Why the name Second Quantization ?

[DaTario]

Why the name "Second Quantization"?

Hi all,

The title said it all. My question is:

How is one to interpret the name second quantization ?

Which specifically is quantized twice ?

Best Wishes



DaTario

 

[Meir Achuz]

Quantum mechanics takes the classical variables p and x that describe the state of a classical system into non-commuting operators that act on a wave function that describes the state of the quantum mechanical system.

For quantum field theory, the old QM wave function is now considered a quantum mechanical operator. This is a second quantization, hence the name. The old QM wave function is an operator that can create and destroy particles. It acts on a new type of state function tht includes the details about the particles.

 

[ZapperZ]

Hi all,
The title said it all. My question is:
How is one to interpret the name second quantization ?
Which specifically is quantized twice ?
Best Wishes

DaTario

Why the name Second Quantization? Because First Quantization was already taken. [A,B] is roughly the First Quantization. Second Quantization doesn't mean something is done twice (or else we would have called it Twice Quantized).

Zz.

 

[marlon]

Hi all,
The title said it all. My question is:
How is one to interpret the name second quantization ?
Which specifically is quantized twice ?
Best Wishes

DaTario

Recently, i have answered that question in this thread

marlon

ps : do not make "strange" quotes on a photon's energy
That is a big no no if you read the forum's guidelines

 

[DaTario]

Please Marlon,

what do you mean by strange quotes on photon's energy ?

By the way, thank you for the indication of the thread.

Zapperz, thank you also for your comments. But I am not sure I have understood well your understanding of the term second quantization.

Anyway, thank you all.

creation and anilillation operators is one of the main products of second quantization formalism while the wave function is one of the main products of First quantization.

 

[dextercioby]

Well, we don't have "first quantization" and "second quantization". We just have QUANTIZATION, or "canonical formalism" if you prefer this terminology. Quantum mechanics is QUANTIZATION applied to classical dynamical systems with FINITELY MANY degrees of freedom, while Quantum field theory is QUANTIZATION applied to classical dynamical systems with INFINITELY MANY degrees of freedom, the latter a.k.a. "classical fields", or "finite dimensional nonunitary irreducible linear representations of the universal covering group of the restricted Poincaré group"...

That's all there is to know...

Daniel.

 

[marlon]

Quantum mechanics is QUANTIZATION applied to classical dynamical systems with FINITELY MANY degrees of freedom,

This is not accurate. Not only do the degrees of freedom need to be finite in the case of QM, they also need to be FIXED !

regards
marlon

 

[dextercioby]

What do you mean by fixed ? If you mean certain requirements upon the rank of the hessian matrix

$$W_{ij}=:\left(\frac{\partial^{2}L}{\partial \dot{q}^{i}\partial \dot{q}^{j}} \right)$$

to be constant in time when evaluated on the surface of all primary constraints, then you're absolutely right and i appologized not to have mentioned this fact in my first posting.

Daniel.

 

[marlon]

This is indeed my point.

You directly gave the techical version but there is a more intuitive explanation on this on to the Wikipedia website (if you want, just look for second quantization)

regards
marlon

 

[George Jones]

a.k.a. "classical fields", or "finite dimensional nonunitary irreducible linear representations of the universal covering group of the restricted Poincaré group"
Daniel.

Actually, infinite-dimensional unitary irreducible linear representations of the universal covering group of the restricted Poincaré group.

Regards,
George

 

[marlon]

To Dexter and George,

I do not know who of you is right but shouldn't we keep the discussion onto a more intuitive level ? I mean, i assure you that 99,9% of the people (including science advisors and mentors) will not consider such an answer to be very clear. Ofcourse this does NOT mean that it is wrong.

Just my opinion.

regards
marlon

 

[Physics Monkey]

George,

I'm a bit confused by your reply. Are not the quantum fields one usually makes use of finite dimensional representations (vector, spinor, etc) of a non-compact group and hence non-unitary?

DaTario,

Part of the problem with the name second quantization is that it is something of a historical misnomer. Early quantum field theorists thought that they were somehow "quantizing" the wavefunction. For example, Dirac proposed his famous equation for the electron as a single particle relativistic wave equation. However, the relativistic quantum field describing the electron obeys the same equation. Confusion arose. I tend to think of it as something analogous to the way we still use the term "electromotive force" for something that isn't a force.

