Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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Electric octupole moment in cartesian coordinates

I'm trying to calculate the symmetric traceless tensor for the octupole moment in cartesian coordinates... I have to deal with the electrostatic potential of the form: $\Phi^{(4)}(\textbf{r})=\int d^{...
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59 views

What is the Difference between the tensors, $T^a\ _{bc}, T_b\ ^a\ _c, T_{bc} \ ^a$?

The title completely expresses the question. I have come across for several times the mixed tensors where the lower index is written closer to the tensor symbol and the upper index is written a bit ...
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52 views

Trace of a Tensor

What is the significance of defining the trace of a tensor as $g^{\alpha\beta} R_{\alpha\beta}$ instead of $R_{\alpha\alpha}$ on a Riemannian manifold?
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Does the entanglement depend on the basis?

Let's say, we have a composite system $A\otimes B$. We take the basis for $A$ as $|i\rangle,|j\rangle...,$ the basis for $B$ as $|\alpha\rangle,|\beta\rangle....$ Then an entangled state is a state ...
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Maxwell's equations in covariant form

Maxwell's equations of electrodynamics in vector calculus form are \begin{align} \nabla \times \mathbf{B} - \partial_t \mathbf{E} & = \mathbf{J} \\ \nabla \cdot \mathbf{E} & = \rho \\ \nabla ...
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Lorentz Transformations in Minkowski space

If $\Lambda$ represents the Lorentz transformation matrix, then the transformation of contravariant components $x^\mu$ is given by $$x'^\mu=\Lambda^{\mu}{}_{\nu} x^\nu$$ and that of the covariant ...
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Are Fock spaces just a special type of tensor algebra?

Are Fock spaces just a special type of tensor algebra? The definitions I am using: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Tensor_algebra From what I can tell, the ...
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Projection of a tensor

Consider the following tensor (abstract index notation, e.g. Wald's) $B_{ab}$ and timelike vector field $X^{a}$ such that $X^aX_a=-1$ and \begin{equation} B_{ab}=\nabla_bX_a \end{equation} Then one ...
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Does $\partial_\mu =\frac{\partial }{\partial x^\mu}$ or $\partial_\mu =\frac{\partial }{\partial x_\mu}$? [migrated]

I am looking at the chain rule with covariant and contravariant vectors. I understand why we have: $$df=\frac{\partial f}{\partial x^\mu} dx^\mu$$ (Please correct me if I am wrong) since even though ...
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Relation between differentiation of one-form basis and Christoffel Symbols

If I want to covariantly differentiate a one form then I can write: $\nabla_\beta \tilde p = \dfrac{\partial p_\alpha}{\partial x^\beta} \tilde \omega^\alpha + p_\alpha \dfrac{\partial \tilde \omega^...
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172 views

Derivation of Christoffel Symbols

So I am reading a book on relativity & differential geometry and in the text, they gave the Christoffel symbols in terms of the metric and its derivatives, but I wanted to derive it myself. ...
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Defining electromagnetic stress tensor for non-linear media

In textbooks, the electromagnetic stress tensor (in vacuum also called Maxwell stress tensor) is usually derived for linear media, implying that $$ \vec D = \epsilon_0 \epsilon_r \vec E$$ My question ...
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Einstein Summation Convention: One as Upper, One as Lower?

My question refers to the often specified rule defining Einstein Summation Notation in that summation is implied when an index is repeated twice in a single term, once as upper index and once as lower ...
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1answer
42 views

What is the spin of the Kalb-Ramond field?

In bosonic string theory the massless states of the closed string are given by a rank 2 tensor, which is divided into its three irreducible spherical tensors: symmetric traceless, antisymmetric and ...
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72 views

Uncommon tensor notation $\partial_{(\mu}\xi_{\nu)}$

I came across this expression for the change in a metric under an infinitesimal gauge transformation $\epsilon\xi^\mu$. $$h_{\mu\nu}' = h_{\mu\nu}+2\epsilon\partial_{(\mu}\xi_{\nu)}$$ What does the $...
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71 views

Four Vectors in SR and QFT

I'm covering both special relativity and quantum field theory in the summer. I'm currently using Spacetime Physics by Taylor and Wheeler to cover SR. Since I'm covering SR on the side with QFT, I'm ...
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36 views

Dirac notation - trace of product of (bipartite) density matrices

I'm getting confused by the Dirac notation. Suppose I have the following two objects. $$\rho = \sum_k p_k (\rho_A \otimes \rho_B) = \sum_k p_k |k \rangle \langle k | \otimes |k\rangle \langle k | ,$$...
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67 views

$SU(N)$ Yang-Mills Theory

Yang-Mills theory is based on the gauge group $G$ which we take to be $SU(N)$. Consider an example; $$\mathcal{L}=-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}-\sum_{j=1}^N\bar{\psi}_j(i\gamma^\mu D_\mu-m)\...
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What kind of product is $\prod^n_{j=1}\sigma^{(j)}_x$?

