In my last post I made
reference to a comment Noson Yanofsky made in his book, The Outer limits of Reason, whereby he responded to a student’s
question on quantum mechanics: specifically, why does quantum mechanics require
complex algebra (√-1)
to render it mathematically meaningful?
Complex numbers always
take the form a + ib, which I explain in detail elsewhere, but it is best
understood graphically, whereby a exists on the Real number line and b lies on
the ‘imaginary’ axis orthogonal to the Real axis. (i = √-1,
in case you’re wondering.)
In last week’s New Scientist (25 January 2014,
pp.32-5), freelance science journalist, Matthew Chalmers, discusses the work of
theoretical physicist, Bill Wootters of Williams College, Williamstown,
Massachusetts, who has attempted to rid quantum mechanics of complex numbers.
Chalmers introduces
his topic by explaining how i (√-1) is not a number as we
normally understand it – a point I’ve made myself in previous posts. You can’t
count an i quantity of anything, and,
in fact, I’ve argued that i is best
understood as a dimension not a number per se, which is how it is represented
graphically. Chalmers also alludes to the idea that i can be perceived as a dimension, though he doesn’t belabour the
point. Chalmers also gives a very brief history lesson, explaining how i has been around since the 16th
Century at least, where it allowed adventurous mathematicians to solve certain
equations. In fact, in its early manifestation it tended to be a temporary
device that disappeared before the final solution was reached. But later it
became as ‘respectable’ as negative numbers and now it makes regular appearances
in electrical engineering and analyses involving polar co-ordinates, as well as
quantum mechanics where it seems to be a necessary mathematical ingredient. You
must realise that there was a time when negative numbers and even zero were treated
with suspicion by ancient scholars.
As I’ve explained in
detail in another post, quantum mechanics has been rendered mathematically as a
wave function, known as Schrodinger’s equation. Schrodinger’s equation would
have been stillborn, as it explained nothing in the real world, were it not
for Max Born’s ingenious insight to square the modulus (amplitude) of the wave
function and use it to give a probability of finding a particle (including
photons) in the real world. The point is that once someone takes a measurement
or makes an observation of the particle, Schrodinger’s wave function becomes
irrelevant. It’s only useful for making probabilistic predictions, albeit very
accurate ones. But what’s mathematically significant, as pointed out by
Chalmers, is that Born’s Rule (as it’s called) gets rid of the imaginary
component of the complex number, and makes it relevant to the real world with
Real numbers, albeit as a probability.
Wootters ambition to
rid quantum mechanics of imaginary numbers started when he was a PhD student,
but later became a definitive goal. Not surprisingly, Chalmers doesn’t go into
the mathematical details, but he does explain the ramifications. Wootters has
come up with something he calls the ‘u-bit’ and what it tells us is that if we
want to give up complex algebra, everything is connected to everything else.
Wootters expertise is
in quantum information theory, so he’s well placed to explore alternative
methodologies. If the u-bit is a real entity, it must rotate very fast, though
this is left unexplained. Needless to say, there is some scepticism as to its
existence apart from a mathematical one. I’m not a theoretical physicist, more
of an interested bystander, but my own view is that quantum mechanics is
another level of reality – a substrate, if you like, to the world we interact
with. According to Richard Ewles (MATHS 1001, pp.383-4): ‘…the wave function Ψ permeates all of space…
[and when a measurement or observation is made] the original wave function Ψ is no longer a valid description of the state of the particle.’
Many physicists also believe that
Schrodinger’s equation is merely a convenient mathematical device, and
therefore the wave function doesn’t represent anything physical. Whether this
is true or not, its practical usefulness suggests it can tells us something important
about the quantum world. The fact that it ‘disappears’ or becomes irrelevant,
once the particle becomes manifest in the physical world, suggests to me that
there is a disjunct between the 2 physical realms. And the fact that the
quantum world can only be understood with complex numbers simply underlines
this disjunction.
