Second Quantization vs Many-Particle QM

[A. Neumaier]

The first one is just many-particle quantum mechanics re-expressed in terms of creation and annihilation operators. The second is field theory in which the field is quantized. Is it just a coincidence that the result is the same, or is there some deeper reason?

The deeper reason is that the first route is just field theory in which one has split the Hilbert space into eigenspaces of the number operator. Each of the eigenspaces is a space with fixed particle number. Hence it is described by ordinary quantum mechanics.

It is like saying there are two routes to the QM of a particle in a 3-dimensional rotationally invariant potential. The first route starts with an anharmonic oscillator, then extends it to account for spin, and then introduces operators that change the spin. The second route starts with the Hilbert space of a 3-dimensional rotationally invariant potential, then decomposes the Hilbert space into a direct sum of spaces with a fixed angular momentum (which has discrete spectrum, like the number operator).

Conceptually, these routes are very different. But the second, more fundamental construction explains why the first works.

Similarly, quantum field theory is the more fundamental setting and explains both routes for modeling it.

 

[Demystifier]

Similarly, quantum field theory is the more fundamental setting and explains both routes for modeling it.

This, indeed, is the standard view. But it is always useful to look at things from alternative perspectives. So one possible alternative perspective is that non-relativistic QM is fundamental, while relativistic QFT is emergent.

Such a perspective is in fact a very natural one in condensed-matter physics, where one starts from non-relativistic QM of atoms (consisting of a nucleus and several electrons), and derives quantum field theory for collective excitations such as phonons. Moreover, the dispersion relation for phonons is Lorentz invariant with respect to the velocity of sound. By analogy with condensed matter, it is possible to speculate that all relativistic "elementary" particles of the Standard Model are really quasiparticles originating from some more fundamental non-relativistic quantum theory. For more details about such ideas see the book by Volovik
https://www.amazon.com/dp/0199564841/?tag=pfamazon01-20

 

[A. Neumaier]

non-relativistic QM is fundamental, while relativistic QFT is emergent.

How does relativistic QFT emerge from non-relativistic QM? I don't have access to the book you mentioned but as you already write, it is probably speculation only. Whereas the other direction - that non-relativistic QM emerges from relativistic QFT (by taking $c\to\infty$ and restricting to a sector with a fixed number of particles) - is standard knowledge, almost at exercise level.

it is always useful to look at things from alternative perspectives.

Speculations are useful only if they lead to something substantial. Can you point out the use in the present context?

 

[Demystifier]

How does relativistic QFT emerge from non-relativistic QM? I don't have access to the book you mentioned but as you already write, it is probably speculation only.

It's not a speculation. You can find it in any advanced textbook on condensed-matter physics.

Speculations are useful only if they lead to something substantial. Can you point out the use in the present context?

For instance, Volovik proposes a solution of the cosmological-constant problem.

 

[A. Neumaier]

It's not a speculation. You can find it in any advanced textbook on condensed-matter physics.

You are contradicting yourself. Please point me to a page in a textbook where it is shown that QED or another part of the standard model emerges from non-relativistic QM. You yourself described this as a speculation!

 

[Demystifier]

You are contradicting yourself. Please point me to a page in a textbook where it is shown that QED or another part of the standard model emerges from non-relativistic QM. You yourself described this as a speculation!

You are not reading carefully what I said. I did not say that QED or any other specific part of the Standard Model has been strictly derived from non-relativistic QM. That's only a speculation, discussed in the Volovik's book.

I said that relativistic QFT has been derived from non-relativistic QM. I hope you understand that the concept of relativistic QFT is much more general than the Standard Model. One can derive a relativistic QFT model without deriving any specific part of the Standard Model.

To be more specific, let me repeat a part of what I said in #37: In condensed-matter physics one starts from non-relativistic QM of atoms (consisting of a nucleus and several electrons), and derives quantum field theory for collective excitations such as phonons. The dispersion relation for phonons is Lorentz invariant with respect to the velocity of sound.

 

[A. Neumaier]

The dispersion relation for phonons is Lorentz invariant with respect to the velocity of sound.

A Lorentz invariant dispersion relation is still very far from a relativistic QFT. One should at least verify (on the usual level of rigor of theoretical physics) that the Wightman axioms hold; in particular, that all correlation functions are Poincare covariant and that a Poincare invariant vacuum state exists. This is very unlikely, as phonons require a crystal structure, which is not translation invariant.

