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18
votes
5answers
680 views

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. ...
16
votes
2answers
354 views

In relativity, can/should every measurement be reduced to measuring a scalar?

Different authors seem to attach different levels of importance to keeping track of the exact tensor valences of various physical quantities. In the strict-Catholic-school-nun camp, we have Burke ...
15
votes
4answers
723 views

Why do Maxwell's equations contain each of a scalar, vector, pseudovector and pseudoscalar equation?

Maxwell's equations, in differential form, are $$\left\{\begin{align} \vec\nabla\cdot\vec{E}&=~\rho/\epsilon_0,\\ \vec\nabla\times\vec B~&=~\mu_0\vec J+\epsilon_0\mu_0\frac{\partial\vec ...
13
votes
1answer
1k views

Mathematically, what is color charge?

A similar question was asked here, but the answer didn't address the following, at least not in a way that I could understand. Electric charge is simple - it's just a real scalar quantity. Ignoring ...
11
votes
4answers
428 views

Is the concept of tensor rank useful in physics?

The term 'tensor rank' is sporadically used in the mathematical literature to denote the minimum number of simple terms (i.e. tensor products of vectors) needed to express the tensor. This is ...
11
votes
1answer
404 views

What are the local covariant tensors one can form from the metric?

Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...
9
votes
1answer
299 views

Contracting Indices

Does anyone know how to get from (1) to (2) in the system $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ ...
9
votes
1answer
287 views

Introduction to spinors in physics, and their relation to representations

First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a ...
8
votes
6answers
1k views

What is a tensor?

I have a pretty good knowledge of physics but couldn't understand what a tensor is. I just couldn't understand it, and the wiki page is very hard to understand as well. Can someone refer me to a good ...
8
votes
5answers
310 views

Motivation for tensor product in Physics

This question is about a mathematical object (the tensor product) but thinking about the motivation that comes from Physics. Algebraists motivate the tensor product like that: "given $k$ vector spaces ...
7
votes
3answers
266 views

Is “entanglement” unique to quantum systems?

My text shows (sections 0.2 and 0.3) that the joint "state space" of a system composed of two subsystems with $k$ and $l$ "bits of information", respectively, requires $kl$ bits to fully describe it. ...
7
votes
1answer
561 views

Diffeomorphisms, Isometries And General Relativity

Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while. Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
7
votes
4answers
2k views

What is the difference between a spinor and a vector or a tensor?

Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?
7
votes
1answer
132 views

Are there cases in which we should consider tensors as equivalence classes?

Usually in texts about Physics that uses tensors defines them as multilinear maps. So if $V$ is a vector space over the field $F$, a tensor is a multilinear mapping: $$T:V\times\cdots\times V\times ...
6
votes
3answers
170 views

From Manifold to Manifold?

Tensor equations are supposed to stay invariant in form wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor ...
6
votes
4answers
260 views

Is the covariance or contravariance of vectors/tensors something that can be “visualized”?

I'm taking an undergrad GR course, and our text (Lambourne) mentions covariant and contravariant vectors and tensors ad-nauseum, but never really gives a formal definition for what they are, and how ...
6
votes
3answers
1k views

What is the physical significance of the off-diagonal moment of inertia matrix elements?

The tensor of moment of inertia contains six off-diagonal matrix elements, which vanishes if we choose the principle axis of the rotating rigid body and the components of the angular momentum vector ...
6
votes
2answers
276 views

Do Dirac Gamma Matrices act like Tensors?

Do Dirac Gamma Matrices act like Tensors? Is it true that $$ \gamma_\mu = \eta_{\mu\nu}\gamma^\nu~? $$ Also what about $\sigma_{\mu\nu}$, where $\sigma_{\mu\nu}$ is defined to be: \begin{align*} ...
6
votes
2answers
180 views

Tensor decomposition under $\mathrm{SU(3)}$

In Georgi's book (page 143), he calculates the tensor components of $3\otimes 8$ under the $\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning ...
6
votes
0answers
69 views

What is the intepretation of the electromagnetic tensor?

Let $A$ be the four-potential, then we know that we can form the electromagnetic tensor as $F=dA$. This is usually done as a way to have a better writing of Maxwell's equations. So, to simplify the ...
6
votes
0answers
101 views

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 ...
5
votes
5answers
229 views

Tensors and rotations

All the tensors that I have studied so far have always appeared with some kind of rotation. For example, spherical tensors rotate as spherical harmonics, tensors in the context of special relativity ...
5
votes
1answer
224 views

Confused about indices of the Ricci tensor

In an intro to GR book the Ricci tensor is given as: $$R_{\mu\nu}=\partial_{\lambda}\Gamma_{\mu \nu}^{\lambda}-\Gamma_{\lambda \sigma}^{\lambda}\Gamma_{\mu \nu}^{\sigma}-[\partial_{\nu}\Gamma_{\mu ...
4
votes
1answer
197 views

Do partial derivatives commute on tensors?

For example; is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
4
votes
2answers
293 views

Understanding the difference between co- and contra-variant vectors

I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear. One ...
4
votes
1answer
217 views

When a variation of a tensor is not a tensor?

