QUANTUM UNCERTAINTY AND FOURIER TRANSFORMATION : Why there is uncertainty at the quantum scale? Explained with the help of classical examples of Fourier Transformation.
The fact that there is fundamental uncertainty at the quantum scale is not that fundamental after all. If we look closely, we have these uncertainties in our classical world as well. This could just be that the power of mathematics has its limits. This can be demonstrated just with the help of Fourier Transformation.
Thus, the basic question here is: Is Uncertainty in quantum mechanics has the same meaning as Uncomputability in mathematics?
I will try to answer this question in this post. However, these are just my intuitions from readings and my personal understandings. They might be wrong but I will try to give a logical and cogent argument for the same.
Take Godel's Incompleteness Theorem for Instance. It says that for a certain number of Axioms (intrinsic properties), there is always a statement that is true but unprovable. However, in science, we can add these unprovable statements to our original Axioms which then gives the true nature of reality. Could it be that the fundamental nature of reality, the uncertainty, is one of that unprovable statement of mathematics? or is it the true nature of reality?
Let's just take the Fourier Transformation of a sound wave. This mathematical model states that any repeating signal can be represented as the sum of various sinusoidal/cosine signals of different frequencies. The more the signal persists in time, the higher the resolution of two nearly similar frequency waves. If the signal is instantaneous or small over time than it would be very hard to resolve the frequencies and the Fourier Transform will spread over frequency plane. Similarly, if the signal persists longer in time, the frequencies can be easily resolved.
Does it mean we have reached the mathematical limitation of our theories?
The logical interpretation of this could be that mathematics is not reality, mathematics is a tool that we use in science to explain reality. These theorems require some axioms which we provide them by our intuitions of reality. Maybe Quantum Mechanics is not the exact theorem to define the behaviour of the smallest but we use it because it works most of the time and because it works we try to make axioms based on the findings that Quantum Mechanics gives us.
That the Heisenberg uncertainty principle can be demonstrated from the mathematics of Fourier Analysis seems to show that it is not the unavoidable interference of the act of observation upon a quantum system that accounts for its Heisenberg uncertainty, but that this uncertainty is iintrinsic and independent of the act of measurement. Heisenberg's gamma ray microscope thought experiment then does not constitute a physical explanation of the uncertainty which bears Heisenberg's name, but is a mechanical metaphor of merely pedagogical value. After all, quantum nonlocality is just a name for the necessary absence of an underlying causal mechanism for distinctly quantum phenomena, e.g., tunneling, superposition, entanglement, etc.
ReplyDelete