Questions tagged [trace]

Use this tag when having questions concerning expressions with the trace of a matrix/operator.

Filter by
Sorted by
Tagged with
2 votes
1 answer
129 views

Problem understanding expectation value of operators defined with density operator in quantum mechanics

I have a problem in understanding why we can write the expectation value of an operator $\hat{O}$ as the trace of $\hat{\rho}\hat{O}$ where $\hat{\rho}$ is the density matrix defined for pure state. ...
user avatar
  • 601
0 votes
0 answers
22 views

Schmidt decomposition manages to write a pure state using just $d$ terms

Suppose $|\psi\rangle$ $\in \mathrm{H_A}\otimes\mathrm{H_B}$ is a pure state and we can write a representation of $|\psi\rangle$ like $|\psi\rangle = \sum_j |\alpha_j\rangle|\beta_j\rangle$, where $|\...
user avatar
0 votes
1 answer
71 views

How do I prove that a reduced density matrix has properties of a density matrix?

The properties of a density matrix are defined as follows: $(1) \ \ \mathrm{Tr}\rho = 1 $ $(2) \ \ \rho^\dagger = \rho $ $(3) \ \ \rho \ge 0 $ $(4) \ \ \mathrm{Tr}\rho^2 \le 1 $ A reduced density ...
user avatar
  • 54
2 votes
2 answers
97 views

Symmetric form of Einstein Field Equations?

So normally, taking $c = 1$ and ${8\pi G = 1}$, and assuming the cosmological constant is negligible, the Einstein field equations read: $$R_{\mu \nu} - \frac{1}2Rg_{\mu\nu} = T_{\mu \nu}.$$ However, ...
user avatar
  • 401
2 votes
3 answers
43 views

Partial trace of local operators applied to maximally entangled states

I was looking at a problem where two invertible local operators were applied to a maximally entangled state, and didn't quite understand how some of it works out. We have local operators $A \otimes \...
user avatar
0 votes
0 answers
30 views

Order in taking density matrix trace

I remember that for infinite dimensional Hilbert space, the trace is not cyclic, for example, for harmonic oscillators, then we have $$Tr(aa^{\dagger})\neq Tr(a^{\dagger}a)$$ Then, when we calculate ...
user avatar
1 vote
1 answer
55 views

An identity for nested commutators

Let $A,B$ be Hermitian operators on an arbitrary Hilbert space. Define nested commutators of $B$ with respect to $A$ as $\text{Ad}_{A}(B) = \left[A,B\right]$, $\text{Ad}_{A}^{2}(B) = \left[A,\left[A,B\...
user avatar
  • 627
0 votes
1 answer
50 views

Trace of a linear operator in Dirac notation

I've been banging my head against a wall trying to find a proof for: $$Tr(𝑋) = ∑_𝑗⟨𝑗|𝑋|𝑗⟩.$$ This is supposedly fundamental knowledge. Can anyone help with the proof or direct me to a resource ...
user avatar
  • 103
1 vote
2 answers
84 views

Trace of the electromagnetic stress-energy tensor

I was looking in wikipedia about the Electromagnetic stress–energy tensor and it's properties, and it is traceless. I was looking at the proof of it, but I do not understand some steps, mainly because ...
user avatar
  • 934
2 votes
2 answers
166 views

How to calculate the trace of six gamma matrices multiplied to $\gamma_5$?

I read from Weinberg that, the gamma matrices have the following property: \begin{equation} \text{Tr}\{\gamma_5 \gamma_\mu \gamma_\nu \gamma_\rho \gamma_\sigma\}=4i\epsilon_{\mu \nu \rho \sigma} \end{...
user avatar
  • 1,091
0 votes
1 answer
63 views

How are functional traces calculated?

I am trying to follow this paper concerning decay rates in QFT. In equations (E.5), (E.6), (E.7), a functional trace is calculated using Feynman diagrams. However, I am struggling to see why $$Tr[(-\...
user avatar
  • 767
0 votes
1 answer
155 views

Partial Trace of Density Operator

To find the reduced density matrix, $\rho_A$, of a composite quantum system with two subsystems A and B, I've seen that the procedure is to take the partial trace of the full density matrix, $\rho_{AB}...
user avatar
  • 99
2 votes
1 answer
73 views

Trying to understand post-measurement density matrices in a state that spans 2 Hilbert spaces

