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 , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  Quaternionic Quantum Mechanics

+ 0 like - 0 dislike
932 views

Would it be judicious to define the Hilbert space of Quantum Mechanics over the quaternion numbers instead of the complex numbers? We can define a tensor product by the Hamilton numbers $\bf Q$:

$H_{\bf Q}={\bf Q} \otimes_{\bf R} H$

with products:

$q(q' \otimes \psi)=(qq')\otimes \psi$

$(q \otimes \psi)q'=(qq')\otimes \psi$

with $H$ the complex Hilbert space. So that we have the brackets $<q \psi|\phi>=\bar q<\psi|\phi>$, $<\psi q |\phi >=<\psi | \bar q \phi>$ and $<\psi |\phi q>=<\psi |\phi>  q$.

The observables would be right linear operators: $A|\psi q>=A|\psi >q$.

asked Aug 12, 2018 in Theoretical Physics by Antoine Balan (-80 points) [ revision history ]
edited Aug 13, 2018 by Antoine Balan

How do you define the trace operator on quaternions HS ? more seriously, it is an intensive domain of reformulation, particularly since the Seiberg-Witten equations and the large symmetry group of their trivial reducible solution. You would find easily deep reformulations and theories ports focusing on the one you are interested ( f.e. Maxwell, QED, QCD ). There are still difficulties with the trace operator and in general with a few original definitions too restricted to ( real and ) complex based HS. An interesting compilation to start : Quaternions and Hilbert Spaces ( Chapters 1 to 4 )

As far as I understand, complex numbers (functions) appeared in QM because QM is a "wave mechanics", which naturally gives quantization in restricted systems (all musical instruments are based on this property). A harmonic wave at a point has an amplitude and a phase - voilà two real numbers (= a complex number) instead of one real number in Classical Mechanics. Thus, QM is formulated in terms of complex functions for this reason. Of course, later on, spinors were introduced, then other group representations like iso-spinors, etc.

There must be a natural necessity in introducing quaternions, a necessity similar to what happened to QM.

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$\hbar$ysicsOverf$\varnothing$ow
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
...