — it's quantum states of that degree of freedom, evolving in such a way, involving a superposition of others, that you can track them over time because they obey approximate equations. With very large, well-localised masses, the rules are the rules of Newtonian gravity or mechanics, very precise in those ways — but the rules could be slightly different, and you can get out Navier–Stokes equations, hydrodynamic equations, Brownian motion. You get out these rules for how states behave over time in accordance with these equations — but it's always states in superposition with other states, similarly evolving over time, obeying definite rules. That's the Everett interpretation, though he never quite put it like that. In his 1957 paper, the rules in question were the measurement protocols — prepare the instrument, measure spin, put the value into memory, reset, measure again — a sequence of steps. He analysed the quantum state evolving unitarily in terms of a sequence of states satisfying that protocol, differing depending on whether it's a spin-up or spin-down state. So it was present in Everett, but it wasn't very vivid.

Another example — and I want to push it, because I think it's quite central to the Everett interpretation. Consider radiation: classical linear theory, just like the Schrödinger equation, Maxwell's equations — everything's fine. You've got the electromagnetic field evolving over time, just like the quantum state evolving over time. Now imagine what's happening in that field when you switch on two torches: you create excitations in the electromagnetic field in the vicinity of the torches, which then propagate through the field, in accordance with well-defined rules, gradually spreading over time, depending on the medium and various other aspects of the setup, as to what exactly happens where and when. But now imagine somebody comes along and says: look, what we've got, for each instant of time, is a superposition of a beam of light here and a beam of light here, and at the next instant a superposition of beam-here and beam-here, and so forth — so what we've got in this evolution is the development of a superposition of a beam of light pointing in two different directions at once, which is a contradiction — therefore you cannot have a beam of light pointing in two directions at once, it makes no sense. To which the answer is: no. It's not a beam of light in a superposition pointing in two different directions — it is two beams of light. I think it would be a madman who'd deny that on being pressed. The same goes for mobile-phone conversations going on all the time — what's happening in the electromagnetic field? There are lots of mobile-phone conversations taking place at the same time, and it's not that there's a superposition of a conversation containing all of those words at once — it's two stories, not a single story saying two things at once.

So this indicates what the Everett interpretation is: this preparedness to recognise, looking back at the quantum state, a sequence of stories — I've got a story here, and a story here, and that amounts to two stories, not one story saying two things at once.

Simon Saunders

I think the issue is: if we're looking just at low-energy quantum mechanics, with a reasonably stable system of degrees of freedom, and you're neglecting things like phonons and certainly photons, and you consider what the quantum state would be for that complex system, you've already suppressed a whole lot of stuff — it will not be an adequate representation of the physics. But imagine this is a world in which phonons have somehow been suppressed, and maybe there is no radiation — a sort of pretend world, and we've just got that quantum state unitarily evolving. Does that give us everything we want? I think the answer is yes — it's an adequate ontology, but not a perspicuous one, and not even that adequate, because it doesn't account for all the other bits and pieces.

You could go further: take the standard model, the quantum state for the entire Hamiltonian as defined by the standard model, this representation of the Lorentz group, and that evolving state — even Fock space isn't really adequate to this, the whole thing is over-precisified in inappropriate ways — but imagine that somehow unitarily evolving. That would be an adequate representation of reality, but again not a perspicuous one. What is perspicuous comes back to the sound waves or electromagnetic waves: a snapshot of the electromagnetic field at each moment of time may be adequate — you've got reality there — but it's not a perspicuous way of showing you what's going on, of getting you to understand what's going on. What's needed to show an understanding of what's going on in quantum mechanics is pretty well the whole Q-number structure too — the quantum operators, the algebraic structures, the group representation theory. You need all of that in place; it's not just all in the Hamiltonian. You're articulating local structures to whatever discrimination you want depending on what you're interested in, and you will need all of those Q-numbers.

One way of framing it: the Q-number structure, and the whole mathematical technology involved, is what allows you to give structure to the quantum state — the quantum state has that structure, but what expresses or articulates it is this technology of Q-numbers. What I find particularly perspicuous, of course, is projection operators and sequences of such — quantum histories — though that's also just a coarse-graining of many different types of path-integral approaches to quantum theory. It's a fairly structured way of breaking down pretty well any quantum theory, relativistic quantum field theory as well, into something like quantum histories, with a measure over quantum histories, and the particular kinds of histories involved, the probability relationships thereby involved, whether decoherence is going on, and if it isn't, then maybe we've got to give up on probability being something involved in the dynamics.

