Any possible vibration of a quantum field can be thought of as a combination of vibrations with different specific wavelengths—just as any particular sound can be decomposed into a combination of various notes with specific frequencies. At first you might think that any possible wavelength is allowed, but actually there are restrictions. The Planck length—the tiny distance of 10-33 centimeters at which quantum gravity becomes important—provides a lower limit on what wavelengths are allowed. At smaller distances than that, spacetime itself loses its conventional meaning, and the energy of the
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