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seen Aug 19 at 12:23

Technically, it was a branch of metamathematics, usually called metamathics. Metamathics; the investigation of the properties of Realities (more correctly, Reality-fields) intrinsically unknowable by and from their own, but whose general principles could be hazarded at.

Metamathics led to everything else, it led to the places that nobody else had ever seen or heard of or previously imagined. . . . It was like a drug; an ultimately liberating, utterly enhancing, unadulterably beneficial, overpoweringly glorious drug for the intellect of machines as far beyond the sagacity of the human mind as they were beyond its understanding.

This was the way the Minds spent their time. They imagined entirely new universes with altered physical laws, and played with them, lived in them and tinkered with them, sometimes setting up the conditions for life, sometimes just letting things run to see if it would arise spontaneously, sometimes arranging things so that life was impossible but other kinds and types of bizarrely fabulous complications were enabled.

Some of the universes possessed just one tiny but significant alteration, leading to some subtle twist in the way things worked, while others were so wildly, aberrantly different it could take a perfectly first-rate Mind the human equivalent of years of intense thought even to find the one tenuously familiar strand of recognizable reality that would allow it to translate the rest into comprehensibility. Between those extremes lay an infinitude of universes of unutterable fascination, consummate joy and absolute enlightenment. All that humanity knew and could understand, every single aspect, known, guessed at and hoped for in and of the universe was a like a mean and base mud hut compared to the vast, glittering cloud-high palace of monumentally exquisite proportions and prodigious riches that was the metamathical realm. Within the infinities raised to the power of infinities that those metamathical rules provided, the Minds built their immense pleasure domes of rhapsodic philosophical ecstasy.

That was where they lived. That was their home. When they weren't running ships, meddling with alien civilizations or planning the future course of the Culture itself, the Minds existed in those fantastic virtual realities, sojourning beyondward into the multidimensional geographies of their unleashed imaginations, vanishingly far away from the single limited point that was reality.

The Minds had long ago come up with a proper name for it; they called it the Irreal, but they thought of it as Infinite Fun. That was what they really knew it as, The Land of Infinite Fun.

It did the experience pathetically little justice.

Iain M. Banks, Excession, 4.III


Aug
16
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@udrv Also, yes, those more general conditions are called Robin boundary conditions.
Aug
16
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
So $n\cdot j = 0$ is equivalent to the condition that the boundary term vanishes when you integrate $\langle \Psi|\Psi^*\rangle$ by parts and that's exactly what you need to have a self-adjoint extension of the Hamiltonian.
Aug
15
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@arivero What do you mean by "teleportation"? Identification of different sides of the domain, a la gluing a square into a torus?
Aug
15
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@udrv So is this the idea? For a domain $D$, there are essentially two ways of fencing in a particle: either impose an infinite potential outside $D$, or impose the condition that the probability current never cross the boundary. Either way, the particle stays confined to the domain.
Aug
13
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@arivero I'm not sure if that intuition extends to higher dimensions. For instance, in a multiply connected domain in the plane, like a thickened figure-8, collapsing the boundary to a point produces a space that's not even a manifold. In 1d, any function on an interval can be extended to a periodic function, which then descends to a function on the appropriate circle, so the analogue in higher dimensions would be functions on the appropriate torus. That's my mathematical intuition, at least; I can't really speak to the physical intuition.
Aug
13
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
Sorry about the sign issues. I added a note to the OP about this. I'm digesting your answer and will be back shortly with more questions, I'm sure.
Aug
13
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@jac What about domains where the boundary is not totally geodesic? For instance, a disk in $\mathbb{R}^2$ where half the boundary is given Dirichlet and the other half is given Neumann?
Aug
13
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@arivero There's not room in the post to add a third column, so I added a postscript instead :)
May
11
comment Why do all the atoms of a radioactive substance not decay at the same time?
I don't understand why you think me saying the derivative of an exponentially decrease is again an exponential decrease is controversial, but I have deleted the contentious section until I have time to rewrite it.
May
11
comment Why do all the atoms of a radioactive substance not decay at the same time?
@user121330 Ah, thank you. My probability-fu was weak: the Poisson distribution measures the probability of decay in a given length of time.
May
10
comment $B$ field around an infinite wire symmetry argument
As a practical example, if I make a very long current-carrying wire in a lab and run a current through it, the field I measure near the center of the wire will be the circulating magnetic field plus $b$ from the Earth's (and the Sun's, etc) magnetic field.
Jul
12
comment Velocity is “Distance/Time” in a particular direction OR simply “Displacement/Time”?
@AmireBendjeddou Displacement indicates direction.
Jul
12
comment Velocity is “Distance/Time” in a particular direction OR simply “Displacement/Time”?
They are the same thing, infinitesimally.