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. The adiabatic Hamiltonian is defined as $$...
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What is the connection between the coordinate transformation properties and graphical representation of covariant and contravariant components?

So right now I am studying General Relativity (in particular tensor analysis), and I have a question regarding covariant and contravariant components of a vector. I was taught how to transform ...
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83 views

Killing tensor in Minkowski space

I'm trying to solve the Killing tensor equation $\nabla_{(a}K_{bc)} = 0$ in Minkowski space. I'd like to generalise the method we use to find Killing tensors in Minkowski space. We can take $\...
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What expression of Ricci tensor should we choose in order to obatin a correct field equations?

I have doubt regarding the choice of the Ricci tensor $R_{ij}$. I have seen many books and papers use the expression $R_{ij}=\Gamma^i_{jp,i}-\Gamma^i_{ji,p}+\Gamma^i_{in}\Gamma^n_{jp}-\Gamma^i_{pn}\...
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Covariant Derivative of Kronecker Delta

I am reading Carroll's book on GR right now, and I ran into a little trouble in his chapter 3 on curvature. He is establishing the properties of the covariant derivative, and claims that the fact that ...
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30 views

If $T_{kl}=\epsilon_{kil}m_i$, how to show $m_i=0.5\epsilon_{ilk}T_{lk}$? [closed]

In a book I am reading (about magnetic dipole), it is given that $T_{kl}=\epsilon_{kil}m_i$. Then, it says since $T_{kl}=-T_{lk}$, it can be shown that $m_i=0.5\epsilon_{ilk}T_{lk}$. I understand that ...
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41 views

Rewriting Maxwell's equation in tensor form [closed]

Suppose $F_{ij}=\epsilon_{ijk}B_k $, how to prove the following: $\partial_iB_i=0$ becomes $\partial_iF_{jk}+\partial_jF_{ki}+\partial_kF_{ij}=0$ $B_iB_i$ becomes $F_{ij}F_{ij}/2$ I can see that ...
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109 views

Usage of tensors in physics [closed]

As I understand it, tensors are multi-linear maps that map vectors (and dual vectors) to real (or complex) numbers, but I'm hoping to gain some intuition as to why they are useful in physics. Is it ...
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33 views

Commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set $$\frac{\...
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62 views

What is the difference between a tensor, vector, and a matrix? [duplicate]

I'm currently going through notes on a physics course and I'm having trouble understanding the difference between a tensor, a vector, and a matrix. I know that a vector is a kind of tensor and that a ...
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Reciprocal Dyadics: Menzel's Mathematical Physics

I am having difficulty justifying a step in Menzel's development of sets of reciprocal spanning vector sets. Here's an abbreviated development. All Fraktur letters represent vectors. Introduce the ...
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Induced metric is a scalar for transformation from $x\to x'$? (Poisson E.A p.62)

I have a (simple) question about the induced metric $h_{ab}$. In Poisson E.A. (a relativist toolkit) it says in p. 62 that the induced metric $$h_{ab}=g_{{\alpha}{\beta}} \frac{\partial x^{\alpha}}{\...
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Flat space Solution of Einstein Field Equation

Does a trace-free energy-momentum tensor $T_{\mu}^{\mu} = 0$ ensure that the Einstein's field equations have a flat space solution?
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Tensor notation of Maxwell's equations

Tensor notation of Maxwell's equation read So when we explicitly try to find the Maxwell's equation from the above tensor equation we only get gauss law and curl of B. The div.B=0 and curl of E are ...
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How to show invariance using the Maxwell tensor?