 

[George Jones]

Ok, I will try and be a bit more clear at least on the infinite-dimensional representation part. To this end, I'm going to forget about untarity, covering spaces, Poincare etc., and I'm going to consider some toy examples.

Take ordinary physical space to be $$\mathbb{R}^3$$, and consider the action of rotations on fields defined in space.

An example of a scalar field defined on space is temperature $$T$$, which associates a temperature $$T \left( \vec{r} \right)$$ with every position $$\vec{r}$$ in space. In other words, $$T$$ is a function, with $$T: \mathbb{R}^3 \rightarrow \mathbb{R}$$, so $$T$$ is a member of the infinte-dimensional function space $$\{f: \mathbb{R}^3 \rightarrow \mathbb{R} \}$$.

Now let a rotation $$R$$ act on the space of scalar fields. Since $$R$$ operates (rotates) on (3-dim) vectors, this action is defined through a representative operator as follows. The representative of $$R$$ acts on a scalar field $$T$$ to give a new scalar field (i.e., another function) $$T'$$ such that the new temperature at the rotated positon $$\vec{r}' = R \vec{r}$$ is the same as the old temperature at the unrotated position $$\vec{r}$$:

$$T' \left( \vec{r}' \right) = T \left( \vec{r} \right).$$

Thus, the function $$T'$$ is defined by

$$T' \left( \vec{r} \right) = T \left(R^{-1} \vec{r} \right).$$

Now consider a field that assigns a vector $$\vec{E} \left( \vec{r} \right)$$ to each position $$\vec{r}$$ in space. $$\vec{E} \left( \vec{r} \right)$$ is an element of a 3-dimensional space (say) $$V$$ (to distinguish $$V$$ from the space of positions), but $$\vec{E}$$ itself is an element of the infinte-dimensional space $$\{ \vec{A}: \mathbb{R}^3 \rightarrow V \}$$ of vector-valued functions of position.

A rotation $$R$$ acts on the space as vector fields as follows. A rotation $$R$$ acts on the vector field $$\vec{E}$$ to give a new vector field $$\vec{E}'$$ such that the new field evaluated at the rotated position $$\vec{r}'$$ field is the same as the old field evaluated at the unrotated position and then rotated $$\vec{r}$$.
This makes sense because $$R$$ can act directly on $$\vec{E} \left(\vec{r} \right)$$, since $$\vec{E} \left(\vec{r} \right)$$ is an element of the 3-dimensional space $$V$$.

$$\vec{E}' \left( \vec{r}' \right) = R \left( \vec{E} \left( \vec{r} \right) \right).$$

Thus, the vector-valued function $$T'$$ is defined by

$$\vec{E}' \left( \vec{r} \right) = R \left( \vec{E} \left(R^{-1} \vec{r} \right) \right).$$

For both scalar and vector fields, finite-dimensional (1-dim for scalar fields, 3-dim for vector fields) have been used to get at the actual required infinite-dimensional representations.

It is an interesting mathematical exercise to show that $$R^{-1}$$, not $$R^$$, is need in the arguments in order to define a representation, i.e., a homomorphism of groups.

It is also interesting to go through this for the action of the Poincare group on "classical" Dirac spinor-valued fields on spacetime. I may try and give a pedagogical exposition of this, including unitarity, in another thread.

Regards,
George

 

[marlon]

:) George :)

Now this is what we call : "a piece of theoretical art"

regards
marlon

ps : i have been thinking to set up some kind of "general introduction to theoretical physics"-thread covering the intro of QFT all together with basic gauge symmetry and the implementation of group theory. I would like to invite you to participate (along with all others that like this idea) and check out some of the texts i already have written in my journal. What do you think ?

 

[marlon]

ps George, we can also cover the difference between QM and QFT , and "why we use fields" (see my journal). I have written a first attempt on this. I provided a link to that text in my first post of this thread. Let me know what you think of it, please. I would really appreciate it

regards
marlon

 

[George Jones]

George,
I'm a bit confused by your reply. Are not the quantum fields one usually makes use of finite dimensional representations (vector, spinor, etc) of a non-compact group and hence non-unitary?

