John C. Baez
@johncarlosbaez.bsky.social
Mathematical physicist
You seem to understand it pretty well!
November 10, 2025 at 9:24 PM
You seem to understand it pretty well!
There are lots of good rock songs that don't rhyme. Rhyme is very constraining, very hard to do well, and only worthwhile if you can pull it off well. Cole Porter could do it well, and even his rhymes sound pretty strained at times - sometimes entertainingly so, sometimes just awkward.
November 9, 2025 at 2:50 PM
There are lots of good rock songs that don't rhyme. Rhyme is very constraining, very hard to do well, and only worthwhile if you can pull it off well. Cole Porter could do it well, and even his rhymes sound pretty strained at times - sometimes entertainingly so, sometimes just awkward.
Is there a similar but more complicated descrlption of coker f when d > 3? Just curious.
November 9, 2025 at 2:48 PM
Is there a similar but more complicated descrlption of coker f when d > 3? Just curious.
Why? It can be psychologically easier to cling to the hope that you've done something great that will make you immortal than to face the realization that you'll never do anything great: instead, you'll go downhill, losing your mental acuity, and then die and be forgotten.
November 8, 2025 at 10:03 AM
Why? It can be psychologically easier to cling to the hope that you've done something great that will make you immortal than to face the realization that you'll never do anything great: instead, you'll go downhill, losing your mental acuity, and then die and be forgotten.
Congratulations!
"Mentoring" sounds vague enough that you can still escape blame when the problems start. 😉 When you start being a thesis advisor, you're stuck - your honor is on the line.
"Mentoring" sounds vague enough that you can still escape blame when the problems start. 😉 When you start being a thesis advisor, you're stuck - your honor is on the line.
November 7, 2025 at 10:16 AM
Congratulations!
"Mentoring" sounds vague enough that you can still escape blame when the problems start. 😉 When you start being a thesis advisor, you're stuck - your honor is on the line.
"Mentoring" sounds vague enough that you can still escape blame when the problems start. 😉 When you start being a thesis advisor, you're stuck - your honor is on the line.
Nice! I don't think that that p^2 - q^2 being the Lagrangian is so important, but we can think about the 1-parameter groups of canonical transformations generated by p^2 - q^2 and qp+qp, and thus why the commutator of these two elements is proportional to p^2 + q^2.
November 7, 2025 at 10:12 AM
Nice! I don't think that that p^2 - q^2 being the Lagrangian is so important, but we can think about the 1-parameter groups of canonical transformations generated by p^2 - q^2 and qp+qp, and thus why the commutator of these two elements is proportional to p^2 + q^2.
Yes, that's why us successful physicists don't look so hot!
November 6, 2025 at 9:35 AM
Yes, that's why us successful physicists don't look so hot!
"Ah I think I found that X=q-p and Y=p^3-q^3 for [q,p]=iI work here." Nice! But my mumbo-jumbo about the symplectic Lie algebra suggests that the harmonic oscillator is the commutator of two *quadratic* expressions in p,q, and quadratics are much better-behaved than cubics.
November 6, 2025 at 9:28 AM
"Ah I think I found that X=q-p and Y=p^3-q^3 for [q,p]=iI work here." Nice! But my mumbo-jumbo about the symplectic Lie algebra suggests that the harmonic oscillator is the commutator of two *quadratic* expressions in p,q, and quadratics are much better-behaved than cubics.
I don't know Lotte Meitner-Graf!
November 4, 2025 at 2:25 PM
I don't know Lotte Meitner-Graf!
I think the harmonic oscillator Hamiltonian can be obtained by differentiating a representation of the metaplectic group and applying it to an element of the symplectic Lie algebra. Since this Lie algebra is sp(2) = sl(2,R), I'm optimistic that every element is a commutator!
November 4, 2025 at 12:22 AM
I think the harmonic oscillator Hamiltonian can be obtained by differentiating a representation of the metaplectic group and applying it to an element of the symplectic Lie algebra. Since this Lie algebra is sp(2) = sl(2,R), I'm optimistic that every element is a commutator!
The Hamiltonian of the harmonic oscillator is an unbounded operator on an infinite-dimensional Hilbert space, so commutators are much more tricky. For example in finite dim you can't have [X,Y] = i but in infinite dim you can. But this is a good question! I don't know!
November 4, 2025 at 12:14 AM
The Hamiltonian of the harmonic oscillator is an unbounded operator on an infinite-dimensional Hilbert space, so commutators are much more tricky. For example in finite dim you can't have [X,Y] = i but in infinite dim you can. But this is a good question! I don't know!
