Trace Definition and 199 Threads

Hi i have problems, to solve task a) Since I have to calculate the trace of the matrix ##Q##, I started as follows: $$\text{trace} (Q)=\sum\limits_{i=1}^{3}\int_{}^{}d^3x'(3x_i^{'2}-r^{'2}) \rho(x')$$ I then calculated further until I got the following form: $$\text{trace}...

Is there a way to trace all the reactions I have had to others' posts in various threads of this forum? Like an index? If only to retrieve all the useful information I've learned from them, but have forgotten already now where it was located. Also, I thought OP meant online participant, but...

What is the general expression for the trace of a fourth rank tensor? Do you sum over possibilities of contractions with some factor? So, for instance, for the Riemann tensor, it is given by: $\eta_{ab}\eta_{cd}R^{acbd}$ due to these being independent contractions due to the symmetry...

Let ##M## be a nonzero complex ##n\times n##-matrix. Prove $$\operatorname{rank}M \ge |\operatorname{trace} M|^2/\operatorname{trace}(M^\dagger M)$$ What is a necessary and sufficient condition for equality?

Let ##V## be a finite dimensional vector space over a field ##F##. If ##L## is a linear operator on ##V## such that the trace of ##L\circ T## is zero for all linear operators ##T## on ##V##, show that ##L = 0##.

From Rand Lectures on Light, we have, in the interaction picture, the equation of motion of the reduced density matrix: $$i \hbar \rho \dot_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle = \Sigma_b \phi_b | \langle V \rho_{AB} | \phi_b...

I'm excitedly close to the end of proofing my latest novel, and as happens, tweaking of passages occurs. In this instance, a fleet of warships travel together in a collective warp space bubble that has inconsistent gravity, especially around the edges. To facilitate shuttle transfers between...

My answer is when the small coil is rotated, there will be change in magnetic flux in it. Initially (before rotated), the trace in c.r.o will be sinusoidal. After it is rotated, the trace will still be a sinusoidal (but I am not sure whether the amplitude will be the same). But the answer key...

Sorry I just typed out my query .For some reason I can't seem to find the buttons for attaching files on this thread. When writing the QED vacuum polarization loop, the numerator ,consisting momenta slashed + m from the fermion propagators and the two gamma matrices, has a trace over all of it...

Even if how careful we are, I think it is hard for us to avoid TRACE amount of unknown chemicals that get into our body unconsciously, through different pathway, our mouth, our nose and our eyes or our wound. Could our body handle these foreign chemicals in trace amount and keep us healthy? Does...

Hi Pf i am reading this article: pillet.univ-tln.fr/data/pdf/KMS-states.pdf I know that the trace cyclicity can be used when there is a product of matrices. But here we have operators (an hamiltonian , an operator which can be the position operators) . the author take the trace of a product. is...

I hate to create a thread for a step in a calculation, by I don't know what else to do. I'm having a lot of trouble reproducing E. Weinberg's index calculation (found here https://inspirehep.net/literature/7539) that gives the dimension of the moduli space generated by BPS solutions in the...

Hello, I am puzzled about the following condition. Assume a matrix A with complex-valued zero-mean Gaussian entries and a matrix B with complex-valued zero-mean Gaussian entries too (which are mutually independent of the entries of matrix A). Then, how can we prove that...

Hi, I was trying to do the following problem. I was able to do the part in pink highlight (please check "My attempt") but the part in orange highlight makes no sense to me. I'd really appreciate if you could help me to solve the part in orange. Thank you! My attempt: The solution presented...

Uranus and Neptune have roughly an 80-19-1 $H_2$-$He_4$-$CH_4$ mixture. (Of course actual percentages vary but those are rough values for sake of this simplification argument -- unless their actual values are critical to the answer.) Uranus temperature is reported as 2K while Neptune is 20K...

I am trying to follow the rule, that is, raising an index and the contract it. Be ##g_{\mu v}## the metric tensor in Minkowski space. Raising ##n^{v \mu}g_{\mu v}## and then, we need now to contract it. Now, in this step i smell a rat (i learned this pun today, hope this mean what i think this...

