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2
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1answer
223 views

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 ...
3
votes
1answer
75 views

On the Lorentz Group representation [on hold]

I am going through the notes on QFT by Srednicki (which is certainly a worth reading on the subject, and can be found online, see http://web.physics.ucsb.edu/~mark/qft.html). When describing ...
0
votes
0answers
37 views

Is there a physical interpretation of the alternating property?

A map from a vector-space to its base field is called "alternating" if each vector with repeated elements is mapped to zero. I've read that symplectic geometry is an important representation of ...
2
votes
2answers
62 views

Tensors and change of basis

When we say that a tensor is an array of numbers that transform according to some formula from one basis to another, can both bases be of the same coordinate system?
0
votes
1answer
55 views

Matrices as second order tensors proof?

I am trying to proof that all matrices are tensors. I have got to a stage where I need to proof that: $$\gamma_{li} \gamma_{kj}= \frac{\partial q_j}{\partial q_k'} \frac{\partial q'_l}{\partial ...
4
votes
2answers
993 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? ...
2
votes
1answer
117 views

Perfect fluid and Cauchy momentum equation

The stress-energy tensor of a perfect fluid is given by $$T^{\mu\nu}=\left(\rho+pc^{-2}\right)u^\mu u^\nu+pg^{\mu\nu}$$ The divergence of the stress-energy tensor is zero: $\nabla_\mu T^{\mu\nu}=0$. ...
4
votes
1answer
178 views

Why are densities not fields?

I have read (in Statistical mechanics of lattice system 2: exact, series and renormalization group methods by D.A. Lavis and G.M. Bell pg 2 ), that intrinsic variables are either fields or densities. ...
0
votes
0answers
23 views

How is the electromagnetic tensor expanded?

The electromagnetic tensor is given by $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$, and it appears in the Lagrangian as $L = -\frac{1}{4}F_{\mu\nu}^2 - A_{\mu}J_{\mu}$. The text I'm ...
4
votes
1answer
354 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 ...
2
votes
0answers
48 views

Non-local gravitational energy tensor

The well-known derivation of the Landau-Lifshitz gravitational energy pseudotensor, relies on several requirements: 1) that it be constructed entirely from the metric tensor 2) that it be index ...
0
votes
1answer
65 views

How do you take the derivative with respect to a rank two tensor?

I am learning classical field theory and am trying to find the momentum density of the electromagnetic lagrangian as part of an example of Noether's Theorem. The derivative I am encountering is: $$ ...
3
votes
1answer
210 views

How to write a generic density matrix for multi qubit system?

I was reading the paper device independent outlook on quantum mechanics. Here the author defines a generic two qubit density matrix as $$\rho=\frac{1}{4}(\;I\otimes ...
1
vote
2answers
70 views

Clarification on meaning of scalar in math and scalar in physics

When a mathematician says something is a scalar, say on the plane, they mean that it associates to points on the plane real numbers. When a physicist says something is a scalar, they mean that if we ...
10
votes
5answers
482 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 ...
0
votes
3answers
116 views

Tricks for evaluating tensor contractions with Levi-Civita symbol

I am trying to evaluate the Lorentz invariant $\epsilon^{\alpha\beta\gamma\delta}F_{\alpha\beta}F_{\gamma\delta}$, where $F_{\mu\nu}$ is the electromagnetic field tensor, $$ F_{\mu\nu} = ...
7
votes
1answer
234 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 ...
2
votes
1answer
59 views

Differentiating between Tensor Networks

I am trying to study tensor networks and their application to quantum phase transitions. However, I had a question concerning the connection between the projected entangled-pair states (PEPS) and the ...
6
votes
1answer
715 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
46 views

Using tensors on Lagrangian and Hamiltonian

We can write the Lagrangian (with $n$ generalized coordinates) using the following expression: ...
3
votes
1answer
137 views

How can I make two separate equations for Christoffel symbols give the same answer?

I have been studying the covariant derivative and I'm confused by the calculation of the Christoffel symbols $\Gamma$. The equation for computing $\Gamma$ is given as: $${\Gamma^c}_{ab} = \frac12 ...
2
votes
0answers
41 views

Traceless Tensors in $SU(3)$, Georgi's Lie Algebras

I'm doing a self-study through Georgi's Lie Algebra's in Particle Physics and there is a ''note without proof'' in the book that I have not managed to see through myself. In Section 10.3, Georgi ...
2
votes
2answers
170 views

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): ...
0
votes
1answer
40 views

Notation for $N$-particle wave functions

If we have one particle we first look at an orthonormal basis of the one-particle Hilbert space $|n\rangle$. Here $n$ is the abbreviation for a compete set of quantum numbers, for example $n = ...
0
votes
1answer
83 views

