All Questions
Tagged with quantum-mechanics differential-geometry
26
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Apparent or real contradiction is in Eberlein's paper?
The questions I pose here concerns the paper "The Spin Model of Euclidean 3-Space" by W. F. Eberlein (The American Mathematical Monthly, Vol. 69, No. 7 (Aug. - Sep., 1962), pp. 587-598) (...
- 2,065
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Motivating spinors via the Dirac equation
I'm trying to motivate spin through Dirac's equation. So far, here is what I understand:
Upon trying to take the "square root" of the space-time Laplacian (i.e. find an operator such that $D ...
- 825
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1
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33
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Definition of the pullback of $L=Im(dS)$ and related questions on defintions
Def: We call a phase function $S: \mathbb{R}^n \rightarrow \mathbb{R}$ admissible if it satisfies the Hamilton-Jacobi equation. The image $L=Im(dS)$ of the differential of an admissible phase function ...
- 371
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Liouville—von Neumann Operator as a member of a Tensor space ( in an Open Quantum System )
$\textbf{Context:}$
I'm posting in Math SE instead of Physics SE because this is really a question about differential geometry and vector spaces.
Let $\mathcal{H}$ be a Hilbert Space, then $|\psi\...
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115
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Spinors and Klein-Gordon Equation
I'm currently working through Chapter 13 of Wald's General Relativity and spinors are being a little illusive to me. The question is pretty much: Using the Klein-Gordon equation in the form: $$\...
2
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1
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373
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What does fibre-wise mean?
I am doing some exercises in Lagrangian systems in the book Quantum Mechanics for Mathematicians. One exercise says:
Let $f$ be a $C^\infty$ function on a manifold $M$. Show that the Lagrangian ...
- 1,334
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239
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Find curvature of Bertrand Curve to twist a log quaternion around a target axle
I have been implementing a system of log-quaternions... This is just (0,A,B,C) where $A$, $B$ and $C$ represent angles/curvatures. For some reason, until the ...
- 111
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The disrtance associated to Fubini-Study metric in projective Hilbert space $\mathbb{P}H$
Suppose $H$ is a complex Hilbert space and $\mathbb{P}H$ be its projective space. $d$ is the distance function on $\mathbb{P}H$ associted to the Fubini-Study metric on $\mathbb{P}H$.
In the proof of ...
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27
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prequantization euclidiean space
In general the prequantum hilbert space of some manifold $M$ will be the sections of the complex line bundle. I ommitted a lot of details of course.
Locally, a section of a complex line bundle ...
- 988
2
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86
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Dual Space to a Lie Algebra. Problem from Takhtajan. Finding the center of a Poisson Algebra
So I am trying to solve Problem 2.19 from the book "Quantum mechanics for mathematicians" by Takhtajan.
The problem is the following:
Let $g$ be a finite-dimensional Lie Algebra with a Lie bracket $[,...
- 294
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107
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Geometrical Quantization and Connections
Really I think this question boils down to what the physical significance of a connection is.
Physically, we can think of a symplectic manifold $(\mathcal{M},\omega)$ as essentially a phase space.
...
17
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1
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733
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What is the essential difference between classical and quantum information geometry?
This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory.
I have a ...
- 7,379
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54
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Topologically proctected twist in the wave function (Chern number)
In the famous TKNN paper and subsequents the authors wisely argue that the transversal conductance in the Integer Quantum Hall Effect has a topological interpretation as the integral of the curvature ...
- 139
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What geometry or topology best embodies the nonlocality of quantum entanglement?
I am a Princeton physics major.
What geometry or topology best embodies the nonlocality of quantum entanglement? https://en.wikipedia.org/wiki/Quantum_entanglement: "Each particle cannot be ...
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860
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Surface integral of the Gaussian curvature covering the vertex of a cone
The Gaussian curvature of a cone is undefinable at the vertex, and vanishes elsewhere on the cone. However, the cone is an ideal surface which must not be excluded from quantum mechanics where the ...
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