# What is quantum theory about?

Gold Member

## Main Question or Discussion Point

Quantum theory is called "quantum" because it is supposed to be about quanta - discrete spectrum of certain quantities which in classical theory have a continuous spectrum. But actually, quantum theory does not need to be about quanta at all, because many quantum quantities actually have a continuous spectrum.

If you think it's confusing, then you've seen nothing. Quantum mechanics is (or at least used to be) about Schrodinger equation, which is a quantum theory of electron. But Schrodinger equation is not really correct, because Schrodinger equation is not relativistic and does not describe the electron spin. Actually, Schrodinger derived this nonrelativistic equation from his relativistic equation, but relativistic Schrodinger equation of electron is not really called Schrodinger equation (but Klein-Gordon equation), does not really describe electron either (because it does not include spin) and is not really consistent (because it does not conserve probability). The consistent relativistic electron equation is Dirac equation, which includes spin and conserves probability. But even this equation is not fully correct, because one needs to second quantize it, after which it turns out that conservation of probability is no longer conservation of probability but conservation of charge. And similarly, it turns out that the Klein-Gordon equation also needs to be second quantized, so that its non-conservation of probability becomes irrelevant for a similar reason as conservation of probability for the Dirac equation. Thus the Dirac equation is not important because it conserves probability, but because it derives spin from linearization of the Klein-Gordon equation. But to derive spin you don't really need to linearize Klein-Gordon equation, because you can also get spin from linearization of the non-relativistic Schrodinger equation (which is simply called Schrodinger equation, and, by the way, also needs to be second quantized), leading to the Pauli equation who actually did not obtain that equation by linearizing the Schrodinger equation. However, the derivation of spin from linearization is not really a true derivation of spin, because the true derivation of spin comes from irreducible representations of the rotation group. More precisely, not of the proper rotation group SO(3), but of its simply connected covering group SU(2).

If you think that's it, you are deeply wrong. Actually it is misleading to say that all these equations above need to be second quantized, because there is only one quantization, but applied to different degrees of freedom. So what we called second quantization, and was really second quantization of particles, is actually first quantization of fields. So fundamental objects are fields, not particles. Or maybe not, because we measure particles, not fields. But not always, because sometimes we really measure fields. Actually only bosonic fields, because fermionic fields cannot be measured even in principle.

Can we at least say that quantum theory is not really about Schrodinger equation? No, because even relativistic quantum field theory, including spin and everything else we seem to need, still can be represented by a Schrodinger equation. But this Schrodinger equation is actually a generalized functional Schrodinger equation, and was not discovered by Schrodinger. But still, not all quantum field theories can be described by such a generalized Schrodinger equation, because quantum gravity is an exception, requiring Wheeler-DeWitt equation instead of the generalized Schrodinger equation. Wheeler-DeWitt equation is a generalized Klein-Gordon equation (but nobody calls it so). Yet, while Klein-Gordon equation is manifestly relativistic covariant, Wheeler-DeWitt equation is not, even though it is still relativistic. Actually, you don't really need Wheeler-DeWitt equation to do quantum gravity; there is also a manifestly relativistic-covariant way to do it. But that relativistic covariant way is not general-relativistic covariant, which a quantum theory of gravity should be.

Further complication comes from use of quantum field theory in condensed-matter physics, where it suggests that fields are not fundamental at all, not even bosonic ones, because the field description is appropriate only at large distances. Particles are more fundamental in condensed-matter physics, so quantum field theory is better called second quantization in condensed-matter community. But the second-quantized particles in condensed-matter physics are actually pseudo-particles (e.g., phonons), not the fundamental particles. In such an approach to condensed-matter physics, the fundamental particles are atoms and molecules, they are not quantized (despite the fact that they are actually quantum objects), and we know that they are not fundamental at all, because they consist of fundamental quarks and electrons, which are fundamentally described by another quantum field theory, which, by being fermionic field theory, describes fermionic fields which cannot be measured even in principle.