 

[Demystifier]

A Lorentz invariant dispersion relation is still very far from a relativistic QFT. One should at least verify (on the usual level of rigor of theoretical physics) that the Wightman axioms hold; in particular, that all correlation functions are Poincare covariant and that a Poincare invariant vacuum state exists. This is very unlikely, as phonons require a crystal structure, which is not translation invariant.

You don't know much about QFT in condensed matter, do you?

QFT in condensed matter is used as an approximative theory. It is used in a long-distance limit, at distances much larger than the lattice spacing. In this approximation, the effective QFT is translation invariant. So in condensed matter physics the "fundamental" theory is nonrelativistic QM, while relativistic QFT is an emergent, effective, approximative theory.

So yes, Wightman axioms etc hold. But in condensed matter the Wightman axioms are properties of an approximative theory, not of "fundamental" theory.

 

[A. Neumaier]

So yes, Wightman axioms etc hold.

I'd like to ask again for a reference to a page in a book or article where this is shown.

 

[Demystifier]

I'd like to ask again for a reference to a page in a book or article where this is shown.

I couldn't find an explicit discussion of Wightman axioms in condensed matter. But for Lorentz invariance see e.g.
http://lanl.arxiv.org/abs/gr-qc/9311028
http://ls.poly.edu/~jbain/papers/ConPhysST.pdf
http://link.springer.com/article/10.1007/BF00672855
http://physics.stackexchange.com/qu...emergent-special-relativity-in-the-superfluid
https://books.google.hr/books?id=ct...orentz invariance in condensed matter&f=false
I am pretty much convinced that Wightman "axioms" can be derived from those results, even if nobody so far cared to do that explicitly.

 

[A. Neumaier]

[...]
http://ls.poly.edu/~jbain/papers/ConPhysST.pdf
I am pretty much convinced that Wightman "axioms" can be derived from those results, even if nobody so far cared to do that explicitly.

Thanks a lot for the references. I read the quoted one (from 2008, by Jonathan Bain), which also discusses Volovik's work. It made interesting, though ultimately somewhat disappointing reading.

Bain carefully qualifies his discussion by saying in the introduction that

It will be seen that these examples possesses limited viability as analogues of spacetime insofar as they fail to reproduce all aspects of the appropriate physics. On the other hand, all three examples may be considered part of a condensed matter approach to quantum gravity

In particular, the metrics derived for acoustic spacetimes are not translation invariant (which, unlike QED or the standard model precludes an interpretation in the Wightman framework) and requires a general relativistic context. But the field equations seem not to be generally covariant, which makes a general relativistic interpretation dubious. Bain concludes on p.8:

I would thus submit that acoustic spacetimes provide neither dynamical nor kinematical analogues of general relativity.

On the other hand (p.11),

the effective Lagrangian for 3 He-A then is formally identical to the Lagrangian for massless (3 + 1)-dim QED in a curved spacetime. [...]
the low-energy EFT of 3 He-A does not completely reproduce all aspects of the Standard Model. Moreover, [...] a low-energy treatment of the 3 He-A effective metric does not produce the Einstein–Hilbert Lagrangian of general relativity.

On p.13, he discusses violations of Lorentz invariance.

I conclude the the condensed matter space-times may be considered as relativistic in a very liberal sense only; they don't resemble (except superficially) those treated in textbooks on quantum field theory or elementary particle theory. Bain's conclusion is

currently an interpretation of spacetime as a low-energy emergent phenomenon cannot be fully justified. However, this essay also
argued that such an interpretation should nevertheless still be of interest to philosophers of spacetime.

I will check out some of the other references at another time but woul be surprised if the above assessment would have to be revised.

 

[A. Neumaier]

There is a very thick survey paper from 2011 called Analogue Gravity by Barceló, Liberati and Visser, which discusses the known models in detail, and (in Section 7) their potential significance as a basis for deriving QFT. On p.105 they write that

It is a well-known issue that the expected relativistic dynamics, i.e., the Einstein equations, have to date not been reproduced in any known condensed-matter system.

