Paul P. Mealing

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Sunday, 16 October 2011

Where does time go? (in quantum mechanics)

For those who are unacquainted with my blog, I’m not a physicist, or academic of any kind; I’m a self-confessed dilettante. I’ve written on this topic before (The enigma we call time, Jul. 2010) and it’s one of my more popular posts, based on an article I read in Scientific American (June 2010).

This time I’ve been inspired by last week’s New Scientist (8 October, 2011), which was a ‘Special issue’ on TIME; The Most Mysterious Dimension of All. They cover every aspect of time, by various authors, from the age of the universe to our circadian rhythms and everything in between and even beyond. But there were 2 essays in particular that caught my attention and led me to revisit this topic from 12 months ago.

Firstly, a discussion on Time’s Arrow by Amanda Gefter, who points out that, whilst the 2nd law of thermodynamics provides our only theoretical link with time’s arrow, because entropy must always increase, it’s not the solution: ‘If only it were so easy. Unfortunately, the second law does not really explain the arrow of time.’

The point is that entropy is statistical, as Schrodinger pointed out in What is Life? (refer my post, Nov. 2009) as is most of physics, but it’s not a deterministic law like, for example, Einstein’s general theory of relativity. So even though we can say that overall entropy doesn’t go backwards any more than time does, we can’t provide a mathematical relationship that derives time from entropy. In my post on time last year I said: ‘It is entropy that apparently drives the arrow of time…’ Gefter’s exposition suggests that I might be overstating the case.

Gefter goes on to say: ‘The only way to explain the arrow of time, then, is to assume that the universe just happened to start out in an extremely unlikely low entropy state.’ I’ve discussed this before when I reviewed Roger Penrose’s book, Cycles of Time (Jan. 2011), who spends a great deal of space expounding on the significance of the second law to the universe’s entire history, including its future. This is a bit off-topic to my intended subject, so I won’t dwell on it, but, as Gefter points out, the standard explanation for the universe’s initial low entropy is inflationary theory. But then she adds this caveat: ‘Inflation seems to solve the dilemma. On closer inspection, however, it only pushes the problem back.’ In other words, inflation itself must have had a low entropy and the standard explanation is that there were multiple inflations creating a multiverse.

Einstein’s relativity theories tell us that time is observer-dependent, yet entropy and its role in the evolution of the universe suggests that there is an ‘entropic’ time that governs the universe’s entire history. Then there is quantum mechanics that has its own time anomalies in defiance of common sense and everyday experience (see below).

I’ve recently started reading Professor Lisa Randall’s book, Knocking on Heaven’s Door; How Physics and Scientific Thinking Illuminate the Universe and the Modern World. This is an excellent book, from what I’ve read thus far, for anyone wanting an understanding of how scientists think and how science works, without equations and esoteric prose. In her introduction, she tells us that after giving guest lectures for college students, the most common question is not about physics, but how old she is. She’s young, blonde and attractive: the complete antithesis of the stereotypical physicist. In her first chapter she explains the importance of scale in physics and how different laws, and therefore different equations, are applied according to the scale of the world one is examining. I’ve written about this myself in a post (May 2009), which is also one of my more popular ones. Obviously, Randall is far better qualified to expound on this than me.

She mentions, in passing, a movie, What the Bleep do we know? to illustrate the error in scaling quantum mechanical phenomena up to human scale and expecting the same rules to apply. I remember when this movie came out and thinking what a disservice it did to science and how it misrepresented science to a scientifically illiterate audience. I remember having to explain to friends of mine, how, despite the credentials of the people interviewed, it wasn’t science at all. It was just as much fantasy as my own fiction, perhaps more so.

And this finally brings me to the second essay I read in New Scientist, titled, Countdown to the Theory of Everything, because it is ‘time’ that creates the conceptual and theoretical hurdle to a scientific marriage between Einstein’s theory of general relativity (gravity) and quantum mechanics. And this is what scientists specifically mean when they refer to a ‘theory of everything’. To quote Amanda Gefter again: ‘…to unite general relativity with quantum mechanics, we need to work with a single view of time. But which one is the right one?’ And then she goes on to quote various exponents on the topic, like Carlo Rovelli at the Centre for Theoretical Physics at Marseilles: “For me, the solution to the problem is that at the fundamental level of nature, there is no time at all.”

