Video: Why we live in 3D
“THE ONLY way is up.” An earnest student of our physical realities might find room to dispute this jollying phrase. There is also down, and, for that matter, left, right, forwards and backwards. Six ways to go. Then again, the further up you go, the less down you are, and similarly for left and right, forwards and backwards. So that’s three independent directions to move in – gravity and local obstacles permitting.
It is a fact so bald that we rarely stop to ask an even balder question: why?
Physicists have wrestled with this perplexing question of space’s essential three-ness for a good while now – not, it must be said, with much success. Our best theories of nature supply no clue as to why space might have three dimensions, rather than two, four or 5.2. Even worse, the drive for ever-grander replacements keeps finding hints that the magic number is anything but three.
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Perhaps we have been trying too hard. The latest attempt to explain the mystery of three suggests an answer is under our nose – hidden in the workings of physics we already know.
For a long while, our understanding of the world progressed rather nicely by ignoring space. Newton first formalised the concept in 1687 when he introduced his theory of gravity. For him, space and time were real enough, but no more than an impassive backdrop against which more interesting stuff such as falling apples and orbiting planets happened. It was Einstein who forced us to buck up our ideas. In his general theory of relativity, formulated in 1916, space becomes a dynamic entity. Interwoven with time into a 4D space-time, it is bent and buckled by matter to create the force we call gravity.
Oddly, though, general relativity says nothing about space’s most obvious quality, its three-ness. With a little tweaking, relativity’s mathematics works fine in any number of dimensions.
Some have filled the gap with “anthropic” reasoning: space must be just so for us to pose questions about it (see “We’re here because we’re here“). Others think the answer is there within physics, we just haven’t found it yet. A theory of quantum gravity would unify general relativity with quantum mechanics, which describes everything that is not gravity. It would be valid in the first, searing moments of the universe when space-time was itself a tiny, jumbled quantum mess. Perhaps the three-dimensionality of space, not to mention the one-dimensionality of time, emerged from physics unknown during this hectic era. “Emergent space-time is the modern view,” says of the University of Cambridge.
It is a difficult one to test. Vast machines such as the Large Hadron Collider at CERN near Geneva, Switzerland, smash particles together at humongous energies to push back our understanding of physics towards those energetic first few moments, but they are still a way off. So far, no experiment has produced any lead in the problem of quantum gravity.
As far as the mystery of three is concerned, it doesn’t help that the most popular theoretical ideas seem to work in anything other than 3D. String theory, for instance, requires at least six extra spatial dimensions (see “This one goes up to eleven“). Causal dynamical triangulations, an alternative approach, depicts a quantum network of 2D elements that evolves into a macroscopic 3D space. Loop quantum gravity, meanwhile, replaces a smooth space-time with a version that is grainy or foamy at the smallest scales – but it takes the number 3 as an input and has done with it, says one of the theory’s originators, of the University of Aix-Marseille in France. “This just does not seem to be at all one of those questions that science has tools for figuring out at the moment,” he says.
“It doesn’t help that most grand theories of nature seem to work in anything other than 3D”
begs to differ. A theoretical physicist at the Perimeter Institute in Waterloo, Ontario, Canada, he thinks we can solve the question of three – and he reckons the answer lies buried in the roots of a theory we already have. Quantum theory certainly describes the physical world phenomenally well, but it is offensive to our conventional views of reality, allowing objects to be in two states or places at once, for instance, and generally denying principles we hold dear, such as clear lines of cause and effect. The traditional response of physicists has been to ignore these little local difficulties – – or to tie themselves in knots with all manner of interpretations, from strange influences of experimenters on experiments to the existence of many bifurcating worlds.
Müller had an idea to sidestep these “philosophical pitfalls”, as he calls them: derive a basis for quantum mechanics from examining what experience allows. More than a century ago, we similarly gained insight into the laws of thermodynamics by observing what could and couldn’t happen in nature – you can’t have a perpetual-motion machine that gives you energy for nothing, for example. “One thinks very carefully about what the formulas mean, and what can actually be done concretely or operationally,” says Müller.
Working with Lluis Masanes of Bristol University in the UK, Müller looked at a situation where a sender and receiver set out to exchange information encoded in quantum states – the basis of the very real, super-secure technique of quantum cryptography. Countless experiments have shown that quantum information held by the sender and receiver is correlated above and beyond what is possible in the classical world. Change the quantum state of photons used to encode the message at one end, say, and you might see spooky, instantaneous changes in the states of photons at the other end.
The duo started with a few “reasonable” assumptions about how the physical universe surrounding such a sender and receiver worked. It must have a certain number of dimensions, including one of time, for example, and there must be some way for information to flow within it. In addition, they assumed that at least some of the physical processes in the world worked randomly, although they did not specify how randomly.
What follows is a dense mathematical proof with a surprising outcome. Not only is quantum theory the only theory that can supply the degree of randomness and correlation seen in nature – but it can only do so if space is 3D, too ().
Roots of reality
That could be just a mathematical coincidence. Quantum states are described not by 1D real numbers, which all lie on a single line, but by 2D complex numbers that represent points on a plane. The way these numbers interact to produce a complete description of objects such as photons that can be in more than one state at once naturally sketches out a 3D sphere describing all those possible states. Perhaps this result is just emphasising how the dimensionality of basic quantum objects and the dimensionality of space happen to be the same.
Müller thinks not: he thinks it points to an inextricable link between space’s geometry and the degree of probability inherent in quantum theory. If so, the roots of relativity and quantum theory would be embedded in the way information is exchanged in the cosmos, suggesting where to look for any unifying theories. “It offers a clue that the notion of information will be an important part of quantum gravity,” says Müller.