 

[ShayanJ]

I don't know whether this is discussed in the above papers or not, but recently I've read about acoustical analogues of BHs that were called "dumb holes". Its even verified that there is an acoustical Hawking radiation. These results seem significant but I'm not sure what role they play in your discussions. It seems you're arguing whether spacetime can be emergent from an underlying system like the condensed matter systems in labs. But if those systems in labs don't give us GR...well...it maybe that the system that spacetime emerges from has some strange features which are much different from the condensed matter systems we can study in our labs. I don't think that we even know a general approach to study those systems we can study in labs, let alone an approach that is general enough to include anything that can lead to a continuum in a limit.

 

[A. Neumaier]

I've read about acoustical analogues of BHs that were called "dumb holes". Its even verified that there is an acoustical Hawking radiation.

This sort of experimental black holes are discussed in the

survey paper from 2011 called Analogue Gravity

They are useful for helping us understand and explore certain features of gravity in the lab, but in interpreting the results one must always remember that these are model situations that share some but not all features with real gravity.

My coauthor Ulf Leonhardt has written some papers about optical analogues and electrical analogues of general relativity.

In my opinion, none of these (or those in the above survey article) imply anything about a nonrelativistic theory underlying relativistic quantum field theory. The reason is that the transition from relativistic to nonrelativistic is a simple limit, consistent with the more fundamental nature of relativistic quantum fields, while the transition form nonrelativistic to relativistic is complicated, and hasn't even approximately reproduced one of the experimentally verified theories - in spite of 20 years of research on it.

 

[ShayanJ]

This sort of experimental black holes are discussed in the

They are useful for helping us understand and explore certain features of gravity in the lab, but in interpreting the results one must always remember that these are model situations that share some but not all features with real gravity.

My coauthor Ulf Leonhardt has written some papers about optical analogues and electrical analogues of general relativity.

In my opinion, none of these (or those in the above survey article) imply anything about a nonrelativistic theory underlying relativistic quantum field theory. The reason is that the transition from relativistic to nonrelativistic is a simple limit, consistent with the more fundamental nature of relativistic quantum fields, while the transition form nonrelativistic to relativistic is complicated, and hasn't even approximately reproduced one of the experimentally verified theories - in spite of 20 years of research on it.

Do your statements include all emergent-gravity/spacetime approaches?(Actually I'm not sure how many approaches exist!)

 

[A. Neumaier]

Do your statements include all emergent-gravity/spacetime approaches?(Actually I'm not sure how many approaches exist!)

They include those that are based on emergence from nonrelativistic fluid or solid models (or their expected sub elementary-particle variants).

Emergence from spin foam or strings is a different matter, though I also believe that these are dead ends (or at least conceptual overkills). My bet is on canonical gravity with a suitable choice of the (infinitely many) renormalization constants. There are other nonrenormalizable theories (the Gross-Neveu models) with infinitely many renormalization constants that become renormalizable when not expanded in terms of free fields but in a different way. I expect something similar to happen to canonical gravity. Thus no search for an exotic emergence is needed.

 

[atyy]

I'd like to ask again for a reference to a page in a book or article where this is shown.

Another way to see it is that lattice gauge theory is carried out at finite lattice spacing, which makes it non-relativistic, yet it is believed to be able to provide a non-perturbative formulation of many parts of the standard model. As I understand it, gravity on a lattice is also not a problem. Naturally, in all of these cases, the Lorentz invariance is only approximate and at low energies. The main problem for a lattice standard model is chiral fermions.

For a "textbook" quote, one can try http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf (p12)
"Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important for our considerations, we replace the continuum of three-dimensional space by a discrete but dense lattice of points."

Also http://arxiv.org/abs/hep-lat/0211036 (p10)
"In principle all known perturbative results of continuum QED and QCD can also be reproduced using a lattice regularization instead of the more popular ones." [I think this is too strong - I don't think the chiral fermion results can be reproduced yet, even in principle]

Try http://arxiv.org/abs/0901.0964 for lattice gravity. The main problem addressed by the paper is high energies, but there as I understand it, there is no problem with low energies.

 

[A. Neumaier]

lattice gauge theory is carried out at finite lattice spacing

But this is quite a different situation. Here the finite-dimensional approximation emerges from the covariant continuum action,
whereas in approaches to emerging gravity or emerging relativistic QFT one starts with an action that has no simple relationship with the target theory.

