Questions tagged [trace]
Use this tag when having questions concerning expressions with the trace of a matrix/operator.
292
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How is the Ricci scalar the trace of the Ricci tensor?
The Ricci scalar is the uncontracted version of the Ricci tensor $R=R^{\mu}_{\mu}=g^{\mu\nu}R_{\mu\nu}$. Carrol describes the Ricci scalar as being the trace of the Ricci tensor, but I do not ...
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The covariance matrix $\mathbf{C}$: Why does $\ln \text{det} \mathbf{C}=\text{Tr}\ln \mathbf{C}$ hold? [migrated]
Prof. Max Tegmark first introduced the Fisher information matrix into cosmology in his paper titled Karhunen-Loeve eigenvalue problems in cosmology: How should we tackle large data sets?
As I read the ...
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(Anti-)Fundamental Representation of $SU(5)$ GUT
In many places, it has been mentioned that the sum of electrical charges of the particles present in $\overline{5}$ of $SU(5)$ is zero since the trace of $SU(5)$ generators is zero. I do not ...
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Trace of stress tensor in 2D average null energy condition
I was looking through Zamolodchikov's derivation of the $c$-theorem and stumbled across an equation which says the following -
$$\Theta = T^\mu_\mu = 4T_{z\bar{z}}.$$
As far as I understand, for two ...
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Generators of ${\rm SU}(n)$ are traceless. Why?
A general element of the Lie group ${\rm SU}(n)$ is written as
$$
g({\vec{\theta}})=e^{-i\sum_a\theta_a T_a}
$$
where $\theta_a$ for $(a=1,2,\ldots,n^2-1)$ denotes $n^2-1$ real parameters. The ...
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Physical interpretation of unbounded trace class linear maps
Quite generally, quantum states are defined to be positive, trace-class linear maps with trace equal to one on a complex separable Hilbert space $\mathcal{H}$. If we require that these trace-class ...
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A limit of a particular Quantum Fidelity
I have the following problem.
Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More ...
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Form of the trace of the energy-momentum tensor in 2D spacetime
I'm going over this article by Davies, in which he derives the form of the energy-momentum tensor (emt) in 2D spacetime assuming a non vanishing trace anomaly.
He considers a metric of the form
$$
ds^...
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Factorization of density matrices
I'm currently reading through the following document about quantum noise and open quantum systems: https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes13.pdf. On page 6 of the document, ...
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Trace of operators
Trace of an operator $A$ can be written in its eigenbasis $|a\rangle$ as
$${\rm Tr}(A)=\sum_{a} \langle a|A|a\rangle.$$
Since trace is base independent, does it imply that we can also write the ...
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Trace of a density matrix that is acted upon by a separable linear operator
If we let $\hat{\rho}$ be the density matrix of a Bell state. I have seen that calculating the trace,
$$
Tr(\hat{A} \otimes \hat{I} \hat{\rho}) = Tr_{\mathcal{H}_A}(\hat{A}\hat{\rho}_{\mathcal{H}_A})
$...
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Proving an identity for the trace of 2 by 2 matrices in terms of a product of traces
In order to prove the $2$ to $1$ homomorphism $SL(2,\mathbb{C}) \rightarrow SO^{+}(1,3)$ I was given the following trace identity for $2 \times 2$ complex matrices $M_{1},M_{2}$:
$$\text{tr}(M_{1}M_{2}...
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Trying to confirm that the trace of the energy-momentum tensor divided by the energy density is NOT invariant
I am analyzing this question in the FRW universe with a perfect fluid.
The trace of the energy momentum tensor $$T^{\mu \nu} g_{\mu \nu} = \rho - 3p $$ is of course an invariant quantity. It does, ...
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It the thermal expectation value of an operator time-dependent?
In Euclidean time $\tau$, $\langle \hat O(\tau )\rangle= Z^{-1}\mbox{Tr}(e^{-\beta \hat H} \hat O(\tau ))=Z^{-1}\mbox{Tr}(e^{-\beta \hat H} e^{\hat H\tau}\hat O(0 )e^{-\hat H\tau})=Z^{-1}\mbox{Tr}(e^{-...
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Variation vs. derivative wrt a symmetric and traceless tensor
Consider a Lagrangian, $L$, which is a function of, as well as other fields $\psi_i$, a traceless and symmetric tensor denoted by $f^{uv}$, so that $L=L(f^{uv})$, the associated action is $\int L(f^{...
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How to transform a Lindblad operator basis?
I'm trying to understand how to perform a unitary transformation on a set of traceless orthonormal Lindblad operators, following chapter 3.2.2 of The Theory of Open Quantum Systems by Breuer and ...
