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### Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
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### Transformation law for field strength tensor [on hold]

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 ...
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### 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 ...
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### 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 ...
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### How to convert $V \otimes W^*$ to a matrix space? [migrated]

Namely let's say we have chosen basises $e_1, e_2, ... e_k$ for $V$ and $j_1, j_2, ... j_n$ for $W$. Now, since we can always just convert them separately, and then add the matrixes, how we represent ...
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### 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 ...
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### 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 $\bf{F}$ ...
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### Hookes Law and Objective Stress Rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
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### Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
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### Tensor Operators

Motivation. I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. ...
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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 $L=I ... 1answer 38 views ### Photon propagator inverse If i have the operator$D^{\mu\nu}=\partial^{\mu}\partial^{\nu}+m\epsilon^{\mu\alpha\nu}\partial_{\alpha}$. What's your inverse$(D^{\mu\nu})^{-1}$? 0answers 53 views ### Need definition of symmetric and antisymmetric tensor representations of a Lie algebra [migrated] I couldn't find a definitive answer online. Suppose we have a representation of a Lie algebra$(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the$k$-th tensor product of ... 1answer 137 views ### Stress Force - Understanding Cauchy Stress Tensor I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ... 2answers 55 views ### How do I represent$A$transpose$A$in indicial notation? I know this question sounds lame, but the book I am following doesn't use the answer I expect and it has been using a similar notations everywhere else which has confused me. I think Q[Any tensor] ... 0answers 28 views ### Eigenstates of operators on constituent systems in tensor product space Suppose I have two entangled physical systems$\mathcal{A}$and$\mathcal{B}$with respective hilbert spaces$\mathcal{H}_{\mathcal{A}}$and$\mathcal{H}_{\mathcal{B}}$. If$A,B$are operators on ... 1answer 51 views ### Bianchi Identity using null tetrad I'm currently looking at the Newman-Penrose Formalism, and trying to understand where there sets of equations come from. For that, I need to know how I can write the second Bianchi identity for the ... 3answers 82 views ### Convention of tensor indices Let$g_{ij}$be the diagonal Minkowski metric tensor diag$(g) = (1,-1,-1,-1)$, then$g^{ij}$is defined to be$(g^{-1})^{ij}$, hence $$g_{ik}g^{kj} = g_i^{\ \ j} = \text{diag}(1,1,1,1)=\delta_i^{\ \ ... 4answers 572 views ### Why is \vec{S}^{(A)} \otimes \vec{S}^{(B)} = \frac{\hbar^2}{4}(\sigma_x \otimes \sigma_x + \sigma_y \otimes\sigma_y + \sigma_z \otimes \sigma_z)? I haven't been taught tensor product in class but they have taught us addition of spin. I looked up online in this link->http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf#page=10 ... 1answer 65 views ### Decomposition of group representation using tensor method I am dealing with the decomposition of the representation 5\otimes5 of SU(5):$$5\otimes5=15\oplus10 $$demonstration:$$u^iv^j=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}(u^iv^j-u^jv^i)=$$... 1answer 187 views ### What does |x⟩|0⟩ actually mean in bra-ket notation? Consider the following quote from Wikipedia's page on Shor's algorithm: Initialize the registers to Q^{-1/2} \sum_{x=0}^{Q-1} \left|x\right\rangle \left|0\right\rangle where x runs ... 0answers 74 views ### Covariant versus “ordinary” divergence theorem Let M be an oriented m-dimensional manifold with boundary. As stated in Harvey Reall's general relativity notes (here) or Sean Carroll's book, the "covariant" divergence theorem (i.e. with ... 1answer 35 views ### Product on Tensor Products I'm trying to understand how products on tensor products work. For instance, in quantum mechanics, you have (x tensor y) times (z tensor a), where x, y, z, a are all operators acting ... 1answer 79 views ### Why is the full eigenfunction a product of eigenfunctions and not a sum? For example suppose there is a two electron system. Why is the full eigenfunction a product of the spatial eigenfunction and spin-wave-function for the two electron system? 2answers 98 views ### Suggested operatonal definition for a tensor [duplicate] The two tensor definitions I'm (newly) familiar with, by transformation rules, and as a map from a tensor product space to the reals, don't tell me what a tensor does, and to the best of my knowledge ... 2answers 57 views ### Proving a relation with Four-velocity tensor [duplicate] I'm trying to show that: U^a_{\space\space;b}U^bU_a = 0 (Where U is four-velocity) and I'm stuck on how to go about it. I tried expanding it out into the Christoffel symbols, but that didn't seem ... 7answers 248 views ### How can a set of components fail to make up a vector? Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ... 