I continue to believe that “quantum field theory” is a concept that we physicists don’t do nearly enough to explain to a wider audience. And I’m not going to do it here! But I will link to other people thinking about how to think about quantum field theory.
Over on the Google+, I linked to an informal essay by John Norton, in which he recounts the activities of a workshop on QFT at the Center for the Philosophy of Science at the University of Pittsburgh last October. In Norton’s telling, the important conceptual divide was between those who want to study “axiomatic” QFT on the one hand, and those who want to study “heuristic” QFT on the other. Axiomatic QFT is an attempt to make everything absolutely perfectly mathematically rigorous. It is severely handicapped by the fact that it is nearly impossible to get results in QFT that are both interesting and rigorous. Heuristic QFT, on the other hand, is what the vast majority of working field theorists actually do — putting aside delicate questions of whether series converge and integrals are well defined, and instead leaping forward and attempting to match predictions to the data. Philosophers like things to be well-defined, so it’s not surprising that many of them are sympathetic to the axiomatic QFT program, tangible results be damned.
The question of whether or not the interesting parts of QFT can be made rigorous is a good one, but not one that keeps many physicists awake at night. All of the difficulty in making QFT rigorous can be traced to what happens at very short distances and very high energies. And that’s certainly important to understand. But the great insight of Ken Wilson and the effective field theory approach is that, as far as particle physics is concerned, it just doesn’t matter. Many different things can happen at high energies, and we can still get the same low-energy physics at the end of the day. So putting great intellectual effort into “doing things right” at high energies might be misplaced, at least until we actually have some data about what is going on there.
Something like that attitude is defended here by our former guest blogger David Wallace. (Hat tip to Cliff Harvey on G+.) Not the best video quality, but here is David trying to convince his philosophy colleagues to concentrate on “Lagrangian QFT,” which is essentially what Norton called “heuristic QFT,” rather than axiomatic QFT. His reasoning very much follows the Wilsonian effective field theory approach.
The concluding quote says it all:
LQFT is the most successful, precise scientific theory in human history. Insofar as philosophy of physics is about drawing conclusions about the world from our best physical theories, LQFT is the place to look.
“At the end of the day, physics is all about formulating a set of rules that enables you to predict the results of experiments.”
This isn’t true at all. If it were, physics would only matter to physicists. And there would be no reason for the physicists to expect us to pay anything for their hobby. Axiomatization is a procedure for understanding, which is what physics (indeed, all science) really is about. As such, axiomatization of a true theory must be desirable. The issue is whether axiomatization is feasible, and if it is, whether QFT is the true theory that should be axiomatized in order to understand. It is doubtful that axiomatization is the only way to understand a theory, so it is doubtful that QFT must be axiomatized as part of validating and interpreting the theory.
There are people working on rigourous quantum field theory who take the Wilsonian viewpoint, eg. Vincent Rivasseau and Kevin Costello.
“Axiomatic QFT is an attempt to make everything absolutely perfectly mathematically rigorous. It is severely handicapped by the fact that it is nearly impossible to get results in QFT that are both interesting and rigorous.” Just as in the theory of the statistical distribution of prime numbers the most important question is the Riemann Hypothesis, the most important question in axiomatic QFT is perhaps the Mass Gap Problem.
The Clay Mathematics Institute offers a one million dollar prize for a mathematically valid solution of the “Yang Mills Existence and Mass Gap problem” for details see
http://www.claymath.org/millennium/Yang-Mills_Theory .
http://en.wikipedia.org/wiki/Yang-Mills_theory
http://en.wikipedia.org/wiki/Gauge_theory
If the Mass Gap Problem has an affirmative solution then M-theory perhaps has a formulation within QFT. If the Mass Gap Problem has a negative solution then M-theory perhaps has a formulation as an approximation to the model qualitatively described by Wolfram in “A New Kind of Science” Chapter 9.
http://en.wikipedia.org/wiki/M-theory
http://en.wikipedia.org/wiki/A_New_Kind_of_Science
The heuristic approach sounds a lot like what Newton did FIrst, he got the thing working and useful, then he let others worry about the axiomatics. Newton’s calculus was extremely useful for at least 200 years before someone figured out why it works. In fact, calculus, and the fact that it was useful, drove a lot of important mathematics that then enabled that more rigorous approach.
