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.

Dr Who 50th Anniversary Special

A bit late, I know, as it was 2 weeks ago, but worthy of a post. Despite my advanced years, I didn’t see Dr Who in my teenage years when it first came to air. I really only became a fan with the resurrection or second coming in 2005, when Russell T Davies rebooted it with Christopher Eccleston as the Doctor. But, personally, I liked David Tennant and then Matt Smith’s renditions and it was a pleasure to see them together in the 50^th Anniversary special, The Day of the Doctor, alongside John Hurt, who was an inspirational casting choice. One should also mention Steven Moffat, who, as chief writer, deserves credit for making the show a monumental success. Writers rarely get the credit they deserve.

I recently re-watched episodes involving David Tennant and Matt Smith, and I particularly liked the narrative involving Martha Jones, played by Freema Agyeman, who, as far as I know, is the first non-white ‘companion’. Arguably, as significant as Halle Berry’s appearance as a ‘Bond girl’. My favourite episode was the ‘Weeping Angels’ because it was so cleverly structured from a time-travel perspective.

I saw the 50^th Anniversary Special in a cinema in 3D (good 3D as opposed to bad 3D) and I’ve since watched it again on ABC’s iview (expires today). It was also great to see Billie Piper recreate her role as Rose Tyler or Bad Wolf, albeit in a subtly different guise. It was one of many clever elements in this special. At its heart it contains a moral dilemma – a la John Stuart Mill – which was mirrored in one of the subplots. The interaction between John Hurt’s Doctor and Billie Piper’s sentient AI conscience is one of the highlights of the entire story, which was reinforced when I watched it for the second time. I know that some people had trouble following the time jumps and plot machinations, but that wasn’t an issue for me. To create a doomsday device to end all doomsday devices and give it a sentient conscience is a stroke of narrative genius. At 1hr 16 mins it’s not quite movie-length, yet it shows that length is not a criterion for quality. I found it witty, clever and highly entertaining, both in story context and execution; suitably engaging for a 50^th Anniversary celebration.

Postscript: I should confess that the Daleks had an influence on ELVENE, which is readily spotted by any fan of popular Sci-Fi culture.

Probability and Causality – can they be reconciled in our understanding of the universe?

In last month’s Philosophy Now (July/August 2013) Raymond Tallis wrote an interesting and provocative article (as he often does) on the subject of probability and its relationship to quantum mechanics and causality (or not). He started off by referencing a talk he gave at the Hay Festival in Wales titled, ‘Has Physics Killed Philosophy?’ According to Tallis, no, but that’s neither the subject of his article nor this post.

Afterwards, he had a conversation with Raja Panjwani, who apparently is both a philosopher and a physicist as well as ‘an international chess champion’. They got to talking about how, in quantum mechanics, ‘causation has been replaced by probability’ unless one follows the ‘many-worlds’ interpretation of quantum mechanics, whereby every causal effect is realised in some world somewhere. One of the problems with the many-worlds view (not discussed by Tallis) is that it doesn’t account for the probability of an event occurring in ‘our world’ as dictated by Schrodinger’s equation and Born’s rule. (I’ve written an entire post on that subject if the reader is interested.)

David Deutsch, the best known advocate of the many-worlds interpretation, claims that the probabilities are a consequence of how many worlds there are for each quantum event, but if there are infinite possibilities, as the theory seems to dictate according to Feynman’s integral path method, then every probability is one, which would be the case if there were an infinite number of worlds. It has to be said that Deutsch is much cleverer than me, so he probably has an answer to that, which I haven’t seen.

Tallis’s discussion quickly turns to coin-tossing, as did his conversation with Panjwani apparently, to demonstrate to ordinary people (i.e. non-physicists) how probabilities, despite appearances to the contrary, are non-causal. In particular, Tallis makes the point, often lost on gamblers, that a long sequence of ‘Heads’ (for example) has no consequence for the next coin toss, which could still be equal probability ‘Head’ or ‘Tail’. But, assuming that the coin is ‘fair’ (not biased), we know that the probability of a long sequence of ‘Heads’ (or ‘Tails’) becomes exponentially less as the sequence gets longer. So what is the connection? I believe it’s entropy.