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Lipschitz constant of Quantum Fisher information
I was looking for the lipschitz costant of the Q.Fischer information in terms of the trace distance, more specifically, I was looking for a bound "like" the one in the image.
I don' t really ...
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Metric Tensor times its inverse (non-zero curvature)
so I am quite confused regarding the spatial metric tensor $g_{ij}$. If I have $g_{ij}g^{ij}$ I essentially get the trace of the metric tensor $g$ right? Or, do I get $\delta^i_i = 3$ instead?
The ...
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Clarifying Bra-Ket Notation: Orthonormal Bases
I was asked to find the trace of $(A \in M_{n \times n})$, the matrix that can be written in the form:$$A=\frac{1}{n} \sum_{r, \, q \, = \, 1}^n (-1)^{r+q}|r \rangle \langle q| \quad ,$$ where {$|r \...
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Trace of Product of Field Operators
I am studying the theory of interacting electrons in path integral formalism from Altand and Simons, Condensed Matter Field Theory. See Equn just above Eq. 6.6, where he finds trace of the product of ...
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What is the physically significance of having trace less than 1 in squared mixed state density matrix? [closed]
I was going through density matrices where I get to know that the trace of squared mixed state density matrix is less than a pure state density matrix. I also went through the proof. But I want to ...
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How to solve this equation involving an exponential of the angular momentum operator? [closed]
Let $\vec a$ be a given vector, and $\vec J=[\hat J_x,\hat J_y,\hat J_z]$ the total angular momentum operator. $\vec \beta$ is an unknown vector, to be solved for, which satisfies
$$
\vec a=\mathrm{...
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Trace and index manipulation
Imagine that I have a quantity $F_{ab}$ multiplying the stress tensor $T^{ab}$:
\begin{equation}
F_{ab} T^{ab}.
\end{equation}
There is also a metric, say $h_{ab}$. If I want to write the above ...
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Why this definition of the trace?
It is common in quantum mechanics textbooks (e.g. Ballentine page 15) to define the trace of an operator as the sum of its diagonal elements in an orthonormal basis in particular. Why is this ...
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Why must the stress-energy tensor be traceless in 2D classical general relativity?
It's easy to show that the Einstein tensor is always traceless in 1+1 spacetime dimensions; this is just a simple algebraic identity that follows directly from the definitions. The Einstein field ...
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Does the trace of the tensor of inertia have any physical significance?
The tensor of inertia is given by
$$I_{ij} = \sum_k m_k \left( r_k^2 \delta_{ij} - x_{k,i} x_{k,j} \right).$$
In my experience, the trace of a physically important tensor tends to also be important. ...
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How to prove $\mathrm{Tr}[(\partial_\mu U)U^\dagger]=0$?
I am studying ChPT by referring to "A Primer for Chiral Perturbation Theory" by Stefan Scherer.
I'm having a problem with the consideration of terms that appear in the Lagrangian.
The ...
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Unitary equivalence between Hermitian operators
Take two (non-zero) Hermitian operators $A$ and $B$. I want to proof that there exists no unitary operator $W$ such that:
$$W^{\dagger}AW = A + B$$
For my research I proved this for some specific case ...
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What does it mean to linearize the trace of Gauss-Codazzi relations?
What does it mean to linearize the trace of the Gauss-Codazzi relations?
The equations referenced above are the following:
Reference to the paper: https://arxiv.org/abs/hep-th/0511096
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How to define the inverse of Dirac Gamma Matrices in QFT?
The Dirac gamma matrices are a set defined by the 16 following matrices:
$$\Gamma^{(a)}=\{I_{4x4},\gamma^\mu,\sigma^{\mu\nu},\gamma_5\gamma^\mu,\gamma_5\}.\tag{2.122}$$
Now, I wish to determine the ...
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Reduced density matrix computation generalization
I just recently completed a problem in which I had a Hilbert space of the form
$$
H = H_1 \otimes H_2 \otimes H_3
$$
and was tasked with finding the reduced density matrix for the system in the ...
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Trace of a Hamiltonian and zero-point energy
In finite-dimensional quantum mechanics, we are free to assume that our Hamiltonian is traceless. We can define $H' = H - \mathrm{tr}(H)\cdot I$, and since
\begin{equation}
\exp(-iH't) = \exp(-iHt)\...
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Inner product in a composite Hilbert space
Take the two Hilbert spaces $ H_1 = H_2 = C^2$
The basis of $H_1$ is : $ \{ | 1 : + \rangle , |1 : - \rangle \} $
and for $H_2$ : $ \{ | 2 : + \rangle , |2 : - \rangle \} $
Forming the ...
