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Questions tagged [trace]

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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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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
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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
29 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 ...
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1answer
66 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
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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
41 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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98 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, ...
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105 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 ...
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1answer
59 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 - ...
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1answer
70 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
96 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 ...
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1answer
72 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
94 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 ...
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Lower bound on quantum relative entropy

In my research this summer, I have become interested in lower bounds on the standard "Umegaki quantum relative entropy". For two non-negative matrices $X$ and $Y$, the Umegaki quantum relative ...
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1answer
276 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
51 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
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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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23 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 ...
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1answer
58 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
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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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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
115 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
133 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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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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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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53 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
64 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
125 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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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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136 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
128 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
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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 ...
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120 views

Expectation value in high temperature limit

Let $$\hat{A} = \exp\left(i \frac{t}{\hbar}( H - \lambda(a + a^{\dagger}))\right)$$ $$\hat{B} = \exp\left(-i \frac{t}{\hbar}( H + \lambda(a + a^{\dagger}))\right)$$ with $H = \hbar \omega(a^{\dagger}a ...
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1answer
118 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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399 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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219 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
51 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 $R=g^{\mu\nu}R_{\mu\nu}$. ...
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Measurement on density operator

Question: A system in a mixed state $\rho$ is measured with the measurement described by a projection operator $P$. What is the probability of the outcome? What is the density operator ...
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1answer
75 views

Einstein Notation over a Single Tensor

If I have a tensor $X^{\mu}{}_{\nu} = \begin{bmatrix} a & b & c \\ d & e& f\\ g&h&i\\ \end{bmatrix}$ then what is $X^{\mu}_{\;\;\mu}$? From what I understand it would be $(a,b,...
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1answer
88 views

$x'^i_j x^j_k = n\delta^i_k$ rather than $1\delta^i_j$?

These are my calculations $$x'^i_j x^j_k = \sum_{j=1}^n \frac{\partial x'^i}{\partial x^j}\frac{\partial x^j}{\partial x'^k} = \sum_{j=1}^n \frac{\partial x'^i}{\partial x'^k} =n \delta^i_k\ne \delta^...
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1answer
101 views

polarized trace

Lets say I want to calculate the following Trace $\mathrm{Tr}[u^{s_1}(p)~\bar{u}^{s_2}(p)~\gamma^{\mu}\not p~ \not q~ \not p~ \gamma^\nu]$ Now if I consider unpolaized case then $s_1=s_2=s$ and I ...
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1answer
565 views

About the proof of the subadditivity of the von Neumann entropy

I'm trying to understand the proof of the so called subadditivity of the von Neumann entropy, $$S(\rho^{AB}\,)\leq{S}(\rho^A)+S(\rho^B)$$ where $S(\rho)=-\mathrm{tr}\{\rho\log\rho\}$. In the proof I ...
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Traceless energy-momentum tensor

I don't think it is clear to me what exactly is the physical meaning of the energy-momentum tensor being traceless.
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1answer
722 views

Partial completeness relation for Dirac spinors

in studying trace techniques to obtain matrix elements, I came across a problem when we treat scattering of neutrinos on protons. Indeed, since those neutrinos are supposedly created in a weak decay, ...
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141 views

Orthogonality of matrix elements, trace techniques

I am currently reading Thomson's "Modern particle physics", and I have trouble understanding a concept that he somewhat gives for granted. Using Feynman rules, I am able to compute the matrix element ...
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210 views

Partial trace of the time evolution density matrix

Suppose we have a bipartite system, represented by the Hilbert spaces $\mathcal{ H}_S \otimes \mathcal{H}_E$ and some initial state vector $|\psi_E\rangle \otimes |\psi_S\rangle$. Let $H = A \otimes B$...
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110 views

How to calculate the trace below?

I am currently reading Peskin's QFT book on my own. Though it introduces the Trace Technology in Section 5.1, the trace calculations in the following sections are still far from clear to me. Here is ...
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1answer
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Reasoning Check: Trace of squared mixed-state density matrix

It's often written in the QI literature that, for a density operator $\rho$, if $\text{Tr}\left[\rho^{2}\right] < 1$, then $\rho$ describes a mixed state. However, I haven't seen any proofs of this ...
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Are the eigenvectors of the Choi-Jamiolkowski state maximally entangled?

Let $\phi: M_n\rightarrow M_n$ be a quantum channel (completely positive trace preserving). Via the Choi-Jamiolkowski isomorphism we can transform this into a state $$J(\phi) = (I_n\otimes\phi)(M) = \...