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2
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0answers
66 views

Traceless Tensors in $SU(3)$, Georgi's Lie Algebras

I'm doing a self-study through Georgi's Lie Algebra's in Particle Physics and there is a ''note without proof'' in the book that I have not managed to see through myself. In Section 10.3, Georgi ...
3
votes
1answer
157 views

How can I make two separate equations for Christoffel symbols give the same answer?

I have been studying the covariant derivative and I'm confused by the calculation of the Christoffel symbols $\Gamma$. The equation for computing $\Gamma$ is given as: $${\Gamma^c}_{ab} = \frac12 ...
0
votes
1answer
40 views

Notation for $N$-particle wave functions

If we have one particle we first look at an orthonormal basis of the one-particle Hilbert space $|n\rangle$. Here $n$ is the abbreviation for a compete set of quantum numbers, for example $n = ...
0
votes
1answer
119 views

Problem understanding Lorentz invariance [duplicate]

So they usually started with "...This is obviously Lorentz invariant, because of the 4-vector character of the quantity,..., (and after a two page long derivation) another quantity is also obviously ...
1
vote
1answer
77 views

Representations of Lorentz algebra

It is well known that the Lorentz algebra can be written as two $SU(2)$ algebras. By defining $$N_i=\frac{1}{2}(J_i+iK_i), \qquad N^{\dagger}_i=\frac{1}{2}(J_i-iK_i)$$ we have ...
1
vote
3answers
190 views

Technical question about 2-forms

A technical question about the electromagnetic tensor, but before that, it is know if, say, instead of being $$F_{\mu\nu}=\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$$it were ...
2
votes
1answer
288 views

Perfect fluid and Cauchy momentum equation

The stress-energy tensor of a perfect fluid is given by $$T^{\mu\nu}=\left(\rho+pc^{-2}\right)u^\mu u^\nu+pg^{\mu\nu}$$ The divergence of the stress-energy tensor is zero: $\nabla_\mu T^{\mu\nu}=0$. ...
0
votes
1answer
33 views

What is the correct dual of antisymmetric tensors?

In some books I find the dual antisymmetric tensor $$\tilde{H}^{ab}=-\frac{1}{2}\epsilon^{abcd}H_{cd}$$ and other times I find it with no minus sign. How can I tell which to use? Is this like that in ...
0
votes
1answer
76 views

Using metric tensor to contract

Can the metric tensor also contract the indices in the $$\epsilon^{\tau\lambda\mu\nu}~?$$ For example, if we have ...
1
vote
0answers
78 views

Decomposition of a tensor under transformations

To illustrate my question I'll take an example from theory of relativity: An arbitrary 4-tensor $A^{ik}$ changes under a general coordinate transformation: $$ A'^{ik} = C^{i}_mC^{k}_n A^{mn} $$ ...
0
votes
1answer
53 views

The significance of the pressure term within the momentum-energy tensor [duplicate]

EDIT: this question is based around my notion regarding the possible role of potential energy in the momentum energy tensor T$_{\mu\nu}$, The answer below resolves the question and I have deleted ...
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 ...
3
votes
1answer
154 views

Can Bosons couple to gravity? Why do we need vielbein?

It is said that In theories such as Supergravity where there are fermions coupled to gravity, one must use an auxiliary quantity, the frame field (vielbein). In supergravity, can a boson be coupled ...
2
votes
2answers
235 views

Relativity question about 4-velocity

Given a 4-velocity $u^0$, how do you find $u_0$? Do you use $u_{\alpha}u^{\alpha} = -1$?
0
votes
2answers
136 views

Partial Measurement and the Math Behind it

$\newcommand{\ket}[1]{\left| #1 \right>}$ $\newcommand{bra}[1]{\left< #1 \right|}$ Talking about the partial measurement the professor defines the state $\ket \psi$ to be $$\ket{\psi} = ...
1
vote
1answer
106 views

How to act an operator on a two-particle spin state?

I'm doing an assignment for my quantum class at the moment and I'm having trouble figuring out how to act a Spin operator on a two-particle state - specifically in finding the eigenvalues - I've spent ...
0
votes
1answer
182 views

The divergence of the Stress Energy Tensor

I have been studying general relativity and I have often seen in textbooks that the divergence of the stress energy tensor is zero. $$T^{\mu\nu}_{;\nu} = 0$$ but is it possible to contract and ...
1
vote
1answer
133 views

Differentiating the Lagrangian to find geodesic equations?

