The necessity of the 2nd quantization?

[weejee]

Is the 2nd quantization physically essential in the description of relativistic fermions obeying Dirac equation?

In the case of the non-relativistic Schrodinger equation, the 2nd quantization is only a matter of convenience and doesn't actually change any physics, for both bosons and fermions. (Interestingly, canonical quantization of the Schrodinger field gives the same result as choosing the occupation number basis and defining creation and annihilation operators appropriately.)

For the relativistic theories, the 2nd quantization has an essential physical meaning at least for bosons. Due to negative energy(or frequency) states, the vacuum isn't well-defined in the first place unless we quantize the field.

However, for relativistic fermions, it seems to me that the 2nd quantization is just a matter of convenience as it is for the non-relativistic particles. Dirac's hole theory looks just fine to me.

Is my opinion right? If not, what is the problem? Is it that the electromagnetic field can't be included properly in Dirac's hole theory, or that the hole theory gives wrong predictions? If so, can anyone explain the details?

Thanks in advance.

[atyy]

In the case of the non-relativistic Schrodinger equation, the 2nd quantization is only a matter of convenience and doesn't actually change any physics, for both bosons and fermions. (Interestingly, canonical quantization of the Schrodinger field gives the same result as choosing the occupation number basis and defining creation and annihilation operators appropriately.)

This was the most intuitive way to approach it for me. Unlike the underlying "true particles", the "fake particles" (modes) of the 2nd quantized representation can be created and destroyed. So if you are in a situation where you need to describe creation and destruction of particles, the 2nd quantized formalism may come in handy (except in the case of the electron field, there don't yet seem to be underlying "true particles").

[Maaneli]

Is the 2nd quantization physically essential in the description of relativistic fermions obeying Dirac equation?

In the case of the non-relativistic Schrodinger equation, the 2nd quantization is only a matter of convenience and doesn't actually change any physics, for both bosons and fermions. (Interestingly, canonical quantization of the Schrodinger field gives the same result as choosing the occupation number basis and defining creation and annihilation operators appropriately.)

For the relativistic theories, the 2nd quantization has an essential physical meaning at least for bosons. Due to negative energy(or frequency) states, the vacuum isn't well-defined in the first place unless we quantize the field.

However, for relativistic fermions, it seems to me that the 2nd quantization is just a matter of convenience as it is for the non-relativistic particles. Dirac's hole theory looks just fine to me.

Is my opinion right? If not, what is the problem? Is it that the electromagnetic field can't be included properly in Dirac's hole theory, or that the hole theory gives wrong predictions? If so, can anyone explain the details?

Thanks in advance.


Barut and co showed that 2nd quantization of the relativistic Dirac equation is not at all necesssary (though it is convenient) for either the wavefunction or the EM field. One can perfectly compute all radiative processes such as Lamb shift, spontaneous emission, etc., if one includes the fermion charge-field self interactions in the Dirac, Pauli, and Schroedinger equations:

Interpretation of self-field quantum electrodynamics
http://prola.aps.org/abstract/PRA/v43/i7/p4060_1

Self-field quantum electrodynamics: The two-level atom
http://prola.aps.org/abstract/PRA/v41/p2284_1

Quantum electrodynamics based on self-fields: On the origin of thermal radiation detected by an accelerating observer
http://prola.aps.org/abstract/PRA/v41/p2277_1

Quantum electrodynamics based on self-energy, without second quantization: The Lamb shift and long-range Casimir-Polder van der Waals forces near boundaries
http://prola.aps.org/abstract/PRA/v36/p2550_1

http://books.google.com/books?id=7wLVvziRRnoC&pg=PA147&lpg=PA147&dq=QED+based+on+self-energy&source=web&ots=fdPNxe7Uve&sig=Nyde7-6Z7fToB68N8zmxbpWfaHw&hl=en&sa=X&oi=book_result&resnum=1&ct=result

Moreover, such an approach is both nonperturbative and finite! Unfortunately, it has been overlooked for various sociological reasons.

[atyy]

So if you are in a situation where you need to describe creation and destruction of particles, the 2nd quantized formalism may come in handy (except in the case of the electron field, there don't yet seem to be underlying "true particles").

Let me clarify this a bit. In the case of a solid, the "true particles" are those making up the solid, the "fake particles" are the phonons. In the case of the electron field, we only know about the "fake particles", which we call electrons. Since no one has yet discovered any "true particles" underlying the electron field, the electron "fake particle" is called an "elementary particle".

[strangerep]

Interpretation of self-field quantum electrodynamics
http://prola.aps.org/abstract/PRA/v43/i7/p4060_1

[...]

For those without a prola subscription, are any review papers
available online for free? (I had a look on the arxiv but failed
to find any.)

TIA.

[Maaneli]

For those without a prola subscription, are any review papers
available online for free? (I had a look on the arxiv but failed
to find any.)

TIA.

Sorry, the papers are not on arXiv. If you have a university library subscription, you can access these journals via their electronic libraries.

[weejee]

Barut and co showed that 2nd quantization of the relativistic Dirac equation is not at all necesssary (though it is convenient) for either the wavefunction or the EM field. One can perfectly compute all radiative processes such as Lamb shift, spontaneous emission, etc., if one includes the fermion charge-field self interactions in the Dirac, Pauli, and Schroedinger equations:

Interpretation of self-field quantum electrodynamics
http://prola.aps.org/abstract/PRA/v43/i7/p4060_1

Self-field quantum electrodynamics: The two-level atom
http://prola.aps.org/abstract/PRA/v41/p2284_1

Quantum electrodynamics based on self-fields: On the origin of thermal radiation detected by an accelerating observer
http://prola.aps.org/abstract/PRA/v41/p2277_1

Quantum electrodynamics based on self-energy, without second quantization: The Lamb shift and long-range Casimir-Polder van der Waals forces near boundaries
http://prola.aps.org/abstract/PRA/v36/p2550_1

http://books.google.com/books?id=7wLVvziRRnoC&pg=PA147&lpg=PA147&dq=QED+based+on+self-energy&source=web&ots=fdPNxe7Uve&sig=Nyde7-6Z7fToB68N8zmxbpWfaHw&hl=en&sa=X&oi=book_result&resnum=1&ct=result

Moreover, such an approach is both nonperturbative and finite! Unfortunately, it has been overlooked for various sociological reasons.

