Tuesday, October 20, 2015

Quantum field theory obeys all postulates of quantum mechanics

In particular, QFT can't contradict the superposition principle

A beginner who tries to learn quantum field theory, a user named Darkblue, asked a question on the Physics Stack Exchange,
Is a single photon always circularly polarized?
Darkblue has promoted – and is still promoting – the thesis that only circular polarizations and not linear polarizations may exist for an individual photon. To strengthen his point, he also refers to a (zero-citation) preprint by two crackpots that proposed to experimentally "test" a similar nonsensical claim.

(Yes, these two authors – proud senior janitors at IEEE – and Darkblue clearly belong to the Šmoitian sect of morons who feel very self-confident whenever they say "we want to make an experiment", regardless of how stupid the experiment and especially its interpretation is.)

Among other gems, the paper claims that a linearly polarized photon moving in the \(z\)-direction should have \(J_z=0\hbar\). Well, the expectation value \(\langle J_z\rangle =0\hbar\) may be zero but the vanishing value is strictly forbidden as a result of a measurement (zero isn't among the eigenvalues): there are no longitudinal or scalar photons.

I have posted an answer which I won't repost here. It says that we must decide what we measure and if we measure the circular polarization of one photon (left or right), which is a maximum package of information we may measure about an individual photon's polarization, we always get one of these two results. So the circular polarization always exists but typically, the predicted circular polarization isn't certain. The observables "which linear polarization" and "which circular polarization" don't commute with each other, just like \(x\) and \(p\) don't, so if we know one of those, we can't know about the other and vice versa.

In other words, we work with superpositions of states. Linearly polarized photons are the most "balanced" superpositions of the circular ones and they're equally possible to emerge in Nature. I wrote something about the classical electromagnetic waves – states with many coherent photons, too. For one photon, a linear polarization and a circular polarization are not strictly mutually exclusive; for the state of many coherent photons in the same state, the linear and circular polarization become mutually exclusive.

Darkblue has been annoyingly stubborn and much of his or her stupidity is idiosyncratic in character. Hopeless beginners like that sometimes decide to get stuck in a certain kind of a stupid delusion that no one else shares. But I wrote this blog post because of the "deja vu" effect. Some of darkblue's argumentation seems very similar to some widespread myth that hasn't been discussed too often on this blog, if ever. The myth is the following:
Myth: Quantum field theory is a modified, from quantum mechanics different, description of Nature in the same sense in which quantum mechanics differs from classical mechanics. So QM and QFT may disagree in their statements.
Well, this opinion is completely wrong. Quantum field theories are a subclass of quantum mechanical theories. All quantum mechanical theories obey the universal postulates of quantum mechanics, the axioms of the QM framework, such as
  • All information about the properties of the objects (the state of the physical system) is measured through the so-called observables.
  • Observables are represented by linear operators.
  • Allowed values resulting from a measurement are eigenvalues of the measured operator (the spectrum).
  • These operators form an associative but non-commutative algebra.
  • A complex Hilbert space is the minimal nontrivial representation of the algebra of these operators. All elements of this vector space may emerge as states after a measurement of a complete/maximal set of commuting observables (the superposition postulate) because each state is an eigenstate of an appropriate observable.
  • Expectation values of projection operators are physically interpreted as the probabilities of the corresponding outcomes of the measurements. The probabilities are basically everything one can predict in physics. In terms of pure states from the minimum representation (the Hilbert space), the probabilities are squared absolute values of the complex amplitudes (Born's rule).
Quantum field theories agree with all of those things. They just have a particular type of the operator algebras – the algebras of operators. In QFT, we assume that the operators may be organized in terms of things like \(\hat\phi_i(x,y,z,t)\) and the Heisenberg equation governing these operators (more precisely, operator distributions) are supposed to be Lorentz-covariant (relativistic). The Hilbert spaces in QFT have a rich structure and we must get used to some mathematical effects not present in the case of finite-dimensional (and simple enough infinite-dimensional) Hilbert spaces, such as the need for renormalization etc.

