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5
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1answer
2k views

Understanding the partial trace and deriving $\langle l|R_{B}|k\rangle = \text{Tr}((\mathbb{I}_{A} \otimes |k\rangle \langle l|)(R_{AB}))$

By definition according to the notes I am looking through: The partial trace $\text{Tr}_A:L(H_A \otimes H_B) \rightarrow L(H_B)$ is the unique map that satisfies: $$\text{Tr}(L_B \cdot \text{Tr}_A(R_{...
4
votes
2answers
176 views

Infinitesimally change a operator in QM

Reading Balian, "From Microphysics to Macrophysics", I've found the following identity: If we change the operator $\hat{{\mathbf{X}}}$ infinitesimally by $\hat{{\delta\mathbf{X}}}$, the trace of an ...
4
votes
2answers
235 views

Traces in different representation

I am actually working with Green-Schwarz anomaly cancellation mechanism in which I have came across a strange formula which relates trace in the adjoint representation (Tr) to trace in fundamental ...
4
votes
1answer
52 views

Should the trace of a product of gamma matrices depend on the convention I use?

I am trying to work out $$\text{Tr}[\gamma_5\gamma_\mu\gamma_\nu\gamma_\alpha\gamma_\beta]$$ using the same convention as J.J. Sakurai (Advanced Quantum Mechanics), what I get is $$\text{Tr}[\gamma_5\...
3
votes
2answers
616 views

Trace in non-orthogonal basis?

Physicists define the trace of an operator $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal basis, and this quantity is ...
3
votes
2answers
2k views

How to take partial trace?

$L$ is a linear operator acting on hilbert space $V$ of dimension $n$, $L: V \to V$. The trace of a linear operator is defined as sum of diagonal entries of any matrix representation in same input ...
3
votes
3answers
474 views

Question about the Dirac notation for partial trace

I saw the following definition for the partial trace operator: $\rho_A=\sum_k \langle e_k|\rho_{AB}|e_k\rangle$, where $e_k$ is basis for the state space of system $B$. From what I know, in the ...
3
votes
1answer
41 views

Some diracology in traces

Suppose I want to evaluate the trace $p_{\alpha} q_{\beta}\text{Tr}(\gamma^{\alpha} \gamma^0 \gamma^{\beta} \gamma^0)$. Using the standard trace formula for four gamma matrices I arrive at $$p_{\...
3
votes
2answers
73 views

Adding a tracer to the surface of a water droplet

I have a 2 mm water droplet generated by a syringe and falling down. I am using two perpendicular cameras to capture simultaneous frames from it. I need to track the droplet during the time and ...
3
votes
2answers
416 views

Density of states via a Laplace transform

Is this formula correct? $$ \frac{-1}{\pi}Im\int_{0}^{\infty}\!dt~\exp(Et)\Theta (t)\exp(i \epsilon t) ~=~ \sum _{n}\delta (E-E_{n}), $$ $$ \Theta (t)~=~ \sum_{n}\exp(-tE_{n}) ,$$ and I have used ...
2
votes
1answer
122 views

Lowering/raising metric indexes

So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing. Consider Minkowski spacetime. The trace of a matrix $A$ can be ...
2
votes
1answer
79 views

Proof that trace is independent of representation [closed]

$$\begin{align} \sum_{a'} \langle a'|X|a'\rangle &=\sum_{a',b',b''} \langle a'|b'\rangle \langle b'|X|b''\rangle\langle b''|a'\rangle \\ &=\sum_{b',b''} \langle b''|b'\rangle \langle b'|X|b''\...
2
votes
1answer
572 views

In terms of covariance matrices, are partial measurement and partial trace equivalent?

Partial measurement and partial trace There is a connection between a measurement of a part of a system and tracing this subsystem out. Say, we have a system composed of subsystems $A$ and $B$ in a ...
2
votes
1answer
125 views

On shell and off shell simultaneously?