I'm stuck pretty much at the first hurdle trying to follow the derivation of the geodesic equations from the Lagrangian ...
1
vote
1answer
716 views

electrical conductivity and resistivity tensor

By definition of the conductivity tensor $\hat{\sigma}$ and the resistivity tensor $\hat{\rho}$, we have \begin{equation*} \begin{split} & j_{\alpha}=\sigma_{\alpha \beta}E_{\beta} \\ & ...
2
votes
1answer
157 views

Density Matrix Renormalization Group (DMRG) Simulation of a String-Net Model

In the following paper, Dr. Xiao Gang-Wen et. al. introduce the idea that string-net condensed states can be represented in terms of tensor product states: http://arxiv.org/pdf/0809.2821.pdf The ...
0
votes
1answer
86 views

Prove $F_{\mu\rho} \tilde F_\nu^{\phantom{\nu}\rho} = \frac14 \eta_{\mu\nu} F_{\rho\sigma} \tilde{F}^{\rho \sigma} $ using Schouten identity [Done] [closed]

How to prove \begin{align} F_{\mu\rho} \tilde{F}_{\nu}^{\phantom{\nu}\rho} = \frac{1}{4} \eta_{\mu\nu} F_{\rho\sigma} \tilde{F}^{\rho \sigma} \end{align} using Schouten identity \begin{align} 0 = ...
2
votes
2answers
222 views

Tensors and change of basis

When we say that a tensor is an array of numbers that transform according to some formula from one basis to another, can both bases be of the same coordinate system?
3
votes
3answers
116 views

Rotation in the x-t plane

I am currently studying special relativity using tensors. My lecture notes (which happen to be publicly accessible, see top of page 99) say that the standard configuration can be viewed as a rotation ...
2
votes
2answers
257 views

Tensor product of operators in QM

If I wanted to find the coefficients of a linear transformation between 2 vectors in the basis for 2 spin $1/2$ paticles (let's say for starters we are not even looking for a unitary transform): ...
2
votes
2answers
91 views

What does a left-right arrow in a tensor formula mean?

I need help with some some notation I've not seen before. Is using the left-right arrow in this formula $$[P^μ,M^{ρσ}]=i\hbar(g^{\mu\sigma}P^\rho-(\rho\leftrightarrow\sigma))$$ equivalent to writing ...
0
votes
0answers
49 views

Finding inverse tensor operator

For example I have such tensor operator: $$ O^{\mu \nu \alpha \beta} = (a^2+m^2)(\eta^{\mu\nu} \eta^{\alpha\beta} + \eta^{\mu\alpha} \eta^{\nu\beta}) + a^\mu a^\nu \eta^{\alpha\beta} + a^\mu ...
2
votes
2answers
182 views

Making sense out of covariance and contravariance

I just read about co- and contravariant vectors and I am not sure that I got it right: If we imagine that we have a n-dimensional manifold $M$ then a tangent space is spanned by the vectors ...
1
vote
1answer
87 views

Physical interpretation of the relative displacement tensor?

I've resolved a relative displacement tensor into a strain tensor and a rotation tensor, where the strain tensor is: $$ \varepsilon_{i,j} =\begin{pmatrix} 0.2 & 0 & 0 \\ 0 & 0.8 ...
1
vote
3answers
211 views

Levi-Civita symbol and Hermitian conjugate

When we take the Hermitian conjugate/dagger of an operator expression which contains a Levi-Civita symbol, do we need to transpose the Levi-Civita symbol? E.g., for the crossproduct ...
0
votes
1answer
76 views

General Relativity - Four Velocity Derivative Question

I am trying to get my head around a small point used in a book I am reading about General Relativity. The book states that because $u_au^a = c^2$ it follows that $u_a \nabla_b u^a = 0 $ The first ...
1
vote
1answer
168 views

Spinor notation in general relativity

I have a somewhat broad/big question, and I know that there are many references for it available out there. However, so far I couldn't find anything that I can really understand, that's why here is my ...
0
votes
1answer
102 views

what is intertia tensor for tapered cylinder (solid and with separate inside and outside radii)

I need the inertia tensor for tapered cylinders, both solid and hollow, and if possible with independent inner and outer radii on the x and y axes (so the cross-sections of the cylinders can be an ...
0
votes
0answers
538 views

metric determinant and its partial and covariant derivative

question : $\nabla_a \nabla_b \sqrt{g} \phi =\partial_a \sqrt{g} \partial_b \phi$ is true ? because $\nabla_a \sqrt{g}=0$ so we can write $\sqrt{g} \nabla_a \nabla_b \phi$ , but because metric ...
1
vote
1answer
107 views

Varying wrt metric [closed]

