My theoretical physics examination is coming up on Monday and the students are hard at working revising for it (or at least they should be) so I thought I’d lend a hand by deploying some digital technology in the form of the following online interactive video-based learning resource on Complex Analysis:Follow @telescoper
Archive for Physics
I’ve only found out this morning that Professor Sir Sam Edwards passed away last week, on 7th May 2015 at the age of 87. Although I didn’t really know him at all on a personal level, I did come across him when I was an undergraduate student at the University of Cambridge in the 1980s, so I thought I would post a brief item to mark his passing and to pay my respects.
Sam Edwards taught a second-year course at Cambridge to Physics students,entitled Analytical Dynamics as a component of Part IB Advanced Physics. It would have been in 1984 that I took it. If memory serves, which is admittedly rather unlikely, this lecture course was optional and intended for those of us who intended to follow theoretical physics Part II, i.e. in the third year.
I have to admit that Sam Edwards was far from the best lecturer I’ve ever had, and I know I’m not alone in that opinion. In fact, not to put too fine a point on it, his lectures were largely incomprehensible and attendance at them fell sharply after the first few. They were, however, based on an excellent set of typewritten notes from which I learned a lot. It wasn’t at all usual for lecturers to hand out printed lecture notes in those days, but I am glad he did. In fact, I still have them now. Here is the first page:
It’s quite heavy stuff, but enormously useful. I have drawn on a few of the examples contained in his handout for my own lectures on related concepts in theoretical physics, so in a sense my students are gaining some benefit from his legacy.
At the time I was an undergraduate student I didn’t know much about the research interests of the lecturers, but I was fascinated to read in his Guardian obituary how much he contributed to the theoretical development of the field of soft condensed matter, which includes the physics of polymers. In those days – I was at Cambridge from 1982 to 1985 – this was a relatively small part of the activity in the Cavendish laboratory but it has grown substantially over the years.
I feel a bit guilty that I didn’t appreciate more at the time what a distinguished physicist he was, but he undoubtedly played a significant part in the environment at Cambridge that gave me such a good start in my own scientific career and was held in enormously high regard by friends and colleagues at Cambridge and beyond.
Rest in peace, Sir Sam Edwards (1928-2015).Follow @telescoper
Once again it’s time for examinations at the University of Sussex, so here’s a lazy rehash of my previous offerings on the subject that I’ve posted around this time each year since I started blogging.
Of College labours, of the Lecturer’s room
All studded round, as thick as chairs could stand,
With loyal students, faithful to their books,
Half-and-half idlers, hardy recusants,
And honest dunces–of important days,
Examinations, when the man was weighed
As in a balance! of excessive hopes,
Tremblings withal and commendable fears,
Small jealousies, and triumphs good or bad–
Let others that know more speak as they know.
Such glory was but little sought by me,
And little won.
It seems to me a great a pity that our system of education – both at School and University – places such a great emphasis on examination and assessment to the detriment of real learning. On previous occasions, before I moved to the University of Sussex, I’ve bemoaned the role that modularisation has played in this process, especially in my own discipline of physics.
Don’t get me wrong. I’m not opposed to modularisation in principle. I just think the way modules are used in many British universities fails to develop any understanding of the interconnection between different aspects of the subject. That’s an educational disaster because what is most exciting and compelling about physics is its essential unity. Splitting it into little boxes, taught on their own with no relationship to the other boxes, provides us with no scope to nurture the kind of lateral thinking that is key to the way physicists attempt to solve problems. The small size of many module makes the syllabus very “bitty” and fragmented. No sooner have you started to explore something at a proper level than the module is over. More advanced modules, following perhaps the following year, have to recap a large fraction of the earlier modules so there isn’t time to go as deep as one would like even over the whole curriculum.
In most UK universities (including Sussex), tudents take 120 “credits” in a year, split into two semesters. In many institutions, these are split into 10-credit modules with an examination at the end of each semester; there are two semesters per year. Laboratories, projects, and other continuously-assessed work do not involve a written examination, so the system means that a typical student will have 5 written examination papers in January and another 5 in May. Each paper is usually of two hours’ duration.
