Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex?
For example, given
\begin{equation}
\begin{split}
\hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...
Spin is a measure of energy: intrinsic angular momentum. Other forms of energy are the Fourier conjugates of variables in position space. It seems theoretically problematic that no position space representation exists for spin. Nor do I understand why the observation that spin is measured as...
I note the general Taylor series for ##a(t)## as:
\begin{equation}
\begin{split}
a(t)&\approx a(t_0) + a'(t_0) (t-t_0) + \frac{1}{2!} a''(t_0) (t-t_0)^2 ....
\end{split}
\end{equation}
which I rewrite as:
\begin{equation}
\begin{split}
a(t)&\approx a(t_0)\left(1 +...
Momentum ##\vec{p}## and position ##\vec{x}## are Fourier conjugates, as are energy ##E## and time ##t##.
What is the Fourier conjugate of spin, i.e., intrinsic angular momentum? Angular position?
I understand what you are saying, and I don't want to make this a debate about definitions. I note that the American Mathematical society considers the Fourier transform of the square root of the normalized Gaussian also to be a the square root of a p.d.f., i.e. both ##f(x)^2## and...
The mathematical uncertainty principle says a lot about the relationship between Fourier conjugates, specifically the relationship between their variances. Again, I am merely trying to see what has been discovered regarding the relationship between their expected values as represented by the...
The mathematical uncertainty principle is a description of the relationship between the variances of Fourier conjugates with normalized Gaussian distributions. In order to calculate both variances, one must interpret both ##f(x)## and ##\hat{f}(k)## as probability functions. See for example...
##f(x)## is not an arbitrary probability distribution. I have defined ##f(x)## as the probability amplitude of the normalized Gaussian. Furthermore, the Fourier transform of a Gaussian probability distribution is also a Gaussian probability distribution (and the Fourier transform of the...
##k## is the Fourier conjugate of ##x##.
My understanding is that ##k## is not completely independent of ##x##. Were ##k## and ##x## completely independent, i.e., ##Cov(x,k) = 0##, then why would ##E(x^2) E(k^2) = \frac{1}{16 \pi^2}##?
I apologize if I was unclear. I am concerned with probability, specifically probability amplitude functions and Fourier transforms. This is not a question about quantum mechanics.
I will restate the question more explicitly. Given the normalized Gaussian probability amplitude in ##x##-space...
Given two variables ##x## and ##k##, the covariance between the variables is as follows, where ##E## denotes the expected value:
\begin{equation}
\begin{split}
COV(x,k)&= E[x k]-E[x]E[k]
\end{split}
\end{equation}
If ##x## and ##k## are Foureir conjugates and ##f(x)## and ##\hat{f}(k)## are...
Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?
I understand that the Uncertainty Principle relates the variances of Fourier conjugates. I am having trouble finding: 1) the mathematical relationship between the expectation values of Fourier conjugates generally; 2) and then specifically for a normalized Gaussian. Any suggestions or insights?
I apologize for the simplicity of the question. I have been reading a paper on the Legendre transform (https://arxiv.org/pdf/0806.1147.pdf), and I am not understanding a particular step in the discussion.
In the paper, Equation 16, where ##\mathcal{H} = \sqrt{\vec{p}^2 + m^2} ##...
Time slicing derivation of the path integral: http://hitoshi.berkeley.edu/221a/pathintegral.pdf
Momentum used as ##m \vec{v}## and ## \hbar \vec{k}## interchangeably, where ##\vec{v} = \frac{d\vec{x}}{dt}##.
My real question is why in Quantum Mechanics it so often seems that velocity as ##\vec{v}=\frac{d \vec{x}}{dt}## and as ##\vec{v}=\frac{d \omega}{d \vec{k}}## seem to be used interchangeably, when it seems so obvious that they cannot be. One blatant example is the time slicing derivation of the...
I apologize ahead of time for the simplicity of the question, but this has really been bothering me.
Given the de Broglie relation, assuming natural units, where ##\hbar = 1##:
\begin{equation}
\begin{split}
\vec{k} &= M \vec{v}
\end{split}
\end{equation}
My question regards...
I apologize but I don't see how you have answered the question.
"Fixing the function at the boundaries" seems exactly the same as assuming ##\delta f(x_a) = \delta f(x_b) = 0##. Stating the the solution must be "on-shell" implies that the result must be the Euler-Lagrangian, which requires...
I have a question about a very specific step in the derivation of Euler-Lagrangian. Sorry if it seems simple and trivial. I present the question in the course of the derivation.
Given:
\begin{equation}
\begin{split}
F &=\int_{x_a}^{x_b} g(f,f_x,x) dx
\end{split}
\end{equation}...
Another assumption is that ##a(t)## is a scale factor associated with expansion. That is also an assumption. The math can be done without assuming a physical interpretation of ##a(t)##.
It just seems both arbitrary and classical to assume that spacetime would expand and even inflate only in its spacial components. GR and SR affect both space and time. Why wouldn’t the spacetime expansion affect both as well? I’m not expecting an answer. I’m just commenting.
I have one other question regarding the equation. Why the assumption that the cosmic expansion affects space but not time? Why not assume that it affects time and not space? Or both equally?