Now you might think that we really need a more fundamental theory of everything to clean up all that mess. So what our best candidate for the theory of everything - string theory - has to say about all this? First, it says that particles are not really particles but little strings. At first sight it does not change much, but at a second one it changes a lot, and at a third one it changes even more. One surprising result is that first quantization of strings is enough, so one does not need second quantization of strings, called also string-field theory. Actually string-field theory exists as well, but it is not consistent, and almost nobody uses it. But still, first-quantized theory of strings is also a quantum field theory - more precisely conformal quantum field theory in 2 dimensions. But it does not mean that strings live in 2 dimensions, because they really live in 10 dimensions. But the 2-dimensional strings (actually 1-dimensional if we don't count time) that live in 10 dimensions are not the end of the story, because the theory contains also branes - objects having more than 2 stringy dimensions. These branes are not really fundamental, because they are only special configurations of classical 10-dimensional fields, while these 10-dimensional fields themselves are not fundamental. Despite of being non-fundamental, these classical 10-dimensional fields are actually quantum 10-dimensional fields, but nobody knows how to quantize them because, as I said, string-field theory is not consistent (and not even needed).

I said that branes are not fundamental, but actually they are. More precisely, they are not fundamental in perturbative string theory, but they are fundamental in non-perturbative string theory. Therefore non-perturbative string theory is not really a string theory, but it is still called string theory. This non-perturbative string theory is more fundamental than perturbative string theory, but there is one little technical problem with it: nobody knows what this non-perturbative string theory is. At least we have a cool name for it - M-theory, but nobody knows what even "M" stands for. (Some good candidates are Membrane-theory, Matrix-theory, Mystery-theory and Witten-theory (where W is reversed M).) Whatever that M-theory might be, at least we know that in this theory strings and branes are equally fundamental (or better to say, equally non-fundamental) and that the theory lives in at least 11 dimensions, which may be actually 12 dimensions, or perhaps the number of dimensions is not fundamental at all.

vortextor

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To me this is more of a philosophical question than a quantitative one. I think that the question here is valid - namely that we should always more or less be checking the consistency of our thinking. Ineed, "quantum theory" is really a collective term which describes a set of mathematical operations. I think the appropriate thing to consider is the process of "quantization"

The typical process of "canonical quantization" involves the transformation of Poisson brackets to commutators and observables are transformed into operators (e.g. momentum or energy). This is more of a mathematical procedure than anything else. The physical interpretation of this in the Schrodinger picture is that "particles" become wavefunctions.

Moving to field, theory, the problem becomes more complex. Many physicists were more or less unhappy that "particles" and "fields" were treated on unequal footing in the Schrodinger picture (classic example ... hydrogen atom in a magnetic field). It was seen as more desirable to have a "field" for the electron AND for the electromagnetic field. The electron "particle" is of course seen as an excitation of the electron field. Then these fields are allowed to interact -- this is the conceptual framework of QED.as I understand it. The problem (until renormalization) was that this engendered an infinite number of degrees of freedom.

The quantum theory of solids has a slightly different motivation, namely that one is usually dealing with many body systems. Also Bloch waves (electronic states in a periodic potential) can be thought of as excitations of a field as well. The need for "second" quantization arises out of a need to handle many particle dynamics in a easy to handle way. The theoretical framework is similar to that of "quantum field theory" and indeed - the theory of solids is a kind of quantum field theory.

In modern physics, I tend to think of these various theories as higly analogous yet distinct in nature. The mathematical framework has risen to meet certain needs which have risen in particle and solid state theories. The topics you are bringing up are well worthy of thinking about - and indeed a book could probably be written on this. This is just a short reflective response motivated by your thining.

kith

bhobba
Mentor
To me this is more of a philosophical question than a quantitative one. I think that the question here is valid - namely that we should always more or less be checking the consistency of our thinking. Ineed, "quantum theory" is really a collective term which describes a set of mathematical operations.
Sorry - cant agree with you there.

QM basically follows from one axiom - see post 137:

The foundational principle is:
An observation/measurement with possible outcomes i = 1, 2, 3 ..... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.