Such an arrangement means a heavy ratio of assessment to education, one that has risen sharply over the last decades, with the undeniable result that academic standards in physics have fallen across the sector. The system encourages students to think of modules as little bit-sized bits of education to be consumed and then forgotten. Instead of learning to rely on their brains to solve problems, students tend to approach learning by memorising chunks of their notes and regurgitating them in the exam. I find it very sad when students ask me what derivations they should memorize to prepare for examinations. A brain is so much more than a memory device. What we should be doing is giving students the confidence to think for themselves and use their intellect to its full potential rather than encouraging rote learning.
You can contrast this diet of examinations with the regime when I was an undergraduate. My entire degree result was based on six three-hour written examinations taken at the end of my final year, rather than something like 30 examinations taken over 3 years. Moreover, my finals were all in a three-day period. Morning and afternoon exams for three consecutive days is an ordeal I wouldn’t wish on anyone so I’m not saying the old days were better, but I do think we’ve gone far too far to the opposite extreme. The one good thing about the system I went through was that there was no possibility of passing examinations on memory alone. Since they were so close together there was no way of mugging up anything in between them. I only got through by figuring things out in the exam room.
I think the system we have here at the University of Sussex is much better than I’ve experienced elsewhere. For a start the basic module size is 15 credits. This means that students are usually only doing four things in parallel, and they consequently have fewer examinations, especially since they also take laboratory classes and other modules which don’t have a set examination at the end. There’s also a sizeable continuously assessed component (30%) for most modules so it doesn’t all rest on one paper. Although in my view there’s still too much emphasis on assessment and too little on the joy of finding things out, it’s much less pronounced than elsewhere. Maybe that’s one of the reasons why the Department of Physics & Astronomy does so consistently well in the National Student Survey?
We also have modules called Skills in Physics which focus on developing the problem-solving skills I mentioned above; these are taught through a mixture of lectures and small-group tutorials. I don’t know what the students think of these sessions, but I always enjoy them because the problems set for each session are generally a bit wacky, some of them being very testing. In fact I’d say that I’m very impressed at the technical level of the modules in the Department of Physics & Astronomy generally. I’ve been teaching Green’s Functions, Conformal Transformations and the Calculus of Variations to second-year students this semester. Those topics weren’t on the syllabus at all in my previous institution!
Anyway, my Theoretical Physics paper is next week (on 18th May) so I’ll find out if the students managed to learn anything despite having such a lousy lecturer. Which reminds me, I must remember to post some worked examples online to help them with their revision.Follow @telescoper
I came across this little video at the Gatsby Charitable Foundation website and thought I would share it here.
The video (or “motion graphic”) makes the point that the impact of innovative thinking and interventions resulted in an increase in the supply of physics teachers until 2012 but since then it has subsequently declined, with serious implications not only for physics but for the country as a whole.
Modelling by the Department for Education (DfE) and the Institute of Physics (IoP) suggests that we need to recruit around 1,000 new physics teachers every year for at least the next decade in order to meet demand. This year, just 661 teachers started physics teacher training, down from a peak of 900 in 2012. The stark reality is that, if we are to meet the demand for physics teachers and ensure that all pupils have access to well-qualified, specialist teachers, we must look at new ways to recruit, train and retain physics teachers.
Indeed. We’re planning a bit initiative here in the Department of Physics & Astronomy at the University of Sussex, of which more anon..
It seems to me that the basic problem is threefold: (a) that there aren’t enough physics students at University in the first place; (b) that good physics graduates are very employable and get snapped up quickly by employers; (c) that teaching doesn’t seem an attractive career option compared to the many others available. Many efforts focus on (c) but the root cause of the problem is actually (a)…
..nevertheless, I will use this opportunity to point out that bursaries of £25K are available to excellent physics graduates wanting to become physics teachers, courtesy of the Institute of Physics. The deadline for the latest round of applications is this Monday (4th May). Here’s a promotional video:Follow @telescoper
Here I am in Easthampstead Park Conference Centre after a hard day being away at an awayday. In fact we’ve been so busy that I’ve only just checked into my room (actually it’s a suite) and shall very soon be attempting to find the bar so I can have a drink. I’m parched.
