Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 transitive v. Physics To limit the possible values of (a magnitude or quantity) to a discrete set of values by quantum mechanical rules.
 transitive v. Physics To apply quantum mechanics or the quantum theory to.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 v. To limit the number of possible values of a quantity, or states of a system, by applying the rules of quantum mechanics
 v. To approximate a continuously varying signal by one whose amplitude can only have a set of discrete values
 v. To shift each beat in a rhythmic pattern to the nearest beat of a given resolution (eighth note, sixteenth note, etc.), or to adjust the frequency or pitch of a note to the nearest perfect tone in a given musical scale
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 v. approximate (a signal varying continuously in amplitude) by one whose amplitude is restricted to a prescribed set of discrete values
 v. apply quantum theory to; restrict the number of possible values of (a quantity) or states of (a physical entity or system) so that certain variables can assume only certain discrete magnitudes that are integral multiples of a common factor
Etymologies
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Examples

“I asked you to use your own words, and quantize is what you came up with?”

Hannes: You can't naively quantize gravity that is true.
Worldwide Campaign Sheds New Light on Nature's "LHC"  Universe Today

Finally, quantize the theory by promoting one variable in the canonical pair to a differential operator.

This is an undecided question that ultimately requires knowing how to quantize gravity.

Radiation exposure can have very long term effects and are often difficult to quantize.

Ted, you've totally missed something critical  you've attempting to quantize social costs.

These loopy ideas don't pass elementary tests AFAIU, such as showing how one can quantize axiomatically or how to extract dynamics.
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Really you have to quantize the electromagnetic field, and then the interaction between charged particles is governed by the exchange of photons, which are excitations of the EM field.
Everything You Ever Wanted to Know About Quantum Mechanics, But Were Afraid to Ask

Now with hbar = 1 if we quantize this Hamiltonian with

So quantize in a Fourier basis on the ndimensional torus.
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