<p>The answer here is deceptively simple, and has very little to do with gravity directly. You only really need to know a bit of quantum mechanics, and the answer comes almost for free.</p>

<p>In any quantum mechanical theory, we have a Hilbert space $\mathcal H$. (Actually, we really want a rigged Hilbert space, but this distinction isn't particularly relevant here.) This can be as simple as the space of a single q-bit, or it can be as complicated as you need it to be (e.g. the Fock spaces which arise in quantum field theory). The Hilbert space describes all the possible configurations of any system described by this theory; an individual vector is a specific configuration. We also have a linear operator $H$ on $\mathcal H$, called the Hamiltonian, which describes the dynamics of such a system via the Schrödinger equation.</p>

<p>The "information" is just knowing what state $| \psi \rangle \in \mathcal H$ your system is in. This state vector tells you the result of every possible measurement, and thus contains all the information of your system. If you really wanted to, it's possible to express any such state uniquely as a sequence of zeros and ones, but that's a very classical way of thinking, and we're dealing with quantum information, which means that the fundamental objects aren't zeros and ones, but state vectors.</p>

<p>So, when we talk about holography, what we're really saying is that we can determine the state $| \psi \rangle$ that our system (e.g. the universe) is in simply from knowing the results of experiments we perform on the boundary of the spacetime (e.g. infinitely far away). The holographic principle alone doesn't say how this reconstruction works, only that it is possible.</p>

<p>Since this is a bit broad, it may be helpful to see how it works in the only really well-understood example, the AdS/CFT correspondence. In this case, we have a theory of quantum gravity  in $d+1$ dimensions with Hilbert space $\mathcal H$ and Hamiltonian $H$, constrained to have spatially-asymptotically AdS$_{d+1}$ metric (AdS is just a special, maximally symmetric solution to the vacuum Einstein field equations which has nice properties that make this work). It turns out that, at least morally speaking (there is ongoing work to understand the full extent of this), when we collect all the observables in this theory and shuffle them around in well-defined ways, we can construct out of them vectors in a different Hilbert space $\mathcal H'$. That is, we have a linear map $T: \mathcal H \rightarrow \mathcal H'$. This map is 1-to-1, meaning that we don't lose information. Strictly speaking, it doesn't need to be onto, but this distinction isn't crucial at first pass, so you can think of $T$ as a 1-to-1 correspondence (i.e. a linear isomorphism) if you like. In addition, the Hamiltonian $H$ can be mapped to $H'$, which describes compatible dynamics to $H$ on $\mathcal H'$ (in the sense that one can first evolve in the original theory and then map to the new one, or first map and then evolve, and the results will be the same). When we look at the pair $\mathcal H', H'$, we recognize this not as a $d+1$-dimensional theory of quantum gravity, but as a $d$-dimensional theory without gravity, but with extra symmetries that turn it into a so-called "conformal field theory" (which come from the extra symmetries of the AdS spacetime). In some sense, this new theory can be thought of as living on the boundary of AdS. So when we say that all the information is contained on the boundary, we mean that given a state $| \psi' \rangle \in \mathcal H'$ which describes the boundary theory observables, we can reconstruct the full state in the quantum gravity theory simply by $T^{-1}(|\psi'\rangle)$. What I've given here is just the 2-minute summary; AdS/CFT is still an active area of research and everything I said above is only approximately and/or morally true.</p>

<p>You might wonder why we need to rely on quantum mechanics. It turns out that (at least in the AdS/CFT correspondence), we can't do it classically. Highly quantum mechanical behavior in one theory is recovered by the classical limit of the other, and vice-versa. This is both a blessing and a curse, but at any rate there's no real classical equivalent. This is suspected to be true in any nontrivial example of the holographic principle, so we're pretty stuck describing things in terms of quantum information.</p>

CC BY-SA 3.0
2014-12-26T04:16:03.363Z
2014-12-26T04:22:12.837Z
2014-12-26T04:22:12.837Z
Logan M
<p>I'm having trouble understanding what exactly "information" is in the context of the holographic principle suggested by string theory. Can it be equated to a matrix of ones and zeros? Does this information have its own laws of physics or are all laws of physics in our universe a result of this information, as in, does information supersede everything? </p>

CC BY-SA 3.0
2014-07-09T03:48:31.890Z
I would say it is everything you need to know if you wanted to reconstruct the universe exactly as it is.
harogaston
CC BY-SA 3.0
2014-12-26T11:13:36.787Z
This question appears to be off-topic because it is possibly about Physics and should be migrated to Physics SE
ProfRob
CC BY-SA 3.0
2014-12-27T00:07:55.053Z
@RobJeffries I don't know what the scope of this site is (the help center makes no distinction between "physics" and "astrophysics" and I've never really understood the difference between the two), nor do I care if this question is migrated, but the holographic principle has applications in astrophysics and cosmology in terms of understanding both black hole physics (arguably where it was first formulated) and large-scale cosmology (where it's still in development). It's of great interest in astrophysics and cosmology; in fact, I just finished reading a paper on AdS/CFT by two astrophysicists.
Logan M
CC BY-SA 3.0
2014-12-27T00:17:01.037Z
@LoganMaingi Fair comment, but there would almost certainly be more interest (and more interest in your thoughtfully written answer) on Physics SE.
ProfRob
CC BY-SA 3.0
2014-12-27T10:07:44.180Z
@RobJeffries The question is older than 60 days old. At this point it can not be migrated even by moderators (I forgot this rule until just now). If it's completely off-topic here feel free to close it, but even if it's only marginally on-topic it's stuck here and can't be moved to Physics SE.
Logan M
CC BY-SA 3.0
2014-12-27T10:54:57.583Z
OP - please could you consider posting this on Physics SE?
ProfRob
<black-hole><supermassive-black-hole><holographic-principle>
In regards to the holographic principle, what exactly is information?
PJS1987
# In regards to the holographic principle, what exactly is information?