The place is very nice. Here’s a picture from outside:
I’m told it is very close to Broadmoor, the famous high-security psychiatric hospital, although I’m sure that wasn’t one of the reasons for choice of venue.
I have to attend quite a few of these things for one reason or another. This one is on the Future and Sustainability of the South East Physics Network, known as SEPnet for short, which is a consortium of physics departments across the South East of England working together to deliver excellence in both teaching and research. I am here deputising for a Pro Vice Chancellor who can’t be here. I’ve enjoyed pretending to be important, but I’m sure nobody has been taken in.
Although it’s been quite tiring, it has been an interesting day. Lots of ideas and discussion, but we do have to distil all that down into some more specific detail over dinner tonight and during the course of tomorrow morning. Anyway, better begin the search for the bar so I can refresh the old brain cells.Follow @telescoper
You’ve probably all heard of the little logic problem involving the mysterious Cheryl and her friends Albert and Bernard that went viral on the internet recently. I decided not to post about it directly because it’s already been done to death. It did however make me think that if people struggle so much with “ordinary” logic problems of this type its no wonder they are so puzzled by the kind of logical issues raised by quantum mechanics. Hence the motivation of updating a post I did quite a while ago. The question we’ll explore does not concern the date of Cheryl’s birthday but the spin of an electron.
To begin with, let me give a bit of physics background. Spin is a concept of fundamental importance in quantum mechanics, not least because it underlies our most basic theoretical understanding of matter. The standard model of particle physics divides elementary particles into two types, fermions and bosons, according to their spin. One is tempted to think of these elementary particles as little cricket balls that can be rotating clockwise or anti-clockwise as they approach an elementary batsman. But, as I hope to explain, quantum spin is not really like classical spin.
Take the electron, for example. The amount of spin an electron carries is quantized, so that it always has a magnitude which is ±1/2 (in units of Planck’s constant; all fermions have half-integer spin). In addition, according to quantum mechanics, the orientation of the spin is indeterminate until it is measured. Any particular measurement can only determine the component of spin in one direction. Let’s take as an example the case where the measuring device is sensitive to the z-component, i.e. spin in the vertical direction. The outcome of an experiment on a single electron will lead a definite outcome which might either be “up” or “down” relative to this axis.
However, until one makes a measurement the state of the system is not specified and the outcome is consequently not predictable with certainty; there will be a probability of 50% probability for each possible outcome. We could write the state of the system (expressed by the spin part of its wavefunction ψ prior to measurement in the form
|ψ> = (|↑> + |↓>)/√2
This gives me an excuse to use the rather beautiful “bra-ket” notation for the state of a quantum system, originally due to Paul Dirac. The two possibilities are “up” (↑) and “down” (↓) and they are contained within a “ket” (written |>)which is really just a shorthand for a wavefunction describing that particular aspect of the system. A “bra” would be of the form <|; for the mathematicians this represents the Hermitian conjugate of a ket. The √2 is there to insure that the total probability of the spin being either up or down is 1, remembering that the probability is the square of the wavefunction. When we make a measurement we will get one of these two outcomes, with a 50% probability of each.
At the point of measurement the state changes: if we get “up” it becomes purely |↑> and if the result is “down” it becomes |↓>. Either way, the quantum state of the system has changed from a “superposition” state described by the equation above to an “eigenstate” which must be either up or down. This means that all subsequent measurements of the spin in this direction will give the same result: the wave-function has “collapsed” into one particular state. Incidentally, the general term for a two-state quantum system like this is a qubit, and it is the basis of the tentative steps that have been taken towards the construction of a quantum computer.
Notice that what is essential about this is the role of measurement. The collapse of ψ seems to be an irreversible process, but the wavefunction itself evolves according to the Schrödinger equation, which describes reversible, Hamiltonian changes. To understand what happens when the state of the wavefunction changes we need an extra level of interpretation beyond what the mathematics of quantum theory itself provides, because we are generally unable to write down a wave-function that sensibly describes the system plus the measuring apparatus in a single form.
