What does the Quantization Surface mean here?
Reference: H. Latal W. Schweiger (Eds.) - Methods of Quantization
Answer
The word quantization surface is not standard terminology. It apparently refer to a (generalized) Cauchy surface. A (generalized) Cauchy surface is a hypersurface on which the initial conditions are given for a well-posed initial value problem. Phrased differently, for given initial conditions on the Cauchy surface, there exists a unique solution for the evolution differential equation in an appropriate bulk region. (For more information, see also hyperbolic PDE and causal structure on Wikipedia.)
Here the evolution parameter of the system is usually referred to as time, although it doesn't have to be actual time. It could also be light cone time $x^{+}$, etc. Similarly, the word initial does not need to refer to actual time.
The evolution differential equation could be e.g. the Schrödinger equation, the Klein-Gordon equation, the Maxwell's equations, the Navier-Stokes equations, the Einstein field equations, the Arnowitt-Deser-Misner (ADM) equations of GR, etc.
In the context of quantization of a classical field theory, it is usually meant that the evolution problem is given in Hamiltonian form. Traditional Hamiltonian formulation (as opposed to manifestly covariant$^1$ Hamiltonian formulation) has a distinguished spacetime coordinate, which then plays the role of evolution parameter.
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$^1$ There are various manifestly covariant Hamiltonian formalisms. See e.g. De Donder-Weyl theory; this Phys.SE post; and the following reference: C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories. Published in Three hundred years of gravitation (Eds. S. W. Hawking and W. Israel), (1987) 676.
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