Application: Quantum mechanics on curved spaces - Lec 26 - Frederic Schuller

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Recommends user169903 · 2018-06 · Mathematics Stack Exchange · on “How to arrive at frame bundle objects from chart calculation on the base manifold”

In this lecture by Fredric Schuller it is said that the wave-function is not a wave-function. He attempts to find an appropriate coordinate independent derivative using "chart calculations" on the level of base manifold. Application: Quantum mechanics on curved spaces - Lec 26 - Frederic Schuller [this video] We are shown that what we call a wave function is indeed a pull back of a section of the associated bundle to the frame bundle or a vector fiber valued function on the frame bundle which is the principal bundle. At the end of the lecture the co-variant derivative is found on the base manifold on a local chart. However, he explains that using this "chart calculation", we should go up in the f

Recommends bloods · 2019-06 · Mathematics Stack Exchange · on “Covariant derivative of a section of a trivial(?) associated bundle”

First I'll briefly outline the physical context in which my question arose: If one tries to do 2D quantum mechanics in polar coordinates one encounters a problem with a property of the covariant derivative $\nabla$ . Namely that the the radial momentum operator $P_r$ defined in terms of it $P_r:=-i\nabla_r=-i\partial_r$ (if the object to be derivated is a function $f:\mathbb{R^2}→ \mathbb{C}$ ) is no longer self adjoint. According to this lecture [this video] one can address this issue by interpreting the wave function $\Psi$ (i.e. the object to be derivated) no longer as a function $f_\Psi:\mathbb{R^2}→ \mathbb{C}$ but as a section $\Psi:\mathbb{R^2}→E$ on a $\mathbb{C}$ -line bundle $E$ over $\

Recommends bloods · 2019-06 · Physics Stack Exchange · on “Wave function as a section of a complex line bundle to do QM in polar coordinates”

If you want to change the coordinates of a Wave function $\Psi$ in 2D QM from cartesian to polar coordinates in a naive way one encounters a problem, namely the (naively defined) radial momentum operator $P_r$ is no longer self adjoint: $$(\Phi,P_r\Psi)=\int r\Phi ^*(-i)\partial _r\Psi drd\theta\neq(P_r\Phi,\Psi)$$ According to this lecture [this video] one can address this issue by interpreting the wave function $\Psi$ no longer as a function $f_\Psi:\mathbb{R^2}→ \mathbb{C}$ but as a section $\Psi:\mathbb{R^2}→E$ on a $\mathbb{C}$ -line bundle $E$ over $\mathbb{R^2}$ . It is claimed that $E$ is also an associated fibre bundle to the frame bundle $L\mathbb{R^2}$ over $\mathbb{R^2}$ which can be

Recommends VVM · 2023-02 · Mathematics Stack Exchange · on “How to push up / lift a connection one-form from the base manifold to the total space?”

I am following this YouTube lecture by Schuller [this video] where he finds the appropriate formalism for the quantum mechanics in the physical curved space. Everything makes sense to me but at the very end I see that we find the pull backed connection one-form on the base manifold. He says to the end of the lecture that later on we will see how we can find the connection one-form on the total space, the prinicipal bundle, i.e., the frame bundle, level. This is not shown, however, in the next video. I wonder how can we calculate the connection one-form on the total space when we only have information about the pulled back of this form on the base manfiold through a section map? If we have a tange

Recommends VVM · 2023-02 · Physics Stack Exchange · on “How to derive connection Lie algebra valued one-form on the frame bundle if given the pulled back of it on the physical space?”

I am following this YouTube lecture by Schuller [this video] where he finds the appropriate formalism for the quantum mechanics in the physical curved space. Everything makes sense to me but at the very end I see that we find the pull backed connection one-form on the base manifold. He says to the end of the lecture that later on we will see how we can find the connection one-form on the total space, the principal bundle, i.e., the frame bundle, level. This is not shown, however, in the next video. I wonder how can we calculate the connection one-form on the total space when we only have information about the pulled back of this form on the base manfiold through a section map? If we have a tangen

Recommends WillG · 2025-01 · Physics Stack Exchange · on “Resource request for quantization on Riemannian manifolds”

I'm looking for more information on the quantization procedure discussed in this [this video] video lecture by Frederic Schuller. Toward the end of the lecture (around 1:03), he describes a procedure for quantizing a particle on a general Riemannian manifold $(M, g)$ using arbitrary coordinates $x^j$ . In particular, inner products are defined as $(\psi, \phi) = \int d^dx \sqrt{\det(g)} \psi^*(x) \phi(x)$ , position operators $Q^j$ are defined as multiplication by $x^j$ , and momentum operators $P_j$ are defined using a covariant derivative $P_j = -i\nabla_j = -i(\partial_j+\omega_j)$ [he then shows that we must have $2\text{ Re }\omega_j = \partial_j(\ln \sqrt{\det g})$ in order to ensure self-a

Cites ProphetX · 2022-01 · Physics Stack Exchange · on “Momentum operator in Geometric Quantization vs momentum operator on arbitrary curved space(time)s”

In the following stack exchange post Momentum Operator in curved spacetime (QFT) [LINK] a general expression for the momentum operator is given for a Riemannian manifold $(M,g)$ . Similarly, Frederic Schuller's lectures explained the geometric aspects of this [this video] . To make sense out of this, we should think of the wave function as a section of a complex line bundle, which is associated to the frame bundle $LM$ of the manifold $(M,g)$ . In this case, the momentum operator will be a covariant derivative, with Christoffel symbols given in the Stack Exchange post, or in the video I linked. At the same time, we can think of a classical system not only in the Lagrangian point of view, where th

Cites Ivan Burbano · 2022-03 · Physics Stack Exchange · on “Lagrangian Formalism: Canonical quantization in supersymmetric quantum mechanics”

It is surprising how difficult it is to build a coherent story about this because it seems that all of the references have a couple of typos. Rather than an answer, this post has a ton of questions that I came up with trying to build a coherent story. Hopefully it will be a starting point to building one, although I am sure it will also have its own typos. Let us start with an action whose time derivatives are $$S=\int\text{d}t\left(\frac{1}{2}g_{ab}\dot{x}^a\dot{x}^b+ig_{ab}\bar{\psi}^a\frac{D}{Dt}\psi^b+\cdots\right).$$ In this form, the conjugate momenum to $x^a$ is $$p_a=g_{ab}\dot{x}^b+i\bar{\psi}^b\Gamma_{bac}\psi^c,$$ while $\psi^a$ and $\bar{\psi}_a$ form a fermionic conjugate pair.