So far this all seems rather similar to the state of a fair coin: it has a 50-50 chance of being heads or tails, but the doubt is resolved when its state is actually observed. Thereafter we know for sure what it is. But this resemblance is only superficial. A coin only has heads or tails, but the spin of an electron doesn’t have to be just up or down. We could rotate our measuring apparatus by 90° and measure the spin to the left (←) or the right (→). In this case we still have to get a result which is a half-integer times Planck’s constant. It will have a 50-50 chance of being left or right that “becomes” one or the other when a measurement is made.
Now comes the real fun. Suppose we do a series of measurements on the same electron. First we start with an electron whose spin we know nothing about. In other words it is in a superposition state like that shown above. We then make a measurement in the vertical direction. Suppose we get the answer “up”. The electron is now in the eigenstate with spin “up”.
We then pass it through another measurement, but this time it measures the spin to the left or the right. The process of selecting the electron to be one with spin in the “up” direction tells us nothing about whether the horizontal component of its spin is to the left or to the right. Theory thus predicts a 50-50 outcome of this measurement, as is observed experimentally.
Suppose we do such an experiment and establish that the electron’s spin vector is pointing to the left. Now our long-suffering electron passes into a third measurement which this time is again in the vertical direction. You might imagine that since we have already measured this component to be in the up direction, it would be in that direction again this time. In fact, this is not the case. The intervening measurement seems to “reset” the up-down component of the spin; the results of the third measurement are back at square one, with a 50-50 chance of getting up or down.
This is just one example of the kind of irreducible “randomness” that seems to be inherent in quantum theory. However, if you think this is what people mean when they say quantum mechanics is weird, you’re quite mistaken. It gets much weirder than this! So far I have focussed on what happens to the description of single particles when quantum measurements are made. Although there seem to be subtle things going on, it is not really obvious that anything happening is very different from systems in which we simply lack the microscopic information needed to make a prediction with absolute certainty.
At the simplest level, the difference is that quantum mechanics gives us a theory for the wave-function which somehow lies at a more fundamental level of description than the usual way we think of probabilities. Probabilities can be derived mathematically from the wave-function, but there is more information in ψ than there is in |ψ|2; the wave-function is a complex entity whereas the square of its amplitude is entirely real. If one can construct a system of two particles, for example, the resulting wave-function is obtained by superimposing the wave-functions of the individual particles, and probabilities are then obtained by squaring this joint wave-function. This will not, in general, give the same probability distribution as one would get by adding the one-particle probabilities because, for complex entities A and B,
in general. To put this another way, one can write any complex number in the form a+ib (real part plus imaginary part) or, generally more usefully in physics , as Reiθ, where R is the amplitude and θ is called the phase. The square of the amplitude gives the probability associated with the wavefunction of a single particle, but in this case the phase information disappears; the truly unique character of quantum physics and how it impacts on probabilies of measurements only reveals itself when the phase information is retained. This generally requires two or more particles to be involved, as the absolute phase of a single-particle state is essentially impossible to measure.
Finding situations where the quantum phase of a wave-function is important is not easy. It seems to be quite easy to disturb quantum systems in such a way that the phase information becomes scrambled, so testing the fundamental aspects of quantum theory requires considerable experimental ingenuity. But it has been done, and the results are astonishing.
Let us think about a very simple example of a two-component system: a pair of electrons. All we care about for the purpose of this experiment is the spin of the electrons so let us write the state of this system in terms of states such as which I take to mean that the first particle has spin up and the second one has spin down. Suppose we can create this pair of electrons in a state where we know the total spin is zero. The electrons are indistinguishable from each other so until we make a measurement we don’t know which one is spinning up and which one is spinning down. The state of the two-particle system might be this:
|ψ> = (|↑↓> – |↓↑>)/√2
squaring this up would give a 50% probability of “particle one” being up and “particle two” being down and 50% for the contrary arrangement. This doesn’t look too different from the example I discussed above, but this duplex state exhibits a bizarre phenomenon known as quantum entanglement.
Suppose we start the system out in this state and then separate the two electrons without disturbing their spin states. Before making a measurement we really can’t say what the spins of the individual particles are: they are in a mixed state that is neither up nor down but a combination of the two possibilities. When they’re up, they’re up. When they’re down, they’re down. But when they’re only half-way up they’re in an entangled state.
