Can we exponentiate d/dx? Vector (fields)? What is exp? | Lie groups, algebras, brackets #4

Part 5:    • Matrix trace isn't just summing the diagon...  

Can we exponentiate vectors? What does e^(d/dx) mean? Does it make sense to exponentiate a whole bunch of vectors? Well yes! While what these exponentials do seem very different at first, they can be recast into the same framework.

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CHAPTERS:
00:00 Introduction
01:03 What is exponentiation?
04:15 Exponentiating vectors
11:23 Exponentiating derivatives
24:04 Exponentiating vector fields

❗Remark❗

1️⃣ I know that many people would be thinking of series expansion of exponentials. I deliberately avoid this because it is not conducive to learning the intuition of the exponential, and more crucially, it does not apply to the exponential of vectors on manifolds. The result is very manifold-dependent, and I will be very impressed if there is a series-like explanation for the exponential map in differential geometry.

2️⃣ However, I want to know: is there a generalisation of the translation operator statement in the video to manifolds? For a flat plane, we have exp(a * nabla) f(x) = f(x + a). And in fact,the exponential map on the flat manifold of R^n gives x + a = exp_x (a). Hence, for flat R^n, we have exp(a * nabla) f(x) = f(exp_x(a)). Can this be generalised to general manifolds? Is it true if I interpret nabla as not a normal gradient, but covariant derivative? Please let me know if you have any ideas for it. I want this to be true because it connects different “exponential” ideas.

📖 Further reading 📖

1️⃣ Exp vectors

Exponential map in Riemannian geometry (if you actually want to know how this is just a generalised exponential map in the usual sense, rather than just having the same “philosophy”, then go to the relationship to Lie theory section - when they say translations, they mean multiply on the left/right by g, a group element of the Lie group): https://en.wikipedia.org/wiki/Exponen...)

Why exponential map in differential geometry is useful: https://en.wikipedia.org/wiki/Normal_...

❗You might also need to learn these first before tackling the link above❗

Riemannian metric: https://www.ime.usp.br/~gorodski/teac...

Connection (the introduction is the most illuminating part): https://en.wikipedia.org/wiki/Affine_...
https://math.stackexchange.com/questi...

Levi-Civita connection (a particular kind of connection that makes the metric invariant): https://en.wikipedia.org/wiki/Levi-Ci...

2️⃣ Exp d/dx

What I have described is kind of solving PDE with the method of characteristics (identifying characteristic curves along which it becomes an ODE): https://en.wikipedia.org/wiki/Method_...

The partial differential equation is part of the wave equation: https://en.wikipedia.org/wiki/Wave_eq...

Translation operator in quantum mechanics: https://en.wikipedia.org/wiki/Transla...)

Time-ordering: solving differential equations of the form ∂f/∂t = X(t) f, where X(t) is a time-dependent differential operator, e.g. t^2*∂^2/∂x^2: https://en.wikipedia.org/wiki/Ordered...

Time-ordering in example in QM: https://en.wikipedia.org/wiki/Dyson_s...

3️⃣ Exp vector field

Vector flow: https://en.wikipedia.org/wiki/Vector_...

This is more related to the video: https://en.wikipedia.org/wiki/Flow_(m...)

Textbook: https://www.worldscientific.com/doi/p...

I actually wanted to say the following, but I think the video is long enough and didn’t include it into the script, but vector field is actually related to (and most often described by) differential operators, and in that sense both exponential of (1st-order) differential operators and exponential of vector fields yield very similar things: https://en.wikipedia.org/wiki/Vector_...

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