But if you haven't gotten to the topics of renormalization at all, I can assure you that your opinion that QFT is "something different" than a QM theory are complete misunderstandings. QFTs are strictly a subset of QM theories. I believe that there exists a simple sociological reason why some beginners tend to think that QFT is a "different kind of a theory" than QM, and it's the following:
The organization of knowledge into many subjects and courses makes the logical structure of physics look more complex and less unequivocal than it is.
What do I mean? Students learn classical mechanics and then electromagnetism and then quantum mechanics, and so on. These subjects mostly cover "non-overlapping" theories. So people assume that the same thing holds for quantum mechanics and quantum field theory and other pairs. But this segregation of the physics knowledge into many boxes or courses completely misses the key point about the "basic logical kinds of theories" we have in physics. The key point is the following one:
There only exist two types of theories in all of physics:
  1. Classical physics
  2. Quantum mechanics
  3. There is simply not third way.
I added the third option in order to emphasize that it doesn't exist.
Up to 1924, all state-of-the-art theories assumed the framework of classical physics. According to classical physics, there is an objective "state of the world". The possible states of the world form a set which is physically referred to as the "phase space". Because physics wants to deal with the continuous evolution (evolution in continuous time), the phase space is a smooth manifold, possibly an infinite-dimensional one. Equations of motion are usually deterministic and may be expressed by arrows along lines on the phase space that determine where a given state evolves after \(\Delta t\).

Since 1925, all state-of-the-art theories in physics had to be quantum mechanical. Schrödinger designed his non-relativistic quantum (or wave) mechanics for one electron in the atom. Multiparticle generalizations were found much like the relativistic Dirac equation. The latter two were extended to Quantum Electrodynamics, the most accurately tested example of a quantum field theory. Many quantum field theories – relativistic and non-relativistic ones – were studied by particle physicists and condensed matter physicists. And string theory is the only new "kind" of a theory that may be, at least in a certain sense, considered to go "beyond" quantum field theory (although in other senses, QFT and ST are the same thing or in the same universality class). QFT and ST are quantum mechanical theories. They respect the general QM framework – just choose different operator algebras or Heisenberg equations or the "technical details". Even hopeless proposed theories such as loop quantum gravity are fully quantum mechanical ones.

It's as simple as that. There is no "third way". The number of truly qualitatively different logical frameworks in physics is just two, a much smaller number than the number of physics courses in a college. The courses differ on the laws of physics that are only distinguished by the choice of the degrees of freedom and equations they obey, not in the basic logic through which the mathematics is connected to the observations!

Darkblue, the user at the Physics Stack Exchange, decided to "only allow" circular polarizations partly because he or she wanted to interpret the decomposition of the electromagnetic field too dogmatically. Darkblue has probably seen an equation like\[

A_\mu(x^\alpha) &= \int \frac{d^3 k}{\sqrt{2E(\vec k)}}\,\sum_\lambda \epsilon_\mu (\vec k)\cdot \hat a_{\vec k, \lambda}\cdot e^{-ik_\nu x^\nu} +\\
&+ \text{Hermitian conjugate}

\] where the sum over polarizations was immediately said to go over \(\lambda\in\{L,R\}\). So Darkblue concluded that only the circular polarizations may exist or may be allowed. But the equation doesn't say anything of the sort. An equation is a way to rewrite something; an equation never explicitly or implicitly says that it is the only way. When we write \(15=10+5\), it doesn't prevent other identities such as \(15=7+8\) from being true as well.

(Just to be sure, it's also true that if you write two random enough long expressions and the \(=\) sign in between, the equation will probably be wrong. But the equality may still hold for infinitely many choices of the right hand side.)

And be sure that the summation over the circular polarizations \(L,R\) is exactly as good as the summation e.g. over the linear polarizations \(x,y\). Well, the axes \(x,y\) are good for a photon moving in the \(z\)-direction while the two transverse axes would have to be called differently for a general direction of \(\vec k\). There is no "canonical" way to choose the two axes replacing \(x,y\) as a function of \(\vec k\).

But this is not a "special" disadvantage of the linear polarizations. In fact, even if we use the circular polarizations, there is a problem: there is no "canonical" choice of the phases of the complex vectors \(\epsilon_{L/R}(\vec k)\). In fact, these two problems – the absence of a "canonical" choice of the two axes for the linear polarization; and the absence of a "canonical" choice of the phase of the complex vectors for the circular polarization – are actually just one and the same problem expressed in two different variables!

It's because the linearly polarized photon is a superposition of the circularly polarized ones such as\[

\ket x &= \frac{\ket L + \ket R}{\sqrt 2} \\
\ket y &= \frac{\ket L - \ket R}{i\sqrt 2}

\] Note that the absolute values of the coefficients in front of \(\ket L\) and \(\ket R\) are the same in both cases. But the relative phase is different – and the relative phase is what determines the axis of the linear polarization. Just to be sure, the inverse relationships look analogous:\[

\ket L &= \frac{\ket x + i\ket y}{\sqrt 2} \\
\ket R &= \frac{\ket x - i\ket y}{\sqrt 2}

\] The Hilbert space of one photon moving in the direction of \(\vec k\) with the frequency \(|\vec k|\) is two-dimensional and \( \{\ket L,\ket R\} \) is as good a basis as \( \{\ket x,\ket y\} \). There is nothing better about the circular polarization basis relatively to the linear polarization basis – or vice versa. Any complex linear superposition is equally "allowed". It may be equally "real" if you prepare the photon in the appropriate way.

