Particles in quantum field theory

[A. Neumaier]

In this thread, I want to discuss the implications of quantum field theory for the interpretation of quantum mechanics. To set the stage I'll import in the next few posts a number of posts from other threads. The latest of these is the following:

Quantum field theory certainly can be used to describe a single electron in a state with a definite spin-up in the z-direction, can't it?

Only if it is the sole particle in the whole universe.

 

[stevendaryl]

Only if it is the sole particle in the whole universe.

Fine. So in that particular case, QFT does not support the claim that quantum uncertainty is due to lack of information. So it seems illegitimate to assume that such a classical notion of uncertainty applies in the case of huge, complex systems, when it clearly doesn't apply to simple systems.

 

[A. Neumaier]

in that particular case, QFT does not support the claim that quantum uncertainty is due to lack of information. So it seems illegitimate to assume that such a classical notion of uncertainty applies in a case of huge, complex systems, when it clearly doesn't apply to simple systems.

In the particular case stated, there is nothing in the universe to observe it, so its interpretation is moot.

Please wait with further discussion until I have added all relevant pieces from earlier threads, so that all background information is there. This may take a few hours. I'll state when the import is completed.

 

[A. Neumaier]

Let me begin with a short overview of the background assumed in this discussion.

Quantum field theory (QFT) is the most fundamental description of Nature that we presently have. It comes in two flavors, nonrelativistic and relativistic.

In both cases, the observables are smeared fields, their derivatives, products, and linear combinations. For simplicity they will be described here in terms of distribution-valued field operators ##\phi(x)## with spacetime arguments ##x## satisfying equal-time causal commutation relations. The free fields have a particle interpretation, whereas the interpretation of interacting fields is difficult to describe in words or images since everything is obscured by renormalization issues. Renormalization is present both in the nonrelativistic and in the relativistic case; in the latter case there are additional issues with divergences. these are outside the scope of the present discussion.

Most relativistic quantum field theory books only discuss ''asymptotic'' aspects of QFT, which are relevant for predicting cross sections of nuclear reactions and collision experiments. Here particles live in
the asymptotic Hilbert spaces at times ##t\to-\infty## (for the input to a scattering experiment) and ##t\to+\infty## (for the output). At times ##\pm\infty##, there is a clear notion of asymptotic particles
(bound states) since the asymptotic Hilbert spaces are Fock spaces with a natural particle interpretation. In this setting, what happens at finite times is considered irrelevant since only the S-matrix counts. It mediates between the free input Hilbert space and the free output Hilbert space and is a unitary matrix which describes according to the Born rule the detection probability of all collision products. Nothing is said about what happens in multiple collisions, since this cannot be modeled in terms of asymptotic times and spaces only.

In the asymptotic Hilbert spaces of quantum field theory, the Fock spaces, there are (apart from the empty vacuum state and single-particle states) only symmetrized (or antisymmetrized) multiparticle states and their linear compbinations. One cannot create any others using creation operators - they are unphysical. In particular, there are no position operators that could tell the position of a particle. The particles are defined asymptotically - not by their position but by their momentum, which gives their
direction of flight and their energy at very large positive or negative times. For the application to real life scattering experiments it is important to realize that experimental time scales are already ''very
large'' in the microscopic units relevant for few particle scattering events, hence approximating them by ##\pm\infty## is a valid approximation.

There is a second, ''global'' branch of quantum field theory, represented in books and papers about nonequilibrium statistical mechanics. Here macroscopic many-particle systems are considered, and
the focus is either on equilibrium, or on a dynamical description at finite times. This global branch of QFT is relevant for a discussion of the measurement process as an actual process involving a tiny quantum
system and a macroscopic detector. It is also the branch relevant for a discussion of the universe as a quantum system, as it is clearly macroscopic. In contrast, the asymptotic branch of QFT describing scattering theory is usually confined to a mini universe consisitng of two ingoing particles (producing the input collision) and their debris after collision.

