Why is the wavelength not quantized? If the wavelength were quantized, a maximum frequency would exist for every string of finite length, and every string of different length would have a different partition function.
A harmonic oscillator of frequency f only able to have f*(k+1/2) energy where k is a non-negative integer because the length of the amplitude is quantized. If the length of the amplitude was not quantized, every frequency could have an energy of 1 or 3/2.
Quantizing space in 1 direction but not the other seems strange.
My question is in response to this video. Timestamp 17:40 presents the restriction of energy levels for a given frequency, which I use as evidence of quantization of length of amplitude. Timestamp 23:40 presents the ground state energy of a string as a sum of the ground states on an infinite amount of frequencies. From this we can conclude the wavelength is not quantized or the string in question is infinite in length.
Quantizing the wavelength which limits possible frequencies would mean that string theory does not need Zeta renormalization.