 

[atyy]

But this is quite a different situation. Here the finite-dimensional approximation emerges from the covariant continuum action,
whereas in approaches to emerging gravity or emerging relativistic QFT one starts with an action that has no simple relationship with the target theory.

Yes, absolutely. There are (at least) two reasons to be interested in non-relativistic theories.

The first is that Bohmian mechanics is easier to formulate for such theories. In this case, a conservative approach is lattice gauge theory.

In the second case, it is just the condensed matter inferiority complex that since they are not doing fundamental physics, no one else can either. Consequently the standard model and string theory should all be emergent :P

For the second case, one textbook that claims to get quite close is Wen's https://www.amazon.com/dp/019922725X/?tag=pfamazon01-20, where in the last chapter he claims to be able to get QED with massless electrons.

 

[A. Neumaier]

Wen's http://www.textbooks.com/Quantum-Field-Theory-of-Many-Body-System/9780199227259/Wen.php,

The link didn't lead me to information about the book.

 

[Demystifier]

The link didn't lead me to information about the book.

You are a victim of this problem:
https://www.physicsforums.com/threads/amazon-links-to-books.834440/

 

[A. Neumaier]

atyy, could you please link directly to the publisher?

 

[A. Neumaier]

You are a victim of this problem:
https://www.physicsforums.com/threads/amazon-links-to-books.834440/

Though the link was to http://www.textbooks.com/ and not to amazon?

 

[Heinera]

Though the link was to http://www.textbooks.com/ and not to amazon?

There is a script on the server that actually replaces links from what the author of the post wrote, to what you see in your browser. In this case from amazon.com to textbooks.com.

 

[Demystifier]

Though the link was to http://www.textbooks.com/ and not to amazon?

As atyy will probably confirm, the link he gave was to amazon.

To find the original link atyy gave, go to google and type amazon + other keywords seen in blue letters of his link. In this case, typing e.g. "amazon wen quantum field" will suffice.

Another trick is to press the reply button on the atyy's post. Then you will see his original link, which you can copy and paste into your browser.

 

[atyy]

atyy, could you please link directly to the publisher?

I did link to Amazon. Does this work?: http://ukcatalogue.oup.com/product/9780198530947.do?

Anyway, it might be easier to read some of their papers, maybe:
[URL]http://arxiv.org/abs/hep-th/0507118[/URL]
http://arxiv.org/abs/hep-th/0507118
Quantum ether: photons and electrons from a rotor model
Michael Levin, Xiao-Gang Wen

 

[A. Neumaier]

Quantum ether: photons and electrons from a rotor model

Thanks; I'll look at the paper - it was not mentioned in the surveys that I linked to; so maybe it is free from defects (or has others).

The new book link works, too.

 

[king vitamin]

Wen claims he can get every "almost everything" from interacting models of qubits or quantum rotors. Here is a Stack Exchange link, which contains links to the above paper and some others, where he lists types of interactions/particles he can get:

http://physics.stackexchange.com/a/164958

So basically every "feature" besides gravitons. Since you asked about QED, the emergence of U(1) lattice gauge theories occurs often in mean-field descriptions of quantum magnets. This led to some of the earliest models for spin liquids.

A good example of an emergent Lorentz-invariant model from qubits which IMO every theorist should learn is the critical transverse-field Ising model in (1+1) dimensions. In the scaling limit it is a theory of free massless relativistic Majorana fermions (known as the c=1/2 minimal model in CFT literature). Although if you're familiar with bosonization, it might not be that surprising that we get fermions in 1+1 dimensions.

 

[atyy]

Wen claims he can get every "almost everything" from interacting models of qubits or quantum rotors. Here is a Stack Exchange link, which contains links to the above paper and some others, where he lists types of interactions/particles he can get:

http://physics.stackexchange.com/a/164958

So basically every "feature" besides gravitons. Since you asked about QED, the emergence of U(1) lattice gauge theories occurs often in mean-field descriptions of quantum magnets. This led to some of the earliest models for spin liquids.

A good example of an emergent Lorentz-invariant model from qubits which IMO every theorist should learn is the critical transverse-field Ising model in (1+1) dimensions. In the scaling limit it is a theory of free massless relativistic Majorana fermions (known as the c=1/2 minimal model in CFT literature). Although if you're familiar with bosonization, it might not be that surprising that we get fermions in 1+1 dimensions.

