Philosophy Beyond Spacetime:
Implications From Quantum Gravity

by Christian Wüthrich, Baptiste Le Bihan, and Nick Huggett, eds.
Cambridge, 2021, 292 pages
reviewed by T. Nelson

It's not often that you find a good non-technical description of anything in science, let alone mathematical things like super­sym­metric field theory, the Ryu-Takanagi conjecture, and quantum gravity. But this book by ten philosophers of science has it.

Many of the articles in this multi-author book are worth reading, but the one from Kerry McKenzie (“The Philosopher's Stone: Physics, Metaphysics, and the Value of a Final Theory”) and the one from Alyssa Ney (“From Quantum Entanglement to Spatiotemporal Distance”) are particularly noteworthy.

Is space quantum entanglement?

One of the most interesting ideas to come out of black hole theory is that entropy and spatio­temp­oral distance are connected. Alyssa Ney's article is clearly written and it's about as nontechnical as you're ever likely to get on this topic.

Juan Maldacena's AdS/CFT theory showed that there is a holographic link between AdS (anti-de Sitter space, a hypothetical entity from string theory) and CFT (conformal field theory, a theory with scaling symmetry acting as its boundary). Ed Witten showed that this theory gives us profound insights about the nature of space.

It was all that thinking about black holes that provided the clue. Jacob Bekenstein and Stephen Hawking showed that the entropy (called S) of a black hole, which is to say the amount of information inside it, is proportional to the area of its event horizon, the two-dimensional surface at its boundary. That is, S_bh = A / 4G, where G is Newton's gravitational constant.

The Ryu-Takanagi conjecture says that ‘entanglement entropy’ follows the same rule: S(ρ_A) = area of boundary / 4G^d+1, where d is the number of dimensions.

What does that mean? The interaction between two states is normally calculated using the tensor product (which gives no entanglement), but if they're in a superposition, which happens when they're quantumly entangled with each other, the Ryu-Takanagi conjecture says they give rise to a geometric spacetime.

What is entanglement entropy and how can I get me some?

It's been known for almost a century that entropy, a measure of disorder, is information. That principle also applies not only to communication channels, but everywhere, including the quantum world. Specifically, physicists use the von Neumann entropy S(A) = −Tr(ρ_A log ρ_A), where ρ_A is a density matrix of region A. Tr means the ‘trace’ of the matrix (the sum of the diagonals, which gives the total probability of all the possible states).

This means that entropy is a good measure of quantum entang­le­ment. It tells us how much information there is in the quantum realm when there's an interaction—not just between two particles, but between two regions of space. The density matrix of a state tells us how entangled a system is.

In a spectacular short paper (“Building up spacetime with quantum entanglement”, Gen. Rel. Grav. 42 (2010) 2323), Mark Van Raamsdonk proposed that quantum entanglement entropy is also closely related to classically connected space-time. The idea is that entropy on the boundary can identify the metric of a space, including non-AdS space, which is the kind of space humans live in:

For specific CFTs, each state of the field theory on a sphere (× time) corresponds to a spacetime in some specific theory of quantum gravity where the asymptotics of the spacetime are the same as global AdS spacetime. . . . It is fascinating that the intrinsically quantum phenomenon of entanglement appears to be crucial for the emergence of classical spacetime geometry.

Conclusion: entanglement entropy has properties of space, which means you may already have it.

Ney cautions us not to get too excited yet: it's not clear whether we really have emergence of space. It's possible that spacetime structure doesn't really emerge from quantum entanglement but merely constrains it. She writes:

One needs to point to arguments in which entanglement entropies are plausibly being used to explain metrical features, without thereby assuming them.

This note of caution is valuable and much needed. Ney has written several books on metaphysics and its connection to quantum mechanics.

The Philosopher's Stone

In The Philosopher's Stone: Physics, Metaphysics, and the Value of a Final Theory, Harry, Ron, and Hermione find a magic rock that makes you immort– . . . no wait, wrong book. In this thought-provoking article, Kerry McKenzie says metaphysical theories can't be ‘true.’ Unlike theories in physics, they don't make progress toward a final theory. So, he asks, what value do they have?

McKenzie discusses Larry Laudan's idea of ‘pessimistic induction,’ which says that because past theories were based on non-existent objects like phlogiston, caloric, and the ether, science can't really be a sequence of better approximations to the truth. Science has repeatedly undermined metaphysical claims based on science that eventually got super­seded. In response, philosophers of science turned to structuralism, which says to forget about the specific objects that science hypothesizes and focus on the equations and structural relationships, where successive theories do make progress because they're increasingly accurate explanations of the same phenomena. Using this strategy, the structuralists say, metaphysics of science can avoid tying themselves to things that don't actually exist, and make progress.

