Hi,I dont understand the holographic principle,how can the entire physics in a space be done on its boundary,arent we supressing a dimension and there by losing the information contained in it?,and is it neccesary that the boundary be closed?
The entropy of a volume of space IS proportional to the enclosed volume...This is the Beckenstein-Hawking bound.
It depends to which formulation of the holographic principle you are referring: If you want to talk about "information in a certain volume" the defintion of a volume reqauires a closed surface. This surface need not be a physical entity but could in principle be a mathematical concept, only. For a generalized gauge/gravity duality where you say that a theory including gravity (like string theory in AdS spacetime) is dual = mathematically equivalent to a lower-dimensional field theory living on the surface I don't think that the surface must be closed. But for this rather general stement we do not have proof, it's still a conjecture.... and is it neccesary that the boundary be closed?
The Bekenstein-Hawking entropy is proportional to the area of the enclosing surface, not to the volume. That's why it is a manifestation of the holographic principle.
Look at quantum mechanics in an L² Hilbert space defined over the reals. This is certainly a continuum theory. But we know that it can be reformulated using harmonic oscillator wave functions which form a complete but discrete set. In addition we can get rid of wave functions at all and use creation and annihilation operators, energy eigenstates etc. That means that a continuum theory is dual to a discrete theory w/o losing any information.