Erwin Schrodinger in his book (series of lectures, actually), What is Life? gives the example of shuffling cards to demonstrate entropy, which also involves probabilities, as every poker player knows. In other words, entropy, which is one of the fundamental laws of the universe, is directly related to probability. To take the classic example of perfume diffusing from a bottle into an entire room, what is the probability of all the molecules of the perfume ending up back in the bottle? Infinitesimal. In other words, there is a much, much higher probability of the perfume being evenly distributed throughout the entire room, gusts of wind and air-conditioning notwithstanding. Entropy is also linked to the arrow of time, but that’s another not entirely unrelated topic, which I may return to.

Tallis then goes on to discuss how each coin toss is finely dependent on the initial conditions, which is chaos theory. It seems that Tallis was unaware that he was discussing entropy and chaos theory, or, if he did, he didn’t want to muddy the waters. I’ve discussed this elsewhere in more detail, but chaos is deterministic yet unpredictable and seems to be entailed in everything from galactic formation to biological evolution. In other words, like entropy and quantum mechanics, it seems to be a fundamental factor in the universe’s evolvement.

Towards the end of his article, Tallis starts to talk about time and references physicist, Carlo Rovelli, whom he quotes as saying that there is ‘a possibility that quantum mechanics will become “a theory of the relations between variables, rather than a theory of the evolution of variables in time.”’ Now, I’ve quoted Rovelli previously (albeit second-hand from New Scientist) as claiming that at the basic level of physics, time disappears. The relevance of that assertion to this discussion is that causality doesn’t exist without time. Schrodinger’s time dependent equation is dependent on an ‘external clock’ and can only relate to ‘reality’ through probabilities. These probabilities are found by multiplying components of the complex equation with their conjugates, and, as Schrodinger himself pointed out, that is equivalent to solving the equation both forwards and backwards in time (ref: John Gribbin, Erwin Schrodinger and the Quantum Revolution, 2012).

So it is ‘time’ that is intrinsic to causality as we observe and experience it in everyday life, and time is a factor, both in entropy and chaos theory. But what about quantum mechanics? I think the jury is still out on that to be honest. The many-worlds interpretation says it’s not an issue, but John Wheeler’s ‘backwards in time’ thought experiment for the double-slit experiment (since been confirmed according to Paul Davies) says it is.

When I first read Schrodinger’s provocative and insightful book, What is Life? one of the things that struck me (and still does) is how everything in the universe seems to be dependent on probabilities, especially on a macro scale. Einstein famously said “God does not play with dice” in apparent frustration at the non-determinism inherent in quantum mechanics, yet I’d say that ‘God’ plays dice at all levels of nature and evolution. And causality seems to be a consequence, an emergent property at a macro level, without which we would not be able to make sense of the world at all.

I support Montreal protesters for right of religious expression

This is such wrong-thinking that I can’t imagine the motivation behind it. I spent 6 weeks in Montreal in 2001 and have very fond memories of it – it has some similarities with my hometown for the past 2 decades, Melbourne, Australia. (Mind you, I was there in July, so I don’t know what it’s like in winter.)

In Melbourne, this legislation would be unthinkable by any major political party and would receive the strongest opposition. We once had a conservative female politician who wanted to band the hijab in schools, but it never gained any political traction.

At the very least, the legislation proposed by Premiere Pauline Marois is discriminatory, and, at worst, it’s Orwellian. In particular, it tacitly encourages prejudice and discrimination against people who hold religious beliefs. In my view, it’s the antithesis of what a secular society stands for.

Moral relativism – a much abused and misconstrued term

The latest issue of Philosophy Now (Issue 97; July/August 2013) has as its theme ‘The Self’ but there are a couple of articles that touch on ethics and morality, including one that looks at moral relativism (Julien Beillard, pp. 23-4). In a nutshell, Beillard claims that moral relativism is ‘unintelligible’, because, to the moral relativist, all moral stances are equally true and equally false, which is patently ‘absurd’. I know it’s unfair to reduce a 2 page argument to a one-liner, but it’s not the direction I wish to take, albeit I think he has a point.

