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| visits | member for | 1 year, 5 months |
| seen | 10 mins ago | |
| stats | profile views | 251 |
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44s |
answered | In coordinate-free relativity, how do we define a vector? |
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9h |
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Do generators belong to the Lie group or the Lie algebra? @joshphysics: closed under the group operation (which would be redundant, but I've heard it used before), closed as in closed manifold (ie compact and without boundary) or topologically closed ;) ; these lecture notes just say Of course, this formula is valid for any matrix Lie group, so it really might be trivial even if I don't see it yet |
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10h |
comment |
Do generators belong to the Lie group or the Lie algebra? @joshphysics: yeah, I don't think it's actually as obvious as I thought - my argument works for matrix groups that are open subsets of $\mathbb{R}^{n\times n}$, but does it work in general? |
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11h |
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Do generators belong to the Lie group or the Lie algebra? ah... so I was missing something obvious, in particular that for matrix groups, $\mathrm{conj}_g$ is a linear map and thus $\frac{d}{dt}\big|_{t=0}\mathrm{conj}_g(\exp tX)=\mathrm{conj}_g(\frac{d}{dt}\big|_{t=0}\exp tX)=\mathrm{conj}_g(X)$; anyway, +1 |
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12h |
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Do generators belong to the Lie group or the Lie algebra? shouldn't $\mathrm{Ad}$ be the differential of conjugation instead of conjugation itself, ie $\mathrm{Ad}_g=\mathrm{T}_e(\mathrm{conj}_g):\mathrm{T}_eG\to \mathrm{T}_eG$, whereas $\mathrm{conj}_g:G\to G$? |
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16h |
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In coordinate-free relativity, how do we define a vector? sadly misses the point - I'm debating whether I should expand my comment into a 2nd answer... |
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17h |
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In coordinate-free relativity, how do we define a vector? re your clarification: it's part of your geometric model - it's up to you to map physical quantites to geometric ones in a meaningful way (and transformation laws follow from that); eg momentum is most naturally a covector (cf the Lagrangian formulation, contraction with a velocity to get an energy, minimal coupling to the em vector potential, ...), the em field strength is the curvature of a principal connection and thus a lie-algebra-valued 2-form, which you can contract with a gauge-independent charge (a co-adjoint orbit, in case of $U(1)$-connections just a number) to get a regular 2-form |
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1d |
answered | In coordinate-free relativity, how do we define a vector? |
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1d |
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Why distinguish between row and column vectors? let us continue this discussion in chat |
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1d |
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Why distinguish between row and column vectors? (sorry for mangling your name in my last comment): the definition of hypersurfaces becomes easier if you already have a concept of tangent vectors, making this definition of cotangent vectors arguably less fundamental; there's actually a more severe problem: unless I'm mistaken, you need additional structure (eg a volume form) to make the hypersurface (or rather its tangent subspace of codimension 1) characterise a unique cotangent vector |
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1d |
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Why distinguish between row and column vectors? @Murphid: tangent and cotangent vectors only appear on equal footing if you use the coordinate-based definition via their transformation laws; there are two other common definitions of tangent vectors (directional derivates and equivalence classes of curves), and the latter one makes a lot of sense from a 'physical' point of view; I'm not aware of an intrinsic definition of cotangent vectors that's equally 'nice' and does not rely on the definition of tangent vectors |
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1d |
answered | Why distinguish between row and column vectors? |
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1d |
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Why distinguish between row and column vectors? Why not define vectors to be things that act on covectors to product numbers instead? Because infinite-dimensional spaces need not be reflexive |
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May 13 |
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Physical interpretation of Poisson bracket properties @dmckee: notation is domain-specific, and it's quite common to use curlies for Poisson brackets, both in introductory and advanced literature |
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May 13 |
revised |
Physical interpretation of Poisson bracket properties added 13 characters in body |
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May 13 |
answered | Physical interpretation of Poisson bracket properties |
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May 10 |
revised |
If particles can find themselves spontaneously arranged, isn't entropy actually decreasing? add rational |
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May 10 |
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If particles can find themselves spontaneously arranged, isn't entropy actually decreasing? @LubošMotl: care to comment on the revision to my answer? |
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May 10 |
revised |
If particles can find themselves spontaneously arranged, isn't entropy actually decreasing? add rational |
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May 10 |
comment |
Are there problems solvable with Newtonian physics, GR and QM? two historically relevant examples would be the perihelion precession of mercury (GR vs Newton) and black body radiation (classical Rayleigh–Jeans law vs quantum Planck's law) |
