# I Can spacetime exist in superposition?

#### Khashishi

Science Advisor
It's well known that a single particle can exist in superposition, but what about the gravity of the particle? Is the gravity also in superposition? I suppose this makes it difficult to write a wavefunction, since we can't express it in terms of a field over a single spacetime. But what if we naively just assume that gravity is never in superposition and we simply calculate the stress energy density directly from the smeared out wavefunction? What problems would that cause?

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#### Khashishi

Science Advisor
I guess a serious problem is that a wavefunction collapse can't be treated covariantly as an event in spacetime. If we have a particle that passes a slit and hits a screen, the wavefunction spreads out as it passes the slit, but then collapses to a small region on the screen, and we don't know where this collapse actually occurred in spacetime.

#### nikkkom

You are trying to marry a quantum object (particle) to a classical one (GR gravity). That's probably why it does not work.

#### JasonWuzHear

This is fun to think about. It really forces you to try and understand, "what is spacetime?", "what is curvature?", "What is a distance?"......

Space-time is a structure we can use to organize distances between events. These distances define the communication time associated with the two points. This makes me wonder, if you were just given a collection of N nodes along with the distances between each node, could you determine curvature? Could you determine dimensionality?

With two nodes (or points, whatever you want to call them), you only have one distance, and dimensionality or curvature can't be determined from that alone.

Makes sense to me for there to be superpositions of distances between things.

#### haael

I don't know the details, but Hawking was doing something called "summing over histories" where he applied QM principles to curved spacetime. He tried to show this way that some singularities cancel out.

So yes, there are theories where spacetime can be superposed (added).

#### JasonWuzHear

I should also say, that Wheeler's superspace is more easily thought of in superpositions than spacetime. Go ahead and look that up.

#### mfb

Mentor
It's well known that a single particle can exist in superposition, but what about the gravity of the particle? Is the gravity also in superposition?
Probably. If an object is in superposition between two states, a nearby particle should couple to it via gravity, and become part of this superposition. Such an experiment would require a relatively large mass kept in a large macroscopic superposition for a long time - and that is very challenging. Current setups cannot avoid decoherence long enough for such an experiment.

#### MPBenowitz

In QM a particle in superposition is a linear combination of complex amplitudes. In QFT particles are point-like excitations of 'substances' we call fields that have a value at every point in space and time. On the one hand, particles are described by rays in Hilbert spaces while on the other hand spacetime is described by the metric tensor of a differentiable manifold (solutions to Einstien's Field Equations). Superposition is a property of Hilbert spaces. It immediately follows that if spacetime can be in a superposition then spacetime cannot be described by a smooth manifold. In other words, Einstien's Field Equations aren't quite right.

When we look at the cosmos at the largest length scales we see a differentiable manifold whose dynamics are governed by Einstein's Field Equations. If we zoom into the macroscopic length scales, the scales of classical mechanics, we see a differentiable manifold described by Hamilton's equations of motion. When we further zoom into the microscopic length scales, the scales of quantum mechanics, the cosmos can no longer be described by a differentiable manifold. Rather at these scales, the cosmos is described by a Hilbert space whose dynamics are governed by the Schrodinger equation. On the one hand, manifolds are metric spaces containing commutative coordinates whereas on the other hand Hilbert spaces are complex metric spaces containing noncommutative coordinates. A natural question then arises. If we zoom into the smallest possible length scales, the Planck scale, would we expect spacetime itself to be commutative or noncommutative? If it's the former there exists a deep asymmetry in the natural order where there exists a special scale in which the cosmos is described by a Hilbert space and not a differentiable manifold. We would be faced with the following question: why does there exist a special length scale that can only be described using noncommutative coordinates?

#### mr_persistance

In QFT particles are point-like excitations of 'substances' we call fields that have a value at every point in space and time. ?
Is space time in QFT discrete or continuous?

#### MPBenowitz

Is space time in QFT discrete or continuous?
QFT is defined in a Minkowski background which is continuous. The creation and annihilation operators of the fields are discrete.

#### tom.stoer

Science Advisor
The question "can spacetime exist in superposition?" can only be answered in a specific context.

In GR there is no superposition of spacetime which causes logical inconsistencies when considering gravitational effects of quantum fields which are in superposition. So it means that quantum field theory plus GR cannot be a fully consistent framework.

In theories like LQG spacetime, i.e. the spin networks are in superposition. This follows from the fact that LQG is structurally nothing else but QM formulated in a Hilbert space.

People subscribing to string thewory may have different answers ...

Please note that superpositions in QM are somehow trivial; it's just an artefact of the Hilbert space basis one uses. It becomes important only in case of measurements; but I can hardly imagine about a quantum gravity experiment showing superpositions - not even in principal.

In my opinion a collapse interpretation is not appropriate in case of quantum gravity (applied to thew whole universe). Either we find something completely new, or we should use many worlds.

