I've just posted a new paper to the ArXiv, "Classical states, quantum field measurement", which comes out of a math bender I've been on for about the last month. I'll be coming out of that, hopefully. When I'm in that state, ideas come and go so fast that I lose track of them, but almost all of them turn out to be nonsense, so there's really no need to keep track. I make so many mistakes, I've seen ideas I've thought absolutely solid crash and burn because of sign errors, or conceptual misunderstandings, or really anything that can go wrong, that I now always prefix anything with "if I haven't made any mistakes". Sometimes I only realize after a few years what every physicist knows. But this paper feels a little different. Something really dropped out that's simple enough (in a mathematician's sense of simple), just a few lines, the whole paper's only 4 pages, that if I've made a mistake there's not many places for it to hide. Either on Saturday or Sunday, so this is really much too soon to feel confident, I tried something and it worked spectacularly.

So what does this paper do? One of the problems in trying to understand quantum field theory is that "quantized Dirac spinor fields" (otherwise called "fermion fields", they're what we use to describe matter in contrast to electromagnetism) are a lot different from classical physics. This paper kinda fixes that, it makes fermion fields look almost as classical as a 19th Century physicist could wish it to be. Not quite, because one doesn't and one doesn't want to get rid of 90 years of history, but if physicists understand it and I've made not too many mistakes, and hopefully no big mistakes, there'll be some change.

So what does this paper do? Enough of the hand waving! The things (operators) that come out of a Dirac field that correspond to what we can measure are the constituents of what is called a Lie algebra; there are other operators that don't correspond to anything we can measure, a bigger algebra that is the heart of what is a lot different about fermion fields, but they're so necessary to the way the theory is constructed that they've really been thought part of the whole package. This paper introduces a new way to construct the same Lie algebra of observables, but using different, almost, very nearly, really all but classical tools to do it. Once a mathematician sees the few lines that set this up, and if they also accept the embedding of the Lie algebra of observables into the new big Lie algebra, really a whole lot becomes possible. Even if it comes to nothing, there's something about having a new perspective that makes everything never the same again. In the light of the new bigger algebra, that old bigger algebra looks a lot more sensible.

I doubt anyone here will want it, but I can't give a link to the paper on the ArXiv until this evening. The abstract is

Manifestly Lorentz covariant representations of the algebras of the quantized electromagnetic field and of the observables of the quantized Dirac spinor field are constructed that act on Hilbert spaces that are generated using classical random fields acting on a vacuum state, allowing a comparatively classical interpretation of the states of the theory.

so that's fun. [Added September 21st, https://arxiv.org/abs/1709.06711.]

This is something of a placeholder, a way to cite a construction that I think deserves to be known widely that I put into an otherwise deservedly unpublished paper, https://arxiv.org/abs/1211.2831.
So, three images from that paper, the last being the only bibliographical reference needed (and very general it is):

The point of this is just that it contrasts with the usual way of talking about summing amplitudes over all possible trajectories, forward and backward in time without limit. The Lagrangian can be said to fix a deformation of the differential equation that is satisfied by the free field. Not to discount taking a path-integral understanding of the Lagrangian seriously as well, as a contrasting point of view, but we can take the interacting field to be constructed as a complex of free field operators that is purely contained in the backward light-cone of a given point. We can think of the action of the interacting field as a consequence of interference between a carefully weighted infinite sea of free field components that is isolated in the backward light-cone of x; or we can think of it as just doing what has to be done to make it look like there's something complicated at x; or ... .
Everything here can be said to follow from the action of the interaction Lagrangian on the free quantum field by time-ordered commutation being effectively the same as the action of the same expression on the free classical field by time-ordered Poisson bracket, in the sense that the same differential equation is satisfied by the interacting quantum field and by the interacting classical field --- up to the usual worries about renormalization.