So I know that sounds like a bit of a fudge — I'm saying yes, but not really: yes, there's only the quantum state, but no, not really, because we need all the other stuff in order to understand it, to express it, to write it down, to make sense of the representation we've ended up with. It's a real collaboration.

Simon Saunders

Oh, yes — pretty well, yes. This structural kind of representation of reality, I take it, has been giving us reality. I think it got taken in various ways that weren't very helpful — one aspect is the role of mathematics. As long as you think mathematics is really ultimately something more like set theory, and the structure in question is a set-theoretic structure, then I'm not a structural realist at all. Part of what I took to be important about structural realism was that it really was the mathematics that came first.

Simon Saunders

No, I don't think that's right — partly because I'd put it in terms of a postulate, and the postulate is very simple. If there's such a thing as physical probability at all — and that's highly contentious; there are lots of people in many-worlds approaches who deny there's any objective probability, other than something for rational agents — but if you think there is anything like physical probability, let it obey the following postulate: you cannot change X by a physically allowed action on Y, when Y is disjoint from X and the action preserves disjointness throughout. It's an extension of a kind of locality principle — a bit like Bell locality. You can't change the physical probability of X by messing around with Y, which is remote; I'm saying you can't change the probability of X by messing around with Y, which is disjoint. If you have that postulate, it forces equal-amplitude states to have equal probability — any analysis of probability that gives probabilities to states, if it satisfies that postulate, must give equal probability to equal-amplitude states, translating disjointness into orthogonality. That's a strict derivation of the Born rule from a physical principle — but it's a new physical principle, and people may find that a step too far. Notice it's a principle obeyed by the Born rule, as long as it's not supplemented by anything like state collapse — equivalently, as long as the actions permitted are unitary. So the original postulate — you cannot change the probability of X by a physical change to Y that is disjoint from it — has to be a physical change, something you can actually do, and if, as in unitary quantum theory, all physical changes are unitary, then equal-amplitude states must have equal probability. As a mathematical derivation, I doubt it's that interesting, but conceptually I think it's interesting — we'll see whether others find that too.

Look, I'm sorry, we've overrun, haven't we — I got the sense we were getting sidetracked in so many interesting ways, but I think that's probably what you do, and you did it brilliantly. I worry the result is going to seem a bit haphazard, visiting one topic after another.

Simon Saunders

No, it's been lovely. I do hope for another occasion.

Simon Saunders

Well, absolutely — and what they do is stretch me very far. I find myself stretched, and it's not always pleasant — I'm not in my comfort zone as a result. That's how the tension is expressed. But I think the stretching is mostly productive, and it's not to be alleviated by ignoring one side or the other — it is to work with it, and it's potentially fruitful. It's possible to make progress on very limited things which may yet illuminate much more greatly. I'd give an example, partly as a result of teaching Leibnizian metaphysics: I found Leibniz's monadology a very interesting way of resolving the mind–body problem — it was entirely devised to that end, that's why Leibniz devised it, it was the only way he could see of resolving the mind–body problem, given all the arguments — Chinese cells, the Chinese room argument, things like that — Leibniz saw it all very clearly.

There's a quantum version of the monadology I find extremely interesting. It's not quite idealism, but it's a framework of thinking in which representation — or, in Leibnizian terms, perception — is fundamental, and the world is built up out of perceptions. That's the monadology. There's a quantum version where these are really correlational structures, though you could call them perceptual structures if you wanted — I think there may be an appropriate shift of language in that way. Might it be a way of seeing physical reality very differently in quantum-mechanical terms? I think there may be some possibilities of that. If something like that were to really make sense, I think it would shift a lot of the things that are under stress — whether it would change the more fundamental tensions, I'm not sure, I suppose I doubt it. But I was struck by one thing: Bertrand Russell, when he read the Monadology, more or less said that in accordance with it there'd be no such thing as absolute simultaneity — something like that, in 1904 or thereabouts.

Simon Saunders

Yeah.

Simon Saunders

Well, thank you so much — it's been a great pleasure, really enjoyed it. I've also got a sense that you had all kinds of really hard questions still to ask.