#### Denis

I guess a serious problem is that a wavefunction collapse can't be treated covariantly as an event in spacetime. If we have a particle that passes a slit and hits a screen, the wavefunction spreads out as it passes the slit, but then collapses to a small region on the screen, and we don't know where this collapse actually occurred in spacetime.
A similar argument is made in https://arxiv.org/abs/0909.1408

It is claimed there that what would be an observable, say, in Newtonian quantum gravity, cannot be predicted by a covariant theory of quantum gravity, because it would require to compute a scalar product of two wave functions defined on different manifolds.

#### tom.stoer

Science Advisor
A rather simple idea is the following: suppose you start with a spherically symmetric spacetime and a spin-0 quantum state; suppose the state (particle) decays into two spin-1/2 particles, which are measured by a spherical symmetric detector array at two specific locations. If we believe in a "realistic collaps" this automatically breaks our initial symmetry, which is both inconsistent with GR as well as with QM. So either we use an instrumentalistic view w/o any ontic interpretation of the quantum state at all, or we believe in quantum superposition of all possible locations and all corresponding spacetimes; this results in a many-worlds-interpretation.

#### mfb

Mentor
Please note that superpositions in QM are somehow trivial; it's just an artefact of the Hilbert space basis one uses. It becomes important only in case of measurements; but I can hardly imagine about a quantum gravity experiment showing superpositions - not even in principal.
Superpositions of what?

You can put an object into a superposition of two locations, and then entangle its position with the position of another object via the electromagnetic interaction. The same should work via gravity, it is just much more challenging experimentally.

#### tom.stoer

Science Advisor
Superpositions of what?
Superposition of spacetime.

OK, you are right. If quantum gravity contains superpositions of quantum spacetime - like LQG - then entangled particles cause entangled quantum spacetime automatically.

What does string theory tell us about this question?

#### MPBenowitz

Superposition of spacetime.

OK, you are right. If quantum gravity contains superpositions of quantum spacetime - like LQG - then entangled particles cause entangled quantum spacetime automatically.

What does string theory tell us about this question?
The Ads/CFT Correspondence suggests a superimposed spacetime geometry is equivalent to some 2-dimensional state. Formally, Ads/CFT tells us that for a 3-dimensional gauge theory of gravity there exists an equivalent Conformal Field Theory in 1 dimension less. This is significant since it is the only known realization of the holographic principle -- which is a prediction that the entropy of black holes are proportional to their surface area and not their volume. The holographic principle + Ads/CFT is without a doubt the most significant result beyond the standard model. It's important to point out that although the Ads/CFT correspondence is a result of stringy theories it doesn't necessarily mean that a string theory of gravity bears the equivalence with a CFT in dim less. It's entirely possible that string theory isn't the the true theory of gravity and that some other theory -- one that can replicate the predictions of the $\Lambda$CDM model of cosmology is a better candidate.

String Theory has major problems. It's the best we've got is an excuse. From camp Kuhn, LQG and String theories are indistinguishable from each other. How long can field continue without producing any new predictions? My guess is that if the next LHC run turns up empty with no detections of supersymmetry string theorists will jump ship. Falsifiability aside, a simple thought experiment (the one in my 1st response) makes an incredibly strong case against stringy theories.

#### Denis

From camp Kuhn, LQG and String theories are indistinguishable from each other. How long can field continue without producing any new predictions? My guess is that if the next LHC run turns up empty with no detections of supersymmetry string theorists will jump ship. Falsifiability aside, a simple thought experiment (the one in my 1st response) makes an incredibly strong case against stringy theories.
Arguments may be simply ignored. Experiments do not matter at all if a theory does not make predictions about them. String physics never existed, and is not necessary at all, once string theorists control enough grants so that they can offer jobs for young physicists. What they will do is stringy mathematics. This will not be mathematics in the usual sense too, because they usually do not prove theorems on the mathematical level, only at the physical level.

But so what, even if neither physics nor mathematics in their classical meaning, it remains scientific enough, at least in comparison with humanities, which completely degenerates.

#### Haelfix

Science Advisor
Superposition of spacetime.
OK, you are right. If quantum gravity contains superpositions of quantum spacetime - like LQG - then entangled particles cause entangled quantum spacetime automatically.
What does string theory tell us about this question?
There are a bunch of issues here that make's answering the question difficult. The first problem, as you point out, is that superpositions are statements about how you choose your basis . Hence you can rotate them away and typically evade paradoxes and things of that nature. It's really entanglement that carries the irreducibly quantum structure, and this can't be done away with. So when we discuss entanglement and gravity, you really need to formulate a precise question about the entanglement structure. But the entanglement structure of *what*?

The problem is that typically, to even start talking about quantum gravity, we need some sort of saddle approximation in a path integral of some underlying theory. But it is precisely this restriction to semiclassical states that seems to cause the paradoxes described in eg the Feynman lectures on gravitation or when you naively try to quantize the Einstein field equations (say by promoting all the various geometric objects into operators). So already at this most basic level, you are in a certain amount of trouble.