That axiom immediately implies much more than mathematics because we have this thing contained in it - namely 'observation/measurement with possible outcomes i = 1, 2, 3 .....'.

Its this mapping that makes it a physical theory - and also such, that by assumption, occur in a common-sense classical world immediately leads us into the deep foundational issues of the theory devoid of the math.

The typical process of "canonical quantization" involves the transformation of Poisson brackets to commutators and observables are transformed into operators (e.g. momentum or energy). This is more of a mathematical procedure than anything else. The physical interpretation of this in the Schrodinger picture is that "particles" become wavefunctions.
That's Dirac's old view - the modern one is based on symmetry - see Chapter 3 Ballentine.

Thanks
Bill

I dont see Diracs old view as being invalid - he was after all one of the founding fathers of the theory (advertise for Ballentine as you may). Canonical quantization appears in many textbooks, and in my view, it is an equivalent expression of the symmetry approach.

It seems that you have an opinion here that there is only one way to explore ideas in QM or QFT. While the POVM formulation is correct, it is not intuitive to those who are not familiar with it. Certainly it seems fair to bring it up, but your tone is authoritative. Finally, as is clear from the tone of my writing, I was not trying to provide a technical response to what I percieved to be a non-technical/philosophical question.

After all -- the thread title is "What is quantum theory about ??" ... not "Write the singular axiom of quantum theory". Anyway, I feel the following quote attributed to Feynman is appropriate here: "If you think you understand quantum mechanics, you don't understand quantum mechanics."

1 person
atyy
Anyway, I feel the following quote attributed to Feynman is appropriate here: "If you think you understand quantum mechanics, you don't understand quantum mechanics."
Well, Feynman didn't understand QM, just as Dirac did not understand QFT :tongue:

Since Wilson we understand QFT, and since Bohm we understand QM. In both cases, the answer is that QFT and QM are only effective theories.

bhobba
Mentor
I dont see Diracs old view as being invalid
Its not invalid.

Most however recognise that from the POR probabilities must be frame independent is more fundamental than the PB assumption.

Even without the POR most would say its pretty obvious probabilities are frame independent - it simply doesn't make sense otherwise. If it wasn't then the outcomes would have to depend on a frame ie if you get spin up or down would depend on your position, orientation etc which would be rather unexpected - but of course not impossible.

A lot of water has gone under the bridge since Dirac wrote that book.

Thanks
Bill

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bhobba
Mentor
It seems that you have an opinion here that there is only one way to explore ideas in QM or QFT. While the POVM formulation is correct, it is not intuitive to those who are not familiar with it. Certainly it seems fair to bring it up, but your tone is authoritative.
Mate you are missing my point.

I was simply giveing a clear reason against your view 'Ineed, "quantum theory" is really a collective term which describes a set of mathematical operations.'

You are entitled to view QM anyway you like, but if you make claims like that then don't be surprised if some examine it within the way they view QM.

That said I also have to eat some crow because there is a view of QM that's very mathematical:
http://arxiv.org/pdf/quantph/0101012.pdf

Also fair comment - after looking at my response it was authoritative where such was not required. I will have to watch that sort of thing.

Thanks
Bill

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Gold Member
Well, Feynman didn't understand QM, just as Dirac did not understand QFT :tongue:

Since Wilson we understand QFT, and since Bohm we understand QM. In both cases, the answer is that QFT and QM are only effective theories.
:thumbs:

vanhees71
Gold Member
2019 Award
Well, I'd rather say, since Bohm we are even more confused about QM. There is empirical justification for Bohm's trajectories. They are sometimes even misleading. There is a paper by Marlan Scully et al on this. I've to look for it in my paper collection.

strangerep
[...] just as Dirac did not understand QFT :tongue:
Have you studied Dirac's Yeshiva "Lectures on QFT" ? :tongue2:

bhobba
Mentor
Have you studied Dirac's Yeshiva "Lectures on QFT" ? :tongue2:
Dirac understood QFT only too well.

He simply rejected renormalisation - even Feynman who also understood it only too well called it a dippy process.

My suspicion is both simply didn't keep up with what Wilson did.