I am considering the following one loop virtual correction in the DIS process: where I have a quark of momentum $p$ coming in, emitting a gluon before interacting with a photon of momentum $q$ to ...
2
votes
1answer
80 views

Different kinds of trace for statistical ensembles

In the chapter 7 of the book "A Modern Course in Statiscal Physics" by L. Reichl, we found $Tr[\hat{\rho}]=1$ for microcanonical ensembles and $Tr_N[\hat{\rho}]=1$ for canonical and grandcanonical ...
2
votes
1answer
57 views

thermodynamics trace calculation [closed]

I'm trying to calculate a trace to get the average energy. The Hamiltonoperator is $H = \sum\limits_k \varepsilon_k a_k^\dagger a_k$ and $N = \sum\limits_k a_k^\dagger a_k$. The trace to calculate is $...
2
votes
0answers
202 views

Trace of the number operator in second quantization

I would like to find the helmholtz free energy of a system $F=-T ln(Tr[e^{\frac{-H}{T}}])$ namely a bcs superconductor (using Annett's notation) $H=\sum E_k (b_{k\uparrow}^\dagger b_{k\uparrow}+b_{...
1
vote
2answers
102 views

Flat space Solution of Einstein Field Equation

Does a trace-free energy-momentum tensor $T_{\mu}^{\mu} = 0$ ensure that the Einstein's field equations have a flat space solution?
1
vote
2answers
2k views

Kronecker delta confusion

I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same): ...
1
vote
2answers
139 views

Why gauge fields are traceless Hermitian?

So I've had a read of this, and I'm still not convinced as to why gauge fields are traceless and Hermitian. I follow the article fine, it's just the section that says "don't worry about this ...
1
vote
1answer
45 views

Trace of operator defined by two state vectors in Quantum Mechanics

I'm studying QM from the book 'Quantum Mechanics. Concepts and Applications' by Zettili. There's an example which gives us two state vectors $$ | \psi \rangle = 9i \ | \phi_1 \rangle + 2 | \phi_2 \...
1
vote
1answer
94 views

Deriving the Spinor Completeness Relation without using a Representation

Reference: DAMTP problem set 3, question 5 but ignore the spinor solutions given. To preface, this has taken up 1 entire day and a further 2 afternoons of work so I will just list the most promising ...
1
vote
1answer
114 views

understanding the oscillating part of the Gutzwiller trace

given the density of states according to Gutzwiller's trace formula $ g(E)= g_{smooth}(E)+ g_{osc}(E) $ i know that the 'smooth' part comes from $ g_{smooth}(E)= \iint dxdp \delta(E-p^{2}-V(x)) $ ...
1
vote
1answer
98 views

Srednicki - computing divergent piece of loop integral

I was reading through Srednicki and didn't quite understand one of the paragraphs in Section $51$ on loop corrections in the Yukawa theory on P.$322$. It's the fermion loop correction to the local ...
1
vote
1answer
131 views

Colour decomposition in QCD

I am looking to compute the matrix element for the process gg -> u ubar at leading order. It is straightforward to calculate the non-colour part of the usual s, t and u channels. I will call these ...
1
vote
2answers
212 views

how to trace light after refraction by a camera lens?

I am a programmer and I am doing a camera simulation, I am stuck in a matter of how to know where arrives every ray of light after traveling through the lens and being refracted. Every point of the ...
1
vote
0answers
27 views

Scale invariance and stress energy tensor

I have seen in a paper [1] that in a quantum field theory scale invariance takes place provided the stress energy tensor is traceless. How this is true? References: "INFINITE CONFORMAL SYMMETRY IN ...
1
vote
0answers
56 views

Contraction of Kronecker delta = 4 [duplicate]

This suggests, as a shortcut notation, the concept of lowering indices; from any vector we can construct a (0, 1) tensor defined by contraction with the metric: $$A_\nu ≡ g_{\mu\nu}A^\mu$$ so that ...
1
vote
0answers
19 views

could we obtain the potential (in one dimension) from the Gutzwiller trace?

to solve and obtain the potential of a 1-D Hamiltonian we must solve an integral equation $$ N(E)= A \int_{0}^{E}\frac{V^{-1}(x)}{\sqrt{E-x}}$$ fro a some constant 'A' , then my question is since ...
0
votes
2answers
79 views

How can I show that $\mathrm{Tr}\left( f(G^\dagger G)\right)=\mathrm{Tr}\left( f(G G^\dagger)\right)$?

I'm slightly stuck on the following question: Prove that: $\mathrm{Tr}\left( f(G^\dagger G)\right)=\mathrm{Tr}\left( f(G G^\dagger)\right)$ where $G$ is any operator. Using the definition of the ...
0
votes
2answers
74 views

Quantum field theory why is it a trace?