What I would like to understand mathematically is the following situation: Prepare a quantum state that spans two Hilbert spaces Operate on one space with observable operator $\hat{O}$. Obtain ...
user avatar
  • 828
0 votes
0 answers
59 views

Trace of gravitational Weyl spinor

If two of the indices of the gravitational Weyl spinor $\Psi^{ABCD}$ is contracted, does it vanish? I mean does it follow from the traceless nature of the Weyl tensor that $\Psi^{A}_{~~BAD}=0$?
user avatar
2 votes
1 answer
171 views

Intuition for the trace-free energy-momentum tensor condition in CFTs

It is a textbook exercise to show that \begin{equation}T^{\mu}_{\,\,\,\mu}=0 \end{equation} is a sufficient condition for there to be a conserved current associated with a dilation symmetry. This ...
user avatar
  • 1,992
0 votes
1 answer
32 views

Trace orthonormality in irreducible reps. of semisimple Lie algebras

I have a question about an excerpt from Peskin & Schröder "Introduction to QFT" (see below). I understand the claim that I have marked as: "Let $\{t^a\}$ be a basis of a Lie ...
user avatar
  • 513
0 votes
1 answer
78 views

Show that Hilbert-Schmidt inner product is an inner product

I have to show that the Hilbert-Schmidt inner product is an inner product for complex and hermitian $d\times d$ Matrices $$(A,B)=Tr(A^\dagger B)$$ I checked the wolfram page for the definition of an ...
user avatar
  • 1,226
3 votes
3 answers
979 views

How it is possible that a ket precedes a bra in a matrix expression?

Is it possible to rewrite $\langle a| M|b\rangle$ as $|b\rangle \langle a|M$?
user avatar
1 vote
3 answers
136 views

How to take trace ${\rm Tr}(\left |a\rangle \langle a|X|b\rangle \langle b \right|) $ in bra ket notation?

Trace ${\rm Tr}(\left |a\rangle \langle a|X|b\rangle \langle b \right|) =$? Where $a$ and $b$ are two orthogonal state and $X$ is any operator.
user avatar
2 votes
1 answer
2k views

What is the significance of the trace of a tensor?

On a Riemannian manifold, the trace $X$ of a tensor $X_{\mu\nu}$ is defined as $$X=g^{\mu\nu}X_{\mu\nu}.$$ In linear algebra, the trace is the sum of the diagonal elements, so a traceless matrix has ...
user avatar
  • 3,825
0 votes
0 answers
34 views

Dimension of Hilbert Space Commutator $[\hat{x},\hat{p_{x}}]$ [duplicate]

We had an entrance exam and in which there was a question that, The smallest dimension of the Hilbert Space in which we can find operators $\hat{x}$, $\hat{p_{x}}$ that satisfy $[\hat{x},\hat{p_{x}}]=...
user avatar
1 vote
1 answer
225 views

Getting a Trace Out of Spinor Contractions in Quantum Field Theory

I am a bit confused by some point in the calculation of the electron-electron scattering amplitude in Zee's QFT in a Nutshell, Section II.6. He extracts this quantify to work it out separately: $$\tau^...
user avatar
0 votes
1 answer
52 views

"Trace" of a distribution?

In quantum field theory we have to sometimes take the "trace" of a distribution $M(x,y)$, $\text{tr}M\sim\int dx M(x,x)$. This happens for instance when we try to expand the determinant of a ...
user avatar
  • 3,294
0 votes
1 answer
74 views

Trace of commutators with flavor indices

I want to explicitly write out the Lagrangian term $$\operatorname{Tr}\bigg( \sum_{I\neq J}[\phi^I,\phi^J]^2\bigg) ,$$ where $I,J$ are flavor indices and $\phi$ is a scalar field. Why doesn't this ...
user avatar
1 vote
1 answer
105 views

Traces in 't Hooft-Veltman scheme

I'm currently looking at the 't Hooft-Veltman regularization scheme and I'm a bit confused on how exactly one calculates traces in this scheme. As far as I understand one has to divide the $D$-...
user avatar
  • 1,079
3 votes
1 answer
45 views

How do I prove the identity for ${\rm tr}_p [e^{-iS\Delta t}(\rho\otimes\sigma)e^{iS\Delta t}]$ in Seth Lloyd's 2014 Quantum PCA Paper?

