@0537 the rank of the tensor is the max number of indices, the type is the (m,n) it belongs to. As for order, I often seen it used interchageably with rank, but I am not sure whether strictly speaking it has a difference

@0celo7 I am not sure, you might end up getting the rules for the type of tensor for gradients and divergence wrong because of how they depend on +-1 of the cotravarient or covarient number in the (m,n) notation

But in practice, I guess it should not matter much since when you actually do the computation, the indices for the tensor notation should be in the correct places…

@Slereah the multiple time dimension paper you referred me earlier seems to deal with n particles with a time variable for each particle. After discussing with Acuriousmind on that topic we only get the operational part of the answer to the question on "why single time instead of n time". He also pointed out that the results of that paper cannot be naively extended to n time single particle

Recently I am interested in investigating n time one particle extensions to quantum mechanics, so I can better understood the physics of n time dimensions, I am currently looking for digestible papers t…

@ACuriousMind
We know from quantum mechanics that $$\langle \psi_1\lvert\hat{O}\rvert\psi_2\rangle$$ is short for $$\int_{-\infty}^{\infty} \psi_1^*(a)\hat{O}\psi_2(a) da$$ for some continuous(?) basis(?) 'a'

Suppose that $\psi_q$ can be expanded as a linear combination of orthonormal functions $\phi_i$ (i.e. $\phi_i$ forms an orthonormal basis), i.e.

$$\langle\psi_q\rvert=\sum_{i=1}^{\infty}c_{qi}\langle\phi_i\rvert$$

for some coefficients $c_{qi}\in \mathbb{C}$

Then

$$\langle \psi_1\lvert\hat{O}\rvert\psi_2\rangle=\sum_{i=1}^{\infty}c_{1i}^*\langle\phi_i\lvert\hat{O}\sum_{k=1}^{\inft…

Because my thought process that leads to the question is that if $@$ and $\psi$ has different physical meaning (i.e. one has none and another has some kind of physical meaning e.g. momentum), then I cannot wrap my head on how to understand how you can get something physically different by simply rearranging the basis vectors in the underlying vector space (e.g. by choosing different components for each basis vector and then take the inner product of the resulting vector formed by the near combination of the basis vectors)

Ok let me phrase it in another way:

Consider how we write out an expectation value under the continuous basis a.

Under the Born probability interpretation

$\psi(a)$ no physical meaning

$\hat{O}$ give a distribution of the observable o after bracketing (i.e. $\langle\lvert\rvert\rangle$)

$\hat{O}\psi(a)$ no physical meaning

$\psi^*(a)$ no physical meaning

$\psi^*(a)\hat{O}$ no physical meaning

$\psi^*(a)\hat{O}\psi(a)$ no physical meaning

$\psi^*(a)\psi(a)$ probability density

(1)

Then suddenly...