If one of them passes through a vertical spin-measuring device we will then know that particle is definitely spin-up or definitely spin-down. Since we know the total spin of the pair is zero, then we can immediately deduce that the other one must be spinning in the opposite direction because we’re not allowed to violate the law of conservation of angular momentum: if Particle 1 turns out to be spin-up, Particle 2 must be spin-down, and vice versa. It is known experimentally that passing two electrons through identical spin-measuring gadgets gives results consistent with this reasoning. So far there’s nothing so very strange in this.
The problem with entanglement lies in understanding what happens in reality when a measurement is done. Suppose we have two observers, Albert and Bernard, who are bored with Cheryl’s little games and have decided to do something interesting with their lives by becoming physicists. Each is equipped with a device that can measure the spin of an electron in any direction they choose. Particle 1 emerges from the source and travels towards Albert whereas particle 2 travels in Bernard’s direction. Before any measurement, the system is in an entangled superposition state. Suppose Albert decides to measure the spin of electron 1 in the z-direction and finds it spinning up. Immediately, the wave-function for electron 2 collapses into the down direction. If Albert had instead decided to measure spin in the left-right direction and found it “left” similar collapse would have occurred for particle 2, but this time putting it in the “right” direction.
Whatever Albert does, the result of any corresponding measurement made by Bernard has a definite outcome – the opposite to Alberts result. So Albert’s decision whether to make a measurement up-down or left-right instantaneously transmits itself to Bernard who will find a consistent answer, if he makes the same measurement as Albert.
If, on the other hand, Albert makes an up-down measurement but Bernard measures left-right then Albert’s answer has no effect on Bernard, who has a 50% chance of getting “left” and 50% chance of getting right. The point is that whatever Albert decides to do, it has an immediate effect on the wave-function at ’s position; the collapse of the wave-function induced by Albert immediately collapses the state measured by Bernard. How can particle 1 and particle 2 communicate in this way?
This riddle is the core of a thought experiment by Einstein, Podolsky and Rosen in 1935 which has deep implications for the nature of the information that is supplied by quantum mechanics. The essence of the EPR paradox is that each of the two particles – even if they are separated by huge distances – seems to know exactly what the other one is doing. Einstein called this “spooky action at a distance” and went on to point out that this type of thing simply could not happen in the usual calculus of random variables. His argument was later tightened considerably by John Bell in a form now known as Bell’s theorem.
To see how Bell’s theorem works, consider the following roughly analagous situation. Suppose we have two suspects in prison, say Albert and Bernard (presumably Cheryl grassed them up and has been granted immunity from prosecution). The two are taken apart to separate cells for individual questioning. We can allow them to use notes, electronic organizers, tablets of stone or anything to help them remember any agreed strategy they have concocted, but they are not allowed to communicate with each other once the interrogation has started. Each question they are asked has only two possible answers – “yes” or “no” – and there are only three possible questions. We can assume the questions are asked independently and in a random order to the two suspects.
When the questioning is over, the interrogators find that whenever they asked the same question, Albert and Bernard always gave the same answer, but when the question was different they only gave the same answer 25% of the time. What can the interrogators conclude?
The answer is that Albert and Bernard must be cheating. Either they have seen the question list ahead of time or are able to communicate with each other without the interrogator’s knowledge. If they always give the same answer when asked the same question, they must have agreed on answers to all three questions in advance. But when they are asked different questions then, because each question has only two possible responses, by following this strategy it must turn out that at least two of the three prepared answers – and possibly all of them – must be the same for both Albert and Bernard. This puts a lower limit on the probability of them giving the same answer to different questions. I’ll leave it as an exercise to the reader to show that the probability of coincident answers to different questions in this case must be at least 1/3.
This a simple illustration of what in quantum mechanics is known as a Bell inequality. Albert and Bernard can only keep the number of such false agreements down to the measured level of 25% by cheating.
This example is directly analogous to the behaviour of the entangled quantum state described above under repeated interrogations about its spin in three different directions. The result of each measurement can only be either “yes” or “no”. Each individual answer (for each particle) is equally probable in this case; the same question always produces the same answer for both particles, but the probability of agreement for two different questions is indeed ¼ and not larger as would be expected if the answers were random. For example one could ask particle 1 “are you spinning up” and particle 2 “are you spinning to the right”? The probability of both producing an answer “yes” is 25% according to quantum theory but would be higher if the particles weren’t cheating in some way.
