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seen Aug 26 at 8:22

Aug
10
comment QM without complex numbers
@Problemania: $U(n)=Sp(2n,\mathbb R)\cap O(2n)\cap GL(n,\mathbb C)$; however, the intersection of any 2 of the groups on the RHS is sufficient, and in particular $U(n)=Sp(2n,\mathbb R)\cap O(2n)$; complexity arises naturally when we deal with compatible symplectic and orthogonal structures; of course it's equally valid to say that symplectic structures arise naturally from compatible orthogonal and complex structures or orthogonal ones from compatible symplectic and complex ones; but complex structures are arguably less well motivated from a physical (or perhaps 'philosophical') point of view
Aug
3
comment Why can't we do some basic algebra in tensor calculus?
assuming a background in basic linear algebra, think in terms of matrices (which are rank-2 tensors): not all matrices are invertible, and those that are generally aren't orthogonal (ie $A^t\not=A^{-1}$)
Jul
31
comment Which function denotes the energy of thermal motion within a system?
$\frac32 NkT$ for translational motion as well as appropriately weighted multiples of $NkT$ for vibrational and rotational motion as long as quantum effects can be neglegted (ie the spacing of the discrete energy levels is much smaller than $kT$)
Jul
27
comment Can statistical mechanics explain the second law completely?
Let us continue this discussion in chat.
Jul
27
comment Can statistical mechanics explain the second law completely?
I also never understood how your 'logical arrow of time' isn't killed by Loschmidt's paradox; in case of Markov processes, it apparently is, but I have yet to read the cited paper in detail
Jul
27
comment Can statistical mechanics explain the second law completely?
@LubošMotl: note that Botzmann's position on entropy evolved over the years in reaction to cricism by others like Poincaré and Zermelo; in Zu Hrn. Zermelo’s Abhandlung "Ueber die mechaniche Erklärung irreversibler Vorgänge", he mentions the idea that the universe started from an improbable low-entropy state and that there might be regions of the universe with opposite directions of the thermodynamic arrows of time (to which the subjective arrow of time would assumably align); Schrödinger (and probably others) argued that this latter case is impossible under certain (mild) assumptions
Jul
26
comment Can statistical mechanics explain the second law completely?
@BenCrowell: arXiv:0809.1304 cites Uffink, "Compendium of the foundations of statistical physics" (sections 7.6, 7.7) and Bacciagaluppi, "Probability and time symmetry in classical Markov processe", which both essentially conclude that there is no such T-breaking without further assumptions (which was also Boltzmann's position)
Jul
21
comment What is “Energy” of a vacuum in the context of quantum theory?
The problem with 'true nothing' is that it is unphysical: Take a volume of space and remove 'everything', and you won't end up with 'nothing', but with the vacuum, which, philosophically speaking, is a 'something'. Krauss goes a step beyond the vacuum and assumes that spacetime as well as the laws of physics as we know them emerge dynamically (eg via symmetry breaking) from the true groundstate of a theory of quantum gravity. This groundstate is what he calls 'nothing'.
Jul
16
comment Hamiltonian Noether's theorem in classical mechanics
the key word to look for would be 'moment map'; see eg ncatlab.org/nlab/show/…
Jul
12
comment If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
But, but, but: [ insert rant about Noether's 2nd theorem, general covariance and the gravitational stress-energy pseudo-tensor ]
Jul
12
comment If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
@brightmagus: the gravitational potential fixes conservation of energy in curved space-time in a similar way to how the centrifugal potential fixes energy conservation in rotating frames of reference; in the latter case, it all works out because there's a globally inertial frame in which energy conservation holds, in the former case, it works out because energy conservation holds trivially in any generally covariant theory; sorry, but that's the best I can do...
Jul
12
comment If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
@brightmagus, it's hard to argue with forces in GR: if you think of dark energy due to a cc as some fluid, you have to wonder why gravitational attraction acts repulsively if you choose the right equation of state (negative pressure), without any gradients being present; I'm unsure what kind of explanation you're looking for beyond "that's what Noether's theorem tells us"; essentially, energy conservation boils down to the first Friedmann equation $\rho+\rho_\Lambda-3\dot a^2/8\pi Ga^2=0$, which many physicists are uncomfortable with, claiming it's not a proper conservation law
Jul
12
comment If empty space has energy, and space is expanding, is this energy equally distributed as space expands?
@brightmagus: Noether's second theorem applies to general relativity and yields a 'trivial' law of energy conservation, where the contribution due to the cosmological constant gets cancelled by the negative potential energy of the gravitational field; this law of conservation of energy can be expressed in terms of a gravitational stress-energy pseudo-tensor, but there's no tensorial expression for gravitational stress-energy as general relativity does not differentiate between gravity and inertia
Jul
8
comment What kind of manifold can be the phase space of a Hamiltonian system?
topological restrictions aren't placed by authors, but by existence of a symplectic form; Wikipedia lists orientability and non-trivial de-Rham cohomology $H^2(M)$ (ruling out all spheres except the 2-sphere)
Jul
8
comment Why does a flat universe imply an infinite universe?
For example the surface of a torus is flat but finite slight correction: the surface of a torus can be flat, but the generic 2-dimensional torus embedded in $\mathbb R^3$ isn't; Wikipedia informs me that it cannot be flat if the embedding is at least $\mathcal C^2$, but an explicit $\mathcal C^1$ embedding has been found rather recently (April 2012); see eg gipsa-lab.fr/~francis.lazarus/Hevea/Presse/index-en.html for some pictures
Jul
5
comment Equivalence between Hamiltonian and Lagrangian Mechanics
so we just vary $v,p$ independently and do the integration by parts on $\delta\dot q$ as usual to end up with $\delta S = \int(\dot q-v)\delta p + (\partial L/\partial q - \mathrm dp/\mathrm dt)\delta q + (\partial L/\partial v - p)\delta v + \text{boundary terms}$; I'll have to meditate on that for a bit...
Jul
5
comment Equivalence between Hamiltonian and Lagrangian Mechanics
note that due to its dependence on $\dot q^i$, $L_E$ is not a function on some (extended) phase space, but a functional(?!)
Jul
4
comment What predictions can a quantum gravity theory make?
Why no mention of BICEP2 - or do you expect their discovery to turn to dust?
Jul
4
comment What are the alternative theories of dark energy? ($w \neq -1$)
Hajdukovic tries to explain dark matter and energy both by gravitational vacuum polarization due to repulsion between virtual matter and anti-matter particles; in arXiv:1201.4594, he calculates a current $w_\text{eff}\approx -0.99$, which will eventually approach $w_\text{eff}=-1/3$
Jul
3
comment Inflation in a closed universe or a stage 1 multiverse?
Just having edited the question, perhaps going with ininite and finite instead of open/non-compact and closed/compact would be even clearer?