If it's that hard to construct a circle, just think how hard it would be to build a bridge.
November 4, 2025 at 12:11 AM
If it's that hard to construct a circle, just think how hard it would be to build a bridge.
Thanks! Yes, the appearance of Wick rotation in such a stripped-down context is really exciting and tantalizing.
But in that paper I thought the number i was needed to express symmetries in terms of observables. I learned something new this weekend:
golem.ph.utexas.edu/category/202...
But in that paper I thought the number i was needed to express symmetries in terms of observables. I learned something new this weekend:
golem.ph.utexas.edu/category/202...
Dynamics in Jordan Algebras | The n-Category Café
golem.ph.utexas.edu
November 3, 2025 at 10:16 PM
Thanks! Yes, the appearance of Wick rotation in such a stripped-down context is really exciting and tantalizing.
But in that paper I thought the number i was needed to express symmetries in terms of observables. I learned something new this weekend:
golem.ph.utexas.edu/category/202...
But in that paper I thought the number i was needed to express symmetries in terms of observables. I learned something new this weekend:
golem.ph.utexas.edu/category/202...
This left adjointness does what you want for representations.
Algebraic groups over a field K are group objects in the category of algebraic varieties over K. Most of the Lie groups you're likely to run into are algebraic groups over ℝ or ℂ.
Much more:
www.jmilne.org/math/CourseN...
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Algebraic groups over a field K are group objects in the category of algebraic varieties over K. Most of the Lie groups you're likely to run into are algebraic groups over ℝ or ℂ.
Much more:
www.jmilne.org/math/CourseN...
(2/2)
www.jmilne.org
November 2, 2025 at 11:37 PM
This left adjointness does what you want for representations.
Algebraic groups over a field K are group objects in the category of algebraic varieties over K. Most of the Lie groups you're likely to run into are algebraic groups over ℝ or ℂ.
Much more:
www.jmilne.org/math/CourseN...
(2/2)
Algebraic groups over a field K are group objects in the category of algebraic varieties over K. Most of the Lie groups you're likely to run into are algebraic groups over ℝ or ℂ.
Much more:
www.jmilne.org/math/CourseN...
(2/2)
Yes, that sounds about right. Here's my guess: there's a category Gp(K) of algebraic groups over any field K, and if k is a subfield there's a forgetful functor Gp(K) → Gp(k), and I bet this has a left adjoint, which is complexification if k = ℝ, K = ℂ.
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en.wikipedia.org/wiki/Algebra...
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en.wikipedia.org/wiki/Algebra...
Algebraic group - Wikipedia
en.wikipedia.org
November 2, 2025 at 11:24 PM
Yes, that sounds about right. Here's my guess: there's a category Gp(K) of algebraic groups over any field K, and if k is a subfield there's a forgetful functor Gp(K) → Gp(k), and I bet this has a left adjoint, which is complexification if k = ℝ, K = ℂ.
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en.wikipedia.org/wiki/Algebra...
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en.wikipedia.org/wiki/Algebra...
Mathematicians working on Lie theory often go the other way! They start with complex Lie algebras and their corresponding complex Lie groups, and then look at "real forms" of those. For example, GL(n,C) has GL(n,R) and U(n) as real forms.
Check it out:
en.wikipedia.org/wiki/Real_fo...
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Check it out:
en.wikipedia.org/wiki/Real_fo...
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Real form (Lie theory) - Wikipedia
en.wikipedia.org
November 2, 2025 at 3:47 PM
Mathematicians working on Lie theory often go the other way! They start with complex Lie algebras and their corresponding complex Lie groups, and then look at "real forms" of those. For example, GL(n,C) has GL(n,R) and U(n) as real forms.
Check it out:
en.wikipedia.org/wiki/Real_fo...
(2/2)
Check it out:
en.wikipedia.org/wiki/Real_fo...
(2/2)
Yes, we complexify Lie groups. It can be done by first complexifying the Lie algebra as you say, or as @barbarafantechi.bsky.social described: defining the group using polynomial equations in real variables and then looking at complex solutions of those. Same result, more or less!
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November 2, 2025 at 3:43 PM
Yes, we complexify Lie groups. It can be done by first complexifying the Lie algebra as you say, or as @barbarafantechi.bsky.social described: defining the group using polynomial equations in real variables and then looking at complex solutions of those. Same result, more or less!
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