Once talking to a visiting professor at my institute, I heard about some simple model that captures the emergence of space coordinates as the eigenvalues of some infinite-dimensional quantum mechanical Hamiltonian in $0+1$ dimensions.

Unfortunately, my attempts to find a reference or explanation inevitably end up in very complicated string theory papers that are far from what I am looking for and in which I am, to say the least, not an expert.

I would love to find some description of how this can be implemented. A toy model is more than enough.

I can offer up something similar to this, which is an isomorphism between something called the Tsirelson bound and the spacetime metric. This is not exactly the emergence of spacetime from quantum mechanics, but it does illustrate how spacetime could be seen as quantum mechanics in diguise.

Suppose we have four operators $A_1, A_2, B_1, B_2$  such that: $$ A_i^2~=~B_i^2~=~1 $$ and $$ [A_i , B_i ]=0 $$ These 4 operators correspond to the observables in Aspect's experiment. A single source of photons emits pairs of photons to the left and right measuring apparatuses. At the measurement station a rapidly moving mirror pushes the photons to be measured for polarization either on direction $A_1$ or $A_2$ for the left detectors, or $B_1$ or $B_2$ for the right detector. The outcomes are $+1$ or $-1$ (thus $A^2 = B^2 ~=~1$). The $A_i$ commute with $B_i$ because they are spatially separated. This set up is diagrammatically the same as used in the Bell theorem as diagrammed below

From http://faculty.virginia.edu/consciousness/new_page_7.htm

Now define an operator $C$ as follows: $$ C~=~A_1 B_1 ~+~A_2 B_1 ~+~A_2 B_2 − A_1 B_2 $$ and it is not hard to show that: $$ C^2~=~4~+~[A_1 ,A_2 ][B_1 ,B_2 ] $$ By using the triangle inequality it is not hard to see that $$ |C^2|~\le~4~+~4,~|C|~\le~ 2\sqrt{2} $$ This is similar to the derivation by Peres for the Tsirelson bound.

The operator $C$ appears very similar to the Lorentz metric, $$ x\cdot y ~=~ -x_0y_0 + x_1y_1 + x_2y_2 + x_3y_3, $$ which is the metric distance with Lorentz geometry or $SO(3,1)$. In the case of the Riemann sphere $\sim \mathbb CP^1$ the set of conformal transformations are linear fractional transformations $$ z~\rightarrow~\frac{az + b}{cz + d} $$ where this transformation is isomorphic to $PSL(2,\mathbb C)$. The heavenly sphere is then the case of the null metric distance, or equivalently the projective light cone. The product space $V$ of $dim ~=~ n$ has the Jordan algebra is the $v^2 ~=~ \langle v, v \rangle$ ($v \in V$, $\langle u, v \rangle$ is the $\mathbb{R}^n$ inner product) so that a spin factor $J(V) ~\sim~ V\oplus\mathbb R$ (think of space plus time) such that $$ (u, \alpha )◦(v, \beta ) ~=~ (\alpha v + \beta u, \langle u,v\rangle - \alpha \beta ). $$ Then $J(V)$ is isomorphic to Minkowski spacetime. This Clifford algebra define on the right the spacetime metric $\langle u,v\rangle~ -~ \alpha\beta$.

So now suppose we let there be the spacetime $V_c$ with the basis elements $$ u_1 ~=~ (A_1 , 0, 0), u_2 ~=~ (0, B_1 , 0), u_3 ~=~ (0, 0,A_2) $$ $$ v_1 ~=~ (B_1 , 0, 0), v_2 ~=~ (0, A_2, 0), v_3 ~=~ (0, 0, B_2) $$ and the real line R containing the two elements $(A_1 , B_2)$, it is then easy to see that the C operator can be expressed according to the Clifford algebra.

The connection between the null condition and the Tsirelson bound might be made by defining the elements of the real line $\mathbb R$ as $(i A_1 + \sqrt{2 \sqrt{2}}, i B_2 ~+~ \sqrt{2 \sqrt{2}})$ so that the product is the real valued part so that this is $-B_2 A_1 ~+~ 2\sqrt{2}$. In that way the modified $|C|^2$ would be zero if it is at the Tsirelson bound, and similar to a choice of metric signature is negative if outside the Tsirelson bound.

This is a somewhat elementary or naive approach. We have though a monoid or magma connection between the Jordan algebra and the $W^*$ algebra. This might be an interesting topic to pursue. It is my thinking that spacetime physics and quantum physics are on a deep level identical. In effect spacetime is probably built up from quantum entanglements.

  • Thanks so much for the very intriguing discussion. I'd vaguely come across this kind of stuff before, but never seen such an accessible treatment illustrating the steps comprising a rigorous approach to the ideas. – John Forkosh Jun 21 '16 at 0:04
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    I got a few votes here. It is a good thing I checked, because I forgot to put in the diagram I wanted for this. This should make is clearer. An operator pertaining to measurements that are not located at the same place leads to an interval that measures a distance between two points, which for the null case is of course zero. This corresponds to the Tsirelson bound. The next step is to look at event horizons. – Lawrence B. Crowell Jun 21 '16 at 2:05
  • Thanks again. I didn't realize who you were until I googled you and came across fqxi.org/community/forum/topic/370 (and similar). But when you get into AdS/CFT, I'm in pretty much the same position as Andrii Magalich describes above:) You have (or can recommend) anything along those lines accessible to ~1st,2nd year graduate? That is, maybe similar to above but more fully developed, and maybe with "gentler" introductory discussion? – John Forkosh Jun 21 '16 at 5:57
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    As for recommending papers along these lines I would definitely read Raamsdonk's papers on the emergence of spacetime from entanglements. They are not terribly heavy with string/M-theory language. – Lawrence B. Crowell Jun 22 '16 at 14:16
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    That is the seminal paper. I think I have watched the UT lecture as well. This is an interesting frontier. One should also think of this in light of Dixon-Bern gauge-gravity, the BCFW method and Arkani-Ahmed's amplituhedron, as well as string/M-theory. Also check out Verlinde [ arxiv.org/abs/1211.6913 ] on error correction codes and see this in light of Duff et al with SLOCC [ arxiv.org/abs/1101.3559 ]. – Lawrence B. Crowell Jun 23 '16 at 12:47

This probably refers to the BFSS matrix model. This is argued to arise in several equivalent ways: as the worldvolume theory of a large number of D0-branes in type IIA, as the KK-compactification of 10d SYM to zero space dimensions, or as a certain non-commutative limit of of the worldsheet action of the M2-brane in M-theory. In any case, it ends up being a quantum mechanical system whose degrees of freedom are a set of 9+1 large matrices. These play the role of of would-be coordinate functions and their eigenvalues may be in interpreted as points in a spacetime thus defined.

In the 90s there was much excitement about the BFSS model, as people hoped it might provide a definition of M-theory. It is from these times that Witten changed the original suggestion that "M" is for "magic, mystery and membrane" to the suggestion that it is for "magic, mystery and matrix". (See Witten's 2014 Kyoto prize speach, last paragraph). But these days it is more quiet around the BFSS model, a fate it shares with many of the ideas that were exciting in the 90s and now lay somewhat more dormant.

Notice that there is also the IKKT matrix model which takes this one step further by reducing one dimension further down (D(-1)-branes). There is a Japanese group who claims that a MonteCarlo computer simulation of the IKKT model which they have set up shows a spontaneous generation of a 10d spacetime with 3+1 large and 6 tiny dimensions (Kim-Nishimura-Tsuchiya 12).

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