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14
votes
6answers
2k 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 ...
15
votes
1answer
2k 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 ...
8
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?
18
votes
6answers
988 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. ...
3
votes
2answers
117 views

“Vectors” (i.e. 1-tensors) their definition and motivation for relativity

I'm reading Einstein Gravity in a Nutshell (by Zee) and here he defines a vector as an object which is invariant under coordinate representation; concretely, if in one coordinate representation, $V$, ...
9
votes
5answers
394 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 ...
13
votes
1answer
463 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 ...
11
votes
2answers
540 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 ...
9
votes
1answer
325 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}+ ...
7
votes
3answers
712 views

Covariant and contravariant vectors

Reading Weinberg's Gravitation and Cosmology, I came across the sentence (p.115, above equation (4.11.8)) The partial derivative operator $\partial/\partial x^\mu$ is a covariant vector, or in ...
2
votes
2answers
77 views

Basic understanding of stress tensors in a fluid

So, after having spent the last 9 hours attempting to understand the basic tenets of stress tensors in fluids, I can honestly say that I think I know less now than when I began. My questions are ...
2
votes
2answers
404 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
0answers
125 views

Stiffness tensor

Let's have a stiffness tensor: $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for an anisotropic body. How does body symmetry (cubic, hexagonal ...
1
vote
0answers
123 views

Einstein +Maxwell 's tensor

Why is it true that we can deduce that Einstein's GR equations coupled with Maxwell's EM equations may be written in the form $$R_{ij}=C(F_{ik}F_j^{\,\,k}-{1\over 4}g_{ij}F_{mn}F^{mn})$$ without ...
0
votes
1answer
97 views

Square of a tensor

I think, $$\sigma_{ij}\sigma^{ij} = \sigma^2.$$ However, on the Wikipedia page on Raychaudhuri equation, It was mentioned: $$\sigma^2=\frac{1}{2}\sigma^{ij}\sigma_{ij}$$ I am confused, but I think ...
17
votes
4answers
1k 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 ...
16
votes
2answers
382 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
2answers
877 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 ...
6
votes
2answers
412 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
4answers
436 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 ...
4
votes
2answers
513 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
443 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, ...
0
votes
0answers
57 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 ...
0
votes
2answers
269 views

What is pseudo tensor?

What is the pseudo tensor in relativity? How do we transform tensor and pseudo tensor under parity?
4
votes
3answers
253 views

How to understand the definition of vector and tensor?

Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having ...
3
votes
2answers
635 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
3answers
6k 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 ...
2
votes
1answer
142 views

Spin tensor and Lorentz group operator in bispinor case

For infinisesimal bispinor transformations we have $$ \delta \Psi = \frac{1}{2}\omega^{\mu \nu}\eta_{\mu \nu}\Psi , \quad \delta \bar {\Psi} = -\frac{1}{2}\omega^{\mu \nu}\bar {\Psi}\eta_{\mu \nu}, ...
2
votes
0answers
417 views

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 ...
2
votes
2answers
804 views

Tensor product notation [closed]

In the image there is a tensor product: $$F_{\mu\nu}F^{\mu\nu}=2(B^2-\frac{E^2}{c^2})$$ It's about how this operation on the co- and contravariant field strength tensors can give one of the ...
1
vote
4answers
198 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 ...
1
vote
1answer
194 views

Finding the metric tensor from the Einstein field equation?

I have have set my self a challenge to learn all the maths behind the Einstein field equation (EFE), and from reading it seems that the Metric tensor is the thing we are trying to find (from the 10 ...
1
vote
1answer
286 views

Symmetry of stress-energy tensor

Why in the general case of classical field theory canonical stress-energy tensor doesn't have symmetry of the permutation of the indices? For explanation, let's have a "derivation" of an expression ...
1
vote
0answers
71 views

Equivalence of simple formulations of qubit entanglement

I'm reading some very elementary treatments of quantum computation and am unsure about the correspondence among "definitions" of qubit entanglement. One definition states that (1) the bits of a ...
1
vote
1answer
598 views

Levi-Civita symbol in Euclidean space

Suppose a component of tensor field is described by $B^k=\varepsilon^{kij} \phi_{ij}$. If we define $B^k$ in an Euclidean space then does the rising or lowering of the indices of the Levi-Civita ...
7
votes
1answer
141 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 ...
3
votes
1answer
113 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
2answers
371 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
207 views

How quantum field transforms in case of some particular spin

Except when a particle is spin-0, field of all particles transforms when frame of reference is changed, and this defines what spin is. The question is, specifically how does the quantum field ...
2
votes
1answer
608 views

Difference between $\partial$ and $\nabla$ in general relativity

I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones. In our lectures we just had $\partial_\mu$ which would have the plain partial ...
1
vote
1answer
313 views

Calculating electromagnetic invariant in matrix form

I'm kind of confused. I want to calculate the electromagnetic invariant $I := F^{\mu\nu}F_{\mu\nu} $, but I'm not sure what is the easiest way to do so. So, I was trying to do it in matrix form, i.e. ...
0
votes
0answers
55 views

Cubic symmetry and a stiffness tensor [duplicate]

Possible Duplicate: Stiffness tensor Let's have a stiffness tensor: $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for an anisotropic body. ...