McKenzie calls this a ‘trickle-down theory’ and says it doesn't work because metaphysics by its very nature resists approx­ima­tion. He calls this ‘the problem of progress.’ Meta­physics takes quantitative properties, he says, and assigns them into sui generis dichotomous categories which are mutually exclusive and impossible to approximate. By this he means it is all-or-nothing, and that poses a problem: how can you increase the accuracy of something that must be either 100% black or 100% white? How does one even debate on what is fundamental when the present theories in science will eventually be superseded?

He uses the contrast between Hume's ‘intrinsic property’ and modern structuralism as an example. Humeanism, he says, requires all fundamental properties to be intrinsic, which means they're inherent in the object, not imposed from outside. Structuralism requires that no fundamental properties are intrinsic. There's no middle ground. No matter which side you take, it's all or nothing, and can't be approximated, so (like religion, I suppose, but for different reasons) it can't evolve.

McKenzie concludes on a note of existentialist despair:

The value of engaging in metaphysical speculation at present emerges as deeply unclear. Given that naturalist [metaphysicians] have spent much of their energy denigrating the metaphysics of others, is strikes me that we need to find some justification that our project . . . enjoys a more elevated status.

It seems to me that pessimistic induction and structuralism are both inadequate. Science progresses not just by explaining things and inventing better theories, but by discovering new phenomena. Natural phenomena are inherently inaccessible to our senses. Our only contact with them is through experiment and logic.

Phlogiston and the ether were not embarrassments; they were theories and steps forward in understanding the underlying phenom­ena. Scientists will tell you that all theories are wrong; even with our best theories, a better theory can appear at any moment and improve our understanding. This is not failure, but progress: the phenomena remain unchanged. Maybe metaphysics could help us understand the apparent contradiction that a fundamentally unknowable world can be understandable. Just a thought.

Taking Up Superspace

Also noteworthy is Tushar Menon's non-technical description of supersymmetric field theory (SUSY), which is a theory that proposes that the quantum fields of bosons (force-carrying particles) and fermions (which are constituents of matter) are related by the principle of symmetry. Describing this in words sounds like an impossible task, and it absolutely is, but Menon makes a heroic attempt to explain the basics in the article “Taking Up Superspace.”

This book is highly recommended to anyone interested in the progress contemporary physics has made in its eternal quest to blow up the universe and the progress philosophers have made in determining whether the universe really exists or not. So, as always happens, when the philosophers manage to prove the universe exists, the physicists go and change it from existing to non-existing.

jul 17, 2024. updated jul 18 2024

The Emergence of Spacetime in String Theory

by Tiziana Vistarini
Routledge, 2019, 142 pages
reviewed by T. Nelson

It's not the end of the world if string theory turns out not to be background-independent. It's been a spectacularly fruitful theory and many powerful ideas, most notably AdS/CFT, have come out of it. Superstring theory is simpler in some ways than quantum field theory (QFT). If those supersymmetric particles really exist, it might even replace it.

Just imagine: no more Feynman diagrams! Woo-hoo!

But the idea these days among physicists is that for a theory of quantum gravity to be successful, it can't just propose teeny weeny little Planck-scale objects floating in an existing space-time, but should also explain how spacetime “emerges” from them. That means whatever they're proposing has to exist in a place where there is no space and no time, and yet somehow produces a real physical four-dimensional space-time in which other particles (and also big collections of particles like humans) can exist, move, and interact with each other.

That's a tall order philosophically as well as physically. Our common idea of “existing” is that something occupies space and moves in time; if these objects and forces do neither, then they are nowhere forever. In other words, by our common definition, they don't exist.

It's not good enough to say a theory has conformal symmetry, that it predicts gravitons, that it has no fixed fields, or even that you can get a classical Einstein field result out of it. There are many other features that must be explained: real spacetime propagates light and radiation, and particles can move around inside it and form atoms and molecules, for a start.

Tiziana Vistarini's expertise is in metaphysics, but she seems to understand string theory reasonably well. The book could be useful to someone looking for an introduction to the basic concepts. But the arguments she makes in favor of background independence are really mostly bald assertions with not enough consideration of alternatives. There's also very little philosophical speculation about what space may be. I recommend Harold Erbin's String Field Theory: A Modern Introduction for a more nuanced discussion.

jul 25, 2024