Questions tagged [trace]

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Taking derivatives of traces over matrix products

I started with evaluating the following derivative with respect to a general element of an $n\times n$ matrix, $$\frac{\partial}{\partial X_{ab}}\left(\mathrm{Tr}{(XX)}\right)$$ I wrote out the ...
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19 views

Expansion of trace in photon self energy

I am studying through the photon self energy $$ i\Pi_{\mu\nu}(q) = \int\frac{d^4 k}{(2\pi)^4}Tr\left[(-ie\gamma_\mu)\frac{i(\require{cancel}\cancel k+m)}{k^2-m^2+i\epsilon}(-ie\gamma_\nu)\frac{i(\...
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0answers
29 views

Normalization of Generators of $SU(N)$

I have given a finite-dimensional matrix-representation of $SU(N)$. In this representation, the generators are denoted by $G^{a}$ for $a=1,\dots N^{2}-1$. I have to show that I can choose the ...
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1answer
48 views

How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless?

How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless? where I represents the identity matrix I know that $$\{\gamma^\mu , \gamma^\nu\}=\gamma^\mu\gamma^\nu + \...
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1answer
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Symmetrized Operators [closed]

Let $A$ be an operator on $H^{n} = H \otimes H \otimes ... \otimes H$ and $W$ be a statistical operator on $H^{n}$. A text books says, if $A$ is symmetric, thenfor any $W$ $Tr(AW) = Tr(AW_{sym})$ ...
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1answer
34 views

How is this Trace equality true?

To split the amplitude into a colour part and a kinematical part , we must start by setting up a Lagrangian. In my notes I am told to start promoting gauge fields $A^\mu$ to be matrices: $$(A_\mu ) ...
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1answer
48 views

Factorizing four generators into two commutators

Let $t_a$ be generators of any given Lie algebra such that $[t_a, t_b]=iC^{c}_{ab}t_c$. Let $A_{a\mu}$ be gauge bosons associated with this Lie algebra. Here $\mu$ is the spacetime index and $a$ is ...
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1answer
88 views

Is the result of the product of metric tensors $\eta^{\mu\nu} \eta_{\mu\nu} = 1$? [duplicate]

Is the result of the product of metric tensors $\eta^{\mu\nu} \eta_{\mu\nu} = 1$? If so how would I prove this? I know that tensors are represented as matrices but I don't know how I'd prove this (...
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1answer
31 views

Path ordering in the expansion of a Maldacena-Wilson line

In $4$d Euclidean space, the Maldacena-Wilson line is defined as: $$\mathcal{W}(C) = \frac{1}{N} \text{Tr} \left\lbrace \mathcal{P} \exp \int_C d\tau \left( i \dot{x}_\mu A_\mu^a(x) + \left| \dot{x} \...
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2answers
93 views

Trace of the Operator

I want to ask a question about the fundamental knowledge of trace of the an operator. The operator $A$ is $$A = v (G_r-G_a)$$ where v is the velocity operator of the Hamiltonian ($v=dH/dk$); $G_r$ and ...
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0answers
21 views

Trace class of the position autocorrelation function

Let us consider a quantum system of $N$ distinguishable particles, and let us tag the configuration $\hat q_j$ of one of those. I am interested in checking whether the position auto-correlation ...
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3answers
91 views

Proving identity $tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{5})=-4i\epsilon^{\mu\nu\rho\sigma}$

Im trying to proof the following identity: $tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{5})=-4i\epsilon^{\mu\nu\rho\sigma}$ when $\gamma^{\mu},\gamma^{\nu},\gamma^{\rho},\gamma^{\...
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1answer
122 views

If $\text{Tr}_B \rho^{AB}$ is almost pure, then $\rho^{AB}$ is almost a product state?

Let $\rho^{AB}$ be a bipartite state, and let $\rho^A$ denote the partial trace. Suppose $$ \lVert \rho^A - |\sigma\rangle\langle\sigma|^A \rVert_1 \leq \varepsilon $$ for some pure state $|\sigma\...
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28 views

Reference for variational characterization of quantum trace distance in infinite dimensions

Consider two density matrices $\rho$ and $\sigma$. It is well known that for finite-dimensional systems, the trace distance $\frac{1}{2}\Vert \rho-\sigma \Vert_1$ has the variational characterization $...
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3answers
125 views

Variation of $\log(-\det g_{\mu\nu})$ for Einstein -Hilbert action

In Einstein-Hilbert variation is needed the variation of the determinant of a metric tensor. For convenience, firstly evaluated the variation of $\log(-\det g_{\mu\nu})$ considering $\det g_{\mu\nu}&...
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3answers
81 views

How am I to interpret $\text{Tr}(\text{ad}_X\text{ad}_Y)$?