The next problem is that thinking operationally, what does it mean to 'zoom' into the Planck scale (as one of the posters asks). What does that mean exactly in a laboratory? It shouldn't be hard to convince yourself that the only way to 'probe' structure at that scale, is to ask questions about scattering experiments. However it is precisely at this scale where all known forces have the same comparable gravitational couplings as gravity does to itself. So you can't just learn about gravity by itself, you have to know everything at once. Worse, it is also at this scale where on dimensional grounds, you start to create energy densities that exceed the classical hoop conjecture, and so you start having to worry about creating black hole horizons. The more energy you stick into the incident particles to probe things, the bigger the horizon is and the less you learn about quantum gravity (and about what is or is not commutative).

Now, for attempts at answering the first question about entanglement, well this is the huge industry that all the famous theorists have been working on these past few years. The Ryu-Takayanagi formula, Tensor networks/MERA (entanglement as the glue of spacetime), entanglement wedge reconstruction, ER=EPR. Really quite fascinating developments, but as usual with this stuff, its still in its infancy and restricted to toy models. The relationship with string theory are there at some fundamental level (since most of this stuff is derived from AdS/CFT) but quite obscure.

#### tom.stoer

Science Advisor
The Ads/CFT Correspondence suggests a superimposed spacetime geometry is equivalent to some 2-dimensional state. Formally, Ads/CFT tells us that for a 3-dimensional gauge theory of gravity there exists an equivalent Conformal Field Theory in 1 dimension less ...
My question is a rather formal one.

If we have superpositions of quantum states of "ordinary" matter, then this should result in superpositions of spacetime.

In LQG there is a kind of Hilbert space structure, i.e.

$$|\text{quantum spacetime}\rangle = \sum\,|\text{spin network state}\rangle$$

In geometrodynamics (asymptotic safety approaches) path integrals are used, so superpositions are encoded in the measure over equivalence classes of spacetime manifolds

$$\int \frac{Dg}{\text{Diff}}$$

In perturbative string theory I've always seen classical background spacetime with strings moving in this spacetime; so this is not quantum spacetime at all (problem of background independence).

There are a bunch of issues here that make's answering the question difficult.
Other approaches do have this formal structure. They may turn out to be wrong, but the structure is there, partially quite precisely defined. All I am asking is how this formal = mathematical structure of quantum spacetime may loook like in string- / M-theory / AdS/CFT related models.

Suppose the above mentioned spin-0 particle decays into a superposition of two-particle states: how does the superposition of quantum spacetimes look like?

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#### Haelfix

Science Advisor
Other approaches do have this formal structure. They may turn out to be wrong, but the structure is there, partially quite precisely defined. All I am asking is how this formal = mathematical structure of quantum spacetime may loook like in string- / M-theory / AdS/CFT related models.

Suppose the above mentioned spin-0 particle decays into a superposition of two-particle states: how does the superposition of quantum spacetimes look like?
Sorry, i'm not clear about the setup here or what sort of approximation scheme we are working in. Talking about semiclassical gravity, the semiclassical spacetime would appear to not be in a superposition at all, b/c by assumption the regime is such that the particle states aren't energetic enough to backreact on the ambient semiclassical spacetime. The particles themselves might be entangled, and note that linearized gravitons propagating in the theory would also be entangled. But where the entanglement lies within the fundamental degrees of freedom that defines the full UV path integral is of course unknown. However I would say your question is more a statement about the inconsistency of keeping a classical spacetime as a fundamental unit, b/c if you include the incredibly weak gravitational field of the particles in the state of the system, you would indeed find that if you could make an objective statement about the gravitational field as required by the classical Einstein equations, you would objectively know whether there was a decay or not of the quantum particle, and that can't be allowed by the rules of quantum mechanics.

Incidentally the form for the path integral measure you mentioned for regular Einstein gravity, seems to hold as well in string theory, except that 'differentiable manifold's is replaced by 'stringy manifolds' (typically Riemann surfaces when working on the worldsheet or some sort of generalized Stringy geometry when working on the target manifold). Background dependance or independance are typically statements about attempts to 'solve' a given theory, they aren't necessarily statements about the fundamental theories themselves.

Note that in the LQG case, as well as the AdS/CFT case there is a well defined HIlbert space, and so it's much easier to talk about entanglement. In the AdS/CFT case in particular (unlike any spinfoam model i've ever seen), there is also a well defined dictionary between the bulk semiclassical space and this boundary Hilbert space, so we can ask questions about what entanglement might look like. Please see note 21 (review measures of entanglement in QFTs in note 19 for notation etc) here:
http://www.hartmanhep.net/topics2015/

#### tom.stoer

Science Advisor
Thank you; I will check the information you recommend.

Just for clarification: I am talking about the full theory w/o any approximation. I mentioned the spacetime background used in perturbative string theory only to make clear that this is not what I expect at the fundamental level.

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