Lattice non-perturbative definition

In summary, the conversation is about a paper by Xiao-Gang Wen titled "A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model." The paper addresses the problem of describing the standard model perturbatively and suggests using a "Hamiltonian quantum theory" to get an exact description by placing it on a finite lattice. The paper also discusses the introduction of a finite dimensional Hilbert space and the use of boundary conditions. The conversation then shifts to a specific question about the mechanism of how mirror fermions acquire masses and the use of solitons in this context. The expert provides a summary of the paper and explains the concept of topological insulators and the use of solitons
  • #1
shiraz
Dear All
I would like to understand a paper for Xiao-Gang Wen " A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model" .
Can anyone help me by suggesting another reference that is a little bit easier.
Thank you
 
Physics news on Phys.org
  • #2
This paper is a merger of a lot of complicated issues. For other readers, the paper is posted on arXiv here: https://arxiv.org/pdf/1305.1045.pdf
Basically the problem they are addressing is that the standard model can only be described perturbatively (as a coupling constant expansion), keeping the first one or two perturbative terms. They look to turn this into a "Hamiltonian quantum theory" (a term I have not heard before, but as they define in the paper a theory which is described by operators and has a time evolution described by a quantity such as [itex]U(t,t') = \mathcal{T}exp(-i\int_{t_0}^tdt'V_I(t')) [/itex]) in order to get an exact description of a SO(10) gauge theory by placing it on a finite lattice.

He also mentions a finite dimensional Hilbert space, but I assume this means they are introducing boundary conditions somewhere. The most helpful places to look are in the appendices, as this is where they show the math. It goes into the Clifford algebra of the hopping elements and how they place this chiral gauge theory on a lattice.

But to answer your first question, you need to clarify yourself. This paper combines many advanced topics, so just asking for a general reference to understand it isn't answerable. What specifically are having trouble understanding in the paper? Also, if your specific difficulty involves standard model questions, you should move over to the high energy forum. If you have questions regarding the lattice model their using and how gauge theories can be described on it, then this is the right forum.
 
  • Like
Likes shiraz
  • #3
DeathbyGreen said:
This paper is a merger of a lot of complicated issues. For other readers, the paper is posted on arXiv here: https://arxiv.org/pdf/1305.1045.pdf
Basically the problem they are addressing is that the standard model can only be described perturbatively (as a coupling constant expansion), keeping the first one or two perturbative terms. They look to turn this into a "Hamiltonian quantum theory" (a term I have not heard before, but as they define in the paper a theory which is described by operators and has a time evolution described by a quantity such as [itex]U(t,t') = \mathcal{T}exp(-i\int_{t_0}^tdt'V_I(t')) [/itex]) in order to get an exact description of a SO(10) gauge theory by placing it on a finite lattice.

He also mentions a finite dimensional Hilbert space, but I assume this means they are introducing boundary conditions somewhere. The most helpful places to look are in the appendices, as this is where they show the math. It goes into the Clifford algebra of the hopping elements and how they place this chiral gauge theory on a lattice.

But to answer your first question, you need to clarify yourself. This paper combines many advanced topics, so just asking for a general reference to understand it isn't answerable. What specifically are having trouble understanding in the paper? Also, if your specific difficulty involves standard model questions, you should move over to the high energy forum. If you have questions regarding the lattice model their using and how gauge theories can be described on it, then this is the right forum.
Dear DeathbyGreen
I am trying to understand how the mechanism of how mirror fermions acquire masses ( gapped) " the first column of page 4" especially when he start talking about solitons.
Thank you
 
  • #4
So the system they are describing is a topological insulator. In an actual example, such as graphene under circularly polarized light (floquet topological insulator) or standard topologically insulating graphene just as a theoretical toy model, the Lagrangian can be described using a QED 2+1 Lagrangian. This is because in graphene you have not only fermion spin degrees of freedom, but also sublattice degrees of freedom (pseudospin). The "psuedospinors" can be described using left and right handed Weyl spinors, each of which would have spin up and down components (think 4 component gamma matrices used in the Dirac equation).

So in their system, they have Weyl fermions which are gapped in the bulk, and gapless on the edges. In order to reflect standard model SO(10) symmetries they introduce 16 Weyl spinors for 16 different introduced field as [itex]\sum_{\alpha}^{16}\phi_{\alpha}[/itex]. They then want to break the symmetry of the left handed spinors ("gap them out") but keep the symmetry of the right handed ones. I think this is to reflect the types of symmetry you have in the standard model, but I can't speak much to this as a condensed matter guy. So what they want is to introduce an interaction which can break symmetries in a portion of the system while keeping an overall symmetry intact. This reminds me of how Haldane originally envisioned using spin orbit coupling to induce a topological phase without violating time reversal symmetry of the system (time reversal symmetry in topological insulators is key to having protected edge states).

