Last week’s

*New Scientist’s*cover declared*SPACE versus TIME; one has to go. But which?*(15 June 2013). This served as a rhetorical introduction to physics' most famous conundrum: the irreconcilability of its 2 most successful theories - quantum mechanics and Einstein’s theory of general relativity - both conceived at the dawn of the so-called golden age of physics in the early 20^{th}Century.
The feature article (pp. 35-7) cites a
number of theoretical physicists including Joe Polchinski (University of
California, Santa Barbara), Sean Carroll (California Institute of Technology,
Pasadena), Nathan Seiberg (Institute for Advanced Study, Princeton), Abhay
Ashtekar (Pennsylvania University), Juan Malcadena (no institute cited) and
Steve Giddings (also University of California).

Most scientists and science commentators
seem to be banking on String Theory to resolve the problem, though both its
proponents and critics acknowledge there’s no evidence to separate it from
alternative theories like loop quantum gravity (LQG), plus it predicts 10
spatial dimensions and 10

^{500}universes. However, physicists are used to theories not gelling with common sense and it’s possible that both the extra dimensions and the multiverse could exist without us knowing about them.
Personally, I was intrigued by Ashtekar’s
collaboration with Lee Smolin (a strong proponent of LQG) and Carlo Rovelli
where ‘Chunks of space [at the Planck scale] appear first in the theory, while
time pops up only later…’ In a much earlier publication of

*New Scientist*on ‘Time’ Rovelli is quoted as claiming that time disappears mathematically: “For me, the solution to the problem is that at the fundamental level of nature, there is no time at all.” Which I discussed in a post on this very subject in Oct. 2011.
In a more recent post (May 2013) I quoted
Paul Davies from

*The Goldilocks Enigma*: ‘[The] vanishing of time for the entire universe becomes very explicit in quantum cosmology, where the time variable simply drops out of the quantum description.’ And in the very article I’m discussing now, the author, Anil Ananthaswamy, explains how the wave function of Schrodinger’s equation, whilst it evolves in time, ‘…time is itself not part of the Hilbert space where everything else physical sits, but somehow exists outside of it.’ (Hilbert space is the ‘abstract’ space that Schrodinger’s wave function inhabits.) ‘When we measure the evolution of a quantum state, it is to the beat of an external timepiece of unknown provenance.’
Back in May 2011, I wrote my most popular post ever: an exposition on Schrodinger’s equation, where I deconstructed the
famous time dependent equation with a bit of sleight-of-hand. The
sleight-of-hand was to introduce the quantum expression for momentum (p

_{x}= -i*d/dx) without explaining where it came from (the truth is I didn’t know at the time). However, I recently found a YouTube video that remedies that, because the anonymous author of the video derives Schrodinger’s equation in 2 stages with the*~~ h~~**time independent**version first (effectively the RHS of the time dependent equation). The fundamental difference is that he derives the expression for p_{x}= i*d/dx, which I now demonstrate below.*~~
h~~
Basically the wave function, which exploits
Euler’s famous equation, using complex algebra (imaginary numbers) is expressed
thus: Ψ = Ae

^{i(kx−ωt)}^{}
If one differentiates this equation wrt x
we get ik(Ae

^{i(kx−ωt)}), which is ikΨ. If we differentiate it again we get d^{2}Ψ/dx^{2}= (ik)^{2}Ψ.
Now k is related to wavelength (λ)
by 2π such that k = 2π/λ.

And from Planck’s equation (E =

*hf*) and the fact that (for light) c =*f*λ we can get a relationship between momentum (p) and λ. If p = mc and E = mc^{2}, then p = E/c. Therefore p =*hf*/*f*λ which gives**p = h/λ**effectively the momentum version of Planck’s equation. Note that p is related to wavelength (space) and E is related to frequency (time).
This then is the quantum equation for momentum based on h
(Planck’s constant) and λ. And, of course, according to Louis de Broglie,
particles as well as light can have wavelengths.

And if we substitute 2π/k for λ we get p = hk/2π which can be reformulated as

k = p/~~h~~ where ~~h~~ = h/2π.

And substituting this
in (ik)~~h~~)

^{2}we get –(p/^{2}{ i^{2}= -1}
So Ψ d~~h~~)

^{2}/dx^{2}=^{ }-(p_{x}/^{2}Ψ
Making p the subject of the equation we get p

_{x}^{2}= -~~ h~~^{2}d^{2}/dx^{2}(Ψ cancels out on both sides) and I used this expression in my previous post on this topic.
And if I take the
square root of p

_{x}^{2}I get p_{x}= i*d/dx, the quantum term for momentum.*~~ h~~
So the quantum version
of momentum is a consequence of Schrodinger’s equation and not an input as I
previously implied. Note that √-1 can be i or –i so p

_{x}can be negative or positive. It makes no difference when it’s used in Schrodinger’s equation because we use p_{x}^{2}.
If you didn’t follow
that, don’t worry, I’m just correcting something I wrote a couple of years ago
that’s always bothered me. It’s probably easier to follow on the video where I found the solution.

But the relevance to
this discussion is that this is probably the way Schrodinger derived it. In
other words, he derived the term for momentum first (RHS), then the time
dependent factor (LHS), which is the version we always see and is the one
inscribed on his grave’s headstone.

This has been a
lengthy and esoteric detour but it highlights the complementary roles of space
and time (implicit in a wave function) that we find in quantum mechanics.

Going back to the

*New Scientist*article, the author also provides arguments from theorists that support the idea that time is more fundamental than space and others who believe that neither is more fundamental than the other.
But reading the
article, I couldn’t help but think that gravity plays a pivotal role regarding
time and we already know that time is affected by gravity. The article keeps
returning to black holes because that’s where the 2 theories (quantum mechanics
and general relativity) collide. From the outside, at the event horizon, time
becomes frozen but from the inside time would become infinite (everything would
happen at once) (refer

**Addendum**below). Few people seem to consider the possibility that going from quantum mechanics to classical physics is like a phase change in the same way that we have phase changes from ice to water. And in that phase change time itself may be altered.
Referring to one of
the quotes I cited earlier, it occurs to me that the ‘external
timepiece of unknown provenance’ could be a direct consequence of gravity, which
determines the rate of time for all objects in free fall.

**Addendum:**Many accounts of the event horizon, including descriptions in a recent special issue of

*Scientific American; Extreme Physics*(Summer 2013), claim that one can cross an event horizon without even knowing it. However, if time is stopped for 'you' according to observers outside the event horizon, then their time must surely appear infinite to ‘you’, to be consistent. Kiwi, Roy Kerr, who solved Einstein's field equations for a rotating black hole (the most likely scenario), claims that there are 2 event horizons, and after crossing the first one, time becomes space-like and space becomes time-like. This infers, to me, that time becomes static and infinite and space becomes dynamic. Of course, no one really knows, and no one is ever going to cross an event horizon and come back to tell us.

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