Now, Darkblue wanted to believe that this superposition principle is just some idiosyncrasy of the course on (non-relativistic) quantum mechanics and it may or should be forgotten in quantum field theory. But this is a total nonsense as well, of course.

First of all, the photons have nothing to do with non-relativistic quantum mechanics per se. Photons are particles of the light which consequently move by the speed of light, \(c\). At speeds that are this high, you need relativity to study the motion. So the correct description of the motion of photons is unavoidably a relativistic theory. A non-relativistic theory of quantum mechanics just isn't good enough for photons.

Second, in practice, when we talk about the (relativistic, as I just explained) quantum mechanical theory describing the photons, we can't use a theory that is "completely analogous" to the theories of the non-relativistic type. We actually need something like a quantum field theory. Quantum field theory is the simplest framework (I say "simplest" because string theory may be sometimes said to be a "more advanced framework") in which the behavior of the photons may be quantitatively described at all.

So when I talked about the observables encoding the polarization of a photon and the measurements done to determine this piece of information, I actually meant quantum field theory: nothing simpler can work. But just because we talk about QFT doesn't mean that we should forget about the superposition principle and the "democracy" between all choices of the bases. The Hilbert space of QFT is a linear vector space, just like in QM. QFT is not just analogous to QM: QFT is a special case of quantum mechanics!

As far as you think that the Hilbert space of a QFT is something "very different" from the Hilbert space of non-relativistic quantum mechanics describing the same or similar physical system, you must have completely misunderstood something essential about at least one of these theories. In reality, the non-relativistic QM is a limit, the non-relativistic limit, of QFT. So if you extract the relevant Hilbert space for a given physical system assuming low speeds and the dynamical equations (e.g. the Heisenberg or Schrödinger equations governing these systems), you must get exactly the same thing up to subleading terms that (relatively) vanish in the \(v/c\to 0\) limit. You may need to rename things a bit and change the notation while you're relating your QM and QFT description of the hydrogen atom, for example. But if you can't manage to do this step, then you haven't understood QFT yet!

Because \(\ket x,\ket y\) are as good ket vectors for one photon as \(\ket L,\ket R\) – the superposition principle or the linearity of the vector space prohibits some "preferred bases" that are qualitatively better than others – and because in QFT, these photons may be created by creation operators, it follows that the creation operators \(a^\dagger_x,a^\dagger_y\) creating photons with a linear polarization are as good as their circularly polarized counterparts \(a^\dagger_L,a^\dagger_R\), too.

After all, these creation operators are linear operators acting on the Hilbert space of the QFT (the Fock space, at least in the case of a free QFT). And one thing you are encouraged to do with the linear operators is to define their (arbitrary) linear combinations. The linear combinations of \(a^\dagger_x,a^\dagger_y\) needed to get \(a^\dagger_L,a^\dagger_R\) or vice versa mimic the structure of the linear combinations we wrote for the one-photon states. It's no coincidence.

(When you define linear combinations of linear operators acting on the same Hilbert space, you want the individual terms to have the same units, the same grading, and belong to the same superselection sector etc. because it's impolite to add apples and oranges. But clearly, all these conditions are satisfied when we talk about linear and circular polarizations of photons. All the coefficients that we need are basically multiples of powers of \(i\) and powers of \(\sqrt{2}\).)

Maybe the reduction of quantum field theory – e.g. a Dirac field – to the non-relativistic quantum mechanics in the \(v\ll c\) limit isn't sufficiently well explained in the basic courses of QFT if it is explained at all.