“The roots of relativity and quantum theory might lie in the way information is exchanged in the cosmos”
Since his and Masanes’s paper was published, Borivoje Dakic and at the University of Vienna, Austria, have used similar arguments from information theory to restrict the dimensionality of space. They, too, show that only in a 3D universe can quantum mechanics hold true – at least in a universe where microscopic objects interact “pairwise” with each other, as they appear to in ours. Relax this restriction so that three or more quantum systems can interact at the same time, and higher-dimensional universes are just fine ().
of Brunel University in London thinks we must treat such arguments with care. They start out with a few assumptions that seem indisputable, but additional ones are often “smuggled in” further on, he says. “Somewhere along the line the rabbit has to enter the hat.”
He thinks the conclusions we draw depend critically on the mathematical language we use. There are consistent representations of quantum mechanics using not 2D complex numbers, but 4D “qܲٱԾDzԲ” and 8D “octonions”. Brody and his colleague Eva-Maria Graefe of Imperial College London showed last year that in the quaternion formulation, five dimensions of space naturally drop out, while octonion quantum mechanics demands nine ().
Back to square one? Not quite. These reformulations predict subtly different outcomes for certain experiments than does conventional quantum mechanics, including different degrees of correlations between particles. If quantum interactions truly are the source of space’s dimensionality, that suggests new ways to pin the answer down in the lab without the huge expenditure on machines such as the Large Hadron Collider, says Brody.
, also of Imperial College, thinks it is an intriguing avenue to investigate, but cautions against jumping on the latest bandwagon. “Remember when chaos theory was meant to solve anything and everything?” he says. His hunch is that the dimensionality of space might be too human a concept for us to explain – a view articulated by the philosopher Immanuel Kant in the 18th century. Kant described space as “a subjective condition of sensibility” whose only significance was in allowing us to specify the relationship between other objects. “Perhaps it is just a variable that these particular monkeys evolved to find useful in their quest for bananas,” says Rudolph.
Newton and Einstein, both proponents of the reality of space, presumably would have approved of the ongoing attempts to explain something so basic as its three-ness. In the end, though, we might be forced to accept it as a brute fact, says of the University of California in San Diego. “It might be an unexplained explainer.”
We’re here because we’re here
Life in fewer than three dimensions would be unimaginably different. Our minds are made of neurons criss-crossing in 3D, cylindrical tubes provide our digestive tracts and 3D helices transmit our genetic blueprints. Two-dimensional life might be too limited to even ask questions about the dimensionality of space.
Other arguments might discount higher numbers. In our world of three spatial dimensions, the gravitational force between two objects falls off with the square of the distance between them. In two dimensions, it would decrease far slower, in proportion to the distance. In a 4D world, it would fall off more steeply as the cube of the distance.
In both cases, planetary orbits would be unstable: the merest asteroid strike would be enough to cause a planet to spiral into its parent star or fly out into deep space. Not that there would be planets or stars – similar stability arguments apply to the electromagnetic force that holds electrons in orbit around atomic nuclei.
The dimensionality of space, then, is one of those numbers in physics, like the constants that determine the strength of electromagnetism and gravity, that dictate whether complex life can evolve. “In my opinion it is the only known explanation,” says physicist of Stanford University in California. Atoms and molecules would not exist in the form that we know them in anything other than three dimensions.
Others are unimpressed by such arguments. “They are nonsense,” says Carlo Rovelli of the University of Aix-Marseille in France. “If we lived in six dimensions, many people would find equally good arguments to show that only six dimensions are possible.” Critics also point out that, unlike constants of nature, the dimensionality of space is a small, perfect integer. It is hard to imagine something that rounded dropping out of some random environmental process.
This one goes up to eleven
String theory says the fundamental units of reality are not point-like, zero-dimensional particles, but one-dimensional strings. The tricky bit is that these strings need at least nine spatial dimensions to move in to preserve the theory’s mathematical consistency. The extra six are assumed to be “compactified” at scales of a billionth of a trillionth of a trillionth of a centimetre.
At least, that was the old-fashioned view. These days, the very notion of dimensionality in string theory is vague, says of Harvard University. In an overarching 11-D mega-idea called M-theory, physics plays out on mathematical objects called branes that have an arbitrary number of dimensions fewer than 10.
In this picture our universe could be bound to a “3-brane” within a higher-dimensional space, floating around with numerous others that are invisible, but right in front of our eyes. Or perhaps not; perhaps our 3D theories are holograms hailing from some higher-dimensional structure. The many different ways to play with string theory’s imaginary dimensions present a “landscape” of innumerable potential universes, and no principle to determine which one is right.
That doesn’t stop people trying. In 2005 of Harvard University and Andreas Karch of the University of Washington in Seattle showed how a universe that starts out with equal numbers of branes and anti-branes – branes with opposite orientations – would evolve into one dominated by 3-branes and 7-branes. Vafa’s own work with Robert Brandenberger of McGill University in Montreal, Canada, suggests that the dimensionality of a string-theory universe is intimately entwined with its expansion: if the universe began as an unimaginably small 9D ball of string that suddenly began to unwind, three would become infinitely large and the rest would remain curled up.
That idea has recently been revived by of Columbia University in New York and his colleagues (). Even so, Vafa admits the case is not closed. “Right now, we cannot claim that string theory predicts the dimensionality of space,” he says.
This article appeared in print under the headline “Seeing triple”