I saw people write $\frac{\partial( F^{ab} F_{ab})}{\partial g^{ef}}$ as $\frac {\partial (g^{ca}g^{db}F_{cd}F_{ab})}{\partial g^{ef}}$ in a way that exposes the dependence on the metric. but ...
4
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0answers
67 views

Transformation law for field strength tensor [closed]

How do I derive the transformation law for the field strength tensor$$F_{\mu\nu}^A = \partial_\mu V_\nu^A - \partial_\nu V_\mu^A - gC_{BC}^A V_\mu^B V_\nu^C$$to show that it transforms like a vector ...
5
votes
4answers
143 views

Physical intuition on $\mathbf{v}\otimes \mathbf{w}$

On Physics there's one very clear intuition on what a vector $\mathbf{v}$ is: they represent things with direction and magnitude (although when no metric is available there's no clear concept of ...
8
votes
1answer
146 views

What does Ricci tensor do with two vectors?

I have found it easier to understand the meaning of a particular tensor if I can find out what does it do if I cancel all its lower indices with vectors in short: $g_{ij} u^i v^j$: dot product of ...
0
votes
2answers
206 views

What does it mean to “contract” a tensor identity?

I'm taking a GR course at the moment, completely stumped on this step here: starting from the Bianchi identity: Then it says "Contracting the Bianchi identity..." How does this work and what ...
2
votes
1answer
486 views

Prove Christoffel Symbol Identity

In a book I am reading, the following identity is claimed and then "left to the reader to prove." $g_{ij}$ is the metric tensor, and $\Gamma$ is the Christoffel symbol of the second kind with the ...
0
votes
1answer
57 views

How do I construct the Maxwell tensor $\bf{^*F}$ from Fadaray one $\bf{F}$ in a non-flat spacetime?

In the book Gravitation (Misner, Throne and Wheeler), it's said that to consider the line element of the flat space on the derivation of Maxwell tensor $\bf{^*F}$ from the Fadaray tensor $\bf{F}$ ...
2
votes
0answers
135 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
3
votes
1answer
175 views

In continuum mechanics, why is the stress vector $T=\sigma\cdot n$ not a covector?

In continuum mechanics, the stress vector (see Cauchy stress tensor) $T=\sigma\cdot n$ is the surface density of a force. Forces are covectors, since they map a displacement vector to a scalar energy. ...
3
votes
1answer
237 views

Tensor components change under rotation-translation

I am currently working on a research project in a non-physics field, where I would like to work on a very constrained 2nd order tensor (3x3, symmetric, traceless). The tensor represents probability of ...
1
vote
0answers
40 views

SR: vector field and change of reference [closed]

If $U$ and $V$ are vector fields, then the derivative of $U$ along $V$ is the vector field $\nabla _V U$ with components $$\nabla _V U^a=V^b \frac{\partial U^a}{\partial x^b}.$$ I would like to verify ...
2
votes
1answer
348 views

Does the velocity vector always point in the same direction as the momentum vector?

I was told that the angular velocity vector does not always have to point in the same direction as the angular momentum vector. This is due to the fact that they are related by the equation $L=I ...
1
vote
1answer
67 views

Photon propagator inverse

If i have the operator $D^{\mu\nu}=\partial^{\mu}\partial^{\nu}+m\epsilon^{\mu\alpha\nu}\partial_{\alpha}$. What's your inverse $(D^{\mu\nu})^{-1}$?
1
vote
2answers
121 views

Index gymnastics and representing bra-kets as covariant and contravariant tensors

I am trying to figure out how to write, in Einstein notation as well as pick out elements of $$\langle A|[\mu]|B\rangle \langle X|[\nu]|Y\rangle$$ where $[\mu] = \begin{bmatrix} \mu_{11} & ...
0
votes
2answers
205 views

Off-diagonal terms in metric for 4D space-time [closed]

Consider a delta between two events in 4D space-time written as a 4-vector, $x^\mu=(dt, dR)$. The time $dt$ is a scalar difference in time. The 3-vector $dR$ points some direction in space. One ...
1
vote
2answers
86 views

How do I represent $A$ transpose $A$ in indicial notation?

I know this question sounds lame, but the book I am following doesn't use the answer I expect and it has been using a similar notations everywhere else which has confused me. I think Q[Any tensor] ...
0
votes
0answers
51 views

Eigenstates of operators on constituent systems in tensor product space

Suppose I have two entangled physical systems $\mathcal{A}$ and $\mathcal{B}$ with respective hilbert spaces $\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{B}}$. If $A,B$ are operators on ...