Spacetime Geometry as Information Geometry

In an entry to the 2013 FQXi essay contest and in an accompanying paper, Jonathan Heckman, a postdoc at Harvard, put forward a scintillating new idea - that one can derive the theory of strings and of gravity starting from nothing more but a Bayesian statistical inference model in which a collective of $N$ agents (representing by points on a $d$-dimensional grid) sample a probability distribution in order to obtain the best fits to a set of parameters $\{y_1,\ldots,y_M\}$. In investigating the statistical mechanics of such a collective, he finds that their dynamics can be described by an effective field theory, which happens to be the non-linear sigma model. Further requiring that the "judgements" of the collective be stable under perturbations implies that the dimension $d$ of the manifold in which the agents are embedded must be equal to two. Furthermore, he notes that conformal invariance of the resulting two dimensional "agent space", leads us to Einstein's theory of gravity and that the effective dynamics of the collective is described by a theory of strings.

At first glance his line of reasoning appears to be impeccable, and it is only the profound nature of his conclusions that might lead one to question whether his approach has any fatal flaws. Disregarding that possibility for the time being, let us proceed towards further interpreting this ground-breaking result.

The collective lives on a two-dimensional manifold which one can naturally identify with the worldsheet swept out by a string moving in an $M$-dimensional spacetime. Moreover, the space of parameters to which the collective performs a fit must also naturally be identified with the background geometry the string is embedded in. This leads to ask, whether it makes sense to identify the points of a spacetime geometry with statistical parameters and, if so, how can one then relate our usual geometrical notions of distance, angles, etc. to information based concepts.

Cosmological Rulers

To begin, let us switch to a simpler setting - that of our usual flat Minkowski $3+1$ dimensional spacetime, within which are embedded at random locations a set of agents which resemble the wireless routers commonly used in homes and offices. Each agent transmits a single tone at fixed time intervals indicating its presence to all the other agents in its vicinity. Each agent also listens for the tone broadcast by other agents, and by accumulating many such events performs an estimate of its distance to each of the other agents. (illustration) These agents have no scales and no way to measure distances and areas. How can a distance scale arise solely from exchanging signals between agents?

First, let us consider the situation when we do have a way to measure distances. Each agent transmits a signal, say in the form of an em wave, which propagates outwards isotropically from the location of the agent. Now, conservation of energy implies that the total flux $\Phi$ through any closed surface enclosing the agent should stay the same (illustration). In particular given two spherical surfaces $S_1$ and $S_2$ of radii $r_1$ and $r_2$ (with $ r_2 > r_1 $), the flux per unit area $I$:

$$ I(r) = \frac{\Phi}{4\pi r^2} $$

is smaller the greater the distance from the agent: $ I_2 (r_2) < I_1 (r_1) $. So, when an agent $A_1$ emits a signal containing $n$-bits, another agent $A_2$ situated a distance $r_{12}$ from the first one can receive at most:

$$ m = n \frac{a}{4 \pi r_{12}^2} $$

bits of the original signal. Here $a$ is a unit of area, which characterizes the size of the "aperture" using which an agent captures signals. Alternatively, we can state that $\mathcal{A}_2$ receives a fraction of the total flux emitted by $\mathcal{A}_1$, given by:

$$ \Phi' = \Phi \frac{a}{4 \pi r_{12}^2} $$

Since all agents are identical - emit identical signals and have apertures of the same area - $\mathcal{A}_2$ can use the value of the received flux to determine the distance from $\mathcal{A}_1$ as:

$$ r_{12} = \sqrt{ \frac{a}{4 \pi} \frac{\Phi}{\Phi'} } = \sqrt{ \frac{\Phi_0}{\Phi_{12}} } $$

where, for WTLOG (without loss of generality), we have set the area of the aperture $ a = 4\pi$. $\Phi_0 = \Phi$ is the flux emitted by .$\mathcal{A}_1$ and since we are assuming all agents are identical, this can be set to a universal value $\Phi_0$. Finally, $\Phi_{12} = \Phi'$ is the flux received by $\mathcal{A}_2$ from $\mathcal{A}_1$

Let us note that since, a priori, we do not have access to any "rulers" we can only measure ratios of distances. This, in fact, is exactly what is done in most modern cosmological observations. There is no way to determine absolute distances to stars and galaxies, without reference to some celestial objects which are used as "standard candles". We observe a given standard candle - say a type Ia supernova - in some distant galaxy and determine the amount $z$ its light is red-shifted by the time it reaches us. Using some other methods we determine the physical distance that $z$ corresponds to. In this way, we map out the large scale structure of our Universe (or at least of our local neighborhood) by comparing the spectra received from various objects with each other.

In this manner, each agent $\mathcal{A}_i$ can determine its distance to any other agent $\mathcal{A}_j$ as:

$$ r_{ij} = \sqrt{ \frac{\Phi_0}{\Phi_{ij}} } $$

And since, we are in a flat background without any dissipation it is safe to assume that $\Phi_{ij} = \Phi_{ji}$ and therefore $r_{ij} = r_{ji}$.

