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The statement of the holographic principle seems very similar to specification of boundary conditions. A differential equation can be uniquely solved if one specifies the boundary conditions. Put differently, one may (uniquely) solve for a field given its values (or that of its derivatives) on the boundary of the manifold. This seems to be an tempting parallel to what the holographic principle states. Except, in the latter case, one specifies a Lagrangian on the boundary and associates it to a Lagrangian in the interior. Is there an obvious flaw in this interpretation?

My motivation for this question is the following query: can we conceive of a mathematical operator that, along with the Lagrangian on the boundary, would allow one to compute the corresponding Lagrangian in the bulk?

roymustang
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    possibly related https://physics.stackexchange.com/questions/528244/is-this-a-holographic-principle –  Oct 06 '21 at 16:12
  • It is partly related, yes. However, that question essentially inquires if Taylor-expansion in one of the variables can be called the holographic principle. – roymustang Oct 08 '21 at 00:01
  • The holographic principle says (very roughly) that we only need to specify data on a surface of codimension $2$ in spacetime, like the (asymptotic) boundary of anti-de Sitter spacetime at a single time, which is the best-understood example. Are you thinking of a surface of codimension $1$ in spacetime, like the whole timelike (asymptotic) boundary of anti-de Sitter spacetime? – Chiral Anomaly Oct 11 '21 at 03:19
  • No, not really. It is a more generic query. – roymustang Oct 15 '21 at 22:26

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