Talk:Lie product formula

Rename

This page needs to be moved under the title, 'Lie-Trotter Product formula'. 146.87.147.235 (talk) 13:18, 1 April 2009 (UTC)

It is also called Lie-Trotter-Kato product formula and Trotter-Kato product formula. I don't know which is the most common name. Probably a few redirects would not hurt (right now there is only Lie-Trotter and Lie product identity). --Marozols (talk) 19:35, 1 April 2009 (UTC)

I have added a few redirects, see http://wiki.riteme.site/wiki/Special:WhatLinksHere/Lie_product_formula --Berland (talk) 06:09, 2 April 2009 (UTC)

I've only known this formula by the name "Trotter Formula" and there's 1000 times more literature that calls it that, than any of the names you suggested above. MathWorld calls it Trotter Formula. It would be a shame if Lie was the first one to use it, but even in this wiki article itself, the reference to Trotter's 1959 paper is the earliest one presented. Unless one can find a reference where Lie used it, then the title should be changed to Trotter formula 129.67.86.185 (talk) 17:20, 12 July 2010 (UTC)

I agree with this, in physics, (with path integrals, path integral molecular dynamics), all the literature refers to this as the "Trotter formula", or "Trotter expansion". Then again, we are using operators on a Hilbert space, whereas apparently Lie only proved it for finite matrices. I'm going to move it to Trotter formula soon and do a small rewrite. Danski14^(talk) 15:55, 14 May 2012 (UTC)

Self-Adjoint?

I was under the impression that the operators needed to be self adjoint. Is this incorrect? —Preceding unsigned comment added by 70.9.91.215 (talk) 21:52, 9 March 2011 (UTC)

Can I ask for some clarification?

$e^{A+B}=\lim _{N\rightarrow \infty }(e^{A/N}e^{B/N})^{N},$

$e^{A+B}=e^{C}=\lim _{N\rightarrow \infty }(e^{A/N}e^{B/N})^{N},$

where C is

(copied from Vector space page on direct Product and direct sum)

Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum ${\textstyle \bigoplus _{i\in I}V_{i}}$ (also called coproduct and denoted ${\textstyle \coprod _{i\in I}V_{i}}$),

Right? Like if you had A and B and added them to get C... performing the infinitesimal transformation of an element[[1]] you end up at the same representation?

What makes me ask?

Covering Map Explorer At least according to this demo... adding in 3 dimensions, A+B forms a covering grid. Have to turn off "Use Polar Map" to get the rectangular grid. This is formed from literally angleA around one axis, angleB around another axis, and added together; if you turn off "Add Composite" then horizontal lines are ${A}\times {B}$ and the other are ${B}\times {A}$ which, if you extend much beyond well even 20 degrees (pi/6) there's quite a bit of deformation where AxB and BxA don't align. But then adding A+B (Add composite) works well... and is the same as an infinitesimal transform.... The grid is then transformed by another orientation indicated the the latitude/longitude and spin sliders. The option OA+OB or O(A+B) is whether the offset(origin) rotation is applied to the added value or on each factor and then added; the later isn't generally useful for my 3D game stuff; but maybe split transforms intersecting is useful in some cases... Near the origin, the distortion is barely notable between the two. The 'Latitude' is the primary slider that makes things change; to start spin and longitude are gimbal locked with latitude 0 (or near 1pi)

The 'Size' slider controls the angular span of the grid. If 'Use Polar Map' is set, then size controls the 'radius' by the angle around the sphere it is. It's by default a covering that reflects the topology of quaternions in polar map mode, and increasing the slider to 180 completes the toroid which is typically projected to (but in rotation space, without a specific covering map other than it's own axis-angle coordinate.

I'm told that 'programming isn't math' And I understand even this interactive thing could just be a 'artists rendition' but it's honestly $(x,y,z)$ rotations around axis-axis-axis which could be scaled to axis-angle. And although I can certainly confirm $A+B=C$ it seems this is a controversial viewpoint. If this is not the math for this, please give me a resource for working with log-quaternions?

I Added an option to do Lie Algebraic Iteration, which can be toggled between simple vector addition, and N steps... 100 steps is almost enough to get pretty close. D3x0r (talk) 18:36, 18 April 2021 (UTC)

The proposed equation cannot be supported by Lie Algebra as defined; It only applies to Rotation Vectors before becoming elements of so(3). Recovering axis-angle from so(3) matricii is impractical if not impossible (https://github.com/d3x0r/STFRPhysics/blob/master/LieProductRule.md). There is no path backwards from Lie Algebra to ...(whatever this math system should be called)... .

For the above reasons, really makes it useless to state within this context.

D3x0r (talk)

D3x0r (talk) 05:39, 18 April 2021 (UTC)