I understand that PCB trace spacing is typically based on a minimum found in certain standards and that is voltage based. If the breakdown strength of the dielectric is based on e-field intensity, wouldn't it be beneficial to actually consider the material properties and make it based on a...

The textbook gives some examples for ##P_1 \left ( cos \theta \right)##, ##P_2 \left ( cos \theta \right)##, and ##P_3 \left ( cos \theta \right)##. The procedure is clear. The key to the problem is to find the symmetric traceless tensor for the proper rank. I first construct a symmetric...

Properties 1) If ##(\gamma^{\alpha}\gamma^{\beta}...\gamma^{\mu}\gamma^{\nu})## contains an odd number of ##\gamma##-matrices $$Tr(\gamma^{\alpha}\gamma^{\beta}...\gamma^{\mu}\gamma^{\nu})=0$$ 2) $$Tr(\not{\!A}\not{\!B})=4AB$$ 3) $$Tr(\not{\!A}\not{\!B}\not{\!C}\not{\!D})=4\Big(...

I've been studying different scattering processes (from Mandl & Shaw QFT's book, chapter 8) and there's always a common step I do not understand: the showing-up of the trace. Let me give two specific examples. Lepton Pair Production ##( e^+ (\vec p_1, r_1) + e^- (\vec p_2, r_2) \rightarrow l^+...

The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?

I am attempting to understand how POVMs fit in with quantum measurement, and I think I am getting tripped up in notation when it comes to multipartite systems. The situation is as follows: System: \rho_A Measurement instrument: \rho_B = |\phi\rangle\langle\phi| (pure state) The multipartite...

I'm working out the quark loop diagram and I've drawn it as follows: where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices. For this diagram I've written...

I’ve just bought this useful looking 100 MHz scope in the hope of repairing it. When I switch to digital mode, the trace disappears off-screen on all four channels. 1 and 3 are up high, 2 and 4 down low. Sometimes the top or bottom of the trace is visible, and it flickers up and down the...

Consider the first qubit (subsystem A): First, the density operator for the system AB is ## \rho ^{AB} =\frac { \left |00 \right > \left <00 \right |+ \left |01 \right > \left < 01\right |+\left | 11\right > \left < 11\right | } 3 ##. Then, the reduced density operator of subsystem A is ##...

Hey! :o Let $K/F$ be a finite Galois extension and let $G= \operatorname{Gal}(K/F)$. For each $\sigma\in G$ we define $V_{\sigma}=\{\sigma (b)-b:b\in K\}$. Show that $V_{\sigma}$ is $F$-subspace of $\ker \operatorname{Tr}_{K/F}$. Show that $K/\mathcal{F} (\langle \sigma \rangle) \cong...

I am sorry for asking this stupid question, but in the Yang-Mills lagrangian, there is a term ##Tr(F^{\mu \nu}F_{\mu \nu})##. Isn't ##F^{\mu \nu}F_{\mu \nu}## a number?

Hi all, I am working on a project at the moment, and I have to evaluate the trace by using the Casimir's trick. The trace form is $$Tr[(\displaystyle{\not} P +M_{0})\gamma^{\mu}(\displaystyle{\not} P^{'} +M^{'}_{0})(\displaystyle{\not} p^{'}_{1} +m^{'}_{1})\gamma^{\nu}(\displaystyle{\not} p_{1}...

For finite matrices ##A## and ##B## we have Tr(AB)=Tr(BA) What happens in case of infinite matrices?

G'Day All, This is my first post so please let me know if I have completed this form incorrectly, or missed a point of etiquette etc... 1. Homework Statement The problem is to determine the pressurisation rate of a tank being filled by a pipe connected to a compressor. Assumptions: Pipe...

Hi all I am trying to reproduce some results from a paper, but I'm not sure how to proceed. I have the following: ##\phi## is a complex matrix and can be decomposed into real and imaginary parts: $$\phi=\frac{\phi_R +i\phi_I}{\sqrt{2}}$$ so that $$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...

