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Questions tagged [trace]

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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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0answers
30 views

Traceless stress tensor

What does it mean, when the viscous (or viscoelastic) stress tensor is traceless $\tau_{rr}+\tau_{\theta \theta}+\tau_{\phi \phi}=0$? Why if the viscoelastic model is linear it is traceless and if ...
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1answer
56 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{...
3
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1answer
63 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 ...
2
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1answer
66 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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0answers
27 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 ...
4
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1answer
49 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
43 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 \...
0
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1answer
142 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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35 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
88 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
82 views

reduced density matrix of state

given a multi particle state I have to calculate the reduced density matrix where I trace out the third particle for this I first calculate the corresponding 2D density matrix with the bra vector of ...
4
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1answer
167 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
50 views

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

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
74 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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0answers
31 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 ...
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1answer
27 views

Evaluating a trace with two factors of $\gamma^5$

In the process of calculating a spin-averaged square amplitude in QFT, I came across the following expression: $$ \text{Tr}\left[\gamma^\mu\gamma^5\gamma^\alpha\gamma^\nu\gamma^5\gamma^\beta\right] $$ ...
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1answer
47 views

Confusion with trace of gamma matrices

Using $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu} \mathbf{1}$, it is easy to show that: \begin{align*} \operatorname{tr} \gamma^\mu \gamma^\nu = 4\eta^{\mu\nu} \end{align*} Now, it is also true that ...
2
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1answer
84 views

Trace over configuration basis

Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
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2answers
50 views

Alternate definitions of Thermal states

The definition of thermal states I'm used to is: $$\tau_{\beta} = \frac{1}{Z}\,e^{-\beta H}$$ where $Z$ is the partition function defined as $Z= \mathrm{Tr}(e^{-\beta H})$, $\beta$ the inverse ...
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1answer
66 views

Why can we write lagrangian for gauge theory without the traces?

I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true: $$-\frac{1}{2}...
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2answers
318 views

Electromagnetic stress tensor is only traceless in 4D?

The electromagnetic stress tensor $F_{\mu \nu}$ is as we all know traceless in 4 dimensions. With $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and $A = (A_0,A_1,A_2,A_3)= (\phi, A_1, A_2, ...
3
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2answers
247 views

Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact ...
2
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1answer
103 views

Proof of strong convexity of trace distance

I'm trying to follow the Nielsen and Chuang proof (equation 9.49 of Chapter 9, page 408). I reproduce it here for completeness. With trace distance defined as $D(\rho, \sigma) = \frac{1}{2}tr(|\rho - ...
2
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1answer
181 views

Is partial trace the inverse operation of Kronecker product?

Computer science student here, who is interested in quantum information theory. Suppose I have these pure states: \begin{bmatrix}1&0\\0&0\end{bmatrix} and \begin{bmatrix}0&0\\0&1\end{...
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1answer
152 views

Quantum map and preservation of trace

I am currently learning about quantum maps, ie maps that transform a density matrix into another one. Assume we are in the Hilbert space : $H_A \otimes H_B$. I call the quantum map on the density ...
1
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1answer
114 views

Why we ignore off-diagonal elements in partition function?

In quantum statistical mechanics, the density operator is $$ \rho = \exp(-\beta H_0)/Z $$ where $$Z = \text{Tr} (\exp(-\beta H_0)) \, .$$ Why do we take the trace over only diagonal elements and ...
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2answers
158 views

Trace of the Riemann Curvature Tensor

Referring to Wald's General Relativity, I have two questions. Let ${R_{abc}}^d$ be the Riemann curvature tensor. The author has never defined what it means by "trace of a tensor" before page 40 of ...
2
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1answer
418 views

Trace of 4 Gell-Mann matrices

Does any one know what would be $tr(t^a t^b t^c t^d)$, where $t^a$ etc are Gell-Mann matrices? This came about when analyzing the color factor for the compton effect for QCD. So, must be pretty ...
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1answer
104 views

Question about the true significance of the partial trace

Consider a composite system whose Hilbert space is $\mathcal{H}_{AB}=\mathcal{H}_A\otimes \mathcal{H}_B$, where $\{|0_A\rangle, |1_A\rangle\}$ and $\{|0_B\rangle, |1_B\rangle\}$ are orthonormal bases ...
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1answer
148 views

Tracing over a Fock space?

Suppose you have a bosonic Fock space with a vacuum $|0\rangle$. A particular state is labeled by the parameter $N \in \mathbb{Z}$. You can construct states like $$ | n_{N} \rangle = \frac{ \left( \...
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0answers
37 views

How to understand the non-abelian DBI action?

I'm reading chapter 7 in "String Theory and M-theory, A Modern Introduction" by Becker, Becker and Schwarz. It says that the understanding of the square root of the determinant is to compute the ...
3
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1answer
66 views

Estimate of trace of powers of density matrix

Given a very generic, lower bounded Hamiltonian, is there a estimate on how $Tr(\rho^{1/k})$ grows as $k>0$ increases? Does this quantity diverge as a function of $N$, the degrees of freedom of the ...
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1answer
196 views

Are density matrices symmetric? [closed]

The context is that I want to simplify an expression like $$ \mathrm{Trace}[\rho_1 \rho_2 \rho_3] + \mathrm{Trace}[\rho_2 \rho_1 \rho_3] $$ (Note that the second term is not a cyclic permutation of ...
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0answers
84 views

Stress-energy tensor of the EM field has zero trace

The stress energy-tensor in EM is defined by $$T^{\mu \nu} = -\frac{1}{4\pi}\left(E^\mu_\rho E^{\nu\rho} - \frac{1}{4} g^{\mu\nu}E_{\rho\sigma}E^{\rho\sigma}\right)$$ I aim to show one of the ...
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1answer
181 views

How can I prove this relation involving gamma matrices?