There are two other details. First, how does a given agent $\mathcal{A}_i$ distinguish between the flux received from two different agents $\mathcal{A}_j$ and $\mathcal{A}_k$, which lie at equal distances from $\mathcal{A}_i$? Second, even with the ability to measure distances to other agents, how does any one agent reconstruct the geometry in its neighborhood? Without some sense of direction, distances alone are not sufficient to allow an agent to distinguish between two equally distant neighbors.

The first problem can be addressed by equipping each agent with a random number generator. The procedure followed by any agent is then as follows:

1. When its first turned on, the agent generates a random number, its unique ID, and transmits that embedded with its default signal.


2. As it receives signals from other agents, it compares the numbers it reads from their signals with its own. If it receives a signal with a number identical to its own, it generates another random number and sets that as its new ID.


3. This process is continued until every ID the agent receives is different from its own, for some minimum specified duration.


4. Once this equilibrium state is reached, the agent uses the measured value of incoming fluxes to associate a distance to each one of its neighbors.


5. In the event of an ID conflict - agent receives flux signal with ID identical to its own - the system resets and starts from step 2.

The second problem, that of being able to distinguish neighbors which are equidistant from a given agent, but not coincident, can be addressed in several ways.  One possible method is

Area and Information Density

 

 

Conclusion

The Measurement Problem, Part 1

The Problem

The measurement problem becomes a problem only when we neglect to specify the nature of the observer's Hilbert space. Postulates I (Systems are described by vectors in a Hilbert space) and II (Time evolution occurs via some given Hamiltonian for a particular system) are fine in that regard. These two postulates deal only with the description of a quantum system. It is the third postulate (Measurement leads to collapse of state vector to an eigenstate) where there is a problem.

A measurement is said to occur whenever one quantum system - the "observer" - described by a Hilbert space ( $ H_{O} $ ) interacts with another system described by a Hilbert space ($H_{S}$). The complete Hilbert space of the system ("observer" and the "observed") is given by:
$$ H_{O+S} = H_{O} \otimes H_{S} $$
To actually realize the dichotomy between an "internal" and "external" observer, the size of the observer's Hilbert space, given by its dimension ($dim(H_O)$), must be comparable to ($dim(H_S)$) - the dimension of the Hilbert space corresponding to the system under observation. Instead, what we generally encounter is ($dim(H_O) \gg dim(H_S)$) as is the case for, say, an apparatus with a vacuum chamber and other paraphernalia which is being used to study an atomic scale sample.

In this case the apparatus is not described by the three states ($\{\ket{ready}, \ket{up}, \ket{down}\}$), but by the larger family of states ($\{\ket{ready;\alpha}, \ket{up;\alpha}, \ket{down;\alpha}$) where ($\alpha$) parametrizes the "helper" degrees of freedom of the apparatus which are not directly involved in generating the final output, but are nevertheless present in any interaction. Examples of these d.o.f are the states of the electrons in the wiring which transmits data between the apparatus and the system.

The initial state of the complete system is of the form:
$$\ket{\psi_i} = \ket{ready;\alpha} (\mu \ket{1} + \nu \ket{0} )$$
When $ H_O$ interacts with $ H_S$ in such a way, that a measurement is said to have occurred, the final state of the composite system can be written as:
$$\ket{\psi_i} = \ket{up;\alpha} (\mu_{up} \ket{1} + \nu_{up} \ket{0}) + \ket{down;\alpha} (\mu_{down} \ket{1} + \nu_{down} \ket{0})$$
In a complete self-consistent theory, one would hope that all paradoxes regarding measurement could be resolved by understanding unitary evolution of the full Hilbert space ($H_{sum}$). This is not quite the case. Consider the case when the system being observed is a spin-1/2 object with a two dimensional Hilbert space ($H_{sys}$) a basis for which can be written as ($\{ \ket{0}; \ket{1} \} $). The Hilbert space of the observing apparatus ($H_{obs}$) is large enough to describe all the possible positions of dials, meters and probes on the apparatus. Let us assume that ($H_{obs}$) can itself be written as a tensor product:
$$H_{obs} = H_{pointer} \otimes H_{res}$$
For some poorly understood reason, when ($N_{obs} \rightarrow \infty$), an interaction between the two systems - observer and subject - causes the state of the subject to "collapse" to one of the eigenstates of the operator (or "property") of the subject being measured ($\ket{\psi_{sub}} \rightarrow \ket{\phi^i_{obs}}$).

When QM was first invented, it was understood that the measuring apparatus is a classical system requiring an infinite number of degrees of freedom for its complete description. Thus the ``collapse'' that occurs is because of something that happens at the interface of the classical measuring apparatus and the quantum system being observed. This ad-hoc separation of the classical from the quantum came to known as the "Heisenberg cut" (or "Bohr cut" depending of your reading of history). Since the quantum description of systems with even a few degrees of freedom appeared to be a great technical feat in those early days, physicists didn't have much reason to worry about systems with large ($N \gg 1$) dimension Hilbert spaces.

Mechanisms for Wavevector Collapse

To address the lack of understanding of state vector collapse in QM, and to get a grasp on the description of systems with large Hilbert spaces, first the many-worlds interpretation (MWI) and later the consistent histories or decoherence framework was constructed.