Thanks
Bill

vanhees71
Gold Member
2019 Award
Perhaps that's the reason why Weinberg is a great "Dirac basher". In nearly any Weinberg textbook you find some nasty remark against Dirac ;-)).

To me this is more of a philosophical question than a quantitative one..

For those who share the same a priori iit's a matter of science, for other it is a philosophical question.

Other question could be : is it an other interpretation of QM or a new formalism of QM ? Interpretation of QM is it philosophical question or physics question ?

Patrick

strangerep
There is empirical justification for Bohm's trajectories. [...]
Umm,... is that what you meant to say?

vanhees71
Gold Member
2019 Award
Argh! I wanted to say the opposite! There is no empirical justification for Bohm's trajectories. The paper by Scully et al is

M. O. Scully, Do Bohm trajectories always provide a trustworthy physical picture of particle motion?, Physica Scripta Volume 1998 T76
http://dx.doi.org/10.1238/Physica.Topical.076a00041

ShayanJ
Gold Member
Well, Feynman didn't understand QM, just as Dirac did not understand QFT :tongue:

Since Wilson we understand QFT, and since Bohm we understand QM. In both cases, the answer is that QFT and QM are only effective theories.
Well, I agree with you about QFT, but not QM. Bohm's theory is not widely accepted(in case you care, not accepted by me too ). As far as I know, mainstream QM is working well and Quantum Decoherence is doing well in solving some of its foundational problems. So I see no point in accepting a theory which is giving the same predictions as QM and is as strange, but is more complicated. I really believe in Occam's razor and so I don't even look at Bohm's theory!

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Gold Member
Well, I agree with you about QFT, but not QM. Bohm's theory is not widely accepted(in case you care, not accepted by me too ). As far as I know, conventional QM is working well and Quantum Decoherence is doing well in solving some of its foundational problems. So I see no point in accepting a theory which is giving the same predictions as QM and is as strange, but is more complicated. I really believe in Occam's razor and so I don't even look at Bohm's theory!
Decoherence alone does not resolve the problem of outcomes. Bohm's theory (combined with decoherence) does.

vanhees71
Gold Member
2019 Award
In how far does Bohm's ideas solve anything which is not solved by quantum theory? Bohm suggests to calculate trajectories that are unobservable in the first place and of no empirical significance. So what do I gain when I calculate them and what does this solve to some (apparent?) problems?

What do you understand under "the problem of outcomes"? There is no problem as soon as you define observables as what they literally are: outcomes of quantitative measurements of some aspect of an object, e.g., the momentum of a particle or a photon.

As any other physical theory quantum theory tries to describe the relations between so defined quantitatively observable aspects of objects in mapping abstract mathematical entities of the theory to the measurable quantities. In quantum theory an observable is represented by an essentially self-adjoint operator on Hilbert space, and the possible values it takes is given by the spectrum of these operators. Further you have the notion of states, which are represented by positive definite self-adjoint trace-class operators (the Statistical Operators), and you can prepare a system completely by determining a complete set of compatible observables (by, e.g., an ideal von Neumann filter measurement-preparation process). Then the Stat. Op. is uniquely determined as the corresponding projection operator built by the common eigenvector of the observables (the details are a bit more subtle when it comes to preparation in a state where you have observables taking values in the continuous spectrum, but this is not the point here). Then you have, according to quantum theory, a state of complete knowledge about the system, and the Stat. Op. is just a book-keeping device about the probabilistic knowledge you have about the system. A further interpretation towards an ontic view leads only to troubles and inconsistencies without further value added to the theory.

Now, when you measure an observable with an ideal measurement apparatus, by definition you find a value. You can then check the validity of quantum theory by measuring a large set of equally prepared systems, in making the usual statistical analyses whether the probabilistic predictions of quantum theory are correct (and even if the postulate that the observables take only values in the spectrum of the operators your associate with them in theory). So where is there a problem with the outcomes of measurements?

That an ontic interpretation of the quantum state is problematic, should not be of so much concern to physicists. Physics describes nature or more precisely, as Bohr put it, it describes what we can objectively know about nature.