So they begin considering interactions which satisfy this condition; as in a typical topological insulator, any magnetic impurity will violate time reversal symmetry and thus remove the protected states. They use a soliton as a description of a magnetic impurity, a type of "solitary" spin wave of magnetic moments. So these are types of impurities they worry about which would remove the symmetry they want to keep. They mention seeing these as gravitational anomalies, but I don't know what that means; clarification on that would best be posed in a different forum.
 
Last edited:
  • Like
Likes shiraz
  • #5
DeathbyGreen said:
So the system they are describing is a topological insulator. In an actual example, such as graphene under circularly polarized light (floquet topological insulator) or standard topologically insulating graphene just as a theoretical toy model, the Lagrangian can be described using a QED 2+1 Lagrangian. This is because in graphene you have not only fermion spin degrees of freedom, but also sublattice degrees of freedom (pseudospin). The "psuedospinors" can be described using left and right handed Weyl spinors, each of which would have spin up and down components (think 4 component gamma matrices used in the Dirac equation).

So in their system, they have Weyl fermions which are gapped in the bulk, and gapless on the edges. In order to reflect standard model SO(10) symmetries they introduce 16 Weyl spinors for 16 different introduced field as [itex]\sum_{\alpha}^{16}\phi_{\alpha}[/itex]. They then want to break the symmetry of the left handed spinors ("gap them out") but keep the symmetry of the right handed ones. I think this is to reflect the types of symmetry you have in the standard model, but I can't speak much to this as a condensed matter guy. So what they want is to introduce an interaction which can break symmetries in a portion of the system while keeping an overall symmetry intact. This reminds me of how Haldane originally envisioned using spin orbit coupling to induce a topological phase without violating time reversal symmetry of the system (time reversal symmetry in topological insulators is key to having protected edge states).

So they begin considering interactions which satisfy this condition; as in a typical topological insulator, any magnetic impurity will violate time reversal symmetry and thus remove the protected states. They use a soliton as a description of a magnetic impurity, a type of "solitary" spin wave of magnetic moments. So these are types of impurities they worry about which would remove the symmetry they want to keep. They mention seeing these as gravitational anomalies, but I don't know what that means; clarification on that would best be posed in a different forum.
Dear DeathbyGreen
Thank you for your help. I understand the main point which is gapping out the the left handed by intriducing an interaction between them and the scalar fields ( note : we have 10 scalar fields and not 16).why did u write "they introduce 16 Weyl spinors for 16 different introduced field scalar fields although we have only 10). and if you please can u explain for me more about pseudo spinnors?
Thank you for your help
 

1. What is the lattice non-perturbative definition?

The lattice non-perturbative definition is a mathematical framework used in quantum field theory to define and study theories that cannot be solved using traditional perturbative methods. It involves discretizing the space and time dimensions of a theory onto a lattice, and using numerical simulations to calculate physical quantities.

2. Why is the lattice non-perturbative definition important?

The lattice non-perturbative definition allows for the study of theories that cannot be solved using traditional perturbative methods, such as quantum chromodynamics (QCD). It also provides a way to calculate physical quantities with high precision, making it an important tool for testing and refining theories.

3. How does the lattice non-perturbative definition differ from perturbative methods?

Perturbative methods involve using a series expansion to calculate physical quantities, assuming that the interaction between particles is small. However, this approach does not work for strongly interacting theories like QCD. The lattice non-perturbative definition, on the other hand, discretizes the theory onto a lattice and uses numerical simulations to calculate physical quantities, making it applicable even for strongly interacting theories.

4. What are the limitations of the lattice non-perturbative definition?

The lattice non-perturbative definition has some limitations, such as the need for high computational power and the difficulty in extrapolating results to the continuum limit. It also requires careful treatment of finite size effects and selection of appropriate lattice spacing and lattice volume. Additionally, it may not be applicable to all theories and may not provide analytical solutions.

5. How is the lattice non-perturbative definition used in practical applications?

The lattice non-perturbative definition has been used in various practical applications, such as calculating the mass of particles, studying phase transitions, and determining the equation of state for nuclear matter. It has also been used to study the properties of the strong nuclear force and the behavior of matter at high temperatures and densities, which has implications for astrophysics and cosmology.

Similar threads

  • Beyond the Standard Models
Replies
1
Views
2K
  • Atomic and Condensed Matter
Replies
5
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
1K
  • Atomic and Condensed Matter
Replies
3
Views
11K
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
858
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
  • Beyond the Standard Models
Replies
3
Views
2K
  • Quantum Physics
3
Replies
87
Views
7K
  • Beyond the Standard Models
Replies
0
Views
862
  • Quantum Physics
Replies
12
Views
2K
Back
Top