Quantum mechanics without complex algebra

 

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 16^th 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.

The Introspective Cosmos

I haven’t written anything meaty for a while, and I’m worried I might lose my touch. Besides, I feel the need to stimulate my brain and, hopefully, yours in the process.

Just before Christmas, I read an excellent book by Noson S. Yanofsky, titled: The Outer Limits of Reason; What Science, Mathematics, and Logic CANNOT Tell Us. Yanofsky is Professor in the Department of Computer and Information Science at Brooklyn College and The Graduate Center of the City of University of New York. He is also co-author of Quantum Computing for Computer Scientists (which I haven’t read).

Yanofsky’s book (the one I read) covers a range of topics, including classical and quantum physics, chaos theory, determinism, free will, Godel’s Incompleteness Theorem, the P-NP problem, the anthropic principle and a whole lot more. The point is that he is well versed in all these areas, yet he’s very easy to read. His fundamental point, delineated in the title, is that it is impossible for us to know everything. And there will always be more that we don’t know compared to what we do know. Anyone with a remote interest in epistemology should read this book. He really does explain the limits of our knowledge, both theoretically and practically. At the end of each section he gives a synopsis of ‘further reading’, not just a list. I found the book so compelling, I even read all the ‘Notes’ in the appendix (something I rarely do).

Along the way, he explains things like countable infinities and uncountable infinities and why it is important to make the distinction. He also explains the difference between computing problems that are simply incomputable and computing problems that are computable but would take more time than the Universe allows, even if the computer was a quantum computer.

He discusses, in depth, philosophical issues like the limitations of mathematical Platonism, and provides compelling arguments that the mathematics we use to describe physical phenomena invariably have limitations that the physical phenomena don’t. In other words, no mathematical equation, no matter its efficacy, can cover all physical contingencies. The physical world is invariably more complex than the mathematics we use to interpret it, and a lot of the mathematical tools we use deal with large scale averages rather than individual entities – like the universal gas equation versus individual molecules.

He points out that there is no ‘fire in the equations’ (as does Lee Smolin in Time Reborn, which I’ve also read recently) meaning mathematics can describe physical phenomena but can’t create them. My own view is that mathematics is a code that only an intelligence like ours can uncover. As anyone who reads my blog knows, I believe mathematics is largely discovered, not invented. Marcus du Sautoy presented a TV programme called The Code, which exemplifies this view. But this code is somehow intrinsic in nature in that the Universe obeys laws and the laws not only require mathematics to quantify them but, without mathematics, we would not know their existence except, possibly, at a very rudimentary and uninformed level.

Yanofsky discusses Eugene Wigner’s famous declaration concerning ‘The Unreasonable Effectivenessof Mathematics’ and concludes that it arises from the fact that we use mathematics to probe the physical world, and that, in fact, leaving physics aside, there is a ‘paucity of mathematics in general science’. But in the next paragraph, Yanofsky says this:

The answers to Wigner’s unreasonable effectiveness leads to much deeper questions. Rather than asking why the laws of physics follow mathematics, ask why there are any laws at all.

In the same vein, Yanofsky gives a personal anecdote of a student asking him why complex numbers work for quantum mechanics. He answers that ‘…the universe does not function using complex numbers, Newton’s formula, or any other law of nature. Rather, the universe works the way it does. It is humans who use the tools they have to understand the world.’ And this is completely true as far as it goes, yet I would say that complex numbers are part of ‘the code’ required to understand one of the deepest and fundamental mysteries of the Universe.

Yanofsky’s fundamental question, quoted above, ‘why are there any laws at all?’ leads him to discuss the very structure of the universe, the emergence of life and, finally, our place in it. In fact he lists this as 3 questions:

1: Why is there any structure at all in the universe?

2: Why is the structure that exists capable of sustaining life?

3: Why did this life-sustaining structure generate a creature with enough intelligence to understand the structure?

I’ve long maintained that the last question represents the universe’s greatest enigma. There is something analogous here between us as individuals and the cosmos itself. We are each an organism with a brain that creates something we call consciousness that allows us to reflect on ourselves, individually. And the Universe created, via an extraordinary convoluted process, the ability to reflect on itself, its origins and its possible meaning.

Not surprisingly, Yanofsky doesn’t give any religious answers to this but, instead, seems to draw heavily on Paul Davies (whom he acknowledges generously at the end of the chapter) in providing various possible answers to these questions, including John Wheeler’s controversial thesis that the universe, via a cosmic scale quantum loop, has this particular life and intelligence generating structure simply because we’re in it. I’ve discussed these issues before, without coming to any definitive conclusion, so I won’t pursue them any further here.

In his notes on this chapter, Yanofsky makes this point:

Perhaps we can say that the universe is against having intelligent life and that the chances of having intelligent life are, say, 0.0000001 percent. We, therefore, only see intelligent life in 0.0000001 percent of the universe.

This reminds me of John Barrow’s point, in one of his many books, that the reason the universe is so old, and so big, is because that’s how long it takes to create complex life, and, because the universe is uniformly expanding, age and size are commensurate.

So Yanofsky’s is a deep and informative book on many levels, putting in perspective not only our place in the universe but the infinite knowledge we will never know. Towards the end he provides a table that summarises the points he delineates throughout the book in detail:

Solvable computer problems                             Unsolvable computer problems

Describable phenomena                                    Indescribable phenomena

Algebraic numbers                                            Transcendent numbers

(Provable) mathematical statements                 Mathematical facts

Finally, he makes the point that, in our everyday lives, we make decisions based primarily on emotions not reason. We seemed to have transcended our biological and evolutionary requirements when we turned to mathematics and logic to comprehend phenomena hidden from our senses and attempted to understand the origin and structure of the universe itself.