(Asked by: username_0 on 2014-07-09)

I'm having trouble understanding what exactly "information" is in the context of the holographic principle suggested by string theory. Can it be equated to a matrix of ones and zeros? Does this information have its own laws of physics or are all laws of physics in our universe a result of this information, as in, does information supersede everything?

**Tags:** black-hole, supermassive-black-hole, holographic-principle

> **Comments:**
> **username_1**: I would say it is everything you need to know if you wanted to reconstruct the universe exactly as it is.  
> **username_2**: This question appears to be off-topic because it is possibly about Physics and should be migrated to Physics SE  
> **username_3**: @RobJeffries I don't know what the scope of this site is (the help center makes no distinction between "physics" and "astrophysics" and I've never really understood the difference between the two), nor do I care if this question is migrated, but the holographic principle has applications in astrophysics and cosmology in terms of understanding both black hole physics (arguably where it was first formulated) and large-scale cosmology (where it's still in development). It's of great interest in astrophysics and cosmology; in fact, I just finished reading a paper on AdS/CFT by two astrophysicists.  
> **username_2**: @LoganMaingi Fair comment, but there would almost certainly be more interest (and more interest in your thoughtfully written answer) on Physics SE.  
> **username_3**: @RobJeffries The question is older than 60 days old. At this point it can not be migrated even by moderators (I forgot this rule until just now). If it's completely off-topic here feel free to close it, but even if it's only marginally on-topic it's stuck here and can't be moved to Physics SE.  
> **username_2**: OP - please could you consider posting this on Physics SE?  

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## 1 answers

### Answer by username_3 on 2014-12-26. Score: 2 | Accepted answer

The answer here is deceptively simple, and has very little to do with gravity directly. You only really need to know a bit of quantum mechanics, and the answer comes almost for free.

In any quantum mechanical theory, we have a Hilbert space $\mathcal H$. (Actually, we really want a rigged Hilbert space, but this distinction isn't particularly relevant here.) This can be as simple as the space of a single q-bit, or it can be as complicated as you need it to be (e.g. the Fock spaces which arise in quantum field theory). The Hilbert space describes all the possible configurations of any system described by this theory; an individual vector is a specific configuration. We also have a linear operator $H$ on $\mathcal H$, called the Hamiltonian, which describes the dynamics of such a system via the Schrödinger equation.

The "information" is just knowing what state $| \psi \rangle \in \mathcal H$ your system is in. This state vector tells you the result of every possible measurement, and thus contains all the information of your system. If you really wanted to, it's possible to express any such state uniquely as a sequence of zeros and ones, but that's a very classical way of thinking, and we're dealing with quantum information, which means that the fundamental objects aren't zeros and ones, but state vectors.

So, when we talk about holography, what we're really saying is that we can determine the state $| \psi \rangle$ that our system (e.g. the universe) is in simply from knowing the results of experiments we perform on the boundary of the spacetime (e.g. infinitely far away). The holographic principle alone doesn't say how this reconstruction works, only that it is possible.

Since this is a bit broad, it may be helpful to see how it works in the only really well-understood example, the AdS/CFT correspondence. In this case, we have a theory of quantum gravity in $d+1$ dimensions with Hilbert space $\mathcal H$ and Hamiltonian $H$, constrained to have spatially-asymptotically AdS$\_{d+1}$ metric (AdS is just a special, maximally symmetric solution to the vacuum Einstein field equations which has nice properties that make this work). It turns out that, at least morally speaking (there is ongoing work to understand the full extent of this), when we collect all the observables in this theory and shuffle them around in well-defined ways, we can construct out of them vectors in a different Hilbert space $\mathcal H'$. That is, we have a linear map $T: \mathcal H \rightarrow \mathcal H'$. This map is 1-to-1, meaning that we don't lose information. Strictly speaking, it doesn't need to be onto, but this distinction isn't crucial at first pass, so you can think of $T$ as a 1-to-1 correspondence (i.e. a linear isomorphism) if you like. In addition, the Hamiltonian $H$ can be mapped to $H'$, which describes compatible dynamics to $H$ on $\mathcal H'$ (in the sense that one can first evolve in the original theory and then map to the new one, or first map and then evolve, and the results will be the same). When we look at the pair $\mathcal H', H'$, we recognize this not as a $d+1$-dimensional theory of quantum gravity, but as a $d$-dimensional theory without gravity, but with extra symmetries that turn it into a so-called "conformal field theory" (which come from the extra symmetries of the AdS spacetime). In some sense, this new theory can be thought of as living on the boundary of AdS. So when we say that all the information is contained on the boundary, we mean that given a state $| \psi' \rangle \in \mathcal H'$ which describes the boundary theory observables, we can reconstruct the full state in the quantum gravity theory simply by $T^{-1}(|\psi'\rangle)$. What I've given here is just the 2-minute summary; AdS/CFT is still an active area of research and everything I said above is only approximately and/or morally true.

You might wonder why we need to rely on quantum mechanics. It turns out that (at least in the AdS/CFT correspondence), we can't do it classically. Highly quantum mechanical behavior in one theory is recovered by the classical limit of the other, and vice-versa. This is both a blessing and a curse, but at any rate there's no real classical equivalent. This is suspected to be true in any nontrivial example of the holographic principle, so we're pretty stuck describing things in terms of quantum information.


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