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Trace of stress energy tensor
How do I prove that $T^\mu{}_\mu = \beta_iO_i$, where the sum runs over all $\beta$-functions in the theory, and $O_i$ is the corresponding vertex operator?
Related to this, how do I prove that $\frac{...
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Why does the trace of the pion operator vanish?
I am working my way through Srednicki's QFT book. Currently I'm in chapter 94.
Between equations 94.20 and 94.21, Srednicki says that the following expression vanishes in the case of two light ...
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1
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Possible error in Hatfield's "Quantum Field Theory of Point Particles and Strings"?
In problem 8.1 of Hatfield's Quantum Field Theory of Point Particles and Strings, the reader is tasked with calculating the trace $$\operatorname {tr}(\gamma_0\frac{(\not p_2 + m)}{2m}\gamma_0\frac{(\...
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Trace manipulation in finding the matrix element squared
Here's the $s$-channel of the scalar-fermion scattering given that $L_{int} = -g\phi\bar\psi\psi$:
And I found the matrix element of this channel is
Then to find $\sum_{s,s'}|\tilde M_1|^2$:
I was ...
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Conformal invariance and tracelessness of the energy-momentum tensor: contradictory statements
Before starting my question, let me define a couple terms to avoid the confusion that usually accompanies this topic:
I define the $c$-number valued energy-momentum tensor as $T^{\mu\nu} = \frac{2}{\...
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Inequality on tensor product in trace
Let $\rho$ and $\sigma$ be density operators and $N$ an integer. We assume that $\sigma>0$. I want to show that the inequality is true:
$$N \cdot \text{Tr}[\rho\log\sigma]- \text{Tr}[\rho^{\otimes ...
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Why is the trace of the outer product of two states equal to the inner product of the two states?
Why is it that, given two quantum states $|\psi_1\rangle$, $|\psi_2\rangle$,
$$\mathrm{Tr}(|\psi_1\rangle\langle\psi_2|) = \langle\psi_2|\psi_1\rangle \quad $$
I went through the equation with the ...
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Trace of product of six Pauli matrices
Using the standard definition of the Pauli matrices with the zeroth included, i.e.
$$ \sigma^{\mu} = (I, \sigma^i) $$
$$ \bar \sigma^{\mu} = (I, -\sigma^i) $$
it's a standard result that
$$ Tr[\sigma^...
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How can one visualize a real, symmetric $3 \times 3$ tensor with zero trace?
I am looking for the simplest way to visualize a real, symmetric 3x3 tensor than has vanishing trace. (All entries are real numbers.)
It cannot be an ellipsoid, because an ellipsoid has three positive ...
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Trace of $SU(N)$ generators and number of spin components
I am reading Peskin's and Schroeder's book "An Introduction to Quantum Field Theory". At Chapter 16.6, the authors write down an expression for the trace of of two generators of $SU(N)$
$$\...
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Lorentz generators trace
I am reading Peskin's and Schroeder's book "An introduction to Quantum Field Theory". At some point in Chapter 3.1, they write down a particular representation for the Lorentz generator of a ...
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Notation of trace in QCD
I have been reading about QCD, and I have found a notation about the trace at Schwinger Dyson equation like $Tr_D[exp]$ and I don not know whats does means. Thank you.
Update reference: equation 27
...
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Linearity of partial trace
In Quantum Processes Systems, and Information by Schumacher and Westmoreland we are given this property of the partial trace
$$ Tr_R G^{RQ} = Tr_R \left( |\alpha^R,\phi^Q \rangle\langle\beta^R,\psi^Q| ...
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The significance of trace of a matrix [duplicate]
Since the determinant and trace of a matrix give information about the solution of the 1D propagation system, i.e. the solution is propagating, scattering or tunneling, etc.
How do we extract ...
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Trace and determinants in QFT's
I'm trying to understand this paper: https://doi.org/10.1103/PhysRevA.46.6490. It's about path integration with defects (theories on submanifolds). Let me here try to explain what in particular I'm ...
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Is the trace of group generators a representation invariant?
This is likely a basic question, but I can't come up with a straightforward (dis)proof that the traces of generators of a Lie group are invariant. The reason I am asking is because the elements of the ...
user346886
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3
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Trace of density matrix square greater than 1?
I learned that if you have a density matrix $\rho$ then
$\mathrm{Tr}(\rho^2)=1 \Rightarrow$ pure state
$\mathrm{Tr}\rho^2<1 \Rightarrow$ mixed state
Can one have $\mathrm{Tr}(\rho^2)>1$?
For ...
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