childofsaturn
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How to take partial trace?
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19 votes

Let $H_A \otimes H_B$ be your Hilbert space, and $O$ be an operator acting on this composite space. Then $O$ can be written has $$ O = \sum_{i,j} c_{ij} M_i \otimes N_j$$ where the $M_i$'s and $N_j$'s ...

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Elliptic genus; What is it within string/M-theory?
11 votes

Below is a summary of my very limited understanding of what the elliptic genus is. I'll first give you the mathematical definition, followed by an explanation of how it appears naturally in physics. ...

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Berry Curvature and Curvature Tensor
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10 votes

Indeed the structure is similar for a very good reason: Both of these are particular examples of a beautiful concept in mathematics. You have some underlying manifold $M$. In addition to the ...

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Canonical momentum density vs. energy-momentum tensor
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7 votes

The two quantities don't correspond because they are conserved quantities corresponding to different symmetries. One is a symmetry from shifting your field, the other from shifting space-time itself. ...

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How to show that the product of a Killing tensor and tangent vector is conserved along a geodesic?
6 votes

One can directly show that it's conserved along geodesics as Dvij has done, but it's also illuminating to note that this is just a special case of Noether's theorem. In particular whenever a metric ...

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Proving Lorentz invariance of Maxwell equations
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6 votes

You aren't "replacing a mathematical proof". What the statements you are referring to mean is that in tensor notation, the proof is immediate, so that nothing needs to be written down. This is because ...

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To prove, $\nabla.(\nabla\phi \times \nabla\psi)$ =0
4 votes

By definition of the cross product and gradient operators, and summing over repeated indices we have, $$\nabla . (\nabla \phi \times \nabla \psi) = \partial_i (\epsilon_{ijk} \partial_j \phi \,\...

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A Quantum Mechanics question that no one can answer
4 votes

Open up the chapter on perturbation theory in any quantum mechanics textbook and pick any Hamiltonian you see. Chances are an exact solution to that quantum system isn't known. Ask your Professor to ...

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Why two different Lagrangians to derive geodesic equations?
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4 votes

Actually the equations of motion one ends up with are not manifestly the same: If I let $$L_1 = \sqrt{g_{\mu \nu} \frac{dx^{\mu}}{dt} \frac{dx^{\nu}}{dt}}$$ one finds that the Euler-Lagrange equation ...

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Tensor product of operators in QM
4 votes

Adding to Lubos' answer, let me address this part more specificially: I am very confused about this - actually - I'm guessing that transformations that can be written $A⊗B$ are a subset of all ...

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The abstract space of metrics in GR
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4 votes

Sure, this kind of thing is well studied. The "space of metrics" of a manifold is usually called its moduli space. More precisely, if we have a manifold $M$, Let $\text{Met}(M)$ be the set of all ...

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Kronecker delta confusion
4 votes

It's useful to know how matrix multiplication is defined: For $n \times n$ matrices, $A$ and $B$, denote the entry in the $i$th row and $j$th column by $A^i_j$ and $B^i_j$ respectively. Then for $C ...

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Exact closed form solution to the quantum harmonic oscillator
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3 votes

First note the following: If $\psi(x,0)$ is any normalizable wave-function in your Hilbert space, then $\psi(x,t) = e^{-iHt} \psi(x,0)$ (I've set $\hbar = 1$) satisfies the time-dependent Schrodinger ...

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GRE practice: Total internal reflection question
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3 votes

Elimination by the use of limit analysis is probably the best way to do it on the actual test, but here's the full solution: Let $\theta'$ by the angle that light enters the cable. By Snell's law ...

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Proof that the Earth rotates?
3 votes

Let us check the claim that the earth is rotating about it's own axis. We may choose this axis to be the $z$-axis. The earth can be approximated by a sphere. Consider a pendulum living somewhere on ...