3answers 81 views ### Tensor product in quantum mechanics In Cohen-Tannoudji's Quantum Mechanics book the tensor product of two two Hilbert spaces (\mathcal H = \mathcal H_1 \otimes \mathcal H_2) was introduced in (2.312) by saying that to every pair of ... 2answers 94 views ### Relation between Vector space V and its dual V^{*} [closed] I asked the same question in Math.SE, but I was suggested to ask it here as well. I am studying relativity, and as you know the theory extensively uses the notion of covariant and contravariant ... 0answers 21 views ### Decomposition into symmetric and antisymmetric form [closed] (a) Given a second-rank tensor Tμν, often viewed as an N \times N matrix (for a space of dimension N), show by explicit construction that one can always decompose T_{\mu\nu} into a symmetric ... 1answer 84 views ### SU(3) irreducible representations with tensor method I am dealing with the tensor product representation of SU(3) and I have some problems in understanding some decomposition. 1) Let's find the irreducible representation of 3\otimes\bar{3} we have ... 1answer 46 views ### What does a colon mean in hydrodynamics equations? In some hydrodynamics book I saw a notation like e:e where e is a matrix (shear stress tensor). This double dot product is in a scalar equation, so the result of this operation must be scalar. I ... 1answer 86 views ### Why tensor product? [duplicate] Let A an B be two discrete observables (like spins). When exactly and why we have to consider their tensor product when talking about the mutual observation of the corresponding phenomena? 1answer 44 views ### What is the dyad corresponding to a stress tensor? (As I understand it ... qualifies every sentence in what follows).. a stress tensor is a rank 2 tensor that maps a unit vector normal to a surface to the stress (or traction) vector corresponding to ... 1answer 95 views ### Inverse Metric Tensor First the setup: Let \mathcal M be a 2-dimensional manifold. Let U_P be some open neighbourhood of a point P \in \mathcal M. Let \mathcal F : U_P \rightarrow \mathbb R \times \mathbb R be ... 1answer 59 views ### Simple question about the electromagnetic tensor written as a 2-form I noticed that the 2 form (Electromagnetic tensor) is written as:$$F= F^{ab}e^a \wedge e^b$$while we know that$$F= F_{\mu\nu}dx^\mu \wedge dx^\nu$$Is there something wrong with the indices ... 2answers 130 views ### Is a vector field not a vector quantity? I'm trying to make sense of Poisson bracket relation$$\{L_i,A_k\}_{PB}~=~\epsilon_{ikl}A_l,\tag1$$where L_i is ith component of angular momentum, A_k is kth component of an arbitrary ... 2answers 38 views ### Is it inevitable to compute the quadruople tensor in components? Why? [closed] I was trying to determine the quadrupole tensor for a given charge distribution in one go from this equation:$$\overleftrightarrow{D}=\int d^3r \varrho(\vec{r})\left(3\vec{r} \circ ... 1answer 36 views ### Electromagnetic tensor notation How do you transform between the electromagnetic tensors$F_{\mu\nu}$and$F^{\mu\nu}$?$$F_{\mu \nu}= \begin{pmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & -B_z & B_y \\ -E_y ... 1answer 101 views ### Physical visualisation of curvature I was wondering-how do you visualise curvature in the context of general relativity. The gravity well and trampoline analogies are quite wrong, so I want a more realistic approach to it (say, the way ... 0answers 23 views ### Quasi-primary fields and usual fields How do i see that the way quasi-primary/primary fields transform contain the transformation rule for fields as we know it (scalar, vector fields) in QFT? 2answers 174 views ### Tensor product of two different Pauli matrices$\sigma_2\otimes\eta_1 $I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product$\sigma_2\otimes\eta_1$where$[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$... 1answer 56 views ### components of mixed tensor with same indices If my tensor$a^{\mu\nu}=$matrix of 4*4 size (let's say, in 1+3 dimensions with mostly negative convention for the metric), what is$a^{\mu}_{\mu}$? Is it the trace or the vector of diagonal ... 2answers 155 views ### Metric tensor in SRT I just read on this webpage that we have (click me)$g_{\alpha \beta} = g_{\alpha}^{\beta} = g^{\alpha \beta}.$Now, although I understand that the first and the last one are equal, I don't think ... 1answer 40 views ### Trouble getting the matrix representation of a 4-state Hamiltonian$\newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\ket}[1]{\left| #1 \right\rangle}$I have a simple 4-state Hamiltonian and am trying to find the matrix representation (in order to determine ... 1answer 81 views ### Naturalness of tensor fields in general relativity? In the third chapter of the book The Large Scale Structure of Space-Time, the authors say regarding the matter fields in general relativity: These fields will obey equations which can be expressed ... 1answer 95 views ### Group notation$\otimes$and$\oplus$used for representations of quarks and mesons I've been trying to figure out this statement from the PDG quark model summary (PDF). Following$\mathrm{SU}(3)$, the nine possible$q\bar{q}′$combinations containing the light$u$,$d$, and$s$... 2answers 210 views ### If$v_{a \dot{b}}$transforms like a four-vector, what does$v_{a}^{\dot{b}}$describe? The$( \frac{1}{2}, 0)$representation of the Lorentz group acts on left-chiral spinors$\chi_a$, the$( 0,\frac{1}{2} )$representation on right-chiral spinors$\chi^{\dot a}$. The$( \frac{1}{2}, ...
How are exactly $u_j\partial_ju_i$ and $u_i\partial_j u_i$ related? And what is their relation to ($\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\boldsymbol{u}\cdot(\nabla\boldsymbol{u})$ ? I ask ...