Both branches rely on the same operator description of the observable field algebra, and utilize it in mutually consistent but otherwise very different way. For a readable introduction to the less well-known
global branch of QFT see, e.g.,

J. Berges,
Introduction to Nonequilibrium Quantum Field Theory,
AIP Conf. Proc. 739 (2004), 3--62.
(preprint version: hep-ph/0409233)

 

[A. Neumaier]

Let me continue with some remarks on the meaning of what is computable in finite-time QFT. The central objects in finite-time QFT are the expectations of products of one or more field operators at different spacetime positions in a given Heisenberg (space-time independent) state. The state characterizes the whole universe under consideration; the expectations are collectively referred to as N-point functions; for N=2 as correlations functions.

Here the term ''universe'' refers to a completely isolated system not interacting at all with anything outside - thus it is a universe in itself. If the universe is microscopically small we have a scattering experiment; if the universe is of macroscopic, human-scale size we have a piece of solid or fluid matter, and if the universe is of cosmological size we have a true universe. In the first two cases, there is an idealization involved since a real system of such a size is not truly isolated; this has to be accounted for outside QFT.

To find out what the N-point functions mean, we turn to the macroscopic limit of quantum field theories. These are treated in a more or less approximate way in typical books on nonequilibrium statistical mechanics, at least for the case of ideal gases. The end result is always a classical description, usually hydromechanic or kinetic. From these limiting descriptions one can infer that the 1-point functions are just the quantum analogues of the classical fields in 4-dimensional space-time, whereas the Wigner transforms of the 2-point functions are the quantum analogues of the classical fields on a 7-dimensional mass shell of an 8-dimensional phase space. Thus not the field operators themselves but the N-point functions with N=1 or N=2 contain the classically observable information. N-point functions with larger N no longer have a direct classical meaning but appear in the BBGKY-like truncation schemes for constructing classical dynamical equations from the quantum description.

Note that the nonlinearity of the quantum field equations directly translate into strong nonlinearities of the macroscopic effective equations (Navier-Stokes, Boltzmann, quantum BBGKY).The above remarks also show that quantum field theory does not only predict probabilities but predicts the whole of macroscopic classical mechanics (though proofs are available only in special cases). Quantum field theory predicts - under the usual assumptions of statistical mechanics, which include local equilibrium - hydrodynamics and elasticity theory, and hence everything computable from it.

Of course it predicts only the general theoretical structure, since all the detail depends on the initial conditions. But it predicts in principle all material properties, and quantum chemists are doing precisely that when they use the Dirac-Fock-Bogoliubov approximation of QED. All items mentioned are essentially exact predictions of QFT, with errors dominated by the computational techniques available rather than the uncertainty due to the averaging. Together with prepared or observed initial conditions it predicts the values of the macroscopic observables at later times. For example, computational fluid dynamics is an essential tool for the optimization of modern aircrafts.

Local equilibrium itself is usually justified in an ad hoc way assuming fast relaxation scales. These can probably be derived, too, but I haven't seen a derivation. But one knows when this condition is not satisfied in practice - namely if the mean free path lenth is too long. This happens for very dilute gases, where the Boltzmann equation must be used instead of hydrodynamic equations (and can be derived from QFT).

Note that the standard properties of expectation values imply intrinsic uncertainty estimates for the accuracy of the observable fields (N-point functions). For application to actual macroscopic measurements we do not need definite values but only values accurate enough to match experimental practice. This is a much less severe condition.

We all know from classical nonequilibrium thermodynamics that the macroscopic local observables are a small set of fields (in the simplest case just internal energy density and mass density). We also know from statistical mechanics in the grand canonical ensemble that these are given microscopically not by eigenvalues but by certain well-defined expectations. Under the assumption of local equilibrium, the fluctuations of the corresponding averaged quantum fields around the expectations are negligible. Thus the values of the macroscopic effective fields (obtained by corresponding small-scale averaging in the statistical coarse-graining procedure) are sharp for all practical purposes.

Mathematically, this becomes exact only in the thermodynamic limit. But for observable systems, which have finite extent, one can estimate the uncertainties through the standard fluctuation formulas of statistical mechanics. One finds that for macroscopic observations at the human length and time scale, we typically get engineering accuracy. This is the reason why engineering was already successful long before the advent of quantum mechanics.

