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The idea of the holographic principle is that all the data about what's inside a volume can be discribed by fields on it's boundary.

But... isn't this just obvious calculus?

e.g. take a field in polar coordinates $\phi(r,\theta,\rho)$ with $r<1$.

The boundary of this volume is at $r=1$. We can define an infinite set of fields:

$$\Phi_n(\theta,\rho) = \frac{\partial^n}{\partial r^n}\phi(r,\theta,\rho)|_{r=1}.$$

These fields live on the boundary. The field inside the boundary can be reconstructed by a simple Taylor series:

$$\phi(r,\theta,\rho) \equiv \sum_{n=0}^{\infty}\frac{1}{n!}(r-1)^n \Phi_n(\theta,\rho).$$

So it's easy to create a set of fields on the boundary that are equivalent to fields on the interior. Assuming the fields are analytic. So what's so special about the holographic principle?

Qmechanic
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  • One would think that to know the state of a system one would have to acquire all the information in the volume instead of the surface alone. – Pagoda Jan 31 '20 at 18:52
  • You answered your own question when you wrote "Assuming the fields are analytic." Analytic fields satisfy Laplace's equation, but other fields could satisfy Poisson's equation, or more general ones. – D. Halsey Feb 11 '20 at 22:50
  • @Halsey, So it seems to me like the holographic principle is saying the fields have a special property akin to being analytic which allows them to be written as fields on a boundary. Still this doesn't seem as mindblowing as first thought. –  Feb 12 '20 at 22:52

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In the holographic principle for quantum gravity the DOFs are encoded on a spacetime codimension-2 Cauchy surface. A conventional hyperbolic system has a spacetime codimension-1 Cauchy surface. A Taylor series $f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$ is typically not viewed as an example of holography. For starters because we usually don't require analyticity of physical fields. In fact, typically only a finite number of derivatives are important in physical theories, cf. e.g. this & this Phys.SE posts. See also this related Phys.SE post.

Qmechanic
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  • True, but the solutions to the wave equation are usually analytic functions. Except at poles. So classical fields are usually discribed by analytic functions. –  Feb 11 '20 at 15:48