# Tagged Questions

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### Tensor indices and row and column labels of corresponding representation matrices

When reading undergraduate GR literature, I often see that the authors represent tensors ${\eta^\alpha}_{\beta}$, ${\eta^\beta}_{\alpha}$, $\eta_{\alpha \beta}$, $\eta^{\alpha \beta}$ as matrices. ...
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### What is the difference between scalar and vector mesons?

My understanding is that vectors and pseudooscalars change sign under parity operation and pseudovectors and scalars do not. However, I don't understand what the difference between a vector and ...
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### 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 ...
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### 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 ...
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### The signature of the metric and the definition of the electromagnetic tensor

I've read the definition of the electromagnetic field tensor to be ...
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### What do physicists mean by ${g^{i}}_j$?

Maybe this is an idiot question, but in relativity I see a lot of ${g^{i}}_j$ for a metric tensor $g$. Is this just $$\delta^i_j ~=~ g(dx^i \sharp, \partial_{ x^j})~?$$
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### Type/Valence of the stress tensor

In classical continuum mechanics, the stress tensor is said to be of type/valence (1,1) and I do not see why. If I am correct, its maps a vector $n$ defined in $\mathbb{R}^3$ (which is the normal to ...
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### Tensor Product of Hilbert spaces

This question is regarding a definition of Tensor product of Hilbert spaces that I found in Wald's book on QFT in curved space time. Let's first get some notation straight. Let $(V,+,*)$ denote a set ...
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### Tensor Introduction

I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, ...
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### What is the mathematical formulation for buckling?

Argument: Buckling is an engineering concept that can only be applied to thin columns with compressive loading. (Is it possible to) Prove the above sentence right or wrong with mathematical ...
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### Fermi-Propagated Jacobi equation in the book The Large scale structure of space-time

On page 81, equation (4.6), the author use the Fermi derivative to write the Jacobi equation \tag{4.6} \frac{{D^2}_\text{F}}{\partial s^2} {}_{\bot}Z^a = ...
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### How should Christoffel symbols be written (in LaTeX)? [closed]

I'm writing a summary of a lecture on relativity, and we've recently introduced the Christoffel symbols. It seems that the upstairs indices are the "leftmost" and the downstairs indices are somewhat ...
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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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### 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 ...
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### How to define tensor contraction without referring to summation?

The textbook defines a tensor to be an element in $(T^*)^k×T^l→R$. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation ...
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### Stress tensor in product of 2D CFTs

I was struggling with a question, hoping someone could point me in the right direction. I'm interested in 2D CFTs on a cylinder. I want to take the tensor product of two CFTs. My questions are these: ...
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### Proper time along path in Minkowski Space

Consider the path $x^\mu(u)$ in Minkowski space; such that: $$t = \frac{a}{c} \sinh(u) , \quad x = a \cosh(u) ,\quad y = 0 ,\quad z = 0$$ where $a$ is a positive constant and $u$ is a parameter ...
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### Are the cylindrical and spherical form of Jeans' equations equivalent?

The question kind of says it all, what I really want to know is are the differences in their forms only due to the co-ordinate transform? And as such should a suitable spherical system satisfy ...
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### Killing Equation, trouble with tensor algebra

I'm attempting to follow a proof that the commutator of two Killing vectors is itself a Killing vector. The source that I've posted is from my course notes. I've highlighted the part I'm stuck on. ...
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### Epsilon Tensor in FeynCalc

A few days ago I started to use the Mathematica package FeynCalc and one thing confuses me: Assume we have a four-vector $p_\mu$ and we contract it with the epsilon tensor. FeynCalc produces ...
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### The connection between classical and quantum spins

I have two questions, which are connected with each other. The first question. In a classical relativistic (SRT) case for one particle can be defined (in a reason of "antisymmetric" nature of ...
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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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### I lost a factor of two in the electromagnetic field tensor

I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-) ...
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### Nature of Fields in QFT

I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere! There are specific types of fields used in physics: scalar fields (i.e. as ...
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### How does the Lorentz transformation $\Lambda^{\mu}{}_{\nu}$ transform?

For example the Four-velocity transforms as $$U^{a'}=\Lambda^{a'}{}_{\nu}U^{\nu},$$ the Faradaytensor as $$F^{a'b'}=\Lambda_{\,\,\mu}^{a'}\Lambda_{\,\,\nu}^{b'}F^{\mu\nu}$$ or in Matrixnotation: ...
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### 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 ...
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### Notation for anti-symmetric part of a tensor

I know that $A_{[a} B_{b]} = \frac{1}{2!}(A_{a}B_{b} - A_{b}B_{a})$ But how can write $E_{[a} F_{bc]}$ like the above? Can you provide a reference where this notational matter is discussed?
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### 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): ...
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### Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
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### What does “transform among themselves” mean?

I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context: An arbitrary rotation of the ...
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### 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 ...
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### Dimension of vector resulting from tensorial product

I'm quoting what I found in a book about quantum computation: Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vectors, then ...
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### Why is the Ricci tensor defined as $R^\mu _{\nu \mu \sigma}$?

The Ricci tensor is defined as the contraction of the Riemann tensor in its upper and the second lower index. I was wondering why it is defined this way. What happens if the Ricci tensor is defined ...
Let us consider three spin-1/2 particles and only focusing on their intrinsic spin $S$. The Hilbert space has then to be $\mathcal H = ℂ^2 ⊗ ℂ^2 ⊗ ℂ^2$. The spin can be described by $V ∈ \text{SU(2)}$ ...
In differential geometry, the differential forms are anti-symmetric tensors. So, why is the anti-symmetric tensor like $d x_1 \otimes dx_2 - d x_2 \otimes d x_1$, more important than the ...