And this led me to contemplate how the dimension of time effectively disappears in many quantum phenomena, and at best, becomes an anomaly. In both quantum tunneling and entanglement, time becomes inconsequential. Also, superposition, which is just as difficult to conceptualise as any other quantum phenomenon, actually makes sense if time does not exist: something exists everywhere at once.

In May this year, I wrote an exposition on Schrodinger’s equation, aimed at physics students, which has since become my most popular post. Now Schrodinger’s equation, in its most common form, is a time-dependent wave function, which belies the ideas that I’ve just outlined above. But Schrodinger’s equation entails the paradox that lies at the heart of quantum mechanics and time is part of that paradox. As Gefter points out: ‘Unlike general relativity, where time is contained within the system, quantum mechanics requires a clock that sits outside the system…’ The time in Schrodinger’s equation is the observer’s time and his equation tells us that the particle or photon that the wave function describes, actually ‘permeates all space’, to quote Richard Elwes in MATHS 1001. And as the standard, or Copenhagen, interpretation of quantum mechanics tells us, the observer is a key participant because it’s their measurement or observation that brings the particle or photon into the real world. At best, Schrodinger’s continuous time-dependent wave function can only give us a probability of its position in the real world, albeit an accurate probability.

Schrodinger realised that his equation had to incorporate complex algebra otherwise it didn’t work. I find it curious that only quantum mechanics requires imaginary numbers (square route of -1, i ). What’s curious is that i is not a number per se: you can’t count or quantify anything with i ; it’s more like a dimension. To quote Elwes again: ‘…our human minds are incapable of visualizing the 4-dimensional graph that a complex function demands.’ This is because the imaginary plane is orthogonal to the real number plane. Schrodinger’s equation only relates to the real world when you square the modulus of his wave function and get rid of the imaginary numbers.

I’ve argued in a previous post (Jun. 2011) that quantum mechanics infers a Platonic world like a substratum to the ‘classical physical’ world that we are more familiar with. Is it possible that in this world time does not exist? Penrose, in Cycles of Time, points out that we need mass for time to exist. This is because photons have zero time, which is why nothing can travel faster than light. However a photon in the real world has energy, which means it has a frequency, which means there must be time. Schrodinger’s equation includes energy times a wave function so all aspects are entailed in the equation. Heisenberg’s Uncertainty Principle also demonstrates that there is a clear relationship between energy and time, in exactly the same way there is between space and momentum. As Richard Feynman points out, translation in space is linked to the conservation of momentum and translation in time is linked to the conservation of energy. So time and energy are linked in the classical world and Heisenberg’s equation tells us that they are linked quantum mechanically as well, but only through a particle’s emergence into the physical world, even if it’s a virtual particle.

But if quantum phenomena are time-dimensionless then photons are the perfect candidate. The entire universe can go by in a photon’s lifetime. The same happens at the event horizon of a black hole. Does this mean that the event horizon of a black hole is the boundary between the classical physical world and the quantum world?

In Nov. 2009, I reviewed Fulvio Melia’s book, Cracking the Einstein Code, which is effectively an exposition and brief biography of Roy Kerr, a Kiwi who used Einstein’s field equations to describe the space-time of a rotating body, which is the norm for bodies in the universe. Kerr’s theoretical examination of a spinning black hole led him to postulate that it would have 2 event horizons, and when a body crosses the first event horizon, the parameters of space and time are reversed: space becomes time-like and time becomes space-like. This is because time freezes at the event horizon for an outside observer and the external time becomes infinite from the inside. Time becomes space-like in that it becomes static and infinite.