Equation (1) in Seth Lloyd's paper on Quantum PCA says: $\text{tr}_{p}\text{e}^{-iS\Delta t} \rho \otimes \sigma \text{e}^{iS\Delta t} = \cos^2(\Delta t)\sigma + \sin^2(\Delta t) \rho - i \sin(\Delta ...
user avatar
0 votes
3 answers
168 views

Metric tensor times its inverse using Kronecker delta

From tensor calculus, we know that \begin{equation} g^{\mu\nu}=\delta_{\lambda}^{\mu}\delta_{\phi}^{\nu}g^{\lambda\phi}.\tag{1} \end{equation} Based on (1), is the following true? \begin{equation} g^{\...
user avatar
  • 107
-4 votes
2 answers
249 views

What is the physics meaning of the "trace" (Tr) and how we can calculate it? [duplicate]

What is the physics meaning of the "trace" (Tr) and how we can calculate it? $$ Z=tr\left \{ e^{\left ( -H \right )} \right \} $$ Where Z is the partition function .
user avatar
-1 votes
1 answer
69 views

Proof of gamma matrix trace identity from Griffiths Introduction to Particle Physics [closed]

Griffiths states that the product of eq (9.8): $ 8[p_1^\mu p_3^\nu + p_1^\nu p_3^\mu -g^{\mu\nu}(p_1 \cdot p_3) -i\varepsilon^{\mu\nu\lambda\sigma}p_{1\lambda}p_{3\sigma}]$ and eq (9.9) $ 8[p_{2\mu} ...
user avatar
1 vote
1 answer
140 views

Trace of Dirac matrices

I was calculating the trace of two Dirac matricies and I used their anti-commutation relations: $$ Tr(\gamma^{\mu} \gamma^{\nu}) = -Tr(\gamma^{\nu} \gamma^{\mu}) - Tr(2\eta^{\mu\nu}) $$ $$ = - Tr(\...
user avatar
1 vote
1 answer
93 views

Trace of two Dirac matrices in 4 dimensions

I want to show that tr($\gamma^\mu \gamma^\nu$) = 4$\eta^{\mu \nu}$. I know that {$\gamma^\mu , \gamma^\nu$} = 2$\eta^{\mu\nu}I_4$ and tr($\gamma^\mu \gamma^\nu$) = tr($\gamma^\nu \gamma^\mu$), so tr(...
user avatar
  • 574
0 votes
0 answers
71 views

$U(1)^{3} $ anomaly, trace of a hypercharge?

I have recently found the definition of the $U(1)^{3}$ anomaly as: $$\mathcal{A} = Tr[Y^{3}]_{L} -Tr[Y^{3}]_{R} $$ Where $Y$ is the hypercharge of the left, $L$ or right, $R$ components. What I don't ...
user avatar
0 votes
1 answer
85 views

Trace evaluation in inverse muon decay

Applying Casimir´s trick when averaging over the the initial and summating over the final spin states in the inverse muon decay yields (Griffiths, example 10.1) among others the following trace $$ \...
user avatar
  • 71
1 vote
1 answer
71 views

How to take trace over group and Dirac indices?

I'm currently reading Pokorski's book "Gauge Field Theories" and in Chapter 13 he discusses, among other things, Fujikawa's method of deriving the chiral current (see page 488 and the ...
user avatar
  • 1,079
1 vote
1 answer
378 views

Proof of $\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=0$

Using $\gamma^{5}\gamma^\mu=-\gamma^\mu\gamma^{5}$ and $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ I obtain \begin{equation}\tag{1} T_{\mu\nu}:=\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=-\mathrm{tr}(\gamma^\...
user avatar
  • 1,244
2 votes
2 answers
221 views

Gauge theory - definition of the trace

I am currently reading Nakahara's book and starting from chapter $10$, some sort of trace constantly shows up in the equations. For example, \begin{equation} S=-\frac{1}{2}\int\mathrm{tr}(F\wedge*F) \...
user avatar
  • 1,244
1 vote
1 answer
262 views

Trace of 4 gamma matrices

Can anyone help me understand how do i execute this trace? $$ Tr(\gamma_\mu(\gamma^\rho P_{1_\rho})\gamma_\nu (\gamma^\sigma K_{1_\sigma})) $$ I know the rule when we have 4 gammas inside the trace, ...
user avatar
  • 934
0 votes
2 answers
295 views

How to understand the trace operation below?