So in perskin when computing the amplitude of $e^+e^-\rightarrow\mu^+\mu^-$ summing up over spin they say \begin{align}\sum_{s,s'}\bar{v}^{s'}_a(p_2)\gamma^\mu_{ab}u^s_b(p_1)\bar{u}^{s}_c(p_1)\gamma^...
0
votes
2answers
88 views

Gamma matrices and trace operator

I'm trying to show that the trace of the product of the following three Gamma (Dirac) matrices is zero, i.e. $$\text{tr}(\gamma_{\mu} \gamma_{\nu} \gamma_{5})=0 \text{.}$$ I attempted to use the fact ...
0
votes
1answer
256 views

taking the trace

could anyone show me the first couple of steps in taking the trace of something like this, im not sure how to start. 'Tr[$\gamma($$\gamma k + $$\gamma p + $$\gamma q + m) $$\gamma ($$\gamma k + $$\...
0
votes
1answer
51 views

Reduced Density operator in matrix form

I already read book of Quantum Computation and Quantum Information by Nielsen and Chuang according to reduced density operator and I already understand how to do the reduced density using Dirac ...
0
votes
1answer
133 views

How can I solve an equation involving partial trace?

I am unable to find the solution to the following equation: Tr$_{2}[U(|\psi\rangle \langle\psi|\otimes \rho)U^{\dagger}]=\rho$ Here $\psi$ is state vector representing a qubit and $\rho$ state of ...
0
votes
1answer
104 views

About the three point function at one loop order

Could someone explain how exactly do you calculate the trace of the three point function of one loop in QED. in the following link the expression from 1. a (2) http://learn.tsinghua.edu.cn:8080/...
0
votes
0answers
27 views

How to do partial trace of three qubit? [on hold]

Good day, $\|A\rangle=\left(\dfrac{i_0}{j_1}\right)$, $\|B\rangle=\left(\dfrac{i_0}{j_1}\right)$, $\|C\rangle=\left(\frac{i_0}{j_1}\right)$, For 2-qubit systems, the $\|AB\rangle\langle AB|$, ...
0
votes
0answers
36 views

Quartic interactions of a complex scalar field

For a quartic self-interaction of a complex scalar field (matrix), one can write the terms: $aTr((\phi^*\phi)^2)$ and $bTr(\phi^*\phi)^2$ ; the trace and the "double" trace term, with two different ...
0
votes
1answer
44 views

Evaluating the trace of an expression with gamma matrices

I am currently reading Srednicki's Quantum field theory Book and am having some troubles with evaluating the trace of some gamma matrix expressions. For instance in Equation 59.19 Srednicki defines ...
0
votes
0answers
50 views

Solving non scalar integrals in loop calculations

Consider the following integral that comes out of a loop calculation along with some fermionic propagators (e.g virtual one loop correction to a $p \gamma^* \rightarrow p'$ process such as in DIS): $$ ...
0
votes
0answers
52 views

Cyclicity of trace with fermionic arguments

I think this is a non-question, but it has me considerably worried. Consider the piece of a Lagrangian density given by, $$\mathcal{L} = \epsilon_{ij}\mbox{Tr}\left(\chi^{i,\alpha}\left[\chi^{j}_\...
0
votes
0answers
37 views

Problem getting a product of traces out of a single trace in a chiral perturbation theory computation

I am stuck at a computation and I would appreciate any help. $U$ is the pion matrix in chiral perturbation theory $$U=e^{i\sigma_a\phi_a/f}$$ where $\sigma_a$ are Pauli matrices, $\phi_a$ are three ...
-1
votes
1answer
82 views

Trace representation of density matrix question [closed]

System $A$ and system $B$ form a composite system. https://en.wikipedia.org/wiki/Partial_trace I wonder why $\rho_{AB}$ cannot be represented as $(\text{tr}_{B}(\rho))\otimes (\text{tr}_{A}(\rho))$....
-2
votes
1answer
113 views

Multiplication properties about trace of two operators [closed]

Consider two operators $A$ and $B$, their functions $e^A$ and $e^B$ and a basis that mutual diagonalizes $A$ and $B$. Can I say that $$Tr\left[e^Ae^B\right]=Tr\left[e^A\right]Tr\left[e^B\right]~?$$ ...