Another point, I'd like to address is the question, in how far are classical states more "ontic" than quantum states? Also there I have a rather abstract mathematical picture of nature, say a system of classical point particles, which you describe most efficiently as a point in its phase space in the sense of Hamiltonian Mechanics. So what is more "ontic" on a point in a symplectic manifold (classical model) as compared to a Statistical operator in Hilbert space?

ShayanJ
Gold Member
Decoherence alone does not resolve the problem of outcomes. Bohm's theory (combined with decoherence) does.
Decoherence isn't compatible with Bohm's theory. Because QM states are in general superpositions of base states and Decoherence is about interference between these base states. But in Bohm's theory, the state of the system is definite and there is no interference, remember it is a hidden variable theory.
In fact decoherence is a consequence of mainstream QM so something perspectively as different from QM as Bohm's theory can't give rise to it!

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stevendaryl
Staff Emeritus
What do you understand under "the problem of outcomes"? There is no problem as soon as you define observables as what they literally are: outcomes of quantitative measurements of some aspect of an object, e.g., the momentum of a particle or a photon.
I would say there's a problem (at least conceptually). If we're all made up of particles and fields described by quantum mechanics, then presumably a measurement is just a type of interaction between one quantum system and another. Saying that the meaning of quantum amplitudes is given by the probabilities of various measurement outcomes seems strange in that light. Why does "a measurement of an electron's spin in the z-direction" have a definite outcome, but "an electron's spin in the z-direction" does not?

ShayanJ
Gold Member
In how far does Bohm's ideas solve anything which is not solved by quantum theory? Bohm suggests to calculate trajectories that are unobservable in the first place and of no empirical significance. So what do I gain when I calculate them and what does this solve to some (apparent?) problems?

What do you understand under "the problem of outcomes"? There is no problem as soon as you define observables as what they literally are: outcomes of quantitative measurements of some aspect of an object, e.g., the momentum of a particle or a photon.

As any other physical theory quantum theory tries to describe the relations between so defined quantitatively observable aspects of objects in mapping abstract mathematical entities of the theory to the measurable quantities. In quantum theory an observable is represented by an essentially self-adjoint operator on Hilbert space, and the possible values it takes is given by the spectrum of these operators. Further you have the notion of states, which are represented by positive definite self-adjoint trace-class operators (the Statistical Operators), and you can prepare a system completely by determining a complete set of compatible observables (by, e.g., an ideal von Neumann filter measurement-preparation process). Then the Stat. Op. is uniquely determined as the corresponding projection operator built by the common eigenvector of the observables (the details are a bit more subtle when it comes to preparation in a state where you have observables taking values in the continuous spectrum, but this is not the point here). Then you have, according to quantum theory, a state of complete knowledge about the system, and the Stat. Op. is just a book-keeping device about the probabilistic knowledge you have about the system. A further interpretation towards an ontic view leads only to troubles and inconsistencies without further value added to the theory.

Now, when you measure an observable with an ideal measurement apparatus, by definition you find a value. You can then check the validity of quantum theory by measuring a large set of equally prepared systems, in making the usual statistical analyses whether the probabilistic predictions of quantum theory are correct (and even if the postulate that the observables take only values in the spectrum of the operators your associate with them in theory). So where is there a problem with the outcomes of measurements?

That an ontic interpretation of the quantum state is problematic, should not be of so much concern to physicists. Physics describes nature or more precisely, as Bohr put it, it describes what we can objectively know about nature.