When studying the density operator, I read the document below. But the following trace calculation confuse me. It seems what he do is $\mathrm{tr}(\langle\psi _{a}|A|\psi _{a}\rangle)=\mathrm{tr}(\...
user avatar
  • 29
3 votes
5 answers
160 views

Confusion about $\partial_\mu x^\mu = 4$

Why is it that $\partial_\mu x^\mu = 4$? I thought that $\partial_\mu x^\mu$ could be expanded as $$\partial_\mu x^\mu = -\partial_1x^1 + \partial_2x^2 + \partial_3x^3 + \partial_4x^4 \\ =-1+1+1+1\\ =...
user avatar
  • 155
1 vote
1 answer
73 views

Computing state overlap from the expectation value of the Ctrl-Z operator

I am trying to understand an algorithm for computing the overlap between two single qubit states, $\left |\psi\right>$ and $\left |\phi\right>$: $$ \left| \left< \psi | \phi \right> \right|...
user avatar
  • 135
1 vote
1 answer
105 views

Gaussian integrals with gamma matrices in their exponents

I should evaluate Gaussian integrals in the 1+1 Minkowski space, which read $$ I_{1}= \int d^{2}k \, {\rm Tr}\big[ \gamma^{5} \gamma^{\eta} \gamma^{\kappa} e^{\alpha k^{\mu}k_{\mu} + \beta \gamma^{\...
user avatar
  • 354
3 votes
1 answer
229 views

Prove that the partial trace preserves density operators

Let $H_A$ and $H_B$ be two finite dimensional Hilbert spaces and $\rho_{AB}$ a density operator acting on $H_A\otimes H_B$. I am to show that $\rho_{A} = \operatorname{tr}_B\rho_{AB}$ is also a ...
user avatar
2 votes
1 answer
153 views

Where does the expression $\mathrm{Tr}(K) = \sum_{j=1}^{n}\langle\psi_j|K|\psi_j\rangle$ for the partial trace come from?

During my studies of composite quantum systems I find some expressions that leave me with a little doubt. For example: Let K be a linear operator defined in the Hilbert space H. Where H is given by $H ...
user avatar
0 votes
1 answer
120 views

Where do the traces come from in Casimir's trick?

I am following the derivation for electron-muon scattering amplitude in Griffiths textbook and got to the section where they use Casimir's trick. I can't see where the traces come from. Equation 7.123 ...
user avatar
0 votes
1 answer
135 views

Trace properties of the gauge potential in non-Abelian gauge theory

I want to proof equation 69.18 in Srednicki's book "Quantum field theory", which reads: \begin{equation} A_\mu^a(x)=2\text{Tr}[A_\mu(x)T^a].\tag{69.18} \end{equation} $A_\mu(x)$ is the non-...
user avatar
0 votes
0 answers
105 views

Fierz Identity calculation

While reading an article, it's said that to simplify the following Dirac structure $$\left(P_Lv_j^d\bar{v}^s_kP_R\right)_{\alpha\beta}\tag{1}\label{1}$$ where $j,k$ are color indices and $d,s$ ...
user avatar
  • 4,210
2 votes
1 answer
82 views

Are these bra-ket manipulations correct?

I'm reading an old paper ("Wigner's Function and Other Distributions in Mock Phase Spaces," Balazs and Jennings, Phys. Rep. 104(6), 1984), and came across the following statement (in which $\...
user avatar
0 votes
1 answer
61 views

How do I trace out the second qubit to find the reduced density operator? [closed]

I'm doing an exercise to trace out the second qubit to find the reduced density operator for the first qubit: $tr_2|11\rangle\langle00| = |1\rangle\langle0|\langle0|1\rangle$ I'm just wondering if I ...
user avatar
  • 463
0 votes
3 answers
289 views

Is the trace of a reduced density operator always equal to 1?

From my understanding, we define a reduced density operator $\rho_A$ of an operator $\rho_{AB} = |a_1⟩⟨a_2|\otimes |b_1⟩⟨b_2|$, as: $$ρ_A=Tr_a(ρ_{AB})=|a_1⟩⟨a_2|Tr(|b_1⟩⟨b_2|)$$ I also know that $Tr(\...
user avatar
  • 127
0 votes
3 answers
488 views

What does it mean to contract the indices of a Lorentz matrix?

The metric tensor in SR obeys the transformation law (I am using Schutz's bar notation for different frame indices): $$\eta_{\bar{\alpha}\bar{\beta}}=\Lambda^\mu_{~\bar{\alpha}}~\Lambda^\nu_{~\bar{\...
user avatar

1
2 3 4 5