# Questions tagged [trace]

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

291 questions
Filter by
Sorted by
Tagged with
1 vote
44 views

• 361
405 views

### 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 ...
• 51
1 vote
82 views

### (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 ...
• 85
47 views

### 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 ...
610 views

### 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 ...
• 11.8k
129 views

### 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 ...
• 689
229 views

### 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 ...
• 90
177 views

• 1,075
571 views

### 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, ...
1 vote
87 views

• 11
268 views

### 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 ...
• 43
232 views

### 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 ...
139 views

• 1,455
114 views

• 23
1 vote
179 views

### 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 ...
• 727
1 vote
135 views

• 1,297
1 vote
64 views

### 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 ...
166 views

### 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 ...
• 401
99 views

### 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 ...
665 views

### 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 ...
• 381
Is there a unitary $U_{AB}$ such that, for any density operator $\rho$, we have {\rm {Tr}}_A \left[U_{AB} \left(\frac{I_A}{2} \otimes \rho_B\right)U_{AB}^{\dagger}\right]= \frac{\rho_B}{2}+\frac{I_B}...