Thus quantum field theory provides a satisfying description of completely isolated quantum systems ranging from the microscopic scattering system to the macroscopic fluid and solid systems of everyday life. Nowhere is an obvious measurement problem, since the systems modeled by QFT are assumed to be perfectly isolated and hence noninteracting with whatever else might exist. Nevertheless, macroscopic fields are predictable (in principle) to engineering accuracy.

 

[A. Neumaier]

The question to be discussed in the following is what this implies for the measurement and analysis of microscopic experiments with particles and entangled states.

If one takes the above facts as given, the universe (including the parts not observable by us) is a single quantum object composed of many material and nonmaterial parts. The material parts are called galaxies, stars, planets, houses, bricks, cells, molecules, atoms, quarks, etc., the nonmaterial parts are called light, electric fields, magnetic fields, gravitational fields, etc..

For simplicity, I only consider two fields - the mass density (material) and the energy density (immaterial) - whenever I want to be a bit more concrete. In quantum field theory, these fields exist everywhere, but where they have zero (or small enough) mass or energy density they have no physical effect and are considered absent. For example, the solar system has an appreciable mass density concentrated on a limited number of bodies only (the Sun, the planets, asteroids, comets, and space-crafts and their debris), but additional tiny mass distribution in interplanetary space.As a quantum system, the universe is described by a Heisenberg state, represented as a density matrix, an operator acting on a huge, universal Hilbert space. The main assumption needed is that the part of the universe observable by us is approximately in local equilibrium. This is amply corroborated by experiment, and provides a very strong constraint on the universal density matrix. Indeed, local equilibrium is just the assumption needed to derive fluid mechanics or elasticity theory from quantum field theory, and for more than a century we describe every macroscopic object in these terms. Thus only those density matrices qualify as typical that satisfy this experimental constraint. In my free online book, I call the corresponding states Gibbs states.

The typical state of a system realized in Nature is given by a density matrix. A density matrix is well-behaved under restriction to a subsystem, and hence can be used to describe systems of any size. In particular, it is consistent to consider each density matrix of a system in our universe as a restriction of the density matrix of the universe. I postulate that the latter (described by a quantum field theory that we don't know yet in detail) is objectively existent in the sense of realism, and objectively determines the density of everything in the universe, and hence in any part of it.

So the universe has a density matrix, and by restriction one can get from it the density matrix of arbitrary parts of it (given a sufficiently well-defined operational definition of which part is meant). For example, one can look at the density matrix of the Sun, the Earth, or the gravitational field in between. Or of a beam of particles, or a detector, or the current Queen of England. (Well, in the last two cases, there will be some ambiguity concerning precisely which part of the universe belongs to the object. But strictly speaking, we have this problem already for objects like the Earth or the Sun, where the atmosphere gets thinner and thinner and one has to make an arbitrary cutoff.)

As a consequence, the density matrix of any subsystem that can be objectively delineated from the rest of the universe is also objective (though its dynamics is partially uncertain and hence stochastic, since the coupling to the environment - the remaining universe - is ignored).
But one cannot consider the density matrix of Schroedinger's cat, since it is not a well-defined part of the physical universe.

On the other hand, our human approximations to these density matrices are subjective since they depend on how much we know (or postulate) about the system. They are only as good as the extent to which they approximate the true, objective density matrix of the system.

For example, a cup of water left alone is after a while in a state approximately described by a density matrix of the form discussed in statistical thermodynamics. This has the advantage that the density matrix can be described by a few parameters only. This suffices to determine its macroscopic properties, and hence is used in practice although the true density matrix is slightly different and would account for tiny, practically irrelevant deviations from thermodynamics.

The more detailed a state description is the more parameters are needed to describe it since a quantum field has infinitely many degrees of freedom in any extended region of space. For more, read Chapter 10 of my online book mentioned above.

 

[A. Neumaier]

OK, I found it far easier to write a new synopsis of the relevant background than to cross-link to all former discussions. My exposition is now reasonably complete and can be discussed, together with everything that has been left open from other threads that touched this topic. If informative you are welcome to add in the discussion corresponding links.