Wow! It is exactly what I wanted to know. (Sadly, I don't really have enough time to read through these papers except for abstracts and intros.) Thank you so much!!

[weejee]

Let me clarify this a bit. In the case of a solid, the "true particles" are those making up the solid, the "fake particles" are the phonons. In the case of the electron field, we only know about the "fake particles", which we call electrons. Since no one has yet discovered any "true particles" underlying the electron field, the electron "fake particle" is called an "elementary particle".

Do the "true particles" possibly mean strings, if the string theory is correct?

Unveiling the true identiy of the electron must be a fantastic job, although it sounds somewhat elusive to me (probably due to my ignorance of such works).

[atyy]

Do the "true particles" possibly mean strings, if the string theory is correct?

Unveiling the true identiy of the electron must be a fantastic job, although it sounds somewhat elusive to me (probably due to my ignorance of such works).

No. As far as I can tell, the first two papers are not controversial. I believe that consensus has not been reached on the third paper, which is an attempt to extend the approach to gravity that is different from string theory.

Origin of Light
Xiao-Gang Wen
http://arxiv.org/abs/hep-th/0109120

Photons and electrons as emergent phenomena
Michael Levin, Xiao-Gang WenMichael Levin, Xiao-Gang Wen
http://arxiv.org/abs/cond-mat/0407140

A lattice bosonic model as a quantum theory of gravity
Zheng-Cheng Gu and Xiao-Gang Wen
http://arxiv.org/abs/gr-qc/0606100

[Maaneli]

Wow! It is exactly what I wanted to know. (Sadly, I don't really have enough time to read through these papers except for abstracts and intros.) Thank you so much!!

You're welcome! If you don't mind me asking, are you in physics? If so, what field?

[Maaneli]

No. As far as I can tell, the first two papers are not controversial. I believe that consensus has not been reached on the third paper, which is an attempt to extend the approach to gravity that is different from string theory.

Origin of Light
Xiao-Gang Wen
http://arxiv.org/abs/hep-th/0109120

Photons and electrons as emergent phenomena
Michael Levin, Xiao-Gang WenMichael Levin, Xiao-Gang Wen
http://arxiv.org/abs/cond-mat/0407140

A lattice bosonic model as a quantum theory of gravity
Zheng-Cheng Gu and Xiao-Gang Wen
http://arxiv.org/abs/gr-qc/0606100

Yeah, those papers by Wen and Levin are really interesting. Interestingly, string and M theory have nothing new to say about the electron!

[Maaneli]

Wow! It is exactly what I wanted to know. (Sadly, I don't really have enough time to read through these papers except for abstracts and intros.) Thank you so much!!

Also, if you like, I can Email you the more general review papers that Barut has written on the self-field approach. This way you don't have to read all of them to get the gist of the theory.

[weejee]

You're welcome! If you don't mind me asking, are you in physics? If so, what field?

Yeah, I'm a physics student and interested in condensed matter theory(involving some QFT). Hopefully going to be working in that field.

[weejee]

No. As far as I can tell, the first two papers are not controversial. I believe that consensus has not been reached on the third paper, which is an attempt to extend the approach to gravity that is different from string theory.

Origin of Light
Xiao-Gang Wen
http://arxiv.org/abs/hep-th/0109120

Photons and electrons as emergent phenomena
Michael Levin, Xiao-Gang WenMichael Levin, Xiao-Gang Wen
http://arxiv.org/abs/cond-mat/0407140

A lattice bosonic model as a quantum theory of gravity
Zheng-Cheng Gu and Xiao-Gang Wen
http://arxiv.org/abs/gr-qc/0606100


I wonder what you actually mean by "not controversial"?
Aren't these models just a possible explanation of the origin of photons and electrons rather than a proven reality?

[reilly]

Second Quantization is not fundamentally necessary at all. Rather, it provides one of many possible descriptions of quantum mechanical states and operators. That is, the Fock Space of second quantization is but a unitary transformation away from, say, diagonal q or p diagonal representations, or a helicity rep, , or an angular momentum basis.. However, Fock space turns out to be particularly useful for problems in which the number of particles is not conserved. In fact, one can do (virtually) all of field theory, relativistic or not, without quantized quantum fields, but it's a very messy approach.

The relation between second quantization Fock space and coordinate space is most easily seen by different approaches to the harmonic oscillator -- Schrodinger gives us Hermite functions for the oscillator. Heisenberg's approach utilizes step up and down ops, known now as creation and destruction operators. Thus there must be a unitary transform between the Schrodinger and Heisenberg approaches to the oscillator -- it's a great homework exercise to work out the details of that transformation.

Regards,
Reilly Atkinson

[atyy]

I wonder what you actually mean by "not controversial"?
Aren't these models just a possible explanation of the origin of photons and electrons rather than a proven reality?

Yes, that's what the authors themselves say, that's why it's not controversial. The other part that's not controversial (I think) is that the maths works, so whether or not real photons and electrons are made of other particles, we can make fake photons and fake electrons in a condensed matter system.