Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,786 comments
1,470 users with positive rep
820 active unimported users
More ...

  Decomposition of the group of Bogoliubov transformations

+ 5 like - 0 dislike
1240 views

Consider the fermion Fock space $\mathcal{F}=\bigoplus_{k\ge 0}\bigwedge^k\mathfrak{h}$ of some finite-dimensional 1-particle Hilbert space $\mathfrak{h}$. The group $\mathrm{Bog}(\mathcal{F})$ of Bogoliubov transformations can be defined as the set of unitary maps $U$ on $\mathcal{F}$ for which there are linear maps $u:\mathfrak{h}\to\mathfrak{h}$ and $v:\mathfrak{h}\to\mathfrak{h}^*$ such that $$Ua^*(f)U^*=a^*(uf)+a(J^*(v(f)))\quad\forall f\in\mathfrak{h},$$ where $a^*,a:\mathfrak{h}\to\mathcal{B}(\mathcal{F})$ denote the usual fermion creation- and annihilation operators and $J:\mathfrak{h}\to\mathfrak{h}^*$ denotes the Riesz isomorphism. It is not hard to see that these $u$ and $v$ define a unitary map $\Phi(U)\in U(\mathfrak{h}\oplus\mathfrak{h}^*)$ commuting with $\mathcal{J}$, where $$\Phi(U):=\begin{pmatrix}u&J^*vJ^*\\ v&JuJ^*\end{pmatrix},\quad\mathcal{J}:=\begin{pmatrix}0&J^*\\J&0\end{pmatrix}.$$

Defining $G:=\{A\in U(\mathfrak{h}\oplus\mathfrak{h}^*)\mid A\mathcal{J}=\mathcal{J}A\}$, it turns out that $\Phi$ defines a short exact sequence of Lie groups $$1\to\mathbb{S}^1\to\mathrm{Bog}(\mathcal{F})\to G\to 1,$$ Now my question is: does this sequence split (or, put differently, is $\mathrm{Bog}(\mathcal{F})\cong\mathbb{S}^1\times G$)?

Note that, if we are working in the category of groups (as opposed to Lie groups), central extensions of the group $G$ by $\mathbb{S}^1$ are classified (upto isomorphism) by the cohomology group $H^2(G,\mathbb{S}^1)$. If this classification is also valid in the Lie group setting, there might be some general result showing that $H^2(G,\mathbb{S}^1)=0$, which would answer my question positively.

This post imported from StackExchange MathOverflow at 2019-03-12 18:54 (UTC), posted by SE-user Robert Rauch
asked Feb 26, 2019 in Mathematics by Robert Rauch (25 points) [ no revision ]
retagged Mar 12, 2019
If i give you a $v : h^* \to h$ could you give me a preimage for the matrice with coeeficients $u=0$ and $v$ ?

This post imported from StackExchange MathOverflow at 2019-03-12 18:54 (UTC), posted by SE-user Bleuderk
Well, yes: given $v:\mathfrak{h}\to\mathfrak{h}^*$ (note the order of $\mathfrak{h}$ and $\mathfrak{h}^*$), then there is a Bogoliubov transformation $U$ on $\mathcal{F}$ with $u=0$ and the given $v$. The main difficulty here is to construct the image $U\Omega$ of the vacuum $\Omega\in\mathcal{F}$. A proof can be found e.g. in Solovejs lecture Notes on Many Particle Quantum Mechanics (Theorem 9.5). Note that in the infinite-dimensional case $V^*V$ is required to be trace class ("Shale Stinespring condition").

This post imported from StackExchange MathOverflow at 2019-03-12 18:54 (UTC), posted by SE-user Robert Rauch

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverflo$\varnothing$
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...