Probably the most famous experiment of this type was done in the 1980s, by Alain Aspect and collaborators, involving entangled pairs of polarized photons (which are bosons), rather than electrons, primarily because these are easier to prepare.
The implications of quantum entanglement greatly troubled Einstein long before the EPR paradox. Indeed the interpretation of single-particle quantum measurement (which has no entanglement) was already troublesome. Just exactly how does the wave-function relate to the particle? What can one really say about the state of the particle before a measurement is made? What really happens when a wave-function collapses? These questions take us into philosophical territory that I have set foot in already; the difficult relationship between epistemological and ontological uses of probability theory.
Thanks largely to the influence of Niels Bohr, in the relatively early stages of quantum theory a standard approach to this question was adopted. In what became known as the Copenhagen interpretation of quantum mechanics, the collapse of the wave-function as a result of measurement represents a real change in the physical state of the system. Before the measurement, an electron really is neither spinning up nor spinning down but in a kind of quantum purgatory. After a measurement it is released from limbo and becomes definitely something. What collapses the wave-function is something unspecified to do with the interaction of the particle with the measuring apparatus or, in some extreme versions of this doctrine, the intervention of human consciousness.
I find it amazing that such a view could have been held so seriously by so many highly intelligent people. Schrödinger hated this concept so much that he invented a thought-experiment of his own to poke fun at it. This is the famous “Schrödinger’s cat” paradox.
In a closed box there is a cat. Attached to the box is a device which releases poison into the box when triggered by a quantum-mechanical event, such as radiation produced by the decay of a radioactive substance. One can’t tell from the outside whether the poison has been released or not, so one doesn’t know whether the cat is alive or dead. When one opens the box, one learns the truth. Whether the cat has collapsed or not, the wave-function certainly does. At this point one is effectively making a quantum measurement so the wave-function of the cat is either “dead” or “alive” but before opening the box it must be in a superposition state. But do we really think the cat is neither dead nor alive? Isn’t it certainly one or the other, but that our lack of information prevents us from knowing which? And if this is true for a macroscopic object such as a cat, why can’t it be true for a microscopic system, such as that involving just a pair of electrons?
As I learned at a talk a while ago by the Nobel prize-winning physicist Tony Leggett – who has been collecting data on this – most physicists think Schrödinger’s cat is definitely alive or dead before the box is opened. However, most physicists don’t believe that an electron definitely spins either up or down before a measurement is made. But where does one draw the line between the microscopic and macroscopic descriptions of reality? If quantum mechanics works for 1 particle, does it work also for 10, 1000? Or, for that matter, 1023?
Most modern physicists eschew the Copenhagen interpretation in favour of one or other of two modern interpretations. One involves the concept of quantum decoherence, which is basically the idea that the phase information that is crucial to the underlying logic of quantum theory can be destroyed by the interaction of a microscopic system with one of larger size. In effect, this hides the quantum nature of macroscopic systems and allows us to use a more classical description for complicated objects. This certainly happens in practice, but this idea seems to me merely to defer the problem of interpretation rather than solve it. The fact that a large and complex system makes tends to hide its quantum nature from us does not in itself give us the right to have a different interpretations of the wave-function for big things and for small things.
Another trendy way to think about quantum theory is the so-called Many-Worlds interpretation. This asserts that our Universe comprises an ensemble – sometimes called a multiverse – and probabilities are defined over this ensemble. In effect when an electron leaves its source it travels through infinitely many paths in this ensemble of possible worlds, interfering with itself on the way. We live in just one slice of the multiverse so at the end we perceive the electron winding up at just one point on our screen. Part of this is to some extent excusable, because many scientists still believe that one has to have an ensemble in order to have a well-defined probability theory. If one adopts a more sensible interpretation of probability then this is not actually necessary; probability does not have to be interpreted in terms of frequencies. But the many-worlds brigade goes even further than this. They assert that these parallel universes are real. What this means is not completely clear, as one can never visit parallel universes other than our own …
It seems to me that none of these interpretations is at all satisfactory and, in the gap left by the failure to find a sensible way to understand “quantum reality”, there has grown a pathological industry of pseudo-scientific gobbledegook. Claims that entanglement is consistent with telepathy, that parallel universes are scientific truths, that consciousness is a quantum phenomena abound in the New Age sections of bookshops but have no rational foundation. Physicists may complain about this, but they have only themselves to blame.