I'm trying to show that the $(2,0)$ Killing tensor is invariant under the $\text{Ad}$ homomorphism: $K(\text{Ad}_A(X),\text{Ad}_A(Y))=K(X,Y),$ with $X,Y\in \mathfrak{g},\hspace{1mm}A\in G,$ and $K(X,Y)...
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96 views

Relation between metric tensors

$g^{ab}$ is the metric tensor (Minkowski) Im trying to understand if this is true: $g^{ab}g_{ab}=?=g^{aa}g_{aa}$ Have a nice day.
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2answers
86 views

Why the partial trace is the unique function?

Sorry for my broken English. On Page 107 in Quantum Computation by Michael Nielsen, in the box 2.6 Suppose M is any observable on system A, and we have some measuring device which is capable of ...
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1answer
37 views

Is Trace cyclic with respect to tensor product?

Two alternative expressions for the expectation value of energy are \begin{align} \langle H\rangle = \langle \psi|H|\psi\rangle \end{align} which holds only for pure state, and \begin{align} \langle ...
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1answer
29 views

Trace in correlations to compute Wigner transform

In the derivation of Wigner-transformed quantum time correlation functions, the following identity is used (in the case of a one-dimensional particle, for simplicity): \begin{align} C(t) &\equiv \...
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2answers
71 views

Electron muon scattering unpolarised cross-section calculation with traces

Following my class on RQM, we wanted to evaluate the unpolarised cross section for the following process $$ e^+ e^- \rightarrow \mu^+ \mu^- $$ In doing so, whenever summing over spinor indices, a ...
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1answer
75 views

When does the identity $\|\mathrm{Tr}_B UXU^\dagger\|_1=\|\mathrm{Tr}_B X\|_1$, with $U$ unitary, hold?

Let $\mathcal{H}_{AB}$ be a bipartite, complex Euclidean space, and let $U\colon\mathcal{H}_{AB}\to\mathcal{H}_{AB}$ be a unitary operator. Define the trace norm as $$ \lVert X\rVert_1 = \text{Tr}(\...
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80 views

Partial trace of matrix product state

I have come accross a formula that puzzles me a bit in the proof of lemma 23 (page 32) of this paper. The authors start from a (translationally-invariant) matrix product state: $$\lvert\psi\rangle := ...
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142 views

Traces of gamma matrices in $d$ dimensions

For $d=4$, some identities of the traces of gamma matrices are: $tr[\gamma_\mu] = 0$ $tr[\gamma_\mu \gamma_\nu ] = 4g_{\mu\nu}$ $tr[\gamma_\mu\gamma_\alpha\gamma_\nu] = 0$ $tr[\gamma_\mu\gamma_\alpha\...
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2answers
50 views

Closed quark loops written as traces

I'm working out the quark loop diagram. I draw 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. ...
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1answer
68 views

Lax Pairs In Integrability

I am working through Dr. Beiserts notes (https://people.phys.ethz.ch/~nbeisert/lectures/IntHS16-Notes.pdf) and have difficulty obtaining the second step in (2.9): $$\{{\rm tr}L^{k},{\rm tr}L^{\ell}\} ...
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1answer
46 views

Trace of the quantum map $ A^n_m (\rho) = \sum_{ij} | i…i \rangle^n \langle i…i|^m \rho | j…j \rangle^m \langle j…j|^n$

We define some quantum map $ A^n_m (\rho) $ and let it act on density matrix $\rho$: $$ A^n_m (\rho) = \sum_{ij} | i...i \rangle^n \langle i...i|^m \rho | j...j \rangle^m \langle j...j|^n.$$ ...
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1answer
114 views

Confusion about trace in the vertex term of Lagrangian

I was reading through Mariano Quirós's lecture notes titled "Finite Temperature Field Theory and Phase Transitions". In Sec. 1.2, the author is calculating the one-loop effective potential at $T=0$. ...
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What happens if I Wick contract a trace operator internally?

In theories such as $\cal{N}=4$ supersymmetric Yang-Mills, we often consider operators such as $\cal{O}(x_1)=$Tr$(\phi(x_1)\phi(x_1))$, with $\phi$ the scalar field(s) of the theory. Then we go on ...
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1answer
60 views

Positive Semi-Definiteness of a Density Matrix - can the eigenvalues be larger than 1?

I know that one of the requirements for a density matrix is that it is positive-semidefinite. This means that the eigenvalues are non-negative (and sum to 1, so we can assign them the meaning of ...
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67 views

Gamma traceless

I read this Under what conditions is a vector-spinor gamma trace free. And also read many papers about higher spin, but no one explains why irreducible spinor is gamma traceless spinor? Can anyone ...
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1answer
163 views

Is there any meaning of tensor contraction?

Is there any meaning behind tensor contraction. Or is it just randomly getting rid of some components by only selecting those with same index and sum them up? For example, I know tensor is ...
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1answer
338 views

Is a partial trace cyclic?

I want to know if a partial trace keeps the cyclic property of the trace. The partial trace is defined as $$ tr_B: \mathcal{B}_1(\mathcal{H}_A\otimes \mathcal{H}_B) \longrightarrow \mathcal{B}_1(\...
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132 views

Why is the stress-energy tensor for electromagnetic radiation traceless?