Quantum measurement and Free Will

John Bell invoked Free Will for experimenters as part of a derivation of inequalities that would have to be satisfied by classical relativistic models\cite[Bell, Chapter 12], a modification of an earlier stipulation that experimental choices should be "at the whim of experimenters"[Bell, Chapter 7].

He more pragmatically requires that the experimenters' choices should be ``effectively free for the purpose at hand'', which suggests some consideration of just how free that might be in the context of quantum measurement.

Consider Alice and Bob running two ends of an experiment. Alice and Bob each have to choose a random sequence of 0s and 1s. If either of them chooses 0 too often or 1 too often, we have to restart the data collection. They're also not allowed to have too many 0000 sequences, too many 01101110 sequences, et cetera; they have to satisfy all the tests here, say, within some pre-agreed limits. They're not allowed to look at the statistics of their past choices to make sure that they don't break any of the rules. A typical experiment might need Alice and Bob each to generate a sequence that contains a few hundred million 0s and 1s that can be certified after the event to be random enough. Furthermore, without conferring, the two lists must not be correlated, again within some pre-agreed limit. Hard to do. Alice and Bob don't seem to be very free at all. Every individual 0 or 1 can be freely chosen, but the statistics are constrained.

Alice and Bob in practice farm out the job either to random number generators or to photon detectors driven by light from stars 600 light years away (arXiv here). No Free Will required.

I don't have much problem with Bell-EPR experiments these days, but the seemingly pervasive idea that Free Will plays a significant part in the discussion seems unsupportable.

The discussion above hints at the stochastic nature of the constraints on Free Will. Suppose that Alice and Bob are both friends of Wigner. They agree that Wigner can construct quantum mechanical models of their brains that predict the statistics of their choices, which is checked while they practice choosing a list that contains millions of random numbers, millions of times. If quantum theory is truly universal, this is just hard to do, even very hard, but it's not in principle impossible. This model doesn't constrain Alice and Bob's Free Will, it just describes where their Free Will has brought them to. If Alice and Bob include observations of stars 600 light years away to decide their 0 and 1 choices, then Wigner has to include a quantum mechanical model of the light from those stars that is accurate enough to describe the statistics of Alice's and Bob's lists. A quantum mechanical model describes the statistics of Alice's and Bob's choices about as much as would a classical stochastic model.

Bell J S 1987 Speakable and unspeakable in quantum mechanics (Cambridge: Cambridge University Press).

I'll do a bit of catching up on newish news. After a conversation with our daughter, I posted a video to YouTube, https://www.youtube.com/watch?v=frSL-BJTh90, that makes a blunt point about quantum mechanics:

Quantum Mechanics: Event Thinking

Published on Feb 18, 2017

To save time, watch the last five seconds, where I write down the word that this is in part a polemic against. That word appears in almost every interpretation of quantum mechanics. In this video, I talk about how to think about quantum mechanics as about events instead of using that word. This isn't a full–blown interpretation of quantum mechanics in 4'26", but it's a way of thinking that I find helpful. Something can be taken from this way of thinking without knowing anything about quantum mechanics, but inevitably the more math you know already the more you'll pick up on nuances (and, doubtless, know why you disagree with many of them).

Thinking about quantum mechanics as about events helps a little, but thinking of quantum field theory as a formalism for doing signal analysis is better, if you can get to that level of mathematics.