In another article, Joel Marks (p.52) expounds on 3 books he’s written on the subject of Ethics without Morals (one of the titles) without actually arguing his case, so I can’t comment, let alone analyse his position, without reading at least one of them. The reason I raise it at all is because he briefly discusses the idea of ‘morality [being] independent of religion’. Marks calls himself an ‘amoralist’, but again, this is not the direction I wish to take.

Moral relativism is one of the most abused terms one finds on blogs like mine, especially by religious fundamentalists. It’s a reflex action for many of them when faced with an atheist or a non-theist. (I make the distinction, because non-theists don’t particularly care, whereas atheists tend to take a much harder stance towards religion in general.) The point is that fundamentalists take immediate refuge in the belief that all atheists must be moral relativists, which is just nonsense. To paraphrase Marks (out of context) they believe that ‘secular moralists …are on much less secure ground than traditional theism, because it purports to issue commands… without a commander (God).’ (parentheses in the original.)

The point I’d make, in response to this, is that people project their morality onto their God rather than the other way round. For example, homophobes have a homophobic God, and they will find the relevant text to support this view, disregarding the obvious point that the text was written by a human, just as mortal as themselves. Others of the same faith, but a different disposition towards homosexuality, will find relevant texts to support the exact opposite point of view. This was recently demonstrated on this TV panel discussion, involving opposing theologians from the Catholic Church, Judaism and Islam (some were audience members and one was video-linked from California). And one has to ask the obvious question, given the context of this discussion: is this moral relativism in action?

The point is that most moral attitudes and beliefs are cultural, and that includes all the religious ones as well. And like all cultural phenomena, morality evolves, so what was taboo generations ago, can become the social norm, and gay marriage is a good example of a social norm in transition. It also highlights the point that conservative voices like to keep the status quo (some even want to turn the clock back) while radical voices tend to advocate change, which we all recognise, politically, as being right and left. But over generations the radical becomes the status quo, and eventually conservatives defend what was once considered radical, which is how morality evolves.

I would argue that no one ever practices moral relativism – I’ve never met one and I never expect to. Why? Because everyone has a moral stance on virtually every moral question. In effect, this is exactly the point that Beillard makes, albeit in a more roundabout way. The real question is where do they get that stance? For conservatives, the answer is tradition and often religion. But there are liberal theologians as well, so religion is very flexible, completely dependent on the subjective perspective of its adherents. In a secular pluralist society, like the one I live in, there are many diverse moral views (on topics like gay marriage) as the abovementioned TV discussion demonstrates. Abortion is another example that can be delineated pretty much between conservatives and liberals. Are these examples of moral relativism? No, they are examples of diverse cultural norms and topics of debate, as they should be. Some of these issues are decided, for the society as a whole, in Parliaments, where democratically elected members can discuss and argue, sometimes being allowed a conscience vote. In other words, they don’t have to follow party lines. Gay marriage is an issue that should be allowed a conscience vote, though one conservative party, in our country, still refuses. As Penny Wong, a gay member of parliament and a mother, says in the above debate: the issue will only be resolved when both major parties allow a conscience vote. This is democracy in action.

So moral relativism has to be looked at in the context of an evolving culture where mores of the past, like abolition and women’s right to vote, have become the accepted norm, even for conservatives. The same will eventually occur for gay marriage, as we are seeing the transition occurring all over the world. There really is no such thing as moral relativism, except as a catch-phrase for religious conservatives to attempt to sideline their philosophical opponents. No one is a moral relativist for as long as they hold a philosophical position on a moral issue, and that includes most of us.

Addendum: This is Penny Wong's speech to parliament that effectively demonstrates the evolvement of social norms. It says a lot about Australian politics that she's effectively talking to an empty chamber.