Note that the density matrix of the universe has a deterministic dynamics; hence anything probabilistic is introduced into our local views of part of the universe as uncertainty about the details both in the state of the part we know and in the rest of the universe - in the same way as a finite subsystem of atoms in a Newtonian universe would be able to form only a stochastic picture of whatever this subsystem can observe or infer about the big universal system it is embedded in.

 

[A. Neumaier]

The book http://www.langtoninfo.com/web_content/9780521641685_frontmatter.pdf Theory by Calzetta and Hu covers most of the above; look especially at Part IV! See also an article by http://journals.aps.org/prd/pdf/10.1103/PhysRevD.53.5799 .

Note that quantum chemists and quantum field theorists usually don't talk about interpretation issues; they are typically practitioners of shut-up-and-calculate. Indeed, they don't need to care about foundations since they work on the basis of what I described in the previous posts, where everything, including the quantum-classical correspondence, takes a very intuitive form.

You want to know the color of gold? It is encoded in the spectrum of an appropriate Hamiltonian, so one can go ahead and calculate it. No probabilities are involved, only a good approximation theory, good basis sets, and perturbation theory. Well, the ''only'' hides a lot of technical machinery, but no philosophy or interpretation is needed. (Why is mercury liquid? Or, why do relativistic effects not get into chemistry textbooks?. Journal of Chemical Education 68 (1991), 110.)

In general, spectral properties are encoded in correlation functions (2-point functions of appropriate fields and currents), and these are the output of appropriate quantum field theory calculations. (For example, see Yndurain's book in case of QCD.)

There is a gradual transition from quantum to classical, utilized in practice by a host of quantum-classical models where the complexity of molecules, say, is mostly handled via classical mechanics, while a few crucial degrees of freedom are treated quantum mechanically. http://publications.mi.fu-berlin.de/75/1/HoSaScSc02.pdf is an example of this.

My modest contribution to all this is just that I noticed the following: What is sufficient for quantum field theory is also sufficient for quantum mechanics (which is in a sense a simplified quantum field theory for systems with a fixed, small number of particles), and gives more intuitive foundations than the traditional interpretations. In later posts I'll show in which sense the traditional quantum mechanical picture arises from quantum field theory.

 

[vanhees71]

OK, I found it far easier to write a new synopsis of the relevant background than to cross-link to all former discussions. My exposition is now reasonably complete and can be discussed, together with everything that has been left open from other threads that touched this topic. If informative you are welcome to add in the discussion corresponding links.

Note that the density matrix of the universe has a deterministic dynamics; hence anything probabilistic is introduced into our local views of part of the universe as uncertainty about the details both in the state of the part we know and in the rest of the universe - in the same way as a finite subsystem of atoms in a Newtonian universe would be able to form only a stochastic picture of whatever this subsystem can observe or infer about the big universal system it is embedded in.

That's a great summary of QFT. You should transfer it to an Insight article.

I've only one trouble in understanding: What's the meaning of the density matrix (statistical operator) of the entire universe? I thought you are a strict ensemble interpreter, and then the idea of a statistical operator of the entire universe doesn't make sense, because you cannot build an ensemble of entire universes except you follow the idea of some kind of many-worlds interpretation. Although I've nothing specifically against this idea, I find it rather mute, because all the parallel universes, which just occur by some observation of an hitherto undetermined observable, are not observable and thus don't form an ensemble you could use to justify the idea of a statistical operator of the entire universe.

This, of course, brings up the general dilemma of cosmology, which you can a bit oversharply put as the question, whether cosmology can in principle be a natural science at all, because you cannot check the hypotheses by experiment in a strict sense. Of course, there's the very amazing piece of physics called cosmology, but strictly speaking this rests on the bold assumption of the cosmological principle, i.e., that the observable part of the universe is representative for any other part of the universe, which we cannot observe in principle. We simply assume that the picture we have in terms of a very coarse-grained view on the universe in terms of the FLRW metric is valid, because it fits quite well all (nowadays also very precise) observations like the CMBR temperature and polarization fluctuations, the redshift-distance relation through standard candles like supernovae etc. All this is of course very impressive but strictly speaking no observational verification of this picture concerning the entire universe.

