I am asked to compute ##[\phi(x), \phi^\dagger(y)]## , with
##\phi = \int \frac{dp^3}{(2\pi)^3}e^{-ipx}\hat{a}(\vec{p})## and with z=x-y a spacelike vector. And show that this commutator does not vanish, which means that for this non-relativsitic field i.e. with ##p^0 = \frac{\vec{p}^2}{2m}##...
The correct answer is:
#P = \int \frac{dp^3}{(2\pi)^3}\frac{1}{2E_{\vec{p}} \big(a a^{\dagger} + a^{\dagger}a\big)#
But I get terms which are proportional to ##aa## and ##a^{\dagger}a^{\dagger}##
I hereunder display the procedure I followed:
First:
##\phi = \int...
-1st: Could someone give me some insight on what a ket-state refers to when dealing with a field? To my understand it tells us the probability amplitude of having each excitation at any spacetime point, but I don't know if this is accurate. Also, we solve the free field equation not for this...
As explained in the summary, it seems that the commutators of some operators (creation and anihilation) can be ignored when quantising the hamiltonian of the Klein Gordon Field. I wonder why we are allowed to do such a thing.
Is that possible because we are solely within a semiquantum...
Let as consider a system ##H = A\otimes B##
I've been said that quantum negativity, i.e. taking the partial transpose w.r.t A or B and summing the magnitude of the negative eigenvalues obtained, is a measure of how entangled are the parties A and B.
First question:
Why is it that we do not...
I am dealing with restricted boltzmann machines to model distributuins in my final degree project and some question has come to my mind.
A restricted boltzmann machine with v visible binary neurons and h hidden neurons models a distribution in the following manner:
## f_i= e^{ \sum_k b[k]...
When I computes the negativity (with the partial transpose) of the density matrix corresponding to the GHZ I obtain zero, no matter what is the partition I choose. I've read somewhere that this is because GHZ's distillable entanglement is zero, which I don't really understand because I haven't...
Let us consider a sphere of a unit radius . Therefore, by choosing the canonical spherical coordinates ##\theta## and ##\phi## we have, for the differential lenght element:
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In order to find the geodesic we need to extremize the...
First, we shall mention that it is known that the covariant derivative of the metric vanishes, i.e ##\nabla_i g_{mn} = 0##.
Now I want tro prove the following:
$$ \nabla_i A_k = g_{kn}\nabla_i A^n$$
The demonstration I encounter takes advantage of the Leibniz rule:
$$ \nabla_i A_k = \nabla_i...
The metric tensor in an inertial frame is ## \eta = diag(-1, 1)##. Where I amb dealing with only 1-D space. The metric tranformation rule after a crtain coordinate chane is the following:
$$ g_{\mu \nu} = \frac{\partial x^\alpha}{\partial x'^{\mu }} \frac{\partial x^\beta}{\partial x'\nu }...
Given a certain manifold in ##R^3## I've been told that at every location ##p## it is possible to encounter a reference frame from which the metric is the euclidean at zero order from that point and its first correction is of second order. This, nevertheless does not match with the following...
Suppose you have a tensor quantity called "B" referenced in a certain locally inertial frame (with four Minkowski components for instance). As far as I know, a parallel transportation of this quantity from a certain point "p" to another point "q" consists in expressing it in terms of the...
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I'm struggling with my Final Degree Project. I would like to perform a quantum simulation and perform quantum tomography for a single-qubit using a resrticted boltzmann machine. In order to do so I'm trying to follow the recipe in the paper "Neural Network quantum state tomography, Giacomo...
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In basic optics, we are given the general solution of the wave equation (massless string of length L) as a linear combination of normal modes, that need to have some of the permitted frequencies due to boundary conditions. In laboratory, we observed that phenomenon. We generated a wave in a...
Here it goes. I have been taught that a finite pulse of light does not have a single frequency. By finite pulse I was given an example of a source of light that has been emitted during a finite amount of time and, consequently, covers a finite region of space. Then I was taught that you can...