But there is one remaining possibility for an interpretation of that has been unfairly neglected by quantum theorists despite – or perhaps because of – the fact that is the closest of all to commonsense. This view that quantum mechanics is just an incomplete theory, and the reason it produces only a probabilistic description is that does not provide sufficient information to make definite predictions. This line of reasoning has a distinguished pedigree, but fell out of favour after the arrival of Bell’s theorem and related issues. Early ideas on this theme revolved around the idea that particles could carry “hidden variables” whose behaviour we could not predict because our fundamental description is inadequate. In other words two apparently identical electrons are not really identical; something we cannot directly measure marks them apart. If this works then we can simply use only probability theory to deal with inferences made on the basis of information that’s not sufficient for absolute certainty.
After Bell’s work, however, it became clear that these hidden variables must possess a very peculiar property if they are to describe out quantum world. The property of entanglement requires the hidden variables to be non-local. In other words, two electrons must be able to communicate their values faster than the speed of light. Putting this conclusion together with relativity leads one to deduce that the chain of cause and effect must break down: hidden variables are therefore acausal. This is such an unpalatable idea that it seems to many physicists to be even worse than the alternatives, but to me it seems entirely plausible that the causal structure of space-time must break down at some level. On the other hand, not all “incomplete” interpretations of quantum theory involve hidden variables.
One can think of this category of interpretation as involving an epistemological view of quantum mechanics. The probabilistic nature of the theory has, in some sense, a subjective origin. It represents deficiencies in our state of knowledge. The alternative Copenhagen and Many-Worlds views I discussed above differ greatly from each other, but each is characterized by the mistaken desire to put quantum mechanics – and, therefore, probability – in the realm of ontology.
The idea that quantum mechanics might be incomplete (or even just fundamentally “wrong”) does not seem to me to be all that radical. Although it has been very successful, there are sufficiently many problems of interpretation associated with it that perhaps it will eventually be replaced by something more fundamental, or at least different. Surprisingly, this is a somewhat heretical view among physicists: most, including several Nobel laureates, seem to think that quantum theory is unquestionably the most complete description of nature we will ever obtain. That may be true, of course. But if we never look any deeper we will certainly never know…
With the gradual re-emergence of Bayesian approaches in other branches of physics a number of important steps have been taken towards the construction of a truly inductive interpretation of quantum mechanics. This programme sets out to understand probability in terms of the “degree of belief” that characterizes Bayesian probabilities. Recently, Christopher Fuchs, amongst others, has shown that, contrary to popular myth, the role of probability in quantum mechanics can indeed be understood in this way and, moreover, that a theory in which quantum states are states of knowledge rather than states of reality is complete and well-defined. I am not claiming that this argument is settled, but this approach seems to me by far the most compelling and it is a pity more people aren’t following it up…
So how loud was the Big Bang?
I’ve posted on this before but a comment posted today reminded me that perhaps I should recycle it and update it as it relates to the cosmic microwave background, which is what I work on on the rare occasions on which I get to do anything interesting.
As you probably know the Big Bang theory involves the assumption that the entire Universe – not only the matter and energy but also space-time itself – had its origins in a single event a finite time in the past and it has been expanding ever since. The earliest mathematical models of what we now call the Big Bang were derived independently by Alexander Friedman and George Lemaître in the 1920s. The term “Big Bang” was later coined by Fred Hoyle as a derogatory description of an idea he couldn’t stomach, but the phrase caught on. Strictly speaking, though, the Big Bang was a misnomer.
Friedman and Lemaître had made mathematical models of universes that obeyed the Cosmological Principle, i.e. in which the matter was distributed in a completely uniform manner throughout space. Sound consists of oscillating fluctuations in the pressure and density of the medium through which it travels. These are longitudinal “acoustic” waves that involve successive compressions and rarefactions of matter, in other words departures from the purely homogeneous state required by the Cosmological Principle. The Friedman-Lemaitre models contained no sound waves so they did not really describe a Big Bang at all, let alone how loud it was.