Problem understanding Lorentz invariance [duplicate]

So they usually started with "...This is obviously Lorentz invariant, because of the 4-vector character of the quantity,..., (and after a two page long derivation) another quantity is also obviously ...
1
vote
1answer
60 views

Representations of Lorentz algebra

It is well known that the Lorentz algebra can be written as two $SU(2)$ algebras. By defining $$N_i=\frac{1}{2}(J_i+iK_i), \qquad N^{\dagger}_i=\frac{1}{2}(J_i-iK_i)$$ we have ...
2
votes
3answers
232 views

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 ...
1
vote
0answers
34 views

Decomposition of a tensor under transformations

To illustrate my question I'll take an example from theory of relativity: An arbitrary 4-tensor $A^{ik}$ changes under a general coordinate transformation: $$ A'^{ik} = C^{i}_mC^{k}_n A^{mn} $$ ...
0
votes
1answer
38 views

The significance of the pressure term within the momentum-energy tensor [duplicate]

EDIT: this question is based around my notion regarding the possible role of potential energy in the momentum energy tensor T$_{\mu\nu}$, The answer below resolves the question and I have deleted ...
3
votes
2answers
353 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 ...
0
votes
1answer
52 views

Partial Measurement and the Math Behind it

$\newcommand{\ket}[1]{\left| #1 \right>}$ $\newcommand{bra}[1]{\left< #1 \right|}$ Talking about the partial measurement the professor defines the state $\ket \psi$ to be $$\ket{\psi} = ...
1
vote
1answer
67 views

How to act an operator on a two-particle spin state?

I'm doing an assignment for my quantum class at the moment and I'm having trouble figuring out how to act a Spin operator on a two-particle state - specifically in finding the eigenvalues - I've spent ...
1
vote
1answer
53 views

electrical conductivity and resistivity tensor

By definition of the conductivity tensor $\hat{\sigma}$ and the resistivity tensor $\hat{\rho}$, we have \begin{equation*} \begin{split} & j_{\alpha}=\sigma_{\alpha \beta}E_{\beta} \\ & ...
3
votes
2answers
214 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 ...
2
votes
2answers
88 views

Making sense out of covariance and contravariance

I just read about co- and contravariant vectors and I am not sure that I got it right: If we imagine that we have a n-dimensional manifold $M$ then a tangent space is spanned by the vectors ...
2
votes
1answer
69 views

Density Matrix Renormalization Group (DMRG) Simulation of a String-Net Model

In the following paper, Dr. Xiao Gang-Wen et. al. introduce the idea that string-net condensed states can be represented in terms of tensor product states: http://arxiv.org/pdf/0809.2821.pdf The ...
3
votes
3answers
96 views

Rotation in the x-t plane

I am currently studying special relativity using tensors. My lecture notes (which happen to be publicly accessible, see top of page 99) say that the standard configuration can be viewed as a rotation ...
1
vote
2answers
143 views

Why does the second Weyl scalar describe electromagnetic radiation?

I've been reading about the null tetrad, the Weyl tensor, and the Newman-Penrose identities, and so I found out about the Weyl scalars. While the zeroth, first, third, and fourth scalars describe ...
2
votes
2answers
48 views

What does a left-right arrow in a tensor formula mean?

I need help with some some notation I've not seen before. Is using the left-right arrow in this formula $$[P^μ,M^{ρσ}]=i\hbar(g^{\mu\sigma}P^\rho-(\rho\leftrightarrow\sigma))$$ equivalent to writing ...
0
votes
0answers
22 views

Physical interpretation of the relative displacement tensor?

I've resolved a relative displacement tensor into a strain tensor and a rotation tensor, where the strain tensor is: $$ \varepsilon_{i,j} =\begin{pmatrix} 0.2 & 0 & 0 \\ 0 & 0.8 ...
0
votes
3answers
91 views

Levi-Civita symbol and Hermitian conjugate

When we take the Hermitian conjugate/dagger of an operator expression which contains a Levi-Civita symbol, do we need to transpose the Levi-Civita symbol? E.g., for the crossproduct ...
0
votes
1answer
32 views

what is intertia tensor for tapered cylinder (solid and with separate inside and outside radii)

I need the inertia tensor for tapered cylinders, both solid and hollow, and if possible with independent inner and outer radii on the x and y axes (so the cross-sections of the cylinders can be an ...
3
votes
0answers
147 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
41 views

Transformation law for field strength tensor [closed]

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 ...
8
votes
1answer
140 views

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 ...
3
votes
1answer
124 views

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 ...
5
votes
4answers
123 views

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 ...
0
votes
1answer
55 views

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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7answers
1k 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. ...
2
votes
1answer
165 views

Does the velocity vector always point in the same direction as the momentum vector?

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 ...