I didn't notice your post until after I made my second post. My second post might shed some light, or it might just roil the waters! My examples are somewhat poor because I have used a compact group, but as I said in that post, I may start a thread and go through this (except for proving the irreducibility!) in some detail for "classical" Dirac fields

Regards,
George

 

[George Jones]

i have been thinking to set up some kind of "general introduction to theoretical physics"-thread covering the intro of QFT all together with basic gauge symmetry and the implementation of group theory. I would like to invite you to participate (along with all others that like this idea) and check out some of the texts i already have written in my journal. What do you think ?

Sounds very interesting.

I would like to participate, but I think and write *very* slowly, so I don't know how much this will happen.

Regards,
George

 

[marlon]

I would like to participate, but I think and write *very* slowly,

No problem, me too.

so I don't know how much this will happen.
Regards,
George

Suggestions can be made here

For example, you already have written a nice text in this thread that can serve our goals.

regards
marlon

 

[George Jones]

Physics Monkey and Marlon (and any others), you might interested in some ramblings of mine on sci.physics.research about the Dirac case.

First http://groups.google.ca/group/sci.physics.research/msg/02c877be496f0897?dmode=source&hl=en"

then http://groups.google.ca/group/sci.physics.research/msg/beaf07d4b1ba624b?dmode=source&hl=en".

I would like to find the time to clean these up, latex them, and post them on Physics Forums.

Regards,
George

 

[Physics Monkey]

Thanks for the reply, George, I can see it was just some confusion over terminology.

 

[dextercioby]

Actually, infinite-dimensional unitary irreducible linear representations of the universal covering group of the restricted Poincaré group.
Regards,
George

If it's unitary, it's no longer classical, by virtue of the Wigner theorem. The representation space is no longer a vector space, but a (rigged) Hilbert space which is typical for a quantum theory.

The group i was implying was $\mbox{SL(2,\mathbb{C})} \rtimes S$. It's interesting that "classical fields" are really nonunitary finite dimensional (ir)reducible representations of the $\mbox{SL(2,\mathbb{C})}$ group, but for the sake of completeness it's the universal covering group of the restricted Poincaré group (and not the restricted Lorentz group) which is important when passing from a classical to a quantum description of fields.

Daniel.

 

[DaTario]

I would like to acknowledge all of you for the answers and for the debate.


DaTario

 

[George Jones]

If it's unitary, it's no longer classical, by virtue of the Wigner theorem. The representation space is no longer a vector space, but a (rigged) Hilbert space which is typical for a quantum theory.

Sorry for the late respopnse - I was (and still am to some degree) snowed under by work, and I have been trying to avoid Physics Forums.

I am having trouble unpacking the above sentences. Which Wigner theorem? Also, I read these sentences as saying that Hibert spaces aren't vector spaces, and (the triple of) space in a rigged Hibert spaces aren't vector spaces.

?

The group i was implying was $\mbox{SL(2,\mathbb{C})} \rtimes S$. It's interesting that "classical fields" are really nonunitary finite dimensional (ir)reducible representations of the $\mbox{SL(2,\mathbb{C})}$ group, but for the sake of completeness it's the universal covering group of the restricted Poincaré group (and not the restricted Lorentz group) which is important when passing from a classical to a quantum description of fields.
Daniel.

But all representations of the Poincare group lift to representations of its universal cover, so if you're saying that classical fields are representations of the Poincare group, then they're also representation of its cover. Of course, things don't work the other way round.

Unitarity is mathematical property that is very useful in quantum theory, but, as a mathematical property, it can also apply to classical representations.

In any case, what I'm interested in is the dimension of the representation. Classical fields are tensor fields, and tensor fields are modules over the ring of scalar fields. Addition conditions (e.g. wave equations) required by calssical fields restric themt to vector spaces, but I'm hard pressed to see how these spaces could be finite-dimensional.

As I hinted at in another post, sloppy physics books sometimes refer to finite-dimensionality, but this refers to the dimension of representations used to construct fields, not to the dimensionality of the actual representation spaces of which fields are members.

The easiest way for you to turn my into a believer in the finite-dimensionality of the spaces of classical fields would be to give an explicit example. In the interests of simplicity of presentation, the example doesn't have to a field that really exits, but it could correspond to reality.

Until I see an explicit example to the contrary, I will continue to believe and state that classical fields are elements of infinite-dimensional representation spaces.

Regards,
George