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1
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2answers
43 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 $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
1
vote
1answer
33 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 ...
5
votes
2answers
129 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 ...
0
votes
1answer
30 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 ...
0
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0answers
53 views

How did Einstein know tensors would be needed in the EFEs? [migrated]

I'd really never studied tensors until I started studying the Einstein Field Equations. Since then, I have realized they are fairly common tool in physics and pretty basic to understanding many areas. ...
3
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1answer
72 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 ...
1
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2answers
61 views

Advection operator

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 ...
9
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2answers
195 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}, ...
8
votes
1answer
73 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 $s$ ...
1
vote
1answer
167 views

What is an “Einstein transformation” in general relativity?

When introducing the vielbein formalism in general relativity, I came across the use of an infinitesimal general transformation, or Einstein transformation. The latter term seems not to be covered on ...
4
votes
3answers
107 views

Is a metric tensor field the same thing as $ds² = -dt² + dx²+ dy² + dz²$?

I am having trouble understanding the nature of the metric tensor field on spacetime manifolds. In particular, a Riemannian manifold $(M,g)$ is defined as a real smooth manifold $M$ equipped with an ...
5
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5answers
440 views

Is there a physical interpretation of a tensor as a vector with additional qualities?

What is a tensor? has been asked before, with the most highly up-voted answer defining a tensor of rank $k$ as a vector of a tensor of rank $k-1$. But if a scalar is defined as a physical quantity ...
0
votes
2answers
57 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^{\ \ ...
3
votes
2answers
60 views

Sign of the totally anti-symmetric Levi-Civita tensor $\varepsilon^{\mu_1 \ldots}$ when raising indices

I am confused with the sign we get when we want to raise or lower all indices of the totally anti-symmetric tensor of any rank. Take the metric to be mostly plus ($-+\ldots+$). Then is it ...
0
votes
1answer
104 views

Einstein notation in Electrodynamics

I have a question about the notation used in Electrodynamics, for the Eq.6.5.10. Can anyone tell me why is the sign of the covariant derivative swapped when it's transformed to a contravariant ...
1
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0answers
36 views

What is the relationship between the formal definition of a tensor and the frequently discussed notion of a “higher order matrix”?

I've been doing some self study on the principles of tensors & manifolds in preparation for a first course in general relativity. I tend to learn better when presented with the full mathematical ...
2
votes
2answers
199 views

Tensor Product vs. Direct Product for three spin-1/2 particles

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 $V ∈ \text{SU(2)}$ ...
1
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1answer
72 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 ...
4
votes
1answer
40 views

How can we see that the Riemann curvature tensor is covariant?

The Riemann curvature tensor, using the conventions of wikipedia, is written in terms of Christoffel symbols as: $$ \tag{1} R^\lambda_{\,\,\mu \nu \rho} = \partial_\nu \Gamma^\lambda_{\,\,\rho \mu} - ...
0
votes
1answer
171 views

Prove $G^{+\rho(\mu}H^{+\nu)}{}_{\rho} = -\frac{1}{4}\eta^{\mu \nu}G^{+\rho \sigma}H^+_{\rho \sigma}$ [closed]

I want to prove the following fact for two antisymmetric tensors: $$ G^{+\rho (\mu} H^{+\nu)}{}_{ \rho} = -\frac{1}{4}\eta^{\mu \nu} G^{+\rho \sigma}H_{\rho \sigma}^{+}. \tag{4.39}$$ (See e.g. ...
3
votes
1answer
84 views

Confusion about the Kronecker delta symbol

I am not sure I understand what the short-hand anti-symmetrization means. I.e. I know that $$\delta_{cd}^{[ab]} ~=~ \frac{1}{2}(\delta_{c}^{a}\delta_{d}^{b} - \delta_{c}^{b}\delta_{d}^{a})$$ but how ...
15
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2answers
946 views

Why is stress a tensor quantity?

Why is stress a tensor quantity? Why is pressure not a tensor? According to what I know pressure is an internal force whereas stress is external so how are both quantities not tensors? I am ...
0
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2answers
44 views

Finding $M_{ij}$ from $J_{i} = -\frac{1}{2}\epsilon_{ijk}M_{jk}$ of the Lorentz group

I cannot understand why if $$J_{i} = -\frac{1}{2}\epsilon_{ijk}M_{jk}$$ then $$M_{ij}= -\epsilon_{ijk}J_{k}.$$ Here $M_{ij}$ is the generator of the 4 dimensional Lorentz algebra (although I have ...
0
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0answers
54 views

Coordinate Symbol confusion in general relativity

In a previous post (Finding the metric tensor from the Einstein field equation?), the equation used lambda, rho mu and nu (not sure of the names of the letters!) for the Ricci tensor and swapped to a, ...
1
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2answers
105 views

Is four-current a vector or a vector density?

According to MTW, $$F^{\alpha\beta}{}_{;\beta} = 4\pi J^\alpha$$ and we can infer that the four-current must be an ordinary vector field because the left side is tensorial. But Wikipedia says that ...
2
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3answers
72 views

What is “a vector of $SO(n)$”?

I'm watching (or trying to watch) this lecture from NPTEL on classical field theory. I've understood everything in the series up till this point, including the first half of the lecture on elementary ...
2
votes
1answer
72 views

Why do derivatives act on vector fields on a worldsheet?

The covariant derivative of a vector $A^{\mu}$ at a point $x$ is defined as $$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma}$$ where Greek symbols are ...
0
votes
0answers
62 views

Covariant Derivative Chain rule?