But at the end, I do think that the main reason why Darkblue and other beginners tend to invent nonsensical fairy-tales such as "QFT bans linear polarizations" is nothing else than the notorious anti-quantum zeal. In an exchange – that was obviously a waste of time because Darkblue is self-evidently a mentally impotent moron – the poster wrote:
@LubošMotl: Thank you, I think I get my mistake. What I thought was QFT, is not QFT. The true QFT picture evolves linear complex combination of pure Fock states. My mental picture evolves pure Fock states with phases. By pure Fock States I mean a state with an integer number of energy quantum and an integer number of LH spin and an integer number of RH spin. (In my mental picture without superposition principle the linearly polarized creation operator is invalid because it's not pure). Conceptually I like my picture more. The paper's experiment could tell if it's wrong. I guess I'll wait.
Darkblue wrote "I think I get it" and then he repeats exactly the same "more likeable" stupidity about the ban on the superpositions which is the root of this whole question he posted and everything surrounding it. The idiotic fallacious would-be argument "a holy experiment may make me right and make you wrong" is added as a bonus. Sorry, it can't. The experiment, its proponents, and its proposed intepretations are as stupid as Darkblue. Darkblue wants to ban the superpositions and only allow some (basis) states – in this case, the occupation number eigenstates in the Fock space – that may be considered "elements of the phase space" i.e. the possible objective states of the world. The whole quantum mechanics would reduce down to classical physics if \(0\) and \(1\) were the only probabilities (and probability amplitudes) that the theory allows.

BTW the simplest experiments with linear polarizers and circular polarizers easily falsify all theories that would propose that one of these kinds of polarizations of photons don't exist. It's flabbergastingly stupid to propose that a "new experiment" could reverse or undo this conclusion.

Also, when I say that people believing that one may ban superpositions in quantum mechanics are imbeciles with a putrefying brain, you may have noticed that Gerard 't Hooft recently attempted to propose theories where the superpositions were banned. Does it mean that Gerard 't Hooft currently belongs to the unprestigious group that I have described? The answer is that my statements speak for themselves.

It's the very point of quantum mechanics that it allows and needs all the complex superpositions. Operators don't commute – the "which polarizations" operators don't commute; \(x,p\) don't commute; no non-constant observable commutes with the Hamiltonian \(H\), and so on. And that's why the eigenstates of some operators – which may result from a measurement – are nontrivial linear superpositions of eigenstates of other operators – eigenstates that may result from another measurement. You can't do any physics while denying this basic omnipresent fact.

Certain people are religiously obsessed anti-quantum zealots who hate everything that makes quantum mechanics quantum. And they're looking for every conceivable and inconceivable opportunity to throw away all the postulates of quantum mechanics such as the superposition principle. The transition from non-relativistic QM to QFT is just another opportunity for them – so they hope that the picture that QFT ultimately creates has to be basically classical, without any superpositions of allowed states and without any uncertainty.

They can't ever be satisfied. Quantum mechanics will never go away because non-quantum ("classical") physics has been falsified and the act of falsification is irreversible. Quantum field theory doesn't undo quantum mechanics; string theory doesn't do such a thing, either. The newer theories don't return physics any closer to the classical way of thinking. And the classical framework for physics has simply been proven wrong for 90 years. You can't learn any modern physics if you're incapable of understanding this fundamental facts and the reasons why competent physicists are certain about it.

If you're determined to keep your psychological problem with the very notion of a superposition of two quantum states, to believe that the Universe is prettier without superpositions, you should give up physics. You have no chance to penetrate it. Get employed as a chimp in your local zoo, as a feminist in the Department of Women's Studies, or as something else that is much closer to apes than to modern physicists.

Thank me very much for my wise advise. You are welcome.

P.S.: What really drives me up the wall about these Darkblue idiots is their verbal pride about "experiments" even though they are obviously retarded when it comes to the relevant experiments, too. When I was 6 years old, I played with the linear polarization filter on top of LCD displays from calculators. In effect, as a very small kid, I've done lots of experiments equivalent to this one:

Now, I could probably find similar videos with the calculator displays, too. Or this longer video showing how the polarizing filters may remove unwanted layers from photographs. But this simpler one above is a "cooler" version of what I did. You can see that the light coming from the display is polarized. If you add another filter, you can send the total amount of light that gets through to zero, depending on the orientation of the extra filter. You may verify that the percentage of the light that gets through is proportional to \(\cos^2 \gamma\) where \(\gamma\) is the angle of the rotation in between the two adjacent filters.

So the video above also plays with a pair of filters on top of each other, rotated relatively to each other etc. You can make numerous observations. Even if you don't measure the "cosine" quantitatively, you may derive many facts about the conditions that are needed for the light intensity to be basically unchanged; or for it to drop to zero.

Because he or she claims that the linear polarization can't exist, this Darkblue idiot clearly couldn't have done any experiments like that, experiments that many kids are familiar with and that are standard at high schools, sometimes at basic schools. But he or she extracts tons of self-confidence from talking about experiments, anyway. I simply can't stand pompous fools of this kind. They are exactly like W*it or Sm*lin – obnoxious caricatures of scientists and piles of stinky šit.

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