31–38 of 38 posts
Earlier quoted context omitted.
> implies fractional applications of the arithmetic operators Does it? Where does this implication come from?
You can’t generate a continuum without log/exp
Real numbers are a field for +-/ (although not a group for / because zero is weird).
A simpler preliminary question would be, why are complex numbers present in non quantum wave mechanics, and then how does this compare and contrast to quantum mechanics.
I'm not sure there's anything deep here. Imaginary exponentials contain sines and cosines and so are the solution to a lot of differential equations, even in the complex solutions have no physical interpretations. See e.g. the case of electromagnetism.
Earlier quoted context omitted.
What do matrices with complex numbers 'mean physically'?
The measurement operators are matrices that come about as a result of assigning real eigenvalues (these are your possible measurement outcomes) to orthonormal vectors (your arbitrary coordinate system). The results are hermitian, complex-valued matrices, because that's just what comes out if you try to engineer a matrix to have those eigenvalues and vectors. The rest follows from that. Trying to fit a real number con…
I just don't see how it links up with something tangible in the real world.
The article appears to suggest that QM phenomena exist because of the mathematical curiosities of the complex numbers applied to linear operators. Its not my place to say whether the universe “is” mathematics, but I do feel comfortable using math as an attempt to describe it, and there may be many choices, but our convention generally is to choose the description that eliminates possibilities that are rejected experimentally, and many such may be required, but our secondary preference is to choose the smallest set of descriptions that contain the observations.
Earlier quoted context omitted.
The measurement operators are matrices that come about as a result of assigning real eigenvalues (these are your possible measurement outcomes) to orthonormal vectors (your arbitrary coordinate system). The results are hermitian, complex-valued matrices, because that's just what comes out if you try to engineer a matrix to have those eigenvalues and vectors. The rest follows from that. Trying to fit a real number con…
Can you elaborate on the physical meaning that follows? I just don't see how it links up with something tangible in the real world.
Earlier quoted context omitted.
Can you elaborate on the physical meaning that follows? I just don't see how it links up with something tangible in the real world.
The complex numbers in the matrices appear as a consequence of trying to do something else. I don't think they have much physical meaning on their own, which is why I am surprised that people ask what it would mean if they had to all be real numbers.
The modulus ("complex absolute value") gives us the amplitude, and the angle encodes phase.
The way we usually encode phase is that it's frequency dependent. π or 180° of phase corresponds to a frequency-dependent amount of time.
At any frequency, sine and cosine waves are at 90 degrees from each other, forming a quadrature.
This 90 degrees has not only frequency and time interpretation, but a vector interpretation. If we chop the signal into samples to make a vector of numbers, the sine and cosine vectors will be at 90 degrees in the sense that their cross product is zero. I.e. actually perpendicular in the N-space they inhabit.
Under Fourier analysis, we are projecting the signal onto these basis vectors: how much of the sin, and how much of the cos.
We can combine the sine and cosine into a complex number thanks to Euler's formula e(ix) = cos(x) + i sin(x). By imagining the signal as being in polar coordinates, where its phase angle is the angle around the complex plane, and amplitude is the modulus, we simplify and condense the math. The two vectors at 90 degrees apart are combined into one vector of complex numbers for us to deal with.
So "how much of this signal correlates with sin(x)" gives us the imaginary component, and "how much of this signal correlates with cos(x)" gives us the real component. We can just add these together to make a complex number. Its argument (angle) gives us the phase, and modulus the amplitude.