What is the essence of quantization?

[massless]

How should I understand the procedure of canonical quantization in quantum field theory. Do we really quantize the field by regarding the field as dynamics variables ?What’s the physical essence of quantization?

 

[Demystifier]

You might find this interesting:
http://xxx.lanl.gov/abs/0705.3542 [Europhys. Lett.85:20003, 2009]

 

[DrFaustus]

I made the mistake of checking the forum during my lunch time and I can already see it's going to be a long one - I simply have to reply to some of the nonsense being spread.

massless -> There are many levels on which you could reply. I'll stick to intuitive physical notions. First why do we need fields at all, classical or quantum? The reason is Einstein's causality expressed as nothing can travel faster than light. Fields (along with hyperbolic field equations) are just a mathematical tool to express this "experimental fact". In particular, when you disturb a field, the disturbance cannot propagate faster than light, and that's achieved with hyperbolic field equations. (That's just a property of hyperbolic partial differential equations, i.e. finite speed propagation of disturbances.) On the other hand we need QM because, well, because we need it. :)

So what you want to achieve is to combine the two, that is, you want to combine the principles of QM with finite speed propagation. And this is done in QFT. And an example of physical questions that you want to answer is, besides the usual particle scattering (which is just a small part of the entire QFT framework), "How does a gas of very hot electrons behave?" So hot, in fact, that the electrons are relativistic particles and not slowly moving ones.

So yes, you do "quantize" fields, but fields have always been dynamical variables, even classical fields are - think of the electromagnetic field. You could say that the physical essence of "quantization" is to have "fields that obey the principles of QM". And one such effect is the emergence of "particles" understood as "disturbances of a quantum field". The photon is one quantum of the electromagnetic field. The electron is one quantum of the Dirac field, and so on. Clearly the question is "how" do you quantize a field, and this is achieved, in the canonical formalism, by imposing the canonical commutation relations. But this raises a million of problems, and some of them have not been solved even after 80 years of QFT. But not because of the "critics" that Demystifier is raising.

Speaking of which...

Demystifier -> I honestly do not understand how that paper of yours got published, as it is a collection of wrong ideas. Actually, more than of wrong ideas, the ideas are based on wrong premises. I do not even know where to start.

From the introduction:

For example, fermionic fields do not really have a classical counterpart and do not represent quantum observables.

I don't really know what is that supposed to mean. But I claim that the action associated to the Dirac Lagrangian is a classical action and the spinor fields that appear in it are classical fields. The fact that in classical physics spinor fields are not frequently encountered does not mean they are non existent. Spinors are actually mathematical objects devoid of any physical meaning, classical or quantum.

You continue with

Hermitian operators representing quantum observables can be constructed by taking products of fields in which the number of fermionic fields is even. An example is the energy-momentum tensor $$T_{\mu \nu}(x)$$. Nevertheless, although such operators correspond to quantities that should be measurable in principle, they are not quantities measured in practice. Instead, practically measurable predictions resulting from QFT are properties of particles.

This is plain wrong. The energy of a system is certainly something you do measure, and this info is encoded in the Hamiltonian. An example is most certainly given by "dark energy" and by "dark matter". No one has ever detected a dark matter particle, but we do measure and observe the effects of its energy on the galaxies. And the effects of the (postulated) dark energy on the Universe.

Are nonrelativistic QM and QFT two independent theories describing different objects?

Yes. QFT describes relativistc particles whereas QM describes slow particles. Yet, the two are closely related.

If, on the other hand, nonrelativistic QM is to be derived from more fundamental QFT, then how to derive the probabilistic interpretation of nonrelativistic $$\psi(x, t)$$ by starting from the axioms of QFT? It seems that the standard orthodox formulation of quantum theory cannot answer these questions.

False. QM can be derived from QFT. In all honesty, I find it utterly arrogant to think that after 80 years of QM and QFT no one has ever thought about deriving QM from QFT. Or that no one has ever pointed out this "deficiency". Thousands of brilliant minds have worked on it, no one ever noticed it.

Finally, and crucially, your equation (1), $$\psi(x_1,\ldots,\x_n) = S_{\{x_a\}}\langle0|\phi(x_1)\dots \phi(x_n)|\Phi\rangle$$ is really not clear, to say the least. And it seems that you base your entire discussion on it.

You say that $$\Phi$$ is a Heisenberg picture state, and then you evaluate the product of fields between two Heisenberg picture states. If you're working in the Heisenberg picture you can only form expressions like $$\langle\Phi|O|\Phi\rangle$$ and not $$\langle\Phi|O|\Omega\rangle$$. Unless, clearly, you can "create" one of the two states from the other by acting on it with some operator. But in Heisenberg QM that is, in general, an ill defined expression. And that's why in QFT you always have expressions like $$\langle\Phi|O|\Phi\rangle$$ at the end of the day.

Most importantly, however, $$\psi$$ is NOT to be interpreted as a wave function as it is not. It does not satisfy the Schroedinger equation and no one ever claimed that that is a wave function. In QFT there are no wave functions. At most a wave functional, that does satisfy a functional version of the Schroedinger equation. But your $$\psi$$ is not such an object.

Talking about QFT you conclude

It is not an orthodox quantum theory because practically measurable quantities resulting from QFT are particle positions, which are not described by hermitian operators.

As I remarked above, this is wrong. Practically measurable quantities are not particle position but an almost interminable series of other physical properties for various physical systems (like the viscosity coefficient in a quark gluon plasma - how is that a particle position?)

It is not a field theory in a naive sense because the fundamental measurable quantities at low energies are particles rather than fields.