It doesn’t make much sense to be concentrating on deriving something from first principles when we have no clue as to what those principles are or what they should be deriving. That sounds seriously under-constrained.
Hi Sean,
One thing I wanted to ask you about QFT was in relation to a few quotes that I read last week on the subject (Im in a philosophy of physics class at Cornell right now so the ontology of different concepts like particle will be discussed). This quote was on the Stanford philosophy page for QFT:
“Not only are sharp trajectories excluded by Heisenberg’s uncertainty relations for position and momentum coordinates which hold for non-relativistic quantum mechanics already. More advanced theoretical examinations in AQFT which will be described and scrutinized below, show that quantum particles which behave according to the principles of relativity theory cannot be localized in any bounded region of space-time, no matter how large, a result which excludes even tube-like tajectories.”
How can a sentence like this be valid if we do in fact observe particle like traces in accelerator experiments? Would this not mean that QFT is lacking in a certain sense because it disallows these types of tight pathways for a particle? Or am I right to think this article is just plain wrong? In addition, I came across this sentence on the wiki page for QFT:
“The “second quantization” procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the electromagnetic field must be a quantum field theory, because it is impossible (for various reasons) to define a wavefunction for a single photon.”
I had thought that all particles can be described by a wave function. What are the “various reasons” why a photon cannot be described by a wavefunction? I am starting to think this might be an error as well, as there is no citation provided for this comment. If true, however, would that not be a problem for our best theory? Might it hint to future refinements of the theory that are necessary? I’m a big proponent of the deterministic/reductionist worldview that CV seems to defend, so I would definitely appreciate if you could shed some light on these questions I keep thinking about.
PeteH– When they say “cannot be localized to any bounded region,” they mean *precisely* localized. They don’t exclude a situation where the probability of finding the particle outside the region is less than 1 in a trillion (or whatever).
Single-particle wave functions are part of non-relativistic QM, but not really part of quantum field theory. The reason is just because there is always some amplitude for creating or destroying other particles. The fields keep track of that bookkeeping, so it’s actually a *prediction* of our best theory.
(Really you should talk about “the wave function of the field,” which is done in something called the “functional Schroedinger approach” to QFT. But it’s rarely discussed.)
Every time I hear an argument of the sort of “Not only are sharp trajectories excluded by Heisenberg’s uncertainty relations for position and momentum coordinates which hold for non-relativistic quantum mechanics already.” I can not avoid thinking about the trajectories of electrons (or any other charged particle in a tracker device).
You can perfectly see the tracks in the detector (http://www.particlephysics.ac.uk/news/picture-of-the-week/picture-archive/tracks-in-a-hydrogen-bubble-chamber/000329_med.jpg for an example) and the trajectories are perfectly well defined once you take into account the magnetic field in the detector. So what is happening? I will refrain to post the answer just to entice that “thinking about quantum…” thingie that this post embraces. But please realize that the statement is, at the very least, misleading.
@32: This is an extremely old question with a very old answer, which has evolved over many decades and which has by now been subsumed into the modern theory of decoherence of quantum systems. One of the early papers dealing with this very question is “The wave
mechanics of alpha-ray tracks” Proc.Roy.Soc. v.A126, pp.79–84 (1929).
@Igor Oh! I shouldn’t have underestimated the efficiency of CV readers! Of course, it is an old question. But one that still seems to baffle quite a bunch of grad students (I have seen it first hand). I actually learned the answer from Park’s book on QM, but now that you pointed to the original article by Neville Mott it turned out to be a remarkably clear one. Thanks for the link.