 

[A. Neumaier]

That's a great summary of QFT. You should transfer it to an Insight article.

I'll do it in a week or so, after the current discussions are over.

All this is of course very impressive but strictly speaking no observational verification of this picture concerning the entire universe.

I consider it to be a reasonably valid extrapolation, of the same kind as we extrapolate from our knowledge to unobservable processes such as what happens in the deep interior of the Earth or sun.

I thought you are a strict ensemble interpreter, and then the idea of a statistical operator of the entire universe doesn't make sense

Yes, if I were. But even though I can put myself into the shoes of a Copenhagen interpreter or an ensemble interpreter, and know how to make each of these interpretations consistent (which I cannot say of Bohmian mechanics or MWI - which to me are both extremely weird although I believe I fully understand their claims), I don't think either the Copenhagen or the ensemble interpretation is the final word. Instead I have my own interpretation, the thermal interpretation of quantum mechanics, which is inspired by statistical mechanics and fits well the present thread. I'll come to it in the course of developing the theme here.

 

[A. Neumaier]

For a readable introduction to the less well-known
global branch of QFT see, e.g.,

J. Berges,
Introduction to Nonequilibrium Quantum Field Theory,
AIP Conf. Proc. 739 (2004), 3--62.
(preprint version: hep-ph/0409233)

See also the book
Calzetta,& Hu, Nonequilibrium quantum field theory. Cambridge University Press (2008).

 

[Demystifier]

Quantum field theory (QFT) is the most fundamental description of Nature that we presently have. It comes in two flavors, nonrelativistic and relativistic.

Sometimes it's not true. In condensed matter, QFT is often an effective theory derived from a more fundamental non-relativistic QM. E.g. by starting from a fixed number of non-relativistic atoms one can derive an effective QFT of phonon creation and destruction.

 

[A. Neumaier]

Sometimes it's not true. In condensed matter, QFT is often an effective theory

This doesn't make my statement non-true. QFT is the most fundamental description of Nature that we presently have, but it has a use that goes far beyond this.

 

[Demystifier]

This doesn't make my statement non-true. QFT is the most fundamental description of Nature that we presently have, but it has a use that goes far beyond this.

This is like saying that action functional is the most fundamental description of Nature that we presently have (because the Standard Model of elementary particles is based on an action functional), but it has a use that goes far beyond this. I would say that action functionals and QFT's are just theoretical tools, which by themselves are neither fundamental nor non-fundamental. The Standard Model of particle physics is fundamental, but QFT, as such, is not.

Likewise, the Standard Model is based on group theory and functional analysis, but does it mean that group theory and functional analysis are "the most fundamental description of Nature that we presently have"?

 

[vanhees71]

First of all in the case of QM with a fixed number of particles (i.e., interactions not changing the number of particles) the usual QM ("1st quantization") and the non-relativistic QFT ("2nd quantization") are exactly equivalent. The reason is that, if particle number is conserved, and you start with a Fock state of determined particle number in the QFT you always stay in this subspace.

Of course, you are right, when it comes to the description in terms of effective quasiparticle models the QFT formulation is mandatory. QFT is simply the more general case.

 

[dextercioby]

I know this looks like nitpicking, but the Standard Model is not based on functional analysis, for if it were, we wouldn't label it as mathematically unsound (I believe that one of the Clay Institute Millenium Problems is to write down a complete mathematical foundation of the Yang-Mills model in QFT. The latter is the theoretical cornerstone of the Standard Model).

 

[Demystifier]

I know this looks like nitpicking, but the Standard Model is not based on functional analysis, for if it were, we wouldn't label it as mathematically unsound (I believe that one of the Clay Institute Millenium Problems is to write down a complete mathematical foundation of the Yang-Mills model in QFT. The latter is the theoretical cornerstone of the Standard Model).

Maybe I should have say functional calculus.