Another point, I'd like to address is the question, in how far are classical states more "ontic" than quantum states? Also there I have a rather abstract mathematical picture of nature, say a system of classical point particles, which you describe most efficiently as a point in its phase space in the sense of Hamiltonian Mechanics. So what is more "ontic" on a point in a symplectic manifold (classical model) as compared to a Statistical operator in Hilbert space?
What you described is right but it doesn't mean there is no problem. There is the problem of measurement that has three parts:
1. The problem of the preferred basis (Sect. 2.5.2). What singles out the
preferred physical quantities in nature—e.g., why are physical systems
usually observed to be in definite positions rather than in superpositions
of positions?
2. The problem of the nonobservability of interference (Sect. 2.5.3). Why
is it so difficult to observe quantum interference effects, especially on
macroscopic scales?
3. The problem of outcomes (Sect. 2.5.4). Why do measurements have out-
comes at all, and what selects a particular outcome among the different
possibilities described by the quantum probability distribution?
(Quoted from the book "Decoherence and Quantum to Classical transition" by Maximilian Schlosshauer)

The first two are solved effectively by Decoherence and about the 3rd, I think it is!!!

Anyway, the main problem is gaining a view of inside of the black box of wave function collapse which Decoherence (alone) is very promising in that.

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atyy
That an ontic interpretation of the quantum state is problematic, should not be of so much concern to physicists. Physics describes nature or more precisely, as Bohr put it, it describes what we can objectively know about nature.

Another point, I'd like to address is the question, in how far are classical states more "ontic" than quantum states? Also there I have a rather abstract mathematical picture of nature, say a system of classical point particles, which you describe most efficiently as a point in its phase space in the sense of Hamiltonian Mechanics. So what is more "ontic" on a point in a symplectic manifold (classical model) as compared to a Statistical operator in Hilbert space?
The problem begins when you are forced to put a cut, and acknowledge an observer/measurement as fundamental. This doesn't happen in special relativity or general relativity. As an analogy, is it really fundamental in special relativity that "the observer measures the speed of light to be the same in all inertial reference frames" or do we consider that short hand for "the laws of physics are Poincare invariant". In general relativity, do we accept as fundamental "a test particle moves on a geodesic" or do we prefer "matter and metric are minimally coupled" (well of course, a test particle is fine if there really are test particles that are distinct from matter that has stress-energy).

A theory with a necessary cut at an unknown location can still be useful. Wilson showed that for QFT. But the cut also points to new physics. We don't yet know what new physics lies above the Planck scale, but we are pretty sure there is. Bohmian Mechanics says the same - its achievement is not that it is right - neither the hidden variables not dynamics are unique. It is similar to there usually being many possible UV completions of an effective QFT. Experiment is needed to decide. In both cases, it is worth asking whether one is really forced to acknowledge new physics. In gravity, we ask whether Asymptotic Safety is possible. In quantum theory, we ask whether Many Worlds is possible.

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Gold Member
Decoherence isn't compatible with Bohm's theory. Because QM states are in general superpositions of base states and Decoherence is about interference between these base states. But in Bohm's theory, the state of the system is definite and there is no interference, remember it is a hidden variable theory.
In fact decoherence is a consequence of mainstream QM so something perspectively as different from QM as Bohm's theory can't give rise to it!
Sorry, but you obviously don't know much about Bohm's theory.

ShayanJ
Gold Member
The problem begins when you are forced to put a cut, and acknowledge an observer/measurement as fundamental. This doesn't happen in special relativity or general relativity. As an analogy, is it really fundamental in special relativity that "the observer measures the speed of light to be the same in all inertial reference frames" or do we consider that short hand for "the laws of physics are Poincare invariant". In general relativity, do we accept as fundamental "a test particle moves on a geodesic" or do we prefer "matter and metric are minimally coupled" (well of course, a test particle is fine if there really are test particles that are distinct from matter that has stress-energy).

A theory with a necessary cut at an unknown location can still be useful. Wilson showed that for QFT. But the cut also points to new physics. We don't yet know what new physics lies above the Planck scale, but we are pretty sure there is. Bohmian Mechanics says the same - its achievement is not that it is right - neither the hidden variables not dynamics are unique. It is similar to there usually being many possible UV completions of a QFT. Experiment is needed to decide. In both cases, there are worthy approaches to pursuing new physics. In gravity, we ask whether Asymptotic Safety is possible. In quantum theory, we ask whether Many Worlds is possible.
Decoherence, which naturally arises from mainstream QM, avoids that cut and smoothly carries QM to macroscopic world. So we still can say there is no need for new physics.