Thanks a lot for the reply Sean and others. Definitely good to get these conceptual questions cleared up. And its even better to see that there are no problems with QFT.
@22 Marko Says:
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If you want to have a theory which doesn’t have such axioms (that limit the applicability of other axioms), then in QFT you may easily run into a problem with the theory being (a) self-contradictory, and/or (b) non-predictive, and/or (c) experimentally incorrect.
[…]
If you are a physicist, you must care about (c) first, and about (a) and (b) only if (c) is true. In the case of QFT, (c) is known to have failed, ie. QFT does have predictions that contradict observations (just calculate anything in QED beyond the 137th order of perturbation).
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1) May you point out to some literature on the failures of QED beyond a given order of approximation? (you said 137th order, I’m just curious about that).
2) BTW, about the general topic, I think that on the contrary of what a physicist may think
(see comment @22 above), a mathematician may be interested in foundational matters of QFT just because the “heuristic” techniques used have proven to be of “some” usefulness in “computing”.
Hence, the mathematician may wonder if anything more fundamental may be grasped by such “heuristic” approaches to be of value in mathematics.
It seems to me that such a meaning is implied by at least some mathematician working on foundational QFT.
The 137th order issue is just about the divergence of asymptotic series. The fine structure constant alpha is approximately 1/137 and the nth term for large n is of the order of alpha^n n!. As a rule (which isn’t always valid), you obtain the best approximation by truncating a divergent asymptotic series by summing until the smallest term. The error is then of order exp(-1/alpha) which lies “beyond all orders of alpha”, so to compute the error term you have to go beyond perturbation theory. But this is still possible within field theory, there exist non-perturbative methods. E.g. the rate at which electron-positron pairs are created in an electric field vanishes to all orders in pertuirbation theory, yet it can be calculated exactly, it’s just non-analytic in alpha.
Also, sometimes you can resum the divergent tail of a perturbation series and estimate the non perturbative contribution quite accurately from that. Borel resummation can sometimes be used, but in QFT this usually doesn’t work.
AQFT has become hidebound insofar as its axioms are not as much discussed as they might be. The replacement of the Wightman axioms by the Haag-Kastler axioms seems to have made the lack of discussion more extreme. Instead, people work within the axioms even though it has so far been impossible to construct realistic models in 3+1 dimensions. It seems reasonable to ask how the axioms might be weakened to allow realistic models as well as to continue trying to construct models of the axioms as they currently exist.
Let me suggest here reconsideration of the existence of a ground state as part of the axioms. There is no ground state in a thermal sector of the free field; instead there is a most symmetric state, effectively the most disordered state in the sector. This change allows a continuum of models. Curiously, this possibility is not mentioned in the most recent assessment of the axioms that I’m aware of, Streater, Rep. Prog. Phys. 38, 771-846 (1975), “Outline of axiomatic relativistic quantum field theory”, though relinquishing uniqueness of the vacuum is discussed.
The effectiveness of LQFT is quite striking because at its heart it takes the interacting field to be unitarily equivalent to the free field of which it is a deformation. That is, LQFT constructs a different representation of the free field that is, presumably, not constrained by all the Wightman axioms. Everything is obscured by a number of quite badly controlled high- and low-frequency limits that then have to be redone (regularization, resummation, renormalization) in a way that gives a finite result. For practical purposes, this hits the spot, but it would seem better to find different ways to construct nontrivial representations of free fields that need less re-… to obtain finite results.
This is the way a layman thinks. We will never understand our physical realty. So our goal is not to understand but to devise way of thinking about it that are useful and correctly predicts physical results. Success is thinking about realty as waves when waves work and thinking about realty as particles when particles work. Do not even bother to wonder whether reality is waves or is reality particles. We can think only in the analogies that we understand or think that we understand. The realities of what we call QFT are way beyond our understanding. Our next step in “talking about it” in an understandable way is to put the observer right in the middle of our thinking analogies. Remember: “Now we see in a mirror dimly, but someday face-to-face”