Second Quantization vs Many-Particle QM

[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 Schrödinger 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.