| Summary: | In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mover accent="true"><mi>P</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">]</mo></mrow><mo>=</mo><mi>i</mi><mi>ħ</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>β</mi><msup><mi>p</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, may have different physical consequences such as having no minimal length at all. These differences depend on how the position and/or momentum operators are modified rather than only on the resulting modified commutator. This provides guidance when constructing GUP models since it distinguishes those GUPs that have a minimal length scale, as suggested by some broad arguments about quantum gravity, versus GUPs without a minimal length scale.
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