A photon gas obeys the equation of state $\rho=P/3$ and hence $T^{\mu}_{\quad\mu}=3P-\rho=0$. (Can also be seen by expressing the stress energy tensor in terms of of the electromagnetic tensor as ...
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1answer
58 views

Derivation of Raychaudhuri equation - Trace

In Wald (Wald: General Relativity on page 218, equation 9.2.10) is stated that $$v^c∇_cB_{ab}=−B^c_bB_{ac}+R^d_{cba}v^cv_d $$ and to continue in order for the equation to be derived one needs to take ...
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0answers
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Open Quantum Systems: Born-Approximation and the preservation of Trace, Hermicity and Positivity

This is related to a previous question of mine. We consider a density matrix $\sigma(t)$ operating on a Hilbert space $\mathscr{H}_{s}\otimes \mathscr{H}_b$ with Hamiltonian $H = H_s \otimes \mathbb{...
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1answer
88 views

Integrating of von Neumann equation for density matrix

Suppose we are given the Hamiltonian $$H=f \frac{\text{Tr}\sigma_x \rho}{\text{Tr}\rho}\sigma_x,$$ where $\rho$ is the density matrix, and $\sigma_x$ is the Pauli matrix $$ \sigma_x= \begin{...
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1answer
108 views

Is tracing out a subsystem always akin to discarding all information about it?

Suppose we have some quantum system with sub-systems A and B. It could be, for example, two qubits or groups of qubits. Is it fair to say that tracing out the sub-system A is always akin discarding ...
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1answer
103 views

Why can the partial trace be written as $\text{Tr}_B(\rho)= \sum_k (1 \otimes \langle k|) \rho (1 \otimes |k \rangle)$?

I don't really understand a notation that I stumbled upon regarding a partial trace. According to the definition I have, partial trace is $$\rho_A=\text{Tr}_B(\rho_{AB}):= \sum_k (1_A \otimes \...
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43 views

Diagrammatic expansion of an operator insertion in path integral for Trace Anomaly calculation

Starting with a scale invariant classical field theory, we can prove that the energy-momentum tensor will be traceless. \begin{equation} \Theta^\mu_{\ \mu }=0 \end{equation} In the context of the ...
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1answer
194 views

Relation between the trace anomaly and the energy-momentum tensor being off-shell

Let's say we have a massless QED theory with a Lagrangian \begin{equation} L=i\bar{\psi}\not{D}\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} \end{equation} The symmetric energy-momentum tensor is \begin{...
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1answer
136 views

What is the trace in the Chern-Simons action

I have been looking at the Chern-Simons Lagrangian in $(2+1)$-dimensional spacetime $M$ in terms of a gauge field $A$: $$ S[A] = \frac{k}{4 \pi}\int_M \text{Tr}(A \wedge \text{d}A+ \frac{2}{3}A \...
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1answer
219 views

How can I prove that the partial trace is well-defined?

When I define the partial trace as below, how can I prove it well-defined? I understand that I have to indicate $Tr_k(\rho)$ does not depend on how to take the ONB of $\mathbb{C}^2$ $$n\in \mathbb{Z}_{...
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107 views

QCD Trace Anomaly and Mass

In the paper in equations 4 and 5, some of the mass of the nucleons comes from the "trace anomaly" of the QCD energy-momentum tensor (as described in the paragraph following these equations). Is there ...
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1answer
131 views

Physics Meaning of Trace Technology in QED [closed]

As it pointed out on page 133 of Peskin and Schroeder, any QED amplitude involving external fermions, when squared and summed or averaged over spins, can be converted to traces of products of Dirac ...
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1answer
188 views

reduced density matrix of state [closed]

given a multi particle state I have to calculate the reduced density matrix where I trace out the third particle $$|\psi\rangle = \frac{1}{\sqrt{3}} \left ( |\uparrow \uparrow \downarrow\rangle + \...
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1answer
338 views

What is a definition of the trace norm?

I have found that (one?) definition of the trace norm is $$\mid\mid A\mid\mid = \sqrt{A^*A} \tag{1}$$ but now I am reading this paper where (on page 4) it says In particular, we will restrict ...
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1answer
96 views

Proving identity $\mathrm{Tr}[\gamma^{\mu}\gamma^{\nu}] = 4 \eta^{\mu\nu}$

In the lecture notes accompanying a course I'm following, it is stated that $$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu} $$ Yet when I try to prove this,...
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1answer
227 views

Double-trace operators in CFT?

What is the conceptual difference between so called "single-trace" and "double-trace" (or "multi-trace") operators e.g., in a Conformal Field Theory?
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55 views

How to Explicitly Calculate z-Component of Berry Curvature?

While numerically playing with the 2-level Haldane model recently, I wondered how I could analytically calculate the z-component of the Berry curvature $F$. I framed my problem as needing an ...