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Matrices in Dirac Notation
2 votes

A simple (but rather non-rigorous) way to think about this: Your Hilbert space $\mathcal{H}$ is formed by square-integrable functions on the real line, that is elements of $\mathcal{H}$ are given by ...

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Schrödinger field operators and their commutation relations
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2 votes

If I understood correctly what I've been taught so far, in QFT one must find some way to quantize the fields obeying the field equation in question. This is correct. In this particular case you start ...

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Boson ladder operator $n+1$ factor
2 votes

Let $$|n\rangle = \frac{(a^{\dagger})^n}{\sqrt{n!}} |0\rangle$$ be the n'th normalized eigenstate and recall the commutation relation for bosonic creation/annihilation operators, $[a, a^{\dagger}] = 1$...

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Equations of motion for Polyakov action
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2 votes

You can either vary the action directly, or apply the classical field theory Euler-Lagrange equations. The latter for a Lagrangian $\mathcal{L}(\phi^{\alpha}, \partial_{\mu}\phi^{\alpha})$ read $$\...

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Equation for null geodesic around schwarzschild metric?
2 votes

As requested in the comments by the user, I will write down the differential equation satisfied by the trajectory of a photon that travels along a radial path (no $\theta$ or $\phi$) dependence. By ...

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Why define four-vectors to be quantities that transform only like the position vector transforms?
2 votes

The reason you're asking this question is because defining a vector or a more general tensor as "a quantity that transforms a certain way" is not very conceptually illuminating, but is often done to ...

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Apparent discrepancy between Lagrange field equation and Maxwell equation
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1 votes

Your sign is wrong when computing $$\frac{\partial{(B^2)}}{\partial{(\partial_{y} A_x)}}.$$ The only term in $B^2$ that contains $\partial_{y} A_x $ is $(B_z)^2 = (\partial_{x}A_y - \partial_{y} ...

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Facing a problem in Katrin Becker, Melanie Becker, John Schwarz's String Theory
1 votes

BBS omit to indicate which space they are tracing over in each step. Let me instead sketch why tracing over only the states created by the $\alpha_{-n}$'s in the last step is the correct thing to do. ...

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Question about scalar product of 2 four-vectors
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1 votes

The proof of this identity is entirely analogous to how one would go about proving the Euclidean formula $\vec{a}.\vec{b} = |a||b| \text{cos}(\theta)$. What one has to do is as follows: Find a ...

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Deriving the Canonical Energy Momentum Tensor
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1 votes

Note that the chain rule in this case, since $\mathcal{L} = \mathcal{L}(\varphi, \partial_{\mu} \varphi)$ reads $$\partial_{\nu}\mathcal{L} = \frac{\partial \mathcal{L}}{\partial \varphi} \partial_{\...

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Converting between (abstract) linear operators and their position representations
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1 votes

This simply follows from a resolution of the identity, written (formally) as $\int dy |y\rangle\langle y| = 1$: $\langle x|HU(t)|x\rangle = \int dy\langle x|H|y\rangle\langle y|U(t)|x\rangle$, which ...

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Tensors and rotations
1 votes

What's important when considering the behavior of tensors aren't specifically rotations, but any arbitrary coordinate transformation (a change of basis of your vector space). In general if you have a ...

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The trajectory of a projectile launched from a hilltop
1 votes

I didn't go through all of your work but to answer the question about how to solve that equation, change $\tan(\phi)$ to $\frac{\sin(\phi)}{\cos(\phi)}$ and take $\cos(\phi)$ as a common denominator. ...

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Topology of parameter space
0 votes

This is something that really does depend on the specific problem you are studying. In general one can only say that we have some "space"/manifold of control parameters $\mathcal{M}$. Three different ...

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Why do the energies of the infinite square well decrease as width of the well increases?
0 votes

Actually this must be so, since in the $a \rightarrow \infty$ limit, you better recover the quantum mechanics of a free particle moving on the real line. If you define $k_n = n\pi/a$, one can see that ...

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