This is again wrong. If nothing else, because in QFT particles are interpreted as excitations of fields. This means "particle = field". The fundamental measurable quantities are properties of the fields.

To conclude, Bohmian QM sure deserves further development and is by no means to be disregaded as meaningless. But trying to attack QFT claiming that it is neither a quantum theory nor a field theory most certainly won't go very far.

 

[massless]

I made the mistake of checking the forum during my lunch time and I can already see it's going to be a long one - I simply have to reply to some of the nonsense being spread.

massless -> There are many levels on which you could reply. I'll stick to intuitive physical notions. First why do we need fields at all, classical or quantum? The reason is Einstein's causality expressed as nothing can travel faster than light. Fields (along with hyperbolic field equations) are just a mathematical tool to express this "experimental fact". In particular, when you disturb a field, the disturbance cannot propagate faster than light, and that's achieved with hyperbolic field equations. (That's just a property of hyperbolic partial differential equations, i.e. finite speed propagation of disturbances.) On the other hand we need QM because, well, because we need it. :)

So what you want to achieve is to combine the two, that is, you want to combine the principles of QM with finite speed propagation. And this is done in QFT. And an example of physical questions that you want to answer is, besides the usual particle scattering (which is just a small part of the entire QFT framework), "How does a gas of very hot electrons behave?" So hot, in fact, that the electrons are relativistic particles and not slowly moving ones.

So yes, you do "quantize" fields, but fields have always been dynamical variables, even classical fields are - think of the electromagnetic field. You could say that the physical essence of "quantization" is to have "fields that obey the principles of QM". And one such effect is the emergence of "particles" understood as "disturbances of a quantum field". The photon is one quantum of the electromagnetic field. The electron is one quantum of the Dirac field, and so on. Clearly the question is "how" do you quantize a field, and this is achieved, in the canonical formalism, by imposing the canonical commutation relations. But this raises a million of problems, and some of them have not been solved even after 80 years of QFT. But not because of the "critics" that Demystifier is raising.

Speaking of which...

Demystifier -> I honestly do not understand how that paper of yours got published, as it is a collection of wrong ideas. Actually, more than of wrong ideas, the ideas are based on wrong premises. I do not even know where to start.

From the introduction:
I don't really know what is that supposed to mean. But I claim that the action associated to the Dirac Lagrangian is a classical action and the spinor fields that appear in it are classical fields. The fact that in classical physics spinor fields are not frequently encountered does not mean they are non existent. Spinors are actually mathematical objects devoid of any physical meaning, classical or quantum.

You continue with This is plain wrong. The energy of a system is certainly something you do measure, and this info is encoded in the Hamiltonian. An example is most certainly given by "dark energy" and by "dark matter". No one has ever detected a dark matter particle, but we do measure and observe the effects of its energy on the galaxies. And the effects of the (postulated) dark energy on the Universe.
Yes. QFT describes relativistc particles whereas QM describes slow particles. Yet, the two are closely related.
False. QM can be derived from QFT. In all honesty, I find it utterly arrogant to think that after 80 years of QM and QFT no one has ever thought about deriving QM from QFT. Or that no one has ever pointed out this "deficiency". Thousands of brilliant minds have worked on it, no one ever noticed it.

Finally, and crucially, your equation (1), $$\psi(x_1,\ldots,\x_n) = S_{\{x_a\}}\langle0|\phi(x_1)\dots \phi(x_n)|\Phi\rangle$$ is really not clear, to say the least. And it seems that you base your entire discussion on it.

You say that $$\Phi$$ is a Heisenberg picture state, and then you evaluate the product of fields between two Heisenberg picture states. If you're working in the Heisenberg picture you can only form expressions like $$\langle\Phi|O|\Phi\rangle$$ and not $$\langle\Phi|O|\Omega\rangle$$. Unless, clearly, you can "create" one of the two states from the other by acting on it with some operator. But in Heisenberg QM that is, in general, an ill defined expression. And that's why in QFT you always have expressions like $$\langle\Phi|O|\Phi\rangle$$ at the end of the day.

Most importantly, however, $$\psi$$ is NOT to be interpreted as a wave function as it is not. It does not satisfy the Schroedinger equation and no one ever claimed that that is a wave function. In QFT there are no wave functions. At most a wave functional, that does satisfy a functional version of the Schroedinger equation. But your $$\psi$$ is not such an object.

Talking about QFT you conclude
As I remarked above, this is wrong. Practically measurable quantities are not particle position but an almost interminable series of other physical properties for various physical systems (like the viscosity coefficient in a quark gluon plasma - how is that a particle position?)
This is again wrong. If nothing else, because in QFT particles are interpreted as excitations of fields. This means "particle = field". The fundamental measurable quantities are properties of the fields.

To conclude, Bohmian QM sure deserves further development and is by no means to be disregaded as meaningless. But trying to attack QFT claiming that it is neither a quantum theory nor a field theory most certainly won't go very far.

Thanks for your reply. I’ve read the paper many times, but I still can’t understand the main idea. There are many assumptions just as you quoted which I think improper either. If QFT is neither an orthodox quantum theory nor a field theory, everything we have done seems meaningless.
I also have two questions about the paper. First, in the paper ,the author argues that “If the Bohmian interpretation is assumed, then strings appear as natural objects”. I only know the main idea of Bohm’s hidden variables and can’t prove the point strictly. Is the author’s point of view true? If so, it seems a good understanding of string theory and we may have low energy testable predictions of string theory. Second, can the particle really have so-called “trajectories” which violates the uncertainty principle?