I want to show the invariance of $E^2-c^2B^2$ under the Lorentz transformations. The obvious way to do this is to show that $$E^2-c^2B^2=E'^2-c^2B'^2,$$ where $E'$ and $B'$ are the Lorentz ...
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Killing tensor and Riemann tensor identity

I know that if we have a Killing vector then it's straightforward to show the identity: $$\nabla_a \nabla_b K_c = R_{cba}^k K_d$$ I'm now trying to show the following identity for a $(0,2)$ Killing ...
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119 views

Covariant derivative of a covariant derivative

I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$. This is something I've taken for granted a lot in calculations, namely I though that by the ...
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77 views

Peculiarity about a system of three electrons

Consider three (or any number bigger than 2) electrons without spatial degrees of freedom, thus the only degree of freedom is the spins. The Hilbert space is then formed by the tensor product of the ...
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Understanding basics of tensors

I am trying to understand tensors to learn General relativity. In the book that I am reading they claim that if the basis of a vector space undergoes a linear transformation $T$ then the components of ...
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Dielectric permittivity tensor of rank 2

In special relativity, one knows the Euclidean and Minkowski metric to raise or lower the index of a covector or vector. The Euclidean metric is a (1,1) tensor and is represented by a 3x3 identity ...
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Staggered Indices ($\Lambda^\mu{}_\nu$ vs. $\Lambda_\mu{}^\nu$) on Lorentz Transformations

I have some open-ended questions on the use of staggered indices in writing Lorentz transformations and their inverses and transposes. What are the respective meanings of $\Lambda^\mu{}_\nu$ as ...
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1answer
95 views

How does the Einstein summation convention apply to the following equation?

This is the equation is in the "mathematical form" section of the following wikipedia article: http://en.wikipedia.org/wiki/Geodesics_in_general_relativity More specifically, the "Full geodesic ...
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Deriving $A^{\mu}_{;\nu}$ from $A_{\mu ; \nu}$

We have a covariant derivative of a covariant tensor: $$ A_{\mu ; \nu} = A_{\mu , \nu} - \Gamma^{\alpha}_{\mu \nu} A_{\alpha} $$ The covariant derivative of a contravariant tensor is: $$ A^{\mu}_{;\nu}...
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What is basic tensor algebra in teleparallel equivalent of general relativity?

Teleparallel gravity represents a viable alternative to general relativity where gravitation comes from torsion rather that curvature. The theory is based on a new modified connection, and the ...
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Question about tensor operators [closed]

I have learned there is a so called tensor product in linear algebra when one is trying to pair vectors (or operators) from two different vector spaces, they use the symbol $\otimes$. But then in QM I ...
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Tensors Contracting Indices

I'm pretty confused regarding the components of a tensor once you take its trace (or contraction). I'll use $B\in T_2^1(V)$ to be specific. Let $V$ be an $n$-dimensional vector space with basis $\{E_i\...
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Tensor product - Addition of angular momenta

In the book Quantum Mechanics - Cohen-Tannoudji, in chapter X, equation (B-5) says $$ \vec{S^2} = (\vec{S_1} + \vec{S_2})^2 = \vec{S_1^2} + \vec{S_2^2} + 2\vec{S_1}\cdot\vec{S_2} $$ and $$ \vec{S_1}\...
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How do we determine if a certain physical quantity is a vector?

For instance in Newtonian physics we treat position of objects, displacements, velocities, forces, momenta, angular velocities etc all as vector quantities (little arrows in space which have a certain ...
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Indicating that indices are equal in Einstein notation

tl;dr: I have an expression like this: (dramatization) $$ R_{\mu\nu} = \begin{pmatrix} B^{00}C_{00} & 0 & 0 & 0 \\ 0 & B^{11}C_{10} & 0 & 0 \\ 0 & 0 & B^{22}C_{20} &...
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35 views

Exponential decay of correlation in PEPS

PEPS (Projected Entangled Pair State) is a tensor network that plays the same role in two dimensional lattice as MPS (Matrix Product State) plays in one dimensional spin chain. A good introduction can ...
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Converting between matrix multiplication and tensor contraction

If I have $A^{\alpha \beta} B_{\beta \gamma}$ then this should be the equivalent to the following matrix multiplication: $AB$ since we're summing over the columns of $A$ and the rows of $B$. By the ...
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Tensor products of Hilberts spaces: definition of outer products and commutators

Suppose one has two single-particle Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ and consider the tensor product of these such that $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is a two-particle ...