Adding a little more thinking in terms of events, imagine that we have a black box that puts out a continuous zero voltage on an output wire, but occasionally something happens inside the box so that the voltage rises sharply to some non-zero voltage for a very short period of time, then the voltage equally sharply returns to zero. We set up a clock so that whenever the voltage rises the time is sent to a computer's memory.
  When we put our event black box into a dark room at 20℃ we see events every now and then; if we change the temperature, the statistics of the events changes a little. Imagine we have a different kind of black box, which has a power cable into it, but no output, however when we introduce this box and turn on its power, the statistics of the events from the first box change, so we call the new kind of box a source of events. If we move the source black box to a different place, the statistics of the events change.
  If we have a number of event black boxes, we can do more sophisticated statistics, including correlations between when events happen. Then we can introduce multiple source black boxes and other apparatus, such as lenses, prisms, waveplates, polarizers, crystals, etc., and see what changes there are in the statistics.
  After many decades, we would have a quite comprehensive list of how statistics change as we change many aspects of the geometrical arrangement of source black boxes, event black boxes, and other apparatus. We would find that how the statistics change obey various equations as we move the pieces around. Eventually we would find that there are different kinds of source black boxes, which affect different kinds of event black boxes differently, and we would characterize the different ways that changes of the geometry change the statistics of the events.
  One thing that would soon become clear is that event black boxes cause statistics associated with other event black boxes to change. We'd like to have event black boxes that cause other statistics to change as little as possible, but we'd be disconcerted to discover that there's a limit to how much we can reduce the changes that an event black box will cause in the statistics of other black boxes' events.

  To return to the real world, which already knew about the electromagnetic field, electrons, atoms, before anyone thought of recording times of events so systematically, there was already a lot of knowledge about different kinds of sources, much of which had to be unlearned when quantum mechanics came along. When we use just light, the equations are provided by quantum optics. There are different equations if we use different types of source black boxes. We know what type of source black box we are using because the statistics change differently as we change the geometry. A lot of the work of quantum mechanical experiment is to characterize newly invented source black boxes using event black boxes we have already characterized with other sources carefully enough that we can use the new source black box to characterize newly invented event black boxes.

  The altogether too difficult question is "what is there between the source black boxes and the event black boxes?" The instrumentalist is quite certain that it doesn't matter, all we need to know is how the statistics change. As I said in the last post, there are so many possibilities that it's worth not worrying about what's between too much so we can do other things. Not quite the old-timers "shut up and calculate", more "we can do some fun stuff until such time as there's something it's useful to say for the sake of doing even more fun stuff". There is, inevitably, a lingering thought that if we better understood what is between we could do more fun stuff, but the regularities will be the same whether we understand or not.

Seven years later

Seven years away from this blog. The biggest change is that I've mostly reconciled myself to quantum theory, which would have been a surprise to me seven years ago but seems quite natural to me now. The name of the blog is probably not as appropriate as it was, but whatever.

Why that change? Mostly because there are so many ways to have something "under" quantum theory. "Stochastic superdeterminism" is possible, faster-than-light can't be ruled out if it has only limited effects at large scales, neither can a myriad of GRW-type or de Broglie-Bohm-type approaches (if one is generous about a few things). All of them are somewhat weird, but how are Buridan's ass and I to choose? Moreover, the statistics of the regularities of nature are the same either way, which will not kill me any faster whether they have something one might call an explanation or not.

In any case, by now I'm mostly happy to say that "quantum field theory is a signal processing formalism". Modern physics comes down to recording in a computer as much as we can fit into a reasonable amount of memory. A typical electrical signal could be recorded as an average voltage every trillionth of a second (a terabyte per second, say), but we don't do that because we don't have enough memory, so we save a very lossily compressed signal, perhaps, and quite commonly, as just the times when the signal changed from a low voltage to a high voltage (which might be only a few kilobytes per second). For that to be possible, we have to engineer the hardware so that the electrical signal does make transitions consistently from one voltage to another, and so that a timer is triggered to send the time to computer memory when the transition happens. The records in computer memory are what have to be modeled and perhaps explained by a quantum theoretical model. Where things get tricky is making those models as easy to use as possible. Specifically, we'd like to use quantum theory for reliable everyday engineering, we don't want to have to spend years figuring out how to make some new piece of apparatus work, so there's a kind of simplicity required. Physicists and engineers have all sorts of rules of thumb that work pretty well for relating new experimental apparatus to quantum theoretical models, and I've become more happy than I was to say that's OK, though knowing everything you need to know about quantum optics alone has become a lot.

Enough for now.