Homework Statement Find the trace of a ##4\times 4## matrix ##\mathbb U=exp(\mathbb A)##, where $$\mathbb A = \begin {pmatrix} 0&0&0&{\frac {\pi}{4}}\\ 0&0&{\frac {\pi}{4}}&0\\ 0&{\frac {\pi}{4}}&0&0\\ {\frac {\pi}{4}}&0&0&0 \end {pmatrix}$$ Homework Equations $$e^{(\mathbb A)}=\mathbb P...

Homework Statement Homework Equations in addition to those provided in the questions, I used the following: Tr(B) = sigma<x_j|B|x_j> purity = Tr(rho^2) The Attempt at a Solution [/B] I find calculating trace and purity very confusing. Am I on the right track with question 1? With...

I want a QM textbook which introduces detail knowledge of density matrix and trace (i.e. the average), who can recommend one for me? Thank you.

Homework Statement [/B] The trace of a matrix is defined to be the sum of its diaganol matrix elements. 1. Show that Tr(ΩΛ) = Tr(ΩΛ) 2. Show that Tr(ΩΛθ) = Tr(θΩΛ) = Tr(ΛθΩ) (the permutations are cyclic) my note: the cross here U[+][/+]is supposed to signify the adjoint of the unitary matrix U...

Hi, I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra). 1. Homework Statement Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##...

Hello, Just wondering if the trace of a matrix is independent of basis, seeing as the trace of a matrix is equal to the sun of the eigenvalues of the operator that the matrix is a representation of. Thank you

I have a question about the use of trace in QFT in general - more specifically the use of trace in the lagrangian in the effective theory concerning chiral symmetry in QCD. I am slowly trying to get a hang of everything, and most things i am able to calculate, but i still have som very specific...

The question is : Is it true that two matrices with the same characteristic polynomials have the same trace? I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...

I read that states are positive operators of unit trace - not elements of a vector space. Is it referring to quantum states or all classical states? I know operators are like minus, plus, square root and vectors are like rays in Hilbert space.. but why can't quantum states be vectors when in...

When preparing a fine powder for trace metal analysis, is ashing prior to digesting unnecessary? If not, or not necessarily, where could I learn more about important considerations to determine whether of not any preparation prior to digestion is required? The one instance I have come across...

Hello, I was just wondering if there is a geometric interpretation of the trace in the same way that the determinant is the volume of the vectors that make up a parallelepiped. Thanks!

Homework Statement Good day, I want to ask the matrix that obtained from below formula and example. $$tr_A(L_{AB})=\sum_i [(\langle i|\otimes id)L_{AB}(|i\rangle\otimes id)]$$ this formula above can be represented as in matrix form below, $$tr_A(L_{AB})=...

Hello, I'm stuck with this exercise, so I hope anyone can help me. It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian ##\mathcal{H}##, which is defined by $$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$ is also representable as...

I've worked through a Stern Gerlach experiment for the Sx and Sz directions using the density matrix formalism to account for the environment. This shows a result which I think is correct but relies on decoherence to give the "actual" value. I'm not confident about the result though. Would...

Homework Statement This isn't a homework problem; it's just something I'm working on and I'm a little confused as to how to go about dealing with what I have. I have several traces of Dirac's gamma matrices, and I know that the trace of an odd number of gamma matrices is zero. So my first...

I know that cathode rays follow a helical pattern if they enter a uniform magnetic field at an angle less than 90 degrees. This leaves me with two questions. 1. If the cathode ray described above, hit a phosphor coated screen, would it show only 1 spot or a circle? 2. If the magnetic field was...

What is the trace of a second rank tensor covariant in both indices? For a tensor covariant in one index and contravariant in another ##T^i_j##, the trace is ##T^k_k## but what is the trace for ##T_{ij}## because ##T_{kk}## is not even a tensor?

Homework Statement In Zettili's QM textbook, we are asked to find the trace of an operator |\psi><\chi| . Where the kets |\psi> and |\chi> are equal to some (irrelevant, for the purposes of this question) linear combinations of two orthonormal basis kets. Homework Equations...

Homework Statement Consider the following experiment: Alice and Bob each blindly draw a marble from a vase that contains one black and one white marble. Let’s call the state of the write marble |0〉 and the state of the black marble |1〉. Consider what the state of Bob’s marble is when Alice...