29,540 reputation
22777
bio website joshphysics.com
location Los Angeles
age 28
visits member for 1 year, 9 months
seen 56 mins ago

New project: phermi.com

Let me know if you know of any hard physics problems with clever solutions. (email listed to the left)

Personal website: joshphysics.com

Currently a lecturer at the UCLA Department of Physics and Astronomy.

Ph.D. theoretical high energy physics, UCLA.

BA/BS in physics/math, UC Berkeley.


2h
comment Group representations as vectors
Also, it's not only physicists that use this terminology, mathematicians abuse the terminology often as well. In fact, if you look on the first page of Fulton and Harris' book on representation theory (a hardcore math book to be sure) you'll find the sentence "When there is little ambiguity about the map $\rho$ (and, we're afraid, even sometimes when there is) we sometimes call $V$ itself a representation of $G$..."
2h
comment Group representations as vectors
Comment on question (v3): A group one considers in physics may not be a Lie group, and even when it is, the matrices that represent group elements don't typically satisfy the structure relations of the Lie algebra, but representations of Lie algebra elements do. One motivation for identifying the vector space with the representation is that if, for example, a representation decomposes as a direct sum if irreps, then the vector space can be thought of as decomposing in a corresponding way into a direct sum of subspaces. In this sense, the vector space isn't entirely "passive."
2h
revised Representations of Lie algebras in physics
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2h
comment Group representations as vectors
"...the vectors in $V$ are almost always quantum states..." If by "almost always" you mean anything like "almost everywhere" in math, then I think this rather severe hyperbole. What about group representations in classical mechanics, GR, any classical field theory for that matter, and even quantum field theory where the vectors are elements of the target spaces of fields?
16h
reviewed Approve suggested edit on Bra-ket notation, Bits, & Superposition
1d
revised Hilbert space of harmonic oscillator: Countable vs uncountable?
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1d
revised Hilbert space of a free particle: Countable or Uncountable?
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2d
awarded  Talkative
Oct
17
comment Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?
@Qmechanic Good point. I added an addendum for stress.
Oct
17
revised Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?
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Oct
17
comment Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?
@DanielSank I know :). Thanks for the positive feedback.
Oct
17
comment Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?
@DanielSank I originally thought that was simply the answer, but if you look at the calculation, you'll see that other normalization choices "make the Fourier transform work" as well, so that's not a particularly compelling (or correct) answer as far as I can tell. In fact, some books normalize both the position and momentum eigenstates to be Dirac orthonormal (no $2\pi$'s). See Cohen-Tannoudji for example.
Oct
17
comment Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?
@user3736508 Sure thing. I added an extra section attempting to motivate the convention, but seeing as how that convention isn't universal, it's clear that it is, in fact, just a convention, so in the end, it's really just an aesthetic choice as far as I can tell.
Oct
17
revised Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?
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Oct
17
revised What is $\Delta t$ in the time-energy uncertainty principle?
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Oct
17
answered Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?
Oct
15
answered Motivating Complexification of Lie Algebras?
Oct
12
revised Why is the canonical partition function the Laplace transform of the microcanonical partition function?
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Oct
10
comment Physics First: Where is Science Education Today?
Does one need to apply this approach to all students learning science? Perhaps certain students find physics easier and more enjoyable while others find that to be true of biology or chemistry. Perhaps encouraging the former group, or students who have solid arithmetic, algebra, or even calculus skills to pursue a physics first or even a physics intensive track would work nicely? On a more fundamental level, maybe physics first helps accomplish certain educational goals but not others, so maybe defining these goals are worth thinking about. Sorry for being a bit tangential.
Oct
8
comment List of known universality classes
@StevenMathey Right: I'm trying to help you out here so that this question doesn't get closed.