Let $$l^{\mu} = l^{\mu}_{\parallel} + l^{\mu}_{\bot}$$ be a D-dimensional vector living in a Minkowskian space; the only non-zero components of $l^{\mu}_{\parallel}$ are the first four, while the only ...
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2answers
144 views

The relation between $T^{\alpha\beta}$ and its trace

I have a simple question. Is it true? $$T^{\alpha\beta}T_{\alpha\beta}=T^2$$ Where $T$ is the trace of $T^{\alpha\beta}.$ I think they are different.
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42 views

What's the result of the following multiplication

The question is simple, what is the result of the following multiplication of traces by the metric? $Tr\left[\gamma^\alpha\gamma^\nu\gamma^\beta\gamma^\tau\gamma^5\right]Tr\left[\gamma^\delta\gamma^\...
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0answers
71 views

How do I compute QFT trace of matrices at the end of a Feynman diagram?

At the end of a Feynman diagram I have (p and p' are two incoming momenta): $$Tr[\not p' \gamma^\mu \not p \gamma^\nu(\frac{1+\gamma^5}{2})]=Tr[\gamma^\mu p_\mu' \gamma^\mu \gamma^\nu p_\nu \gamma^\nu(...
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0answers
113 views

Proving a General Result for a Trace of $n$ Gamma Matrices

I am attempting to prove a set of results for the products of gamma matrices and traces of products of gamma matrices, but got stuck on this particular one. $$Tr(\gamma^{\mu_1}...\gamma^{\mu_n})=g^{...
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1answer
97 views

Signature of trace of Dirac Matrices

I came across this question in my problem set: Let $\gamma^\mu$, $\mu=0,1,2,3$ be the Dirac matrices, satisfying: \begin{eqnarray} \gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2\eta^{\mu\nu}I, \:\:\:...
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1answer
156 views

Manipulations with Traces: Saddle point integration in Large-$N$ model

For reference I am trying to work out the derivation in this paper, in which the partition function for an Ising model is approximated by replacing the Ising variables $\sigma_i$ with $N$ component ...
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1answer
88 views

Is this posible in GR $g_{ab}g^{ab}=1$? [duplicate]

Metric tensor multiplied by its inverse. I always see this with different indices.
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1answer
189 views

Trace of $2n$ gamma matrices

To proof $$\mathrm{Tr}(\gamma_{\mu_1}\cdots\gamma_{\mu_{2n}}) =\mathrm{Tr}(\gamma_{\mu_{2n}}\cdots\gamma_{\mu_1}),$$ I use $\gamma_\mu^\dagger=\gamma^0\gamma_\mu\gamma^0$ and get $$\cdots=\mathrm{Tr}...
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1answer
203 views

Trace involving a logarithm of a Klein-Gordon operator

Calculating effective potentials in QFT one ussually finds traces like $$\text{Tr}\ln(\Box+m)$$ Peskin (page 374) argues that the trace of the operator is the sum over its eigenvalues $$\text{Tr}\...
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3answers
247 views

What is the role of determinant and trace of matrices in physics? [closed]

There is vast area of physics where we have to use matrices.It is not only to do the mathematical problems in physics but also to produce a physical realization of an operation. I think matrices carry ...
3
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1answer
161 views

How is tracing out a physical operation?

Suppose $\rho_{AB}$ denotes the density matrix of a bipartite system. Reduced density matrix of A ($\rho_A$) is obtained by tracing out B $$\rho_A\equiv\sum_{i}\langle i_B |\rho_{AB}|i_B\rangle$$ ...
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1answer
595 views

Trace of Gamma Matrices [closed]

If I have: $Tr(\gamma^{\mu}\gamma^{\alpha}\gamma^{\nu}\gamma^{\beta}\gamma^{\rho}\gamma^{\gamma}\gamma^{\sigma}\gamma^{\delta})$ and I want to get it re-ordered like $Tr(\gamma^{\alpha}\gamma^{\...
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1answer
264 views

Trace technology with polarisation vectors

Consider $d$-dimensional gamma matrix structures. I have an expression like $$ \sum_{h_2=\pm}\text{Tr}(\not{\xi}_2\not{p}_3\bar{\not{\xi}}_2\not{p}_1), $$ where $\not p=p^\mu \eta_{\mu\nu}\gamma^\nu$ ...
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1answer
58 views

Ricci tensor $R_{\mu\nu}=\Phi g_{\mu\nu}\implies \Phi = \frac1n R$

Say we are in an $n$-dimensional pseudo-Riemannian manifold for which $$R_{\mu\nu}=\Phi g_{\mu\nu},$$ where $\Phi$ is a scalar field and $R_{\mu\nu}$ is the Ricci tensor, and $R=g^{\mu\nu}R_{\mu\nu}$. ...