This post can be put down to the meanderings of an under-educated yet intellectually curious individual. If time does not exist in the quantum world then it actually makes sense of things like superposition, quantum tunneling and entanglement, not to mention time-reversal as expounded by Feynman in his book, QED, using his unique Feynman diagrams. John Wheeler also postulated a thought experiment in which a measurement taken after a photon passes through a slit in Young’s famous experiment can determine which one it passed through. I believe this has since been confirmed with a real experiment. It would also suggest that a marriage between the quantum world and Einstein’s general theory of relativity may be impossible.

Addendum 1: Earlier this year (May 2011) I reviewed John D. Barrow’s latest book, The Book of Universes, and I happened to revisit it and find this extremely relevant reference to the Hartle-Hawking universe devised by James Hartle and Stephen Hawking, using Feynman’s quantum integral method on a wave function for the whole universe.

What’s relevant to this post is that ‘[Hartle and Hawking] proposed an initial state in which time became another dimension of space.’

To quote Barrow’s interpretation:

‘Time is not fundamental in this theory. It is a quality that emerges when the universe gets large enough for the distinctive quantum effects to become negligible: time is something that arises concretely only in the limiting non-quantum environment. As we follow the Hartle-Hawking universe back to small sizes it becomes dominated by the Euclidean quantum paths. The concept of time disappears and the universe becomes increasingly like a four-dimensional space. There is no beginning to the universe because time disappears.’

What’s more: ‘…the transformation that changes time into another dimension of space corresponds to multiplying it by an imaginary number…’

This gives rise to what ‘Hartle and Hawking called the “no Boundary” state for the origin of the universe… Its beginning is smooth and unremarkable… In effect, the no-boundary condition is a proposal for the state of the universe if it appears from nothing in a quantum event. The story of this universe is that once upon a time there was no time.’

The only problem with this scenario, as Barrow points out, is that there is no big bang singularity, nevertheless it fits the idea that the quantum world doesn’t need time and the very early universe must have been a quantum universe at the very least.

Addendum 2: Also from Barrow (same book), I found this mathematical joke:

The number you have reached is imaginary. Please rotate your phone through ninety degrees and try again. (Answer-phone message for imaginary phone numbers)


Paul P. Mealing said...

I've also added an Addendum to my post, Where does time go?

It's very relevant to the post's overall thesis.

Regards, Paul.

Eli said...

Does this have anything to do with the thing about the universe being a holographic projection...or something? I admit that I can't wrap my mind around this stuff basically at all, but it seems like they might possibly maybe be related.

Paul P. Mealing said...

Hi Eli,

Interestingly, Barrow makes no mention of the ‘holographic’ universe in his book and neither does Paul Davies in his book, The Goldilocks Enigma, though they both discuss the Hartle-Hawking universe.

I did find something in New Scientist, 23 July 2011, p.31 (The Existential Issue).

‘In 1972, Jacob Bekenstein at the Hebrew University of Jerusalem showed that the information content of a black hole is proportional to the 2 dimensional surface area of its event horizon. Later, string theorists managed to show how the original star’s information could be encoded in tiny lumps and bumps on the event horizon, which could then imprint it on the Hawking radiation departing the black hole.’

Leonard Susskind and Gerard ‘t Hoof then postulated: ‘if a 3 dimensional star could be encoded on a black hole’s 2 dimensional event horizon, maybe the same could be true of the whole universe.’ If one imagines that the whole universe is effectively a massive black hole expanding from the big bang with its event horizon 42 billion light years away then ‘this 2D “surface” may encode the entire 3D universe that we experience.’

So whilst the idea originated from the idea of Hawking radiation it has nothing to do with the Hartle-Hawking model of the universe.

Regards, Paul.

Paul P. Mealing said...

By the way, Davies points that there is still a Big Bang in the Hartle-Hawking universe, but there's no singularity.

It's usually shown graphically as a 'shuttle-cock' shape with the boundary at the beginning being 'rounded' semi-spherically. The classic Big Bang is shown as a point origin with the evolving universe a cone.

Regards, Paul.

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