In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a ...
4
votes
2answers
463 views

What's the idea behind the Riemann curvature tensor?

The Riemann curvature tensor can be expressed using the Christoffel symbols like this: $R^m{}_{jkl} = \partial_k\Gamma^m{}_{lj} - \partial_l\Gamma^m{}_{kj} + \Gamma^m{}_{ki}\Gamma^i{}_{lj} ...
4
votes
2answers
299 views

Irreducible Representations of SO(n) tensors

My interest is purely in $\text{SO}(n)$ tensors and how one works out their irrep decomposition. For instance, for rank 2 tensors we simply split into an antisymmetric part, a traceless symmetric part ...
4
votes
1answer
270 views

Irreducible decomposition of higher order tensors

I am familiar with the notion of irreps. My question refers simply to tensor representations (not tensor products of representations) and how can we decompose them into irreducible parts? For example, ...
4
votes
0answers
89 views

Tensor fields and vector bundles [migrated]

Let $M$ be a differentiable manifold, $TM$ and $T^*M$ a tangent and cotangent bundle of $M$ and let $\Gamma (TM),\ \Gamma (T^*M)$ be spaces of smooth sections of $TM$ and $T^*M$. Let $T_s^r (M)$ ...
4
votes
0answers
82 views

Lie derivative of a scalar and PDE

I am reading about differential geometry, and in particular the Lie derivative and its relation to (relativistic) hydrodynamics. In particular, I was wondering if, given two scalar functions ...
4
votes
0answers
231 views

Tensor equations in General Relativity

In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent. Question: why is this true? I'm looking for a ...
3
votes
3answers
4k views

What does this quote about the four dimensional divergence of an antisymmetric tensor mean?

In the beginning, God said that the four dimensional divergence of an antisymmetric second rank tensor equals zero and there was light. Can someone explain what is the meaning of this quote by ...
3
votes
1answer
290 views

Can one raise indices on covariant derivative and products thereof?

Can the following be true? $g^{\sigma\rho}\nabla_{\rho}\nabla_{\mu} = \nabla^{\sigma}\nabla_{\mu}$ $g^{\sigma\rho}\nabla_{\nu}\nabla_{\sigma} = \nabla_{\nu}\nabla^{\rho}$ ...
3
votes
1answer
164 views

Sign crazyness on the stress energy tensor?

I would like to know on what depends the sign of the stress energy tensor in the following formula : $T_{\mu\nu}=\pm(\rho c^2+P)u_{\mu}u_{\nu} \pm P g_{\mu\nu}$ In my case the metric is equal to ...
3
votes
2answers
172 views

Element of area in 4-dimensional space-time

How would you proof that $$ \mathrm {Tr} (\mathbf{S\cdot \bar S })=0$$ where $\mathbf S$ is an element of area delimited for the 4-vectors $\mathbf u$ and $\mathbf v$ given by $$S^{\alpha \beta}\equiv ...
3
votes
2answers
108 views

Tensor contra- and covariance concept

Imagine we have $p$-contravariant and $q$-covariant tensor, such that $$ t~=~t^{i_1, \dots, i_p} _{j_1, \dots, j_q} e_{i_1}\otimes \dots \otimes e_{i_p} \otimes e^{j_1} \otimes \dots \otimes ...
3
votes
2answers
251 views

What are $\partial_t$ and $\partial^\mu$?

I'm reading the Wikipedia page for the Dirac equation: $\rho=\phi^*\phi\,$ ...... $J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$ with the conservation of probability ...
3
votes
2answers
185 views

The signature of the metric and the definition of the electromagnetic tensor

I've read the definition of the electromagnetic field tensor to be ...
3
votes
2answers
330 views

Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct? ...
3
votes
1answer
100 views

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 ...
3
votes
1answer
48 views

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: ...
3
votes
2answers
284 views

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 ...
3
votes
1answer
220 views

Symmetrical Spinors and Symmetrical Tensors

In Quantum Electrodynamics by Landau and Lifshiz there is the following: The correspondence between the spinor $\zeta^{\alpha \dot{\beta}}$ and the 4-vector is a particular case of a general ...
3
votes
1answer
161 views

Riemann tensor notation and Christoffel symbol notation

In paper by Barnich and Brandt Covariant theory of asymptotic symmetries, conservation laws and central charges they defined the Riemann tensor like this: $$R_{\rho\mu\nu}^{\quad \ \ ...
3
votes
1answer
145 views

Interpretation of rank 2 spinors

While inspecting the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group and defining a right-handed spinor with upper dotted index and a left-handed spinor with lower undotted index and ...
3
votes
0answers
112 views

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 ...
3
votes
0answers
59 views

Expectation of 2-form field $B_{MN}$ in string theory

In the context of string theory, in particular when we're dealing with a low energy effective action, if we have an effective action of the form: $$S_{eff} \sim S^{(0)} + \alpha S^{(1)} + (\alpha)^2 ...
2
votes
2answers
198 views

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?
2
votes
3answers
122 views

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 ...