Apart from gravitons, it's also not clear he can get the chiral interaction. http://blog.sina.com.cn/s/blog_aed08fb70101lhwt.html

 

[A. Neumaier]

Since you asked about QED, the emergence of U(1) lattice gauge theories occurs often in mean-field descriptions of quantum magnets.

But this in itself is not very interesting. It leaves the main problem of QED, namely the existence of a good continuum limit, untouched. Without this, one could just propose a fundamental length scale and then assume that QED only exists on the corresponding lattice. Thus speculations about an underlying structure need to wait for confirmation by experiment to be an improvement over the present situation.

 

[king vitamin]

Apart from gravitons, it's also not clear he can get the chiral interaction. http://blog.sina.com.cn/s/blog_aed08fb70101lhwt.html

Interesting, I wondered about that since his book claimed he couldn't get them yet.

But this in itself is not very interesting. It leaves the main problem of QED, namely the existence of a good continuum limit, untouched. Without this, one could just propose a fundamental length scale and then assume that QED only exists on the corresponding lattice. Thus speculations about an underlying structure need to wait for confirmation by experiment to be an improvement over the present situation.

Sure, just answering your question about QED emerging from non-relativistic QM since you seemed interested. Of course there is no claim that this solves issues with triviality or anything.

 

[atyy]

Interesting, I wondered about that since his book claimed he couldn't get them yet.

Yes, he couldn't get them at the time of his book.

The paper rejected by PRL and discussed in the blog post linked above is much more recent. It's http://arxiv.org/abs/1305.1045.

For reference, another claim of an emergent standard model is http://arxiv.org/abs/0908.0591.

Both Wen and Schmelzer include claims to solve the lattice chiral fermion problem. I think there is general caution about these claims because there are also some recent wrong claims by very distinguished scientists on the lattice chiral fermion problem: https://www.physicsforums.com/threads/status-of-lattice-standard-model.823860/.

 

[pavsic]

Apparently, there are two different routes to get to quantum field theory from single-particle quantum mechanics: (I'm going to use nonrelativistic quantum mechanics for this discussion. I think the same issues apply in relativistic quantum mechanics.)

Route 1: Many-particle quantum mechanics
Start with single-particle QM, with the Schrodinger equation: $- \frac{1}{2m} \nabla^2 \psi = i \frac{\partial}{\partial t} \psi$

1. Now, extend it to many (initially, noninteracting) particles: $\Psi(\vec{r_1}, \vec{r_2}, ..., \vec{r_n}, t)$
2. Introduce creation and annihilation operators to get you from (a properly symmetrized) $n$-particle state to an $n+1$-particle state, and vice-verse.

Route 2: Second quantization
Once again, start with the single-particle wave function.

1. Instead of viewing $\psi$ as a wave function, you view it as a classical field.
2. Describe that field using a Lagrangian density $\mathcal{L} = i \psi^* \dot{psi} - \frac{1}{2m}|\nabla \psi|^2$
3. Derive the canonical momentum using $\pi = \dfrac{\partial}{\partial \dot{\psi}}$
4. Impose the commutation rule: $[\pi(\vec{r}), \psi(\vec{r'})] = -i \delta^3(\vec{r'} - \vec{r})$

Conceptually, these routes are very different. The first one is just many-particle quantum mechanics re-expressed in terms of creation and annihilation operators. The second is field theory in which the field is quantized. Is it just a coincidence that the result is the same, or is there some deeper reason?

In my paper "A Theory of Quantized Fields Based on Orthogonal and Symplectic Clifford Algebras", Advances in Applied Clifford Algebras, 22 (2012) 449-481, http://dx.doi.org/10.1007/s00006-011-0314-4, [http://arxiv.org/abs/arXiv:1104.2266] , it is shown that those two routes are conceptually not so different. The essence is in distinguishing between the components and the basis vectors of an object, which can be a vector or a multivector. In infinite dimensions, basis vectors behave as quantized fields.. Namely, in Geometric Algebras, based on Clifford algebras, basis vectors are generators of a Clifford algebra, which can be either orthogonal or symplectic. In the case of an orthogonal Clifford algebra, the generators, if transformed into the so called Witt basis, satisfy the fermionic anticommutation relations, whereas in the case of a symplectic Clifford algebra, the generators satisfy the bosonic commutation relations.