I want to prove that a covariant derivative of a vector $A^{\mu}(x(z))$ at the point $x(z)$ in general would be defined as $$D_z ...
2
votes
0answers
37 views

Magnetic Multipole Tensor

When the electric scalar potential is expanded into spherical coordinates, one gets \begin{align} \phi (\vec r) = \frac{1}{4\pi\varepsilon_0} \sum_{l=0}^{\infty} \sum_{m=-l}^l ...
0
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0answers
32 views

Needed small explanation of the notation in this paper [duplicate]

There is this paper: http://digital.csic.es/bitstream/10261/43192/1/p6374_1.pdf In equation (36) of this, the second line ($\omega^{+ij} =\dots$), there is a term ...
2
votes
2answers
112 views

Derivation of the Riemann tensor confusion

I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ...
2
votes
1answer
39 views

Ambiguity in ordering of isospin states for Clebsch-Gordan coefficients

In studying isospin for nuclear physics, I am confused a bit by an ambiguity I found. If a process that goes from $K^- + p \rightarrow \Sigma^0+ \pi^0$, I can write the isospin for the left hand side ...
2
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0answers
70 views

How to get result $3 \otimes 3 = 6 \oplus \bar{3}$ for $SU(3)$ irreducible representations?

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that $$ 3\otimes 3 =6 \oplus \bar{3}~? $$ I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...
1
vote
1answer
82 views

Traceless multipole moments vs non-traceless moments

There are two different possibilities to define the electric quadrupole tensor: On the one hand, one can define \begin{align}Q_{kl} = \int \rho(\mathbf r') \cdot r'_k \, r'_l d^3r',\end{align} while ...
2
votes
4answers
136 views

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: ...
2
votes
0answers
37 views

Definition of Irreducible Tensor Parts in an Exercise

I am addressing exercise 23.9 on http://www.pma.caltech.edu/Courses/ph136/yr2011/1023.1.K.pdf. The exercise says that a fluid flowing through spacetime $\vec u(\mathcal P)$ can have its gradient ...
1
vote
1answer
92 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 ...
3
votes
2answers
146 views

The definition of transpose of Lorentz transformation (as a mixed tensor)

In the appendix of the textbook of Group Theory in Physics by Wu-Ki Tung, the transpose of a matrix is defined as the following, Eq.(I.3-1) $${{A^T}_i}^j~=~{A^j}_i.$$ This is extremely confusing for ...
11
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4answers
403 views

Is partial derivative a vector or dual vector?

The textbook(Introduction to the Classical Theory of Particles and Fields, by Boris Kosyakov) defines a hypersurface by $$F(x)~=~c,$$ where $F\in C^\infty[\mathbb M_4,\mathbb R]$. Differentiating ...
1
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0answers
57 views

Learning how to use Levi-Civita symbol

I've recently started my second course in Quantum Theory and am now often required to prove more complex commutation relations. I'm aware that the Levi-Civita symbol often makes this sort of thing a ...
2
votes
2answers
102 views

Physical interpretation of order of tensor indices

Using positional index notation with tensors is common. For example, the following simple equation from Carroll's Spacetime and Geometry text (eq. 3.146): $$ R = R^\mu_{\,\,\mu} = ...
1
vote
1answer
105 views

Proof that terms in decomposition of a tensor are symmetric and antisymmetric

Any tensor of rank 2 can be rewritten as: $$A_{bc} = \frac{1}{2}(A_{bc} + A_{cb}) + \frac{1}{2}(A_{bc}-A_{cb})$$ I can understand how that works. My question is: Prove that (independently): ...
3
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1answer
122 views

Vector fields and tensors in E&M

I'm confused by a very basic property of electric fields. The electric field is a vector field. Vectors are tensors. Wikipedia has the following statement in the article about the electromagnetic ...
4
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2answers
72 views

Tensors of rotations about an arbitrary vector in C^2

I'm trying to solve the following equation: $$e^{-i\theta/2 \sigma_{\vec{i}}^A} \otimes e^{-i\theta/2 \sigma_{\vec{i}}^B} |\Psi\rangle_{AB} = e^{i\phi} |\Psi\rangle_{AB} $$ where $e^{i\phi}$ should ...
13
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3answers
521 views

How to tell that the electromagnetic field tensor transforms as a tensor?

Is any matrix a tensor in special relativity? My question is inspired by the definition of the electromagnetic field tensor in Carroll's Spacetime and Geometry book. In equation (1.69), he defines a ...
3
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1answer
76 views

Tensor notation

I'm trying to understand the Maxwell Stress tensor notation. I'm given that each element in the tensor is given by ...
2
votes
1answer
58 views

Why eigenvector points to principal stress plane?

I can represent a tensor by a matrix. Suppose we are talking about a 2nd order tensor, and the matrix is therefore 3x3. If I find one eigenvector of that matrix; that vector represents normal vector ...
2
votes
1answer
93 views

Riemann Curvature Tensor Symmetries Proof

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies ...
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0answers
47 views

Relation involving the Lorentz transformation and the inverse of its transpose

The relation I was referring to in the title is $${\Lambda_a}^b= \eta_{ac} {L^c}_d \eta^{db}$$ where ${\Lambda_a}^b$ is the inverse transpose of $L$, the Lorentz transformation. I was wondering ...
2
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4answers
206 views

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 ...