However, as I have blogged about before, newer versions of the Big Bang theory do contain a mechanism for generating sound waves in the early Universe and, even more importantly, these waves have now been detected and their properties measured.
The above image shows the variations in temperature of the cosmic microwave background as charted by the Planck Satellite. The average temperature of the sky is about 2.73 K but there are variations across the sky that have an rms value of about 0.08 milliKelvin. This corresponds to a fractional variation of a few parts in a hundred thousand relative to the mean temperature. It doesn’t sound like much, but this is evidence for the existence of primordial acoustic waves and therefore of a Big Bang with a genuine “Bang” to it.
A full description of what causes these temperature fluctuations would be very complicated but, roughly speaking, the variation in temperature you corresponds directly to variations in density and pressure arising from sound waves.
So how loud was it?
The waves we are dealing with have wavelengths up to about 200,000 light years and the human ear can only actually hear sound waves with wavelengths up to about 17 metres. In any case the Universe was far too hot and dense for there to have been anyone around listening to the cacophony at the time. In some sense, therefore, it wouldn’t have been loud at all because our ears can’t have heard anything.
Setting aside these rather pedantic objections – I’m never one to allow dull realism to get in the way of a good story- we can get a reasonable value for the loudness in terms of the familiar language of decibels. This defines the level of sound (L) logarithmically in terms of the rms pressure level of the sound wave Prms relative to some reference pressure level Pref
(the 20 appears because of the fact that the energy carried goes as the square of the amplitude of the wave; in terms of energy there would be a factor 10).
There is no absolute scale for loudness because this expression involves the specification of the reference pressure. We have to set this level by analogy with everyday experience. For sound waves in air this is taken to be about 20 microPascals, or about 2×10-10 times the ambient atmospheric air pressure which is about 100,000 Pa. This reference is chosen because the limit of audibility for most people corresponds to pressure variations of this order and these consequently have L=0 dB. It seems reasonable to set the reference pressure of the early Universe to be about the same fraction of the ambient pressure then, i.e.
The physics of how primordial variations in pressure translate into observed fluctuations in the CMB temperature is quite complicated, because the primordial universe consists of a plasma rather than air. Moreover, the actual sound of the Big Bang contains a mixture of wavelengths with slightly different amplitudes. In fact here is the spectrum, showing a distinctive signature that looks, at least in this representation, like a fundamental tone and a series of harmonics…
If you take into account all this structure it all gets a bit messy, but it’s quite easy to get a rough but reasonable estimate by ignoring all these complications. We simply take the rms pressure variation to be the same fraction of ambient pressure as the averaged temperature variation are compared to the average CMB temperature, i.e.
Prms~ a few ×10-5Pamb.
If we do this, scaling both pressures in logarithm in the equation in proportion to the ambient pressure, the ambient pressure cancels out in the ratio, which turns out to be a few times 10-5. With our definition of the decibel level we find that waves of this amplitude, i.e. corresponding to variations of one part in a hundred thousand of the reference level, give roughly L=100dB while part in ten thousand gives about L=120dB. The sound of the Big Bang therefore peaks at levels just a bit less than 120 dB.
As you can see in the Figure above, this is close to the threshold of pain, but it’s perhaps not as loud as you might have guessed in response to the initial question. Modern popular beat combos often play their dreadful rock music much louder than the Big Bang….
A useful yardstick is the amplitude at which the fluctuations in pressure are comparable to the mean pressure. This would give a factor of about 1010 in the logarithm and is pretty much the limit that sound waves can propagate without distortion. These would have L≈190 dB. It is estimated that the 1883 Krakatoa eruption produced a sound level of about 180 dB at a range of 100 miles. By comparison the Big Bang was little more than a whimper.
PS. If you would like to read more about the actual sound of the Big Bang, have a look at John Cramer’s webpages. You can also download simulations of the actual sound. If you listen to them you will hear that it’s more of a “Roar” than a “Bang” because the sound waves don